Towards a Bezout-type theory of affine varieties...Introduction 0.1 Introduction The most prominent example of complete spaces in algebraic geometry is perhaps the complex projective

Towards a Bezout-type theory of affine varieties

by

Pinaki Mondal

A thesis submitted in conformity with the requirementsfor the degree of Doctor of PhilosophyGraduate Department of Mathematics

University of Toronto

Copyright c© 2010 by Pinaki Mondal

Abstract

Towards a Bezout-type theory of affine varieties

Pinaki Mondal

Doctor of Philosophy

Graduate Department of Mathematics

University of Toronto

2010

We study projective completions of affine algebraic varieties (defined over an algebraically

closed field K) which are given by filtrations, or equivalently, integer valued ‘degree like

functions’ on their rings of regular functions. For a polynomial map P := (P1, . . . , Pn) :

X → Kn of affine varieties with generically finite fibers, we prove that there are com-

pletions of the source such that the intersection of completions of the hypersurfaces

{Pj = aj} for generic (a1, . . . , an) ∈ Kn coincides with the respective fiber (in short,

the completions ‘do not add points at infinity’ for P ). Moreover, we show that there

are ‘finite type’ completions with the latter property, i.e. determined by the maximum

of a finite number of ‘semidegrees’, by which we mean degree like functions that send

products into sums. We characterize the latter type completions as the ones for which

ideal I of the ‘hypersurface at infinity’ is radical. Moreover, we establish a one-to-one

correspondence between the collection of minimal associated primes of I and the unique

minimal collection of semidegrees needed to define the corresponding degree like function.

We also prove an ‘affine Bezout type’ theorem for polynomial maps P with finite fibers

that admit semidegrees corresponding to completions that do not add points at infinity

for P . For a wide class of semidegrees of a ‘constructive nature’ our Bezout-type bound

is explicit and sharp.

ii

Dedication

To my father,

who would have been very happy.

iii

Acknowledgements

The first person I must thank is my advisor Pierre Milman, who continued to support

me as a friend, philosopher and guide all these years, in the midst of times I would have

given up all the hope if I were in his place. In mathematics he inspired with brilliant

insights and pure enthusiasm and in other worldly matters he was always there like a

father.

I do not know what I would do but for the friends I have found in Toronto. Najma

Ahmad, Masrour Zoghi, Louis Leung, Tony Huynh, Fernando (the irrational) Espinosa

- these are people who fundamentally altered the way I see the world (and therefore are

to be hold accountable in case of any consequences). There were also friends from the

past I could not get away from - Ayan Chowdhury, Ayanabha Sarkar, Ashikur Bhuiya,

Farhana Hossain - they made me part of their lives and I am so very happy for it.

And then there was Ida Bulat. I could write another thesis on how she changed my

(and my fellow students’) life for the better.

I thank Professor Askold Khovanskii for his suggestions. Finally, I offer my gratitude

to the faculty and staff of the Department of Mathematics at University of Toronto for

their time, commitment and support.

iv

Contents

0 Introduction 1

0.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

0.2 Historical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

0.3 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1 Intersection preserving completions 11

1.0 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.0.1 Affine Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.0.2 Projective Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.0.3 Graded Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.1 Filtrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.3 Preserving Intersections at Infinity . . . . . . . . . . . . . . . . . . . . . 26

2 Filtrations with nilpotent-free graded rings 42

2.0 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.0.1 Lattice points and Toric Varieties . . . . . . . . . . . . . . . . . . 43

2.0.2 Divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.1 Semidegree and Subdegree . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.2 Properties of Subdegree . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

v

2.2.1 Uniqueness of minimal presentations of subdegrees by semidegrees

and a characterization via absence of nilpotents in their graded rings. 53

2.2.2 For finitely generated subdegrees, the semidegrees of the minimal

presentation do not take −∞ value. . . . . . . . . . . . . . . . . . 62

2.2.3 Finitely generated subdegrees with associated filtrations

preserving intersections at ∞. . . . . . . . . . . . . . . . . . . . . 67

3 Affine Bezout-type theorems 76

3.0 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3.0.1 Valuations and convex bodies . . . . . . . . . . . . . . . . . . . . 76

3.1 Bezout Theorem for Semidegrees . . . . . . . . . . . . . . . . . . . . . . 78

3.2 Iterated Semidegree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

3.3 A Formula for Subdegrees . . . . . . . . . . . . . . . . . . . . . . . . . . 90

A Non-degeneracy condition in Bernstein’s theorem 95

Bibliography 101

vi

Chapter 0

Introduction

0.1 Introduction

The most prominent example of complete spaces in algebraic geometry is perhaps the

complex projective space Pn(C). The usual way of constructing Pn(C) is to view it as

the space of lines in Cn+1 that pass through the origin. Equivalently, via Hilbert’s Null-

stellensatz, it can be identified with the maximal homogeneous ideals of the polynomial

ring C[x0, . . . , xn] that do not contain the irrelevant ideal generated by x0, . . . , xn. In

algebraic geometry jargon, the preceding sentence states that Pn(C) is the Proj of the

graded C-algebra C[x0, . . . , xn], which is graded by degree of polynomials. C-algebra

C[x0, . . . , xn] is, on the other hand, isomorphic to the graded C-algebra which is the

direct sum of vector subspaces of polynomials in (x1, . . . , xn) of degree less than or equal

to d over all non-negative integers d. Identifying polynomials in (x1, . . . , xn) with regular

functions on Cn, this gives us an algebraic way of constructing Pn(C) as a completion of

Cn. It is well known that this construction can be generalized: let X be an arbitrary

affine variety and F = {Fd : d ≥ 0} be a filtration on the coordinate ring C[X] of X

(which in our case means F0 ⊆ F1 ⊆ F2 ⊆ · · · is a sequence of vector subspaces of C[X]

such that C[X] =⋃d≥0 Fd, and FdFe ⊆ Fd+e). Then Proj

⊕d≥0 Fd is a completion of X,

1

Chapter 0. Introduction 2

i.e. a complete variety which contains X as a dense open subset.

Giving a filtration F on C[X], on the other hand, is equivalent to defining a degree

like function δ : C[X]→ Z which satisfies the following properties:

1. δ(f + g) ≤ max{δ(f), δ(g)} for all f, g ∈ C[X], with < in the preceding equation

implying δ(f) = δ(g).

2. δ(fg) ≤ δ(f) + δ(g) for all f, g ∈ C[X].

The vector spaces Fd := {f ∈ C[X] : δ(f) ≤ d} define a filtration on C[X] associated with

δ and from that filtration, one constructs a completion of X as in the first paragraph.

For X = Cn and δ equal to the usual degree of polynomials, the completion of Cn we get

via this construction is Pn(C). If we take δ to be a more general weighted degree, we end

up with the corresponding weighted projective space.

Both the usual and weighted degrees satisfy property 2 with exact equality instead of

the inequality. We call the degree like functions which have this property semidegrees.

In the theory of toric geometry, one associates a normal n-dimensional projective variety

to a convex integral polytope of dimension n. Each facet (i.e. codimension one face) of

such a polytope P defines a weighted degree on C[x1, x−11 , . . . , xn, x

−1n ]. It turns out (cf.

Example 2.1.3) that the toric variety XP associated to P is precisely the completion of

the n-torus (C∗)n corresponding to the degree like function δP which is the maximum of

the weighted degrees corresponding to the facets of P . One of the guiding principle of

this thesis is the conviction that the completion of Cn coming from a semidegree should

‘behave in the same way’ as the weighted projective spaces, which are completions of Cn

corresponding to weighted degrees. As the analogue of the δP corresponding to a polytope

P , the same principle leads us to consider subdegrees, i.e. degree like functions which are

the maximum of finitely many semidegrees. As a manifestation of this principle we prove

that in the completion X of an arbitrary affine variety X corresponding to a subdegree

δ on C[X], the irreducible components of the hypersurface at infinity X∞ := X \X have

a canonical one-to-one correspondence with the semidegrees associated to δ, in the same


way that the irreducible components of XP \ (C∗)n correspond to the facets of P (see

Theorem 2.2.1 and Proposition 2.2.12). Given a completion X ↪→ Z determined by a

degree like function δ, we introduce a ‘normalization’ procedure to produce a δ which is a

subdegree (Theorem 2.2.16). When X is normal, the corresponding completion Z is the

normalization of Z. The construction of δ from δ generalizes the well known procedure

of constructing the normalization of a non-normal toric variety (determined by a finite

subset of a lattice) by ‘filling the holes’ [6, Theorem 3.A.5].

This work started as a project to understand ‘affine Bezout type’ theorems. By affine

Bezout type theorems we mean those which estimate the number of solutions of a system

of equations on an affine variety (as opposed to the usual version of Bezout theorem,

which gives a formula for the number of solutions of n polynomials on the projective

space Pn(C)). The most famous theorems of this kind, apart from the Bezout theorem,

are perhaps those of Kushnirenko [20] and Bernstein [2]. Bernstein’s theorem is more

general, and it states the following: let ∆1, . . . ,∆n be convex polytopes in Rn with vertices

in Zn. Then for generic Laurent polynomials f1, . . . , fn such that the Newton polytope of

fi lies inside ∆i, the number of solutions in (C∗)n of the system f1(x) = · · · = fn(x) = 0

is equal to n! times the mixed volume of ∆1, . . . ,∆n. There have been numerous works

which generalize this theorem and provide formulae for the number of solutions of systems

of equations on affine varieties, the systems being generic in some suitable sense (see,

e.g. [15], [16], [21], [31], [30], [32]). Curiously, most of these constructions have one

property in common - they involve some polynomial map f := (f1, . . . , fq) : X → Cq of

affine varieties with finite fibers and completions X ↪→ Z of X which satisfy the following

property:

for generic a := (a1, . . . , aq) ∈ Cq, H1(a) ∩ · · · ∩ Hq(a) ∩ (Z \X) = ∅, (∗)

where Hi(a) := {x ∈ X : fi(x) = ai} and Hi(a) is the closure of Hi(a) in Z for all

1 ≤ i ≤ q. This observation suggests the following


Definition 0.1.1.

• A completion ψ : X ↪→ Z preserves the intersection of subvarieties V1, . . . , Vk of X

at ∞ if V1 ∩ · · · ∩ Vk ∩X∞ = ∅, where X∞ := Z \X is the set of ‘points at infinity’

and Vj is the closure of Vj in Z for every j.

• Given a collection f := {fi : 1 ≤ i ≤ q} of polynomials on X, a := (a1, . . . , aq) ∈ Cq

and a completion ψ of X, we say that ψ preserves f over a at ∞ if ψ preserves the

intersection of the hypersurfaces Hi(a) := {x ∈ X : fi(x) = ai}, i = 1, . . . , q; we

simply say completion ψ preserves f at ∞ if ψ preserves f over a at ∞ for a in a

Zariski open subset of Cq.

Remark - definition 0.1.2. Our results throughout this work remain valid with the

property of preservation of a collection f at ∞ by completion ψ being replaced by a

stronger property as follows:

• If for any linear coordinate change (x1, . . . , xq) 7→ (∑q

i=1 c1ixi, . . . ,∑q

i=1 cqixi) on

Cq, completion ψ preserves {∑q

i=1 cjifi}1≤j≤q over points (∑q

i=1 c1iai, . . . ,∑q

i=1 cqiai)

at ∞ (for a fixed point a ∈ Cq), we say that completion ψ preserves map f :=

(f1, . . . , fq) : X → Cq over a at ∞, and simply say completion ψ preserves map f

at ∞ if ψ preserves map f over a at ∞ for a in a Zariski open subset of Cq.

In this thesis we prove that given any polynomial map f : X → Cn with generically

finite fibers, there exists a degree like function δ on the coordinate ring of X such that δ

is a subdegree and the corresponding completion of X preserves map f at ∞ (Corollary

2.2.19). When δ is itself a semidegree, we derive an affine Bezout type formula for the

size of generic fibers of f in terms of degree D of a d-uple completion of the resulting

variety (Theorem 3.1.1). Using recent work of Kaveh and Khovanskii [17], we provide a

description of D in terms of the volume of a convex body determined by δ (Proposition

3.1.5). For X = Cn, we also describe an iterative procedure (starting with a weighted

degree) of a construction of a semidegree and provide a simple algebraic formula for D

(Theorem 3.2.7). Finally, we consider more general subdegrees δ and derive an upper


bound for the size of generic fibers of f in terms of the degree of an appropriate completion

of X we construct from δ (Theorem 3.3.2).

I express my gratitude to my advisor Professor Pierre Milman for posing the questions

and guiding me throughout this work. I would also like to thank Professor Khovanskii for

helpful suggestions, e.g. he pointed out the possibility of a connection between semide-

grees and ‘orders of poles at infinity’ [18] and indicated the idea of the proof of theorem

1.3.5.

0.2 Historical Background

The history of Bezout’s theorem can be traced at least as far back as 1680. Newton had

observed by that time that the abscissas of the intersection points of two algebraic plane

curves of degrees m and n are given by a polynomial equation of degree less than or equal

to mn [29]. In 1720 Maclaurin’s book ‘Geometrica Organica’ contained the theorem that

two plane algebraic curves of degree m and n intersect at at most mn points, unless

they have infinitely many points in common [22]. But it seems that a correct proof of

the theorem was missing until Etienne Bezout came up with one in 1764, and he also

generalized it to an arbitrary dimension n ≥ 2 ([3], [4], [5]). There was a lot of work

on this theorem in the 20th century. One major stream was about understanding the

concept of multiplicity of intersections and extending it to different algebraic settings.

We mention [35] as an example of a work in this direction. It also has an excellent survey

- the little bit of history contained in this paragraph was taken from there.

On the other hand, our motivation was to understand the developments in “affine

Bezout type” theorems. The statement of Bezout theorem when viewed from this per-

spective is as follows:

Theorem (Bezout). Let f := (f1, . . . , fn) : Cn → Cn be a polynomial map such that


#f−1(0) is finite. then

#f−1(0) ≤n∏i=1

deg(fi), (1)

where #f−1(0) is the size of f−1(0) counted with multiplicity. (1) holds with equality iff

the following condition holds:

the leading homogeneous components of fi have only one common

zero, namely the origin.(E1)

There is also a weighted version of Bezout’s theorem which is more or less well known

(see, e.g. [7]).

Theorem (Weighted Bezout). Let f := (f1, . . . , fn) : Cn → Cn be a polynomial map

such that #f−1(0) is finite. Let δ be a weighted degree on C[x1, . . . , xn] that assigns

weight di to xi. Then

#f−1(0) ≤∏n

i=1 δ(fi)∏ni=1 di

, (2)

where #f−1(0) is the size of f−1(0) counted with multiplicity. (2) holds with equality iff

the following condition holds:

leading weighted homogeneous components of fi have only one com-

mon zero, namely the origin.(E2)

In mid 1970’s Kushnirenko discovered a remarakable formula for the number of so-

lutions of n Laurent polynomials (i.e. elements of C[x1, x−11 , . . . , xn, x

−1n ]) on the torus

(C∗)n in terms of the volume of the Newton polytope NP(fi) of the fi’s. Recall that the

Newton polytope of an element g :=∑aαx

α ∈ C[x1, x−11 , . . . , xn, x

−1n ] is the convex hull

(in Rn) of all α such that aα 6= 0.

Theorem (Kushnirenko [20]). Let A be a finite subset of Zn and P be the convex hull

of A in Rn. Let fi :=∑ai,αx

α ∈ C[x1, x−11 , . . . , xn, x

−1n ] such that NP(fi) ⊆ P for

i = 1, . . . , n. Then

#f−1(0) ≤ n! Vol(P), (3)


where #f−1(0) is the number of the isolated roots of f1, . . . , fn on (C∗)n counted with

multiplicity, and Vol(P) is the n-dimensional Euclidean volume of P. (3) holds with

equality iff the following condition holds:

for all α ∈ Zn, f1,α, . . . , fn,α have no common solutions in (C∗)n, (E3)

where fi,α is the weighted leading form of fi corresponding to the weight α. In other

words, fi,α :=∑

β∈Aα ai,βxβ, where Aα := {β ∈ A : 〈α, β〉 ≥ 〈α, γ〉 for all γ ∈ A}.

The observation crucial to our work is that conditions (E1) and (E2) are equivalent

to the condition that an appropriate completion of Cn preserves {f1, . . . , fn} at ∞ over

the origin, the completion being the usual projective space Pn(C) in the case of Bezout’s

theorem and the weighted projective space Pn(C; 1, d1, . . . , dn) in the case of weighted

Bezout theorem. Similarly, when dim(P) = n, condition (E3) is equivalent to the con-

dition that the toric completion XP of (C∗)n preserves {f1, . . . , fn} at ∞ over 0 (when

dimP < n, variety XP is not a completion of (C∗)n, but of (C∗)m for some m < n;

nonetheless there is a map φP : (C∗)n → XP and condition (E3) is equivalent to having

empty intersection at ∞ of the closure in XP of the image of V (fi) under φP). We refer

the reader to section 2.0.1 for a brief introduction to toric varieties.

Shortly after Kushnirenko proved his theorem, Bernstein extended it to a more general

setting in which he fixed n convex integral polytopes P1, . . . ,Pn and concluded that for

all f1, . . . , fn with NP(fi) ⊆ Pi, the number of isolated roots of f1, . . . , fn on (C∗)n is

bounded by n! times the mixed volume of P1, . . . ,Pn. He also shows that his estimate

is exact for generic f1, . . . , fn satisfying NP(fi) ⊆ Pi for each i. After a far reaching

generalization by A. Khovanskii, e.g. as in monograph [19], this estimate is now known

as the “BKK” bound (where BKK stands for Bernstein, Kushnirenko and Khovanskii).

The BKK bound was later refined by a number of authors including Huber and Sturmfels

([15], [16]), Li and Wang [21], Rojas ([31], [30], [32]). But as it happens with the three

theorems stated in this section, the estimates we get in most of these works for the size of


a fiber of a map are exact if and only if an appropriate completion preserves {f1, . . . , fn}

at∞ over the corresponding point. This is what motivated us to study the phenomenon

of the preservation of the component hypersurfaces of a map at∞, which resulted in this

thesis.

In the appendix, together with a precise statement of Bernstein’s theorem, we provide

a proof of our interpretation of Kushnirenko-Bernstein’s non-degeneracy condition (E3)

as

the completion (C∗)n ↪→ XP preserves {f1, . . . , fn} at ∞ over 0, (E)

where XP is the toric variety associated with the Minkowski sum P of the Newton

polytopes of components of f .

0.3 Organization

Throughout this thesis we work over an arbitrary algebraically closed field K. In chapter

1 we set up the basic theory of the completions that we study in the rest of the thesis. In

section 1.1 we define the notion of a filtration and give a characterization of the comple-

tions of an affine algebraic variety over K which come from filtrations on its coordinate

ring. In section 1.2 we give several basic examples of filtrations and corresponding com-

pletions. At the end of this section we also give an example of a projective completion of

an affine variety (which is a plane cubic minus a non-torsion point) which does not come

from any filtration. On the other hand, we ask the following

Question. Does Kn admit a projective completion not induced by a filtration?

Section 1.3 is where we consider the question: “given a collection of subvarieties

of an affine variety X with finite intersections, when does a completion of X preserve

their intersection at ∞?” We prove in this section that given a map f : X → Kn

with generically finite fibers, there is a filtration on K[X] such that the corresponding


completion preserves map f at ∞. When pondering about a completion ψ preserving

fibers of a map f , it is natural to try to understand how large in f(X) is the set Sψ defined

as the set of all points a in f(X) such that ψ preserves f−1(a) at ∞. The structure of

Sψ, in general, is a mystery to us. We show by means of examples that Sψ can be a

proper Zariski closed or a proper Zariski open subset of f(X). In the special class of a

semidegree, we show later in section 3.1 that Sψ = f(X) if Sψ is not empty.

We introduce degree like functions corresponding to filtrations in section 2.1 and

present examples of completions determined by semidegrees and subdegrees. In particular

we show that normal projective toric varieties come from subdegrees on the ring of

Laurent polynomials. In section 2.2 we prove our main results on projective completions

determined by subdegrees. Our first theorem classifies the filtrations determined by

semi- and subdegrees. As a corollary we deduce that for a completion ψ : X ↪→ X

given by a subdegree δ, the irreducible components of X∞ := X \ X are in a one-to-

one correspondence with the unique minimal collection of semidegrees defining δ. We

also show that X is nonsingular in codimension one at infinity, i.e. for each irreducible

component V of X∞ the local ring OV,X of X along V is regular and hence a discrete

valuation ring; moreover, the semidegree associated to V is proportional to the valuation

associated to OV,X . Given an arbitrary completion which comes from a filtration and

preserves the intersection at ∞ of a collection of subvarieties, we show in our (main)

existence theorem that there is a completion determined by a subdegree which preserves

the intersection at ∞ of the subvarieties in the collection. We make use of the latter to

conclude the existence of completions determined by subdegrees which preserve a given

generically finite polynomial map at ∞. Then we show that ‘subdegree’ can not be

replaced by a ‘semidegree’ in the preceding result: we provide an example of a quasifinite

map f : K2 → K2 which does not admit any semidegree on K[x1, x2] such that the

corresponding completion of K2 preserves {f1, f2} at ∞ over any point in K2.

In section 3.1 we prove the Bezout theorem for a semidegree δ in terms of the degree D


of the resulting projective variety in an appropriate ambient space. We give a description

of D in terms of the volume of a convex body in Rn using [17]. We describe in section 3.2

an explicit construction of iterated semidegrees. We then present a very simple formula for

D in the case that X = Kn and δ is an iterated semidegree on K[x1, . . . , xn] constructed

by means of finitely many iterations starting with the case of a weighted degree as the

initial semidegree. In section 3.3 we give an (implicit) Bezout-type bound for a wide class

of subdegrees: if f : X → Kn is a polynomial with generically finite fibers and δ is a

subdegree such that each of its associated semidegrees takes positive value on each of the

components of f , then we derive a formula for the size of generic fibers of f in terms of

the degree of a completion of X constructed from f and δ. (We expect the latter degree

to be the mixed volume of n convex sets for which we have an explicit construction; but

this is a work in progress.)

In the appendix, together with a precise statement of Bernstein’s theorem, we prove

that the non-degeneracy condition in his theorem is equivalent to the condition that the

completion (C∗)n ↪→ XP preserves {f1, . . . , fn} at∞ over 0, where XP is the toric variety

associated with the Minkowski sum P of the Newton polytopes of components of f .

Chapter 1

Intersection preserving completions

determined by filtrations

We start this chapter by describing how filtrations on the coordinate ring of a given

affine variety X determine projective completions of X. ‘Preserving intersections’ in

the title stands for the intersection of completions being the same as the completion of

the intersection for a family of subvarieties of X. We show that there exist projective

completions of X (determined by filtrations) which preserve the intersection of a given

finite family of subvarieties of X with a finite intersection. For a polynomial map P :=

(P1, . . . , Pn) : X → Kn with generically finite fibers, we prove that there are completions

of the source which preserve the intersection of the component hypersurfaces {Pj = aj}

for generic (a1, . . . , an) ∈ Kn. Moreover, we show that there are completions of X which

preserve the intersection of component hypersurfaces of ξ ◦ P for any non-degenerate

linear transformation ξ of Kn.

1.0 Background

In this section we recall definitions and theorems from basic algebraic geometry that we

use throughout the text.

11

Chapter 1. Intersection preserving completions 12

1.0.1 Affine Varieties

Let K be an algebraically closed field. The affine n-space over K is the set Kn. Let

x1, . . . , xn be the coordinate functions on Kn. A Zariski closed subset of Kn is a set of

solutions of a collection of polynomials in (x1, . . . , xn). The Zariski topology on Kn is

the topology for which the closed sets are precisely the Zariski closed subsets of Kn. A

Zariski closed subset X of Kn is an affine algebraic variety or, in short an affine variety

if it is also irreducible, i.e. it is not the union of two proper Zariski closed subsets. Let

φ : X → Y ⊆ Km be a map between affine varieties. φ is called a morphism or a regular

map if it is the restriction to X of a polynomial map from Kn to Km. The coordinate ring

K[X] of X is the ring of the K-valued functions on X which are restrictions of polynomials

in (x1, . . . , xn). Equivalently, K[X] is the quotient of the polynomial ring K[x1, . . . , xn]

by the ideal of polynomials vanishing on all points of X. The starting point of modern

algebraic geometry is Hilbert’s Nullstellensatz, which gives a one-to-one correspondence

between the Zariski closed subsets of an affine algebraic variety and radical ideals of its

coordinate ring in the following way:

Zariski closed subsets of a variety X ←→ Radical ideals of K[X]

V −→ I(V ) := {f : f(x) = 0 for all x ∈ V }

V (I) := {x : f(x) = 0 for all f ∈ I}} ←− I

In particular, the points of X correspond to the maximal ideals of K[X], and subvari-

eties of X correspond to prime ideals of K[X]. A corollary of this is the following useful

result:

Theorem 1.0.1 ([14, Theorem I.3.5]). The correspondence X → K[X] gives an equiv-

alence of the category of affine algebraic varieties over K and regular maps with the

category of finitely generated domains over K and K-algebra homomorphisms.

In particular, X and Y are isomorphic as affine varieties if and only if K[X] and K[Y ]

are isomorphic as K-algebras, and choosing different sets of generators of K[X] gives


different embeddings of X into possibly different affine spaces.

Let A be a finitely generated domain over K. Let SpecA be the space {p : p is a

prime ideal of A} together with the Zariski topology, in which closed sets are of the form

V (a) := {p : p ⊇ a} for some ideal a of A. As f varies in A, the complement of V (f),

which we denote by D(f), form a basis for the Zariski topology on SpecA. For each

f ∈ A, let Af := {a/fk : a ∈ A, k ∈ N} be the localization of A with respect to f . Note

that for each f, g ∈ A, there is a natural map Af → Afg which maps a/fk 7→ agk/(fg)k.

Since D(f)∩D(g) = D(fg) for all f, g ∈ A, it follows that there is a unique sheaf OSpecA,

called the structure sheaf, on SpecA such that its ring of sections over each D(f) is Af .

Moreover, it follows from theorem 1.0.1 that up to an isomorphism there is a unique affine

variety X, such that X together with the sheaf OX of regular functions on X completely

determines SpecA and OSpecA. By an abuse of notation we will write SpecA also for

this X, whenever the meaning will be clear from the context.

1.0.2 Projective Varieties

The projective n-space over K, denoted Pn(K), or simply Pn, is the quotient of Kn+1 \{0}

under the equivalence relation given by (a0, . . . , an) ∼ (λa0, . . . , λan) for all λ ∈ K \ {0}.

Denote the equivalence class of (a0, . . . , an) by the homogeneous coordinates [a0 : · · · : an].

Let Ui := {[a0 : · · · : an] : ai 6= 0}. For each i, identify Ui with Kn via the mapping

[a0 : · · · : an] 7→ (a0

ai, . . . ,

aiai, . . . ,

anai

).

The Zariski topology on Pn is defined by declaring a subset X to be closed if and only if

X ∩ Ui is Zariski closed in Ui for each i. As in the affine case, a Zariski closed subset X

of Pn is a projective variety if X is also irreducible.

Let x0, . . . , xn be the coordinates on Kn+1. Let f =∑aαx

α be an element in

K[x0, . . . , xn], where α = (α0, . . . , αn) varies over (Z+)n+1, and we use the multi-index

notation that xα := xα00 · · ·xαnn . Recall that f is called homogeneous of degree d if


∑ni=0 αi = d for each α such that aα 6= 0, and an ideal of K[x0, . . . , xn] is homogeneous

if it has a set of homogeneous generators. It turns out that a subset X of Pn is Zariski

closed if and only if there are homogeneous polynomials f1, . . . , fk ∈ K[x0, . . . , xn] such

that X = {[a0 : · · · : an] : fi(a0, . . . , an) = 0 for all i}. Let I(X) be the homogeneous

ideal of K[x0, . . . , xn] generated by the homogeneous polynomials which vanish at all

points of X. Then X is irreducible, i.e. a projective variety, if and only if I(X) is prime.

The quotient S(X) := K[x0, . . . , xn]/I(X) is called the homogeneous coordinate ring of

X. The projective version of Nullstellensatz gives a one-to-one correspondence, defined

exactly as in 1.0.1, between subvarieties of X and radical homogeneous ideals properly

contained in the irrelevant ideal S(X)+ of S(X), where S(X)+ is the image in S(X) of

the ideal of K[x0, . . . , xn] generated by x0, . . . , xn.

A quasi-projective variety is a Zariski open subset of a projective variety. A quasi-

projective variety X is called complete if for every quasi-projective variety Y the projec-

tion map p : X × Y → Y is closed, i.e. p maps Zariski closed sets onto Zariski closed

sets. Complete spaces are algebraic analogue of compact spaces - if X is a ‘reasonable’

topological space, say completely regular or with a countable basis of open sets, one can

show that X is complete in the above sense if and only if X is compact. The fundamen-

tal theorem of elimination theory says that Pn is complete for all n. It follows that any

projective variety, being a Zariski closed subset of some Pn, is also complete. A projective

completion of an affine variety Y is an open immersion ψ : Y ↪→ X of Y into a projective

variety X, where ‘ψ is an open immersion’ means that ψ(Y ) is a Zariski open subset of

X and ψ is an isomorphism of algebraic varieties between Y and ψ(Y ). Whenever the

map ψ is clear from the context, we will simply say that X is a projective completion of

Y .


1.0.3 Graded Rings

A graded ring is a ring S, together with a decomposition S =⊕

d∈Z Sd into a direct

sum of abelian groups Sd such that for any d, e ∈ Z, Sd · Se ⊆ Sd+e. The grading is

non-negative if Sd = 0 for all d < 0. An element of Sd is called a homogeneous element of

degree d, and an ideal a of S is a homogeneous ideal if a =⊕

d∈Z(a ∩ Sd). The irrelevant

ideal of S is the homogeneous ideal generated by S+ :=⊕

d≥1 Sd.

As in subsection 1.0.1, we now associate to every graded ring S a topological space,

together with a sheaf of rings. Mimicing the situation of a projective variety in subsection

1.0.2, define ProjS to be the set of all homogeneous prime ideals of S which do not

contain all of S+. Endow ProjS with the Zariski topology, where closed sets are of the

form V (a) := {p ∈ ProjS : p ⊇ a} for homogeneous ideals a of S. As f varies over the

homogeneous elements of positive degree in S, the sets D(f) := {p ∈ ProjS : f 6∈ p}

form a basis of the Zariski topology on ProjS. Let f be a homogeneous element of degree

d > 0 in S and let S(f) := {s/fk : s ∈ Skd, k ∈ Z} be the subring of elements of degree 0

in the localized ring Sf . As in the affine case, there is a unique sheaf OProjS on ProjS

such that S(f) is the ring of sections over D(f).

Theorem 1.0.2 ([14, Theorems II.2.5]). Let S be a graded ring which is also a finitely

generated domain over K (in particular, each Sd is a K-vector space). Assume the grading

of S is non-negative.

1. Let f be a homogeneous element of positive degree in S. Then there is an iso-

morphism SpecS(f)∼= (D(f),OProjS|D(f)), where the isomorphism is that of ringed

spaces, which means that there is a pair (φ, φ]) of a homeomorphism φ : D(f) →

SpecS(f) and an isomorphism of sheaves φ] : OSpecS(f)→ φ∗(OProjS|D(f)).

2. Up to an isomorphism there is a unique quasi-projective variety X such that X

together with the sheaf OX of regular functions on X completely determines ProjS

and OProjS. By an abuse of notation, this X will also be denoted by ProjS.

3. If in addition S0 = K, then ProjS is a projective variety.


Example 1.0.3. Let d0, . . . , dn be positive integers and let R be the polynomial ring

K[x0, . . . , xn] endowed with the grading given by the weighted degree which associates

weight di to xi. In other words, R =⊕

d≥0Rd with Rd being the K-vector space generated

by monomials xα00 · · ·xαnn such that

∑αjdj = d. Then ProjR is the n-dimensional

weighted projective space Pn(K; d0, . . . , dn). As a set it can be identified with the quotient

of Kn+1 \ {0} under the equivalence relation given by (a0, . . . , an) ∼ (λd0a0, . . . , λdnan)

for all λ ∈ K \ {0}. In particular, for d0 = · · · = dn = 1, ProjR is the usual projective

space Pn(K).

Example 1.0.4. Let d0 = 1 and R, d1, . . . , dn be as in example 1.0.3. Then R(x0)∼=

K[y1, . . . , yn] via the mapping yi 7→ xi/(x0)di . It follows via theorems 1.0.1 and 1.0.2 that

the Zariski open subset Dx0 of ProjR is isomorphic to Kn as algebraic varieties - in other

words ProjR is a projective completion of Kn.

Example 1.0.5. Let S be as in assertion 3 of theorem 1.0.2 (in particular, S0 = K). Let

f0, . . . , fk be a set of generators of S as a K-algebra, with deg(fi) = di > 0. Let R be as

in example 1.0.3. Then there is a surjective homomorphism φ : R → S of graded rings

which maps xi 7→ fi, and therefore S ∼= R/I, where I := kerφ. Let X be the subvariety of

the weighted projective space Pn(K; d0, . . . , dk) corresponding to the homogeneous ideal

I. It follows that the homogeneous coordinate ring of X is isomorphic to S and hence

ProjS ∼= X.

Example 1.0.6. Let S, fi and di, 0 ≤ i ≤ k, be as in example 1.0.5 and d > 0. Let the

d-th truncated ring S[d] ⊆ S be defined by S[d] :=⊕

d|k Sk =⊕

k≥0 Skd. Then p 7→ p∩S[d]

gives a homeomorphism between underlying sets of ProjS and ProjS[d], and moreover,

for each homogeneous f ∈ S+, S(f)∼= (S[d])(fd). It follows that ProjS ∼= ProjS[d]. As in

the preceding example, choosing a set of generators of S[d], one can embed ProjS as a

subvariety of an appropriate weighted projective space. Such an embedding is called a

d-uple embedding. Moreover, there exists d such that S[d] is generated as a K-algebra by


(S[d])1 = Sd (e.g. it suffices to take d = (k + 1)d′, where d′ is the least common multiple

of d0, . . . , dk [27, Lemma in section III.8]). In that case the d-uple embedding embeds

ProjS into a usual projective space.

1.1 Filtrations

Throughout this section A will be a finitely generated algebra over an algebraically closed

field K.

Definition. A filtration F on A is a family {Fi : i ∈ Z} of K-vector subspaces of A

such that

1. Fi ⊆ Fi+1 for all i ∈ Z

2. 1 ∈ F0 \ F−1

3. A =⋃i∈Z Fi and

4. FiFj ⊆ Fi+j for all i, j ∈ Z.

Remark. We introduce condition 1 6∈ F−1 in order to exclude the trivial filtration, i.e.

filtration {Fi := A}i∈Z.

Filtration F is called non-negative if Fi = 0 for all i < 0. Associated to each filtration

F there are two graded rings:

AF :=⊕i∈Z

Fi and

grAF :=⊕i∈Z

(Fi/Fi−1).

We denote a copy of f ∈ Fd in the d-th graded component of AF by (f)d. AF is given

the structure of a graded K-algebra with multiplication defined by:

(∑d

(fd)d)(∑e

(ge)e) :=∑k

∑d+e=k

(fdge)k.

F is called a finitely generated filtration if AF is a finitely generated K-algebra.


Let t be an indeterminate over A. Then there is an isomorphism

AF ∼=∑i∈Z

Fiti ⊆ A[t, t−1] (1.1)

which maps (1)1 7→ t. The following property of AF is a straightforward corollary of this

isomorphism.

Lemma 1.1.1. AF is a domain if and only if A is a domain.

It follows from theorem 1.0.2 that ProjAF is a projective, and hence complete, variety

if F is non-negative, finitely generated and F0 = K. In view of this fact we make the

following

Definition. A filtration F = {Fd}d∈Z on A is called complete if it is non-negative, finitely

generated and F0 = K.

The following proposition connects filtrations on A with projective completions of

SpecA. Recall from section 1.0.3 that the d-th truncated subring of a graded ring S =⊕k∈Z Sk is S[d] :=

⊕k∈Z Skd.

Proposition 1.1.2. If A is a K-algebra and F is a non-negative filtration on A, then

there is an open immersion ψF of SpecA onto a dense open subvariety of ProjAF . The

complement of SpecA in ProjAF is a hypersurface. Conversely, given any non-negatively

graded K-algbera S =⊕

i≥0 Si and an open immersion φ : SpecA ↪→ ProjS such that

SpecA is dense in ProjS and ProjS \ SpecA is a hypersurface, there is a non-negative

filtration F on A and a commutative diagram as follows:

SpecA

ψF��

∼= // SpecA

φ

��ProjAF // ProjS

where the top horizontal map is the identity and the bottom horizontal map is a closed

immersion induced by an isomorphism of AF with S[d]/I for some d > 0 and some ideal

I of S[d] contained in the nilradical of S[d].


Remark.

• By a hypersurface in ProjS for an arbitrary graded ring S we just mean a closed

subvariety given by V (f) for some homogeneous f ∈ S.

• The first part of the proposition is well known.

Proof. Recall from section 1.0.3 that basic open sets in ProjAF are given by D(G) =

{Q ∈ ProjAF |G 6∈ Q} = Spec(AF)(G), where G ranges over the homogeneous elements

in AF , and (AF)(G) is the subring of the local ring (AF)G consisting of degree zero

homogeneous elements. Identify AF with∑

i∈Z Fiti via (1.1). It follows that (AF)(t) =

{gtd/td : g ∈ Fd ⊆ A, d ≥ 0} ∼= {g : g ∈ Fd ⊆ A} = A as a subring of A[t, t−1]; hence

SpecA ∼= Spec(AF)(t)∼= D(t) and ProjAF \ SpecA = V (t). But by lemma 1.1.1 AF is

a domain and it is a general fact that for any graded ring S and any non zero-divisor

G ∈ S, ProjS \ V (G) is dense in ProjS. Therefore SpecA is dense in ProjAF and we

proved the first assertion of the proposition.

Let S =⊕

i≥0 Si be a graded K-algebra and φ : SpecA→ ProjS be an open immer-

sion of SpecA onto a dense open subvariety of ProjS of the form D(f) for some homoge-

neous f ∈ S. Let the degree of f be d. Recall from example 1.0.5 that ProjS ∼= ProjS[d].

Hence replacing S with S[d], we may assume that degree of f is 1. By theorem 1.0.1

the isomorphism of varieties φ : D(f) ∼= SpecS(f) induces an isomorphism of rings

φ∗ : S(f)∼= A. Now, for each g ∈ A, (φ∗)−1(g) = a/fk for some k ≥ 0 and a ∈ Sk. Define

Fk := φ∗(Sk/fk) = {g ∈ R| (φ∗)−1(g) ∈ Sk/fk}.

Then it is easy to see that F = {Fi}i≥0 is a filtration on A. By means of this filtration

we construct, as usual, the ring AF :=⊕

d≥0 Fd. Map φ∗ : S(f) → A induces a map

Φ∗ : S → AF which sends each g ∈ Sd to (φ∗(g/fd))d, i.e. to the copy of φ∗(g/fd) in

Fd ⊆ AF . Consequently Φ∗ is a surjective homomomorphism of graded rings, so that the

induced map Φ : ProjAF → ProjS is a closed immersion [14, Exercise II.3.12(a)] and

ProjAF ∼= Proj S, where S := S/ ker Φ∗.


It remains to show that with I := ker Φ∗, ideal I ⊆ η(S), where η(S) is the nilradical of

S. Indeed, let g ∈ Sd such that Φ∗(g) = (φ∗(g/fd))d = 0 ∈ AF . Then φ∗(g/fd) = 0 ∈ A.

Since φ∗ : S(f) → A is an isomorphism, it follows that g/fd = 0 ∈ S(f) so that there is

some k ≥ 0 such that fkg = 0 ∈ S. But then every prime ideal in S has to contain f

or g, i.e. D(f) ∩ D(g) = ∅. Since D(f) (∼= SpecA) is dense in ProjS, it follows that

D(g) = ∅, so that g ∈ η(S).

To see that the isomorphism ProjAF ∼= Proj S is the identity map when restricted to

SpecA, note that maps φ : SpecA→ ProjS and ψF : SpecA→ ProjAF are completely

determined by the corresponding pull back maps φ∗ : S(f) → A and ψ∗F : (AF)(t) → A

on the localizations of the respective graded K-algebras, where as before t := (1)1 ∈ AF .

The latter two maps give rise to the following commutative diagrams:

A1A // A

S(f)

φ∗

OO

Φ∗ // (AF)(t)

ψ∗F

OO φ∗(g/fd) � 1A // φ∗(g/fd)

g/fd_φ∗

OO

� Φ∗ // (φ∗(g/fd))dtd

_ψ∗F

OO

Since the top horizontal map and both of the vertical maps are isomorphisms, it

follows that the bottom horizontal map is an isomorphism as well, which concludes the

proof of proposition 1.1.2.

Before we proceed further, let us examine the structure of the complement of SpecA

in ProjAF more closely. From the proof of proposition 1.1.2, we see that ProjAF \SpecA

is the closed subspace V (t) := {P ∈ ProjAF | t ∈ P}. Consider the natural projection

π : AF → grAF . Map π is a surjective homomorphism of graded rings and therefore

induces a closed immersion π∗ : Proj(grAF)→ ProjAF , given by p 7→ π−1(p).

Claim. kerπ = 〈t〉, where 〈t〉 is the ideal generated by t in AF .

Proof. Since t := (1)1 and 1 ∈ F0 , it follows that t ∈ kerπ, so that kerπ ⊇ 〈t〉. Also,

π is a homomorphism of graded ring. Hence kerπ is a homogeneous ideal, i.e. ker π

is generated by homogeneous elements from AF . Pick a homogeneous element G in


the d-th graded component of kerπ. Then G = (g)d for some g ∈ Fd−1. But then

G = (g)d = (g)d−1(1)1 = (g)d−1t, so that G ∈ 〈t〉. Therefore, ker π ⊆ 〈t〉.

It follows that π∗(Proj(grAF)) = V (t) [14, Exercise II.3.12(b)]. Thus we have proven

the following

Proposition 1.1.3. ProjAF \ SpecA = Proj(grAF).

We combine the results of 1.1.1, 1.1.2 and 1.1.3 in

Theorem 1.1.4 (see [25, Theorem 1.1] and [26, Theorem 1.1.4]). Let A be a K-algebra

which is a domain and F be a filtration on A. If F is complete, then ProjAF is a

projective variety over K and the map

ψF : p→⊕d≥0

p ∩ Fd

gives an open immmersion of SpecA onto a dense open subscheme of ProjAF . The

complement of SpecA in ProjAF is the hypersurface V ((1)1), which also is the image

of the closed immersion π∗ : Proj(grAF) ↪→ Proj(AF) induced by the natural homomor-

phism π : AF → grAF . Conversely, given any graded ring S and an open immersion

φ : SpecA ↪→ ProjS such that SpecA is dense in ProjS and ProjS \ SpecA is a hyper-

surface, there is a filtration F on A and a commutative diagram as follows:

SpecA

ψF��

∼= // SpecA

φ

��ProjAF // ProjS

where the top horizontal map is the identity and the bottom horizontal map is a closed

immersion induced by an isomorphism of graded rings AF ∼= S[d]/η(S[d])) for some d > 0,

where η(S[d]) is the nilradical of S[d].

Recall from section 1.0.3 the concept of the homogeneous coordinate ring of a closed

subvariety of a n-dimensional weighted projective space. Choosing a set of generators


of AF gives an embedding of ProjAF into a weighted projective space such that the

homogeneous coordinate ring of the image is AF . The following corollary is a rewording

of theorem 1.1.4 in terms of these coordinates.

Corollary 1.1.5. Let X be an affine variety over K and F be a complete filtration on

K[X]. Pick f1, . . . , fN ∈ K[X] such that (1)1, (f1)d1 , . . . , (fN)dN generate K[X]F as a

K-algebra. Then the map φ : X → KN given by φ(x) := (f1(x), . . . , fN(x)) is a closed

immersion, and there exists a commutative diagram as follows:

X

ψF��

φ // KN

ι��

Proj K[X]FΦ // PN(K; 1, d1, . . . , dN)

where ι is the natural injection of KN into the weighted projective space

PN(K; 1, d1, . . . , dN), and Φ is a closed immersion that maps Proj K[X]F isomorphi-

cally onto the closure X of X in PN(K; 1, d1, . . . , dN). In fact, morphism Φ is in-

duced by an isomorphism between K[X]F and the homogeneous coordinate ring of X.

Conversely, given any closed immersion φ : X ↪→ KN , and any N positive inte-

gers d1, . . . , dN , there exists a complete filtration F on K[X] and a closed immersion

Φ : Proj K[X]F ↪→ PN(K; 1, d1, . . . , dN) such that the above diagram is commutative.

Moreover, morphism Φ is induced by a graded ring isomorphism between K[X]F and the

homogeneous coordinate ring of X in PN(K; 1, d1, . . . , dN).

1.2 Examples

In this section we work out some examples of complete filtrations on the coordinate rings

of affine varieties, and determine the corresponding completions. Throughout this section

X will denote an affine variety, A will denote its coordinate ring, F = {Fd : d ≥ 0} will

be a filtration on A and XF := ProjAF will denote the corresponding completion of X.

For the first four examples we set X = Kn and A = K[x1, . . . , xn].


Example 1.2.1. Define F on A by letting Fd be the set of polynomials of degree less

than or equal to d. Identification of Fd with the set of homogeneous polynomials in

x0, . . . , xn of degree d gives an graded ring isomorphism AF ∼= K[x0, . . . , xn], where the

latter is graded by the usual degree. Hence XF is the usual projective space Pn(K).

Example 1.2.2. Let d1, . . . , dn be any n positive integers. Let Fd be the K-linear span

of all the monomials xα11 x

α22 · · ·xαnn such that

∑αidi ≤ d. Set d0 := 1. Then Fd can

be identified with the set of weighted homogeneous polynomials in x0, . . . , xn of weighted

degree d, where weight of xi is di for each i. Thus AF is again isomorphic to K[x0, . . . , xn],

but the grading in this case is induced by the weighted degree (d0, d1, . . . , dn), and XF is

the weighted projective space Pn(K; d0, d1, . . . , dn) which we have already seen in example

1.0.4.

Example 1.2.3. Let Fk be the set of polynomials of degree less than or equal to dk, where

d is a fixed positive integer. Then XF is the d-uple embedding of Pn(K) in Pm−1(K),

where m =(n+dn

)= number of all monomials in n variables of degree at most d. In

particular, for n = 1, XF is the rational canonical curve of degree d in Pd(K). To see

this, note that Fk = (F1)k for all k, which means that AF =⊕

i≥0 Fi is generated by F1

as a K-algebra. Therefore {(xα)1 : |α| ≤ d} is a set of K-algebra generators of AF , where

|α| := α1 + . . . + αn for all α = (α1, . . . , αn) ∈ (Z+)n. It follows by corollary 1.1.5 that

XF is the closure in Pm−1(K) of the image of Kn under the map given by all monomials

in x1, . . . , xn of degree less than or equal to d. But the latter closure is identical with the

image of Pn(K) under the map given by all monomials in x0, x1 . . . , xn of degree equal to

d (since they agree on a dense open set), and this is precisely the d-uple embedding of

Pn [14, Exercise I.2.12].

Example 1.2.4. X is again Kn as above. Assume n ≥ 2. Fix an integer k with

1 ≤ k < n. Let F1 be the K-linear span of all monomials of degree less than or equal to

two excluding those of the form xixj with i ≥ j > k. Let Fd = (F1)d for d ≥ 1. Then


XF is isomorphic to the variety resulting from blowing up Pn(K) along the subspace

V := V (x0, . . . , xk). To show this, let Y be the blow up of Pn(K) along V . Then variety

Y is the closure in Pn(K) × Pk(K) of the image of Kn under the map (x1, . . . , xn) 7→

([1 : x1 : · · · : xn], [1 : x1 : · · · : xk]). Embedding Pn(K)× Pk(K) into Pkn+k+n(K) via the

Segre map, we see that Y is isomorphic to the closure in Pkn+k+n(K) of the image of X

under the map

(x1, . . . , xn) 7→ [1 : x1 : · · · : xn : x1 : x21 : · · · : x1xn : · · · : xk : xkx1 : · · · : xkxn] .

Composing with an automorphism of Pkn+k+n(K), we may assume that the map is

(x1, . . . , xn) 7→ [1 : x1 : · · · : xn : x21 : x1x2 : · · · : x1xn : x2

2 : x2x3 : · · · : x2xn : · · · : x2k :

xkxk+1 : · · · : xkxn : 0 : · · · : 0]. Projecting onto the first N := (n+1)+n+· · ·+(n−k+1)

coordinates, we see that Y is isomorphic to the closure in PN−1(k) of the image of Kn

under the map given by all monomials in F1. But by corollary 1.1.5 this is precisely XF !

Example 1.2.5 (see [26, Example 1.2.5]). Let X be a normal affine variety with trivial

divisor class group ClX (e.g. Kn, (K∗)n, or any X whose coordinate ring is a unique

factorization domain), see section 2.0.2 for a brief introduction to class group and divisors.

Let X ⊆ PL(K) be any normal projective completion of X. We now show that if X∞ :=

X \X is irreducible, then the embedding X ↪→ X arises from a filtration. Without loss

of generality we may assume that X does not lie in any proper hyperplane in PL(K). Let

the homogeneous coordinates of PL(K) be [x0 : . . . : xL]. Let D be the Cartier divisor on

X corresponding to the hyperplane section by x0, i.e. the equation for D on X \ V (xi)

is x0/xi. The Weil divisor associated to D is of the form [D] = a0[X∞] +∑m

j=1 aj[Vj],

where for each j, Vj is the closure in X of a codimension one subvariety Vj of X. Since

ClX = 0, there is a rational function f on X such that the principal divisor of f on X

is [divX(f)] =∑m

j=1 aj[Vj]. Let [D′] := [D]− [divX(f)], where [divX(f)] is the principal

divisor of f on X. But [divX(f)] =∑m

j=1 aj[Vj] + a[X∞] for some a ∈ Z. It follows that

[D′] = a′[X∞] for some a′ ∈ Z. But then a′ = degD′ = degD = deg X > 0 [14, Exercise


II.6.2], and therefore by the definition of the line bundle associated to a Cartier divisor

[10, Subsection B4.4], 1 ∈ Γ(X,OX(D′)). Recall that by our assumption X does not

lie in any proper hyperplane in PL(K). It follows that dimK Γ(X,OX(D)) = L + 1, and

hence dimK Γ(X,OX(D′)) is also L + 1. Let g1, . . . , gL ∈ K(X) such that 1, g1, . . . , gL is

a vector space basis of Γ(X,OX(D′)). Since (gi) + D′ ≥ 0 for all i = 1, . . . , L, it follows

that none of the gi’s has any pole on X, and hence each gi ∈ K[X]. Let φ : X → KL be

the closed immersion which sends x ∈ X to (g1(x), . . . , gL(x)). Then X is precisely the

closure of φ(X) in PL(K) and (by corollary 1.1.5) the embedding X ↪→ X ⊆ PL(K) is

induced by a filtration on K[X].

Example 1.2.6 (see [25, Remark 1] and [26, Example 1.2.6]). Not all projective comple-

tions of affine varieties are determined by a filtration. Our example below is a variation

of an example by Mike Roth and Ravi Vakil considered in [33]. Let X ′ be a nonsingular

cubic curve in P2(K). Let O be one of its 9 inflection points. Recall that X ′ can be

given the structure of an abelian group with O as the origin and that φ : P → [P ]− [O]

gives an injective group homomorphism of X ′ into its class group ClX ′ [14, Example

II.6.10.2]. There are only countably many points P ∈ X ′ such that P is a torsion point,

i.e. k · P = 0 for some k > 0 [34, Section III.3.4]. Pick any non-torsion point P of X ′.

Then X := X ′ \ {P} is an affine variety [12, Proposition 5]. We claim that there is no

homogeneous polynomial f in K[x0, x1, x2] such that V (f) ∩X ′ = {P}. Indeed, assume

to the contrary that there is a homogeneous polynomial f of degree d > 0 such that

V (f) ∩ X ′ = {P}. By Bezout’s theorem [div(f)] = 3d[P ], where [div(f)] is the divisor

on X ′ associated to f . Let l be the equation of the tangent line of X ′ at O. Since O

is an inflection point, it follows that V (l) ∩ X ′ = {O} and thus the divisor of l on X ′

is [div(l)] = 3[O]. Then f/ld is a rational function on X ′, and its corresponding divisor

on X ′ is [div(f/ld)] = 3d[P ] − 3d[O]. Since [div(f/ld)] is a principal divisor, it follows

that [div(f/ld)] = 0 ∈ ClX ′. But then φ(3d · P ) = 3d[P ]− 3d[O] = [div(f/ld)] = 0, and

hence 3d · P = 0, since φ is one-to-one. Therefore P is a torsion point, and we have a


contradiction. Thus the claim is true and there is no homogeneous polynomial f such

that V (f)∩X ′ = {P}. It follows by proposition 1.1.2 that there is no integer d > 0 such

that the embedding of X into the image of the d-uple embedding of X ′ is induced by a

filtration!

1.3 Preserving Intersections at Infinity

Let X be an affine variety over K. Recall that for subsets V1, . . . , Vm of X, a completion

ψ : X ↪→ Z is said to preserve the intersection of V1, . . . , Vm at∞ if V 1∩· · ·∩V m∩X∞ = ∅,

where X∞ := Z \X is the set of ‘points at infinity’ and V j is the closure of Vj in Z for

every j.

Lemma 1.3.1. Let F = {Fd : d ≥ 0} be a complete filtration on A := K[X], and

ψF : X ↪→ XF := ProjAF be the corresponding completion.

1. For each ideal q of A, let qF :=⊕

d≥0(q∩Fd) ⊆ AF . Then the closure of V (q) ⊆ X

in XF is V (qF).

2. Let V1, . . . , Vm be Zariski closed subsets of X with Vi = V (qi) for ideals qi ⊆ A for

each i. Let I be the ideal of AF generated by qF1 , . . . , qFm and (1)1. Then ψF preserves

the intersection of V1, . . . , Vm at ∞ iff the√I ⊇ AF+, where AF+ :=

⊕d>0 Fd is the

irrelevant ideal of AF .

Proof. 1. Recall (example 1.0.6) that there exists d > 0 such that (AF)[d] :=⊕

k≥0 Fkd

is generated by Fd as a K-algebra. Define a new filtration G := {Gk : k ≥ 0} on A by

Gk := Fkd. Let {1, g1, . . . , gm} be a K-vector space basis of G1. Then AG ∼= (AF)[d] and

by corollary 1.1.5, XG := ProjAG is the closure in Pm(K) of φ(X), where φ : X → Km is

defined by: φ(x) := (g1(x), . . . , gm(x)).

Let q be an ideal of A and V := V (q) be the Zariski closed subset of X defined by

q. Let p := kerφ∗ and r := (φ∗)−1(q), where φ∗ : K[y1, . . . , ym] → A is the pull back by

means of φ. Identify X with V (p) and V with V (r) in Km. Then XG and the closure VG


of V in XG are the Zariski closed subsets of Pm(K) determined by the homogenizations

p of p and, respectively, r of r with respect to y0.

Moreover, the closed embedding Φ : XG ↪→ Pm(K) is induced by the surjective

homomorphism Φ∗ : K[y0, . . . , ym] → AG which maps y0 7→ (1)1 and yi 7→ (gi)1 for

1 ≤ i ≤ m. Therefore VG

in XG is defined by the ideal Φ∗(r). But the d-th graded

component of Φ∗(r) is

Φ∗((r)d) := {Φ∗(f(y0, . . . , ym)) : f ∈ r, f homogeneous, deg(f) = d}

= {f((1)1, (g1)1, . . . , (gm)1) :

f homogeneous in y0, . . . , ym, deg(f) = d, f(1, y1, . . . , ym) ∈ r}

= {(f(1, g1, . . . , gm))d :

f homogeneous in y0, . . . , ym, deg(f) = d, f(1, g1, . . . , gm) ∈ q}

= {(f(g1, . . . , gm))d :

f polynomial in y1, . . . , ym, deg(f) ≤ d, f(g1, . . . , gm) ∈ q}

= {(g)d : g ∈ q ∩Gd},

where the last equality is a consequence of AG = K[(1)1, (g1)1, . . . , (gm)1]. Then Φ∗(r) =⊕d≥0 Φ∗((r)d) =

⊕d≥0{(g)d : g ∈ q ∩ Gd} = qG. Now recall (example 1.0.5) that XF

and XG are isomorphic and this isomorphism is induced by the inclusion AG ⊆ AF .

Since qF ∩ AG = qG, it follows that the Zariski closed subset of XG determined by qG is

isomorphic to the Zariski closed subset of XF determined by qF .

2. ψF preserves the intersection of V1, . . . , Vm at ∞ iff V 1 ∩ · · · ∩ V m ∩ X∞ = ∅.

By part 1 V j = V (qFj ) for each j, and by theorem 1.1.4 X∞ = V ((1)1). Therefore

V 1∩· · ·∩V m∩X∞ is determined by the ideal of AF generated by (1)1, qF1 , . . . , q

Fm, which

is precisely the definition of I. Then the projective version of Nullstellensatz (see section

1.0.2) implies V (I) = ∅ iff AF+ ⊆√I.

Theorem 1.3.2 (see [25, Theorem 1.2(1)] and [26, Theorem 1.3.1]). Let V1, . . . , Vm be

Zariski closed subsets in an affine variety X such that ∩mi=1Vi is a finite set. Then there


is a complete filtration F on K[X] such that ψF preserves the intersection of the Vi’s at

∞.

Proof. Let X ⊆ Kn and the ideals in K[x1, . . . , xn] defining X, V1, . . . , Vm be respectively

p, q1, . . . , qm with qj ⊇ p for each j.

Claim. For each i = 1, . . . , n, there is an integer di ≥ 1 such that

xdii = fi,1 + . . .+ fi,m + gi (1.2)

for some fi,j ∈ qj and a polynomial gi ∈ K[xi] of degree less that di.

Proof. If V1 ∩ . . . ∩ Vm = ∅, then by Nullstellensatz 〈q1, . . . , qm〉 is the unit ideal in

K[x1, . . . , xn], and the claim is trivially satisfied with gi := 0 for each i. So assume

V1 ∩ . . . ∩ Vm = {P1, . . . , Pk} ⊆ Kn,

for some k ≥ 1. Let Pi = (ai,1, . . . , ai,n) ∈ Kn. For each i = 1, . . . , n, let

hi := (xi − a1,i)(xi − a2,i) · · · (xi − ak,i).

By Nullstellensatz, for some d′i ≥ 1, hd′ii ∈ 〈q1, . . . , qm〉, i.e. h

d′ii = fi,1 + . . .+fi,m for some

fi,j ∈ qj. Substituting hi =∏

j(xi− aj,i) in the preceding equation we see that the claim

holds with di := kd′i.

Below for S ⊆ K[X] we denote by K〈S〉 the K-linear span of S, and for an element

g ∈ K[x1, . . . , xn], we denote by g the image of g in K[X] = K[x1, . . . , xn]/p. Fix a set of

fi,j’s satisfying the conclusion of the previous claim. Then define a filtration F on K[X]

as follows: let

F0 := K,

F1 := K〈1, x1, . . . , xn, f1,1, . . . , fn,m〉,

Fk := F k1 for k > 1,

F := {Fi : i ≥ 0}.


Clearly F is a complete filtration. We now show that this F satisfies the conclusion

of the theorem. By lemma 1.3.1 this is equivalent to showing that√I ⊇ K[X]F+, where

I is the ideal generated by qF1 , . . . , qFm and (1)1 in K[X]F .

From the construction of F it follows that K[X]F+ is generated by the elements

(1)1, (x1)1, . . . , (xn)1, (f1,1)1, . . . , (fn,m)1. Note that fi,j ∈ qj for each i, j, so that

(fi,j)1 ∈ qFj ⊆ I. Moreover, (1)1 ∈ I. So, all we really need to show is that (xi)1 ∈√I

for all i = 1, . . . , n.

Reducing equation (1.2) mod p, we have (xi)di = fi,1 + . . . + fi,m + gi ∈ K[X] for all

i = 1, . . . , n. Let gi =∑di−1

j=0 ai,jxji . Then in K[X]F ,

((xi)1)di = ((1)1)di−1((fi,1)1 + . . .+ (fi,m)1) +

di−1∑j=0

ai,j((xi)1)j((1)1)di−j.

All of the summands in the right hand side lie inside I, hence ((xi)1)di ∈ I for all

i = 1, . . . , n, as required.

Recall that given a polynomial map f = (f1, . . . , fq) : X → Kq, a = (a1, . . . , aq) ∈

Kq and a completion ψ of X, ψ preserves {f1, . . . , fn} at ∞ over a if ψ preserves the

intersection of the hypersurfaces Hi(a) := {x ∈ X : fi(x) = ai}, i = 1, . . . , q.

Example 1.3.3. Consider map f : K2 → K2 given by f(x, y) := (x, y + x3). For

a := (a1, a2) ∈ K2,

H1(a) = {(a1, y) : y ∈ K},

H2(a) = {(x, a2 − x3) : x ∈ K}.

We claim that in the usual completion P2(K) of K2, the closures of H1(a) and H2(a)

intersect at a point P at infinity for each a ∈ K2, and hence P2(K), as the natural

completion of K2, does not preserve {f1, . . . , fn} at ∞ over any point of K2.

Indeed, write the homogeneous coordinates of P2(K) as [z : x : y] and identify K2

with P2(K) \ V (z). Let a ∈ K2. When K = C, the ‘infinite’ points in H i(a) can be


described as the limits of points in Hi(a). Therefore, the points at infinity of H1(a) are

lim|y|→∞[1 : a1 : y] = lim|y|→∞[1/y : a1/y : 1] = [0 : 0 : 1]. Similarly, the infinite part of

H2(a) is lim|x|→∞[1 : x : a2 − x3] = lim|x|→∞[1/(a2 − x3) : x/(a2 − x3) : 1] = [0 : 0 : 1],

and hence the claim is true with P := [0 : 0 : 1].

To verify the claim for an arbitrary K, one has to apply lemma 1.3.1 with X = K2

and calculate H i(a) ∩ X∞ = V (qFi (a), (1)1), where qi(a) is the ideal of Hi(a). A

straightforward calculation shows: the graded ring K[X]F corresponding to the em-

bedding K2 ↪→ P2(K) is isomorphic to K[x, y, z] where z plays the role of (1)1, and

qFi (a) is the homogenization of qi(a) with respect to z. Then q1(a) = 〈x − a1〉 and

q2(a) = 〈y + x3 − a2〉, so that qF1 (a) = 〈x− a1z〉 and qF2 (a) = 〈yz2 + x3 − a2z3〉. There-

fore H1(a) ∩ X∞ = V (x − a1z, z) = V (x, z) = {[0 : 0 : 1]}. Similarly H2(a) ∩ X∞ =

V (yz2 + x3 − a2z, z) = V (x, z) = {[0 : 0 : 1]} and the claim is valid with the same P as

in K = C case.

Because it is simpler to describe, we will from now on frequently use only the limit

argument (valid only for K = C) in order to find the points at infinity of various subvari-

eties of a given X. In all these cases, the analogous results also follow over an arbitrary

algebraically closed field K by means of straightforward calculations (and if char K = 0,

by Tarski-Lefschetz principle).

We now find, following the proof of Theorem 1.3.2, a completion of K2 which preserves

{f1, . . . , fn} at ∞ over 0. In the notation of theorem 1.3.2, q1 = 〈x〉 and q2 = 〈y + x3〉.

Observe that x ∈ q1, and y satisfies

y = −x3 + (y + x3),

with x3 ∈ q1 and y + x3 ∈ q2. Let filtration F := {Fi : i ≥ 0} on K[x, y] be defined

as follows: F0 := K, F1 := K〈1, x, y, x3〉, and Fk := (F1)k for k > 1. Then as in the

proof of theorem 1.3.2, completion ψF preserves {f1, . . . , fn} at ∞ over 0. Let us now

show this directly. By corollary 1.1.5, the corresponding completion XF is isomorphic

to the closure in P3(K) of the image of φ : K2 → P3(K), where φ(x, y) := [1 : x : y : x3].


Then φ(H1(a)) = {[1 : a1 : y : a31] : y ∈ K} and limit limy→∞[1 : a1 : y : a3

1] =

limy→∞[1/y : a1/y : 1 : a31/y] = [0 : 0 : 1 : 0], for a ∈ K2, so that the only point at infinity

of H1(a) is [0 : 0 : 1 : 0]. Similarly, φ(H1(a)) = {[1 : x : a2 − x3 : x3] : x ∈ K} and

limx→∞[1 : x : a2 − x3 : x3] = limx→∞[1/x3 : 1/x2 : (a2 − x3)/x3 : 1] = [0 : 0 : −1 : 1].

Therefore H2(a) also has only one point at infinity and it is [0 : 0 : −1 : 1]. It follows

that H1(a) ∩H2(a) ∩X∞ = ∅ for all a, i.e. XF preserves {f1, . . . , fn} at ∞ over every

point of K2.

Example 1.3.4. Let f(x, y) := (x, y) on K2. Then for each a = (a1, a2) ∈ K2, H1(a) =

{(a1, y) : y ∈ K} and H2(a) = {(x, a2) : x ∈ K}. Consider filtration F on K[x, y]

defined by: F0 := K, F1 := K〈1, x, y, xy, x2y2〉, and Fk := (F1)k for k ≥ 2. By corollary

1.1.5, XF is the closure of the image of K2 under the map φ : K2 ↪→ P4(K) defined by:

φ(x, y) = [1 : x : y : xy : x2y2]. Then φ(H1(a)) = {[1 : a1 : y : a1y : a21y

2] : y ∈ K}.

If a1 = 0, then φ(H1(a)) = {[1 : 0 : y : 0 : 0] : y ∈ K}, and hence the only point at

infinity in φ(H1(a)) is [0 : 0 : 1 : 0 : 0]. But if a1 6= 0, then dividing all coordinates

by a21y

2, we see that the point at infinity in φ(H1(a)) is [0 : 0 : 0 : 0 : 1]. Similarly,

φ(H2(a)) = {[1 : x : a2 : y : a2x : a22x

2] : x ∈ K} and the only point at infinity in

φ(H2(a)) is [0 : 1 : 0 : 0 : 0] if a2 = 0, and [0 : 0 : 0 : 0 : 1] if a2 6= 0. Therefore XF

preserves {f1, . . . , fn} at ∞ over a iff a1 = 0 or a2 = 0, i.e. iff a belongs to the union of

the coordinate axes.

Let f : X → Y be a generically finite map of affine varieties of the same dimension.

Given any y ∈ Y such that f−1(y) is finite, theorem 1.3.2 guarantees the existence of a

projective completion of X that preserves {f1, . . . , fn} at∞ over y. But as the preceding

example shows, the completion might fail to preserve {f1, . . . , fn} at∞ over ‘most of the’

points in the image of f . This suggests that we should look for a completion ψ which

preserves {f1, . . . , fn} at ∞ over y for generic y ∈ Y , i.e. ψ preserves {f1, . . . , fn} at ∞.

We will demonstrate two ways to accomplish this goal - we start with a simpler-to-prove

theorem 1.3.5 and will present a stronger version in theorem 1.3.7 following (cf. [25,


Theorem 1.2] and [26, Theorem 1.3.4]).

Theorem 1.3.5. ∗ Let f : X → Y ⊆ Kq be a generically finite map of affine va-

rieties of same dimension. Include Kq into (P1(K))q via the componentwise inclusion

(a1, . . . , aq) 7→ ([1 : a1], . . . , [1 : aq]). Let φ : X ↪→ Z be any completion of X. Define X

to be the closure of the graph of f in Z × (P1(K))q. Then X preserves {f1, . . . , fn} at

∞. If φ comes from some filtration on K[X], then there is a filtration F on K[X] and a

commutating diagram as follows:

X

~~}}}}""DDDD

X∼= // XF

Proof. Let π := (π1, . . . , πq) : Z × (P1(K))q → (P1(K))q be the natural projection.

Then π maps X onto the closure Y of Y in (P1(K))q. Let V := Z \ X. Then V :=

(V × (P1(K))q)∩ X is a proper Zariski closed subset of X. Since Z is complete, it follows

that π(V ) is a proper Zariski closed subset of Y . We now show that for all y ∈ Y \π(V ),

X preserves {f1, . . . , fn} at ∞ over y.

Pick an arbitrary y := (y1, . . . , yq) ∈ Y such that X does not preserve {f1, . . . , fn} at

∞ over y. It suffices to show that y ∈ π(V ). As usual, let Hi(y) := {x ∈ X : fi(x) = yi}.

By assumption there is x ∈ H1(y)∩ · · · ∩ Hq(y)∩ (X \X), where for each k, Hk(y) is the

closure of Hk(y) in X. Fix a k, 1 ≤ k ≤ q. Note that πk(Hk(y)) = {yk}. By continuity of

πk it follows that πk(Hk(y)) = {yk}. But then πk(x) = yk. It follows that π(x) = y and

hence x is of the form (z, y) for some z ∈ Z. We claim that z does not lie in X. Indeed,

if z ∈ X, it would imply x ∈ (X × Y ) ∩ (X \ ψ(X)), where ψ : X ↪→ X is the inclusion.

Consider the chain of inclusions: ψ(X) ⊆ X × Y ⊆ Z × (P1(K))q. Note:

1. ψ(X) is the graph of f in X×Y , and hence is a Zariski closed subvariety of X×Y .

2. X × Y is Zariski open in Z × (P1(K))q.

3. If T ⊆ U ⊆ W are topological spaces such that T is closed in U and U is open in

∗The idea of looking at the construction of theorem 1.3.5 is due to Professor A. Khovanskii.


W , then T ∩ U = T , where T is the closure of T in W .

Since X is by definition the closure of ψ(X) in Z × (P1(K))q, it follows via the above

observations that X ∩ (X × Y ) = ψ(X), so that (X × Y ) ∩ (X \ ψ(X)) = ∅. This

contradiction proves the claim. It follows that z ∈ Z \X = V . Then x ∈ V . Therefore

y = π(x) ∈ π(V ) and the first claim of the theorem is proved.

As for the last claim, note that if φ comes from a filtration, then by corollary 1.1.5

we may assume that Z ⊆ Pp(K) for some p and φ(x) = [1 : g1(x) : · · · : gp(x)] for

some g1, . . . , gp ∈ K[X]. Hence the inclusion ψ : X ↪→ X is of the form: ψ(x) =

([1 : g1(x) : · · · : gp(x)], [1 : f1(x)], . . . , [1 : fq(x)]). Let l := (p + 1)2q − 1 and let

us embed Pp(K) × (P1(K))q ↪→ Pl(K) via the Segre embedding s which maps w :=

([w0 : · · · : wp], [w1,0 : w1,1], . . . , [wq,0 : wq,1]) to the point s(w) whose homogeneous

coordinates are monomials of degree q + 1 in w of the form wiw1,j1w2,j2 · · ·wq,jq where

0 ≤ i ≤ p and 0 ≤ jk ≤ 1 for each k. The component of s ◦ ψ corresponding to

i = j1 = · · · = jq = 0 is 1 and hence s ◦ ψ maps x ∈ X to a point with homogeneous

coordinates [1 : h1(x) : · · · : hl(x)] for some h1, . . . , hl ∈ K[X]. Then corollary 1.1.5

implies that there is a filtration F on K[X] such that the closure X of s ◦ψ(X) in Pl(K)

is isomorphic to XF via an isomorphism which is identity on X. Morphism s being an

isomorphism completes the proof.

Remark 1.3.6. Let f : X → Y be a map of n-dimensional affine varieties with gener-

ically finite fibers and ψ be a completion of X. Define Sψ := {a ∈ f(X) : ψ preserves

{f1, . . . , fn} at∞ over y}. It will be interesting to know if Sψ has any intrinsic structure.

In example 1.3.4 Sψ was the union of two coordinate axes in K2, and hence a proper closed

subset of f(X). On the other hand, theorem 1.3.5 shows that there are completions ψ of

X such that Sψ contains a dense open subset of f(X). We now give an example where

Sψ is indeed a proper dense open subset of f(X), namely:

Let X = Y = C2 and f : X → Y be the map defined by f1 := x31 + x2

1x2 + x1x22 − x2

and f2 := x31 +2x2

1x2 +x1x22−x2. It is easy to see that f is quasifinite. Let φ : X ↪→ P2(C)


be the usual completion, and let ψ : X ↪→ X be as in theorem 1.3.4. Let the coordinates

of P2(C) be [Z : X1 : X2]. Identify X with P2(C) \ V (Z), so that xi = Xi/Z for i = 1, 2.

Then X 3 (x1, x2)ψ7→ ([1 : x1 : x2], [1 : f1(x)], [1 : f2(x)]). We claim that f(X) \ Sψ is the

line L := {(c, c) : c ∈ C}.

Indeed, let a := (a1, a2) ∈ Y . Define, as usual, Hi(a) := {x ∈ X : fi(x) = ai} for

i = 1, 2. Let Ci(a) be the closure in P2(C) of Hi(a) for each i. It is easy to see that

P := [0 : 0 : 1] ∈ C1(a) ∩ C2(a). Choose local coordinates ξ1 := X1/X2 and ξ2 := Z/X2

of P2(C) near P := [0 : 0 : 1]. Equations of fi,a in (ξ1, ξ2) coordinates are:

f1,a = ξ31 + ξ2

1 + ξ1 − ξ22 − a1ξ

32

f2,a = ξ31 + 2ξ2

1 + ξ1 − ξ22 − a2ξ

32

(1.3)

It follows that for each i, Ci(a) is smooth at P (in particular, each has only one branch

at P ) and both admit parametrizations at P of the form

γi,a(t) := [t : t2 + o(t3) : 1], (1.4)

where o(t3) means terms of order t3 and higher. In (x1, x2) coordinates the parametriza-

tions are of the form: (x1(t), x2(t)) = (t+o(t2), 1/t) for t 6= 0. Since f2(x) = f1(x)+x21x2,

it follows that for t 6= 0,

ψ(γ1,a(t)) = (γ1,a(t), [1 : a1], [1 : a1 +(t+ o(t2))2

t])

= (γ1,a(t), [1 : a1], [1 : a1 + t+ o(t2)])

Therefore limt→0 ψ(γ1,a(t)) = (P, [1 : a1], [1 : a1]). Since f1(x) = f2(x) − x21x2, the same

argument also gives limt→0 ψ(γ2,a(t)) = (P, [1 : a2], [1 : a2]).

To summarize, we proved that if a ∈ L, then (P, [1 : a1], [1 : a1]) ∈ H1(a)∩H2(a)∩X∞,

where as usual Hi(a) is the closure of Hi(a) in X and X∞ := X \ X. It follows that

L ⊆ f(X) \ Sψ.

To prove the other inclusion, assume a ∈ f(X) \ Sψ. Pick z ∈ H1(a) ∩ H2(a) ∩X∞.

Then z = (Q, [1 : a1], [1 : a2]) for a point Q ∈ P2(C). Therefore it follows that Q ∈


(C1(a) \H1(a))∩ (C2(a) \H2(a)), where curves Ci(a) are the closures in P2(C) of Hi(a),

i = 1, 2. But the only possible choice for such point Q is point P . Therefore limit

limt→0 ψ(γ1,a(t)) = z = limt→0 ψ(γ2,a(t)), which implies that (P, [1 : a1], [1 : a1]) =

(P, [1 : a2], [1 : a2]). Therefore a1 = a2 and a ∈ L, as claimed.

Let f : X → Y ⊆ Kq be as in theorem 1.3.5. In the theorem following we find

completions with even stronger preservation property at ∞, namely completions that

preserve map f at ∞ (remark-definition 0.1.2).

Theorem 1.3.7 (cf. [25, Theorem 1.2(2)] and [26, Theorem 1.3.4]). Let f : X → Y ⊆ Kq

be a generically finite map of affine varieties of the same dimension. Then there is a

complete filtration F on the coordinate ring of X such that ψF preserves map f at ∞.

Proof. Choose a set of coordinates x1, . . . , xp (resp. y1, . . . , yq) of X (resp. Y ). Since f

is generically finite, it follows that the coordinate ring of X is algebraic over the pullback

of the coordinate ring of Y . In particular, each xi satisfies a polynomial of the form:

ki∑j=0

gi,j(y)(xi)j = 0 (1.5)

for some ki ≥ 1 and regular functions gi,j(y) on Y such that gi,ki 6= 0 ∈ K[Y ]. In abuse

of notation, but for the sake of convenience, we implicitly identified in (1.5) variables

yk with polynomials fk for each k. We continue to do so throughout this proof. Let

gi,j(y) =∑

α ci,j,αyα be an arbitrary representation of gi,j in K[Y ]. For each i, j with

1 ≤ i ≤ p and 0 ≤ j ≤ ki, let di,j := degy(∑

α ci,j,αyα) := max{|α| : ci,j,α 6= 0}, where

α = (α1, . . . , αq) ∈ (Z+)q and |α| := α1 + · · · + αq. Let d0 := max{di,0 : 1 ≤ i ≤ p} and


k0 := max{ki : 1 ≤ i ≤ p}. Define a filtration F := {Fi : i ≥ 0} on K[X] as follows:

F0 := K,

F1 := K〈1, x1, . . . , xp, y1, . . . , yq〉 + K〈yβ : |β| ≤ d0〉 +

+ K〈xiyβ : |β| ≤ di,1, 1 ≤ i ≤ p〉,

Fk :=

∑k−1

j=1 FjFk−j + K〈(xi)kyβ : |β| ≤ di,k, 1 ≤ i ≤ p〉 if 1 < k ≤ k0,∑k−1j=1 FjFk−j if k > ki ∀i.

Let g :=∏m

i=1 gi,ki and U := {a ∈ Y : g(a) 6= 0}. Then U is a non-empty Zariski open

subset of Y . Let ξ := (ξ1, . . . , ξq) : Kq → Kq be an arbitrary linear change of coordinates

of Kq. It suffices to show that ψF preserves the components of ξ ◦ f at ∞ over ξ(a) for

a := (a1, . . . , aq) ∈ U . For each a ∈ Y , let Hj(a) := {x ∈ X : (ξj ◦ f)(x) = ξj(a)} and let

qj(a) be the ideal of Hj(a), i.e. the ideal of K[X] generated by ξj(y)− ξj(a). By lemma

1.3.1, ψF preserves the components of ξ◦f at infinity over ξ(a) iff√I(a) = K[X]F+, where

I(a) is the ideal of K[X]F generated by qF1 (a), . . . , qFn (a) and (1)1. Note the following:

(a) Since ξ is a linear change of coordinate, so is ξ−1. Therefore, for all d ≥ 0, the K-span

of {yβ : |β| ≤ d} in K[Y ] is equal to the K-span of {(ξ−1(y))β : degy((ξ−1(y))β) ≤ d}.

(b) If we replace f by ξ ◦ f , and hence y by ξ(y), then gi,j(y) changes to gξi,j(y) :=

gi,j(ξ−1(y)) =

∑α ci,j,α(ξ−1(y))α. But replacing

∑α ci,j,αy

α by∑

α ci,j,α(ξ−1(y))α does

not change its degree di,j in y.

(c) Let gξ :=∏m

i=1 gξi,ki

. Then gξ(ξ(a)) 6= 0 iff g(a) 6= 0.

In view of the latter observations and the construction of F it follows that F does

not change if we replace f by ξ ◦ f . Moreover, the following two claims are equivalent

due to properties (a), (b) and (c) of the preceding paragraph.

(1) ψF preserves the components of f at ∞ over a ∈ U , and

(2) ψF preserves the components of ξ ◦ f at ∞ over a ∈ ξ−1(U).

Therefore it suffices to prove (1) and we may without loss of generality assume ξ

to be the identity. Note that K[X]F+ is generated as a K-algebra by elements (1)1,

(x1)1, . . . , (xp)1, (y1)1, . . . , (yq)1, the (yβ)1’s that appear in the definition of F1, and all


those ((xi)kyβ)k that we inserted in the definition of all Fk’s. Therefore

√I(a) = K[X]F+

iff some power of each of these generators lies in I(a).

Lemma 1.3.7.1. Let a be an arbitrary point in Y .

1. Let β ∈ (Z+)q be such that yβ ∈ F1. then

(a) (y − a)β also lies in F1, and

(b) ((y − a)β)1 ∈ I(a).

2. Let 1 ≤ i ≤ p. Pick k with 1 ≤ k ≤ ki and β ∈ (Z+)q such that xki yβ ∈ Fk. Then

(a) xki (y − a)β lies in Fk, and

(b) if in addition β 6= 0, then (xki (y − a)β)k ∈ I(a).

Proof. 1. Pick β ∈ (Z+)q such that yβ ∈ F1. Expanding (y − a)β in powers of

y1, . . . , yq, we see that (y − a)β =∑|γ|≤|β| cγy

γ for some cγ ∈ K and γ ∈ (Z+)q. By

construction, F1 contains each of the yγ appearing in the preceding expression. It follows

that F1 also contains (y − a)β, which proves assertion 1a. As for 1b, note that if β = 0,

then ((y − a)β)1 = (1)1 ∈ I(a). Otherwise, there exists j, 1 ≤ j ≤ q, such that the j-th

coordinate of β is positive. Then (y−a)β ∈ qj(a), and hence ((y−a)β)1 ∈ qFj (a) ⊆ I(a),

which completes the proof of assertion 1.

2. Let 1 ≤ i ≤ p. Pick k, β such that 1 ≤ k ≤ ki and xki yβ ∈ Fk. As in the proof

of assertion 1, expanding (y − a)β in powers of y1, . . . , yq, we see that xki (y − a)β =∑|γ|≤|β| cγx

ki y

γ for some cγ ∈ K. By construction, Fk contains xki yγ for each |γ| ≤ |β|. It

follows that Fk also contains xki (y − a)β, and this proves assertion 2a. For 2b, note that

if β 6= 0, then there exists j, 1 ≤ j ≤ q, such that the j-th coordinate of β is positive.

Then xki (y − a)β ∈ qj(a), and hence (xki (y − a)β)k ∈ qFj (a) ⊆ I(a), which completes the

proof of the lemma.

We now return to the proof of theorem 1.3.7. Let a be any point in Y and β be

such that yβ ∈ F1. Expanding yβ in powers of y1 − a1, . . . , yq − aq, we see that yβ =∑|γ|≤|β| c

′γ(y−a)γ for some c′γ ∈ K. By assertion 1a of lemma 1.3.7.1, each (y−a)γ in the


preceding expression lies in F1. Therefore, in K[X]F element (yβ)1 =∑|γ|≤|β| c

′γ((y−a)γ)1.

But assertion 1b of lemma 1.3.7.1 implies that ((y − a)γ)1 ∈ I(a) for all |γ| ≤ |β|.

Therefore (yβ)1 ∈ I(a).

Now expand the polynomial of the left hand side of equation (1.5) (as a polynomial

in y := (y1, . . . , yq)) in powers of y1 − a1, . . . , yq − aq. This leads to an equation of the

form

ki∑j=0

gi,j(a)xji +∑j≤kiβ 6=0

hi,j,β(a)(y − a)βxji = 0 , (1.5′)

where terms (y−a)βxji , for β 6= 0 and j ≤ ki, that appear in (1.5′) are such that yβxji ∈ Fj,

and due to assertion 2a of lemma 1.3.7.1, (y− a)βxji are in Fj ⊆ Fki . Therefore equation

(1.5′) implies the following equality in K[X]F :

gi,ki(a)((xi)1)ki = −ki−1∑j=0

gi,j(a)((xi)1)j((1)1)ki−j −∑j≤kiβ 6=0

hi,j,β(a)((y − a)βxji )ki . (1.5′′)

Since (1)1 ∈ I(a), each of the terms under the first summation of the right hand side of

(1.5′′) lies in I(a). Moreover, by assertion 2b of lemma 1.3.7.1, every ((y−a)βxji )ki under

the second summation of the right hand side of (1.5′′) also belongs to I(a). It follows

that gi,ki(a)((xi)1)ki ∈ I(a), and therefore (xi)1 ∈√I(a) if gi,ki(a) 6= 0. Hence for all

a ∈ U and i, 1 ≤ i ≤ p, elements (xi)1 ∈√I(a).

Let a ∈ U and 1 ≤ i ≤ p. Pick (xi)kyβ ∈ Fk with 1 ≤ k ≤ ki. Expanding yβ in powers

of y1 − a1, . . . , yq − aq, we see that (xi)kyβ =

∑|γ|≤|β| c

′γ(xi)

k(y − a)γ for some c′γ ∈ K.

By assertion 2a of lemma 1.3.7.1, each (xi)k(y − a)γ in the preceding expression lies in

Fk. Therefore in K[X]F element ((xi)kyβ)k =

∑|γ|≤|β| c

′γ((xi)

k(y− a)γ)k. If |γ| ≤ |β| and

γ 6= 0, assertion 2b of lemma 1.3.7.1 implies that ((xi)k(y−a)γ)k ∈ I(a) and if γ = 0, then

((xi)k(y − a)γ)k = ((xi)

k)k = ((xi)1)k ∈√I(a) due to the conclusion of the preceding

paragraph. Therefore for xki yβ ∈ Fk, 1 ≤ k ≤ ki, it follows that ((xi)

kyβ)k ∈√I(a).

Consequently√I(a) = K[X]F+ for all a ∈ U , which completes the proof.


Example 1.3.8. Let X = Y = C2 and f := (f1, f2) : X → Y be the map defined by

f1 := x31 + x2

1x2 + x1x22 − cx2 and f2 := x3

1 + x21x2 + x1x

22 − x2 for any complex number

c 6= 0 or 1. Note that this map is a minor variation of the map considered in remark

1.3.6. Let ψ : X ↪→ X ⊆ P2(C)×P1(C)×P1(C) be the completion considered in theorem

1.3.5 and remark 1.3.6, i.e. ψ(x1, x2) := ([1 : x1 : x2], [1 : f1(x1, x2)], [1 : f2(x1, x2)]) for

all (x1, x2) ∈ X. Below we show that ψ does not satisfy the preservation property of

theorem 1.3.7.

For each λ := (λ1, λ2) ∈ C2 let

fλ := λ1f1 + λ2f2 = (λ1 + λ2)x31 + (λ1 + λ2)x2

1x2 + (λ1 + λ2)x1x22 − (λ1c+ λ2)x2 .

Fix λ1, λ2 ∈ C2 \ (L1 ∪ L2 ∪ L3) where L1 is the x1-axis, L2 is the x2-axis and L3 is the

line {(x1, x2) ∈ C2 : x1 + x2 = 0}. Fix an a := (a1, a2) ∈ Y . Let bi := λi1a1 + λi2a2,

i = 1, 2. Every Hi(a) = {x ∈ X : fλi(x) = bi} is as in theorem 1.3.7, i = 1, 2. Let Ci(a)

be the closure in P2(C) of Hi(a) for i = 1, 2. Then P := [0 : 0 : 1] ∈ C1(a) ∩ C2(a),

where as in remark 1.3.6, we choose coordinates [Z : X1 : X2] on P2(C) and identify X

with P2(C)\V (Z). Choose local coordinates ξ1 := X1/X2 and ξ2 := Z/X2 on P2(C) near

P := [0 : 0 : 1]. Equations of fλi(x1, x2)− bi in (ξ1, ξ2) coordinates are:

fλi(x1, x2)− bi = (λi1 + λi2)ξ31 + (λi1 + λi2)ξ2

1 + (λi1 + λi2)ξ1 − (λi1c+ λi2)ξ22 − aiξ3

2 .

Since λi1 + λi2 6= 0, it follows (similarly to the implication (1.3) ⇒ (1.4) of remark 1.3.6)

that for each i, Ci(a) is smooth at P and has a parametrization at P of the form

γi,a(t) := [t :λi1c+ λi2λi1 + λi2

t2 + o(t3) : 1] .

In (x1, x2) coordinates the parametrizations are: (x1(t), x2(t)) = (λi1c+λ

i2

λi1+λi2t+ o(t2), 1/t) for

t ∈ C∗. A straightforward calculation shows that for t ∈ C∗

f1(γi,a(t)) =λi2(1− c)λi1 + λi2

1

t+ o(t2) and

f2(γi,a(t)) =λi1(c− 1)

λi1 + λi2

1

t+ o(t2) ,

(1.6)


i = 1, 2. By our choice of c 6= 0, 1 and λi’s off L1 ∪L2 ∪L3 it follows that the coefficients

at 1t

in the expressions in (1.6) are non-zero. It follows that limt→0 |fj(γi,a(t))| = ∞ for

each i, j, and therefore

limt→0

ψ(γi,a(t)) = limt→0

([1 : γi,a(t)], [1 : f1(γi,a(t))], [1 : f2(γi,a(t))])

= ([0 : 0 : 1], [0 : 1], [0 : 1]) .

Hence ([0 : 0 : 1], [0 : 1], [0 : 1]) is in the closure of Hi(a) in X for i = 1, 2 and every

a ∈ Y . It follows that ψ does not preserve (fλ1 , fλ2) at∞ over any point in Y , as claimed.

Finally, we follow the proof of theorem 1.3.7 to find a completion which preserves

map f at ∞. The algebraic equations satisfied by x1 and x2 over C[f1, f2] are:

x31 +

1

1− c(f1 − f2)x2

1 +1

(1− c)2(f1 − f2)2x1 −

1

1− c(f1 − cf2) = 0, and

x2 −1

1− c(f1 − f2) = 0.

In the notations of the proof of theorem 1.3.7, d1,2 = d1,0 = 1, d1,1 = 2 and d2,0 = 1. Let

F := {Fd : d ≥ 0} be the filtration defined by:

F0 := K, F1 := K〈1, x1, x2, f1, f2, x1f1, x1f2, x1f21 , x1f1f2, x1f

22 〉, Fd := (F1)d for d ≥ 2 .

Construction of the proof of 1.3.7 yields that XF preserves map f at∞. Indeed, ψF(x) =

[1 : x1 : x2 : f1(x) : f2(x) : x1f1(x) : x1f2(x) : x1f21 (x) : x1f1(x)f2(x) : x1f

22 (x)] ∈ P9(C)

for x ∈ C2 and therefore applying (1.6) it follows that limt→0 ψF(γi,a(t)) =

limt→0

[1 :

(λi1c+ λi2λi1 + λi2

t+ o(t2)

):

1

t:

(λi2(1− c)λi1 + λi2

1

t+ o(t2)

):

(λi1(c− 1)

λi1 + λi2

1

t+ o(t2)

): o(1) :

o(1) :

((1− c)2(λi2)2(λi1c+ λi2)

(λi1 + λi2)3

1

t+ o(t2)

):

(−(1− c)2λi1λ

i2(λi1c+ λi2)

(λi1 + λi2)3

1

t+ o(t2)

):(

(1− c)2(λi1)2(λi1c+ λi2)

(λi1 + λi2)3

1

t+ o(t2)

)]

= [0 : 0 : 1 :λi2(1− c)λi1 + λi2

:λi1(c− 1)

λi1 + λi2: 0 : 0 :

(1− c)2(λi2)2(λi1c+ λi2)

(λi1 + λi2)3:

− (1− c)2λi1λi2(λi1c+ λi2)

(λi1 + λi2)3:

(1− c)2(λi1)2(λi1c+ λi2)

(λi1 + λi2)3] .


If λ1 and λ2 are linearly independent, then it follows that limits limt→0 ψF(γi,a(t)) are

different for i = 1, 2, and therefore, completion ψF preserves {fλ1 , fλ2} at ∞ over every

a ∈ Y = C2.

Chapter 2

Intersection preserving filtrations

with nilpotent-free graded rings

We show that in the main theorem of chapter 1 it suffices to take a restricted class of

filtrations, namely those with nilpotent-free graded rings. A filtration F := {Fd : d ∈ Z}

on the coordinate ring A of an affine variety is, in general, uniquely determined by the

corresponding degree like function δ : A \ {0} → Z via Fd = {f : δ(f) ≤ d}. The

second main theorem of this chapter (which appears first!) states that if the graded ring

grAF :=⊕

Fd/Fd−1 has no nilpotents, then the associated degree like function, which

we then call a subdegree, is the maximum of a finite collection of degree like functions

that send products into sums. We also show that completions determined by subdegrees

have various other properties resembling toric completions.

2.0 Background

In section 2.0.1 we briefly introduce projective toric varieties, which will form a large

part of the examples in this chapter. Here we mostly follow the exposition in [11]. We

describe in section 2.0.2 the notion of Cartier and Weil divisors on arbitrary varieties

following [10].

42

Chapter 2. Filtrations with nilpotent-free graded rings 43

2.0.1 Lattice points and Toric Varieties

The affine variety (C∗)n is a group under component-wise multiplication. It is actually

an algebraic group, meaning the group multiplication map (C∗)n × (C∗)n → (C∗)n and

the inverse map (C∗)n → (C∗)n are regular maps of varieties. An n dimensional torus

is an affine variety T isomorphic to (C∗)n, where T inherits a group structure from the

isomorphism. A complex algebraic variety X is a toric variety if it contains a torus T as

a Zariski open subset and the action of T on itself extends continuously to an action of

T on X.

Example 2.0.1. (C∗)n is an affine toric variety, so is Cn for all n. The weighted projective

spaces are examples of projective toric varieties.

In this text, a lattice is a free abelian group of finite rank and a lattice homomorphism

is a homomorphism of abelian groups between two lattices. An affine transformation

between lattices L1 and L2 is a map φ : L1 → L2 which is of the form φ(·) = ψ(·) +α for

a lattice homomorphism ψ and a fixed α ∈ L2. Let A := {α0, . . . , αk} be a finite subset of

ZL for some L ≥ 0. The affine hull of A is aff(A) := {∑k

i=0 aiαi : ai ∈ Z ,∑k

i=0 ai = 1}.

There is a unique sublattice L of ZL such that aff(A) = L + α for any α ∈ aff(A).

Dimension of aff(A) is the rank of L as a free abelian group. Let φA : (C∗)L → Pk(C)

be the map defined by φA(x) := [xα0 : · · · : xαk ] and let XA be the closure of φA((C∗)L)

in Pk(C).

Theorem 2.0.2.

1. XA is a toric variety. The torus T of XA is precisely φA((C∗)L). Dimension of

XA is the same as the dimension of aff(A), say n. In particular, T is isomorphic

to (C∗)n. The action of T on XA is given by coordinatewise multiplication, i.e.

[t0 : · · · : tk] · [x0 : · · · : xk] := [t0x0 : · · · : tkxk].

2. For any K ≥ 0, if B is a finite subset of ZK and φ : ZK → ZL is an affine

transformation such that φ|B : B → A is an isomorphism of sets, then XB is


isomorphic to XA as toric varieties.

3. Let P be the convex hull of A in RL. There is a one-to-one correspondence between

the faces of P and torus orbits of XA. The orbit corresponding to a face Q of

P is OQ := {[z0 : . . . : zk] ∈ XA : zi = 0 iff αi 6∈ Q}. The closure of OQ is

VQ := {[z0 : . . . : zk] ∈ XA : zi = 0 if αi 6∈ Q}. VQ is a subvariety of XQ which is

isomorphic to XA∩Q and its dimension is the same as the Euclidean dimension of

Q. The correspondence Q ←→ VQ is inclusion preserving, i.e. Q ⊆ R iff VQ ⊆ VR.

It follows that XA \ T =⋃{VQ : Q is a facet (i.e. codimension one face) of P}.

4. For each m ≥ 1, define Am := mP ∩ ZN . Then the toric varieties XAm are

isomorphic for m ≥ n − 1. Let XP denote any representative of the isomorphism

class of these varieties. Then XP is the normalization of XA; in particular, it is

normal.

Example 2.0.3. Let A := {(0, 0), (1, 0), (0, 1), (2, 1)} ⊆ Z2. The convex hull P of A is

drawn in the picture below, with its faces and vertices marked.

Embedding φA : (C∗)2 3 (x, y) 7→ [1 : x : y : x2y] ∈ P3(C). Variety φA((C∗)2) is a

group under coordinatewise multiplication, and map φA : (C∗)2 → φA((C∗)2) is a group

isomorphism with the inverse being [W : X : Y : Z] 7→ (XW, YW

). Thus we see that XA is

indeed a toric variety with its torus T := φA((C∗)2) ∼= (C∗)2.

In homogeneous coordinates [W : X : Y : Z] on P3(C), subvariety XA coincides with

V (W 2Z − X2Y ), and a straightforward calculation shows that T = XA \ V (WXY Z).

Therefore XA\T = V (W 2Z−X2Y,WXY Z) = V (Y, Z)∪V (Z,X)∪V (X,W )∪V (W,Y ).


Note that each of the four components are 1 dimensional (in fact isomorphic to P1(C))

and they are preserved under the action of T . In the notation of assertion 3 of theorem

2.0.2 these components are precisely VF1 , . . . , VF4 (the correspondence being in the same

order as listed). The zero dimensional orbits, or fixed points of the action of T , are

Vv1 = V (Y, Z,X), Vv2 = V (Z,X,W ), Vv3 = V (X,W, Y ) and Vv4 = V (W,Y, Z).

Note that XA is not normal. Indeed, let ξ1 := W/Z, ξ2 := X/Z and ξ3 := Y/Z

be coordinates of UZ := P3(C) \ V (Z). Then XA ∩ UZ = V (ξ21 − ξ2

2ξ3). Therefore the

coordinate ring of XA ∩ UZ is A := C[ξ1, ξ2, ξ3]/〈ξ21 − ξ2

2ξ3〉 ∼= C[ξ1, ξ2, ξ21/ξ

22 ]. But then

(ξ1/ξ2)2 ∈ A, even though ξ1/ξ2 6∈ A. It follows that A is not integrally closed. In

example 2.0.6 we will calculate XP and show that it is the normalization of XA.

2.0.2 Divisors

Let A be a commutative ring. A chain of prime ideals of a ring A is a finite increasing

sequence p0 ( p1 ( · · · ( pn, where each pi is a prime ideal of A; the length of the chain

is n. The Krull dimension of A, which we will refer to simply as dimension of A, is the

supremum of the lengths of all chains of prime ideals in A. Any field has dimension 0;

the ring of integers has dimension 1. The dimension of the coordinate ring of any affine

algebraic variety X over an algebraically closed field K coincides with the dimension of

X (which is the transcendence degree over K of the field of K(X) of rational functions

on X).

A chain of submodules of an A-module M of length n is an increasing sequence

M0 ( M1 ( · · · ( Mn, where each Mi is an A-submodule of M . The supremum of

the lengths of all chains of submodules of M is called the length of M . Let R be a

one dimensional Noetherian domain and K be the quotient field of R. For any nonzero

element r ∈ R, R/〈r〉 has finite length as an R-module. Denote this length by ordR(r).

Proposition 2.0.4 ([10, Appendix A3]). Map ordR : r → ordR(r) extends to a group

homomorphism from the multiplicative group K∗ := K \ {0} to Z. Let ordR denote the


latter extension as well. Then ordR(a/b) := ordR(a)−ordR(b) for every non-zero a, b ∈ R.

Let V be a codimension one subvariety of a variety Z. The local ring R := OV,Z of Z at

V is a one-dimensional local domain. For a nonzero r ∈ K(Z), define ordV (r) := ordR(r),

where ordR is defined as above. A Weil divisor on Z is a finite formal sum∑ni[Vi], where

Vi are the codimension one subvarieties of Z, and ni are integers. Its support is the union

of all Vi such that ni 6= 0, and it is effective if ni ≥ 0 for all i. A Cartier divisor D on

Z is defined by data (Uα, fα), where {Uα}α is an open covering of Z and fα are non-zero

elements in K(Z), subject to the condition that fα/fβ is a unit (i.e. a nowhere vanishing

regular function) on Uα ∩ Uβ. For each α, fα is called the local equation of D on Uα. If

in addition each fα is a regular function on Uα, then D is an effective Cartier divisor.

If V is a subvariety of Z of codimension one, let ordV (D) := ordV (fα) for any α such

that Uα ∩ V 6= ∅. The Weil divisor associated to D is [D] :=∑

ordV (D)[V ], where the

sum is over all codimension one subvarieties V of Z; this sum is finite, since there are

only finitely many V with ordV (D) 6= 0. If D is an effective Cartier divisor, then [D] is

an effective Weil divisor. Let Supp(D) :=⋃{V : ordV (D) 6= 0} denote the support of

divisor D.

Two Cartier divisors D and D′ are linearly equivalent if there exists a nonzero rational

function r on Z such that for all z ∈ Z, the quotient of the local equations of D and of

D′ at z is r. The Picard group Pic(Z) of Z is the group of Cartier divisors on Z modulo

linear equivalence. On the other hand, every nonzero rational function r on Z defines

a Weil divisor [div(r)] :=∑

ordV (r)[V ] called the principal divisor corresponding to r.

The quotient of the group of Weil divisors of Z by the subgroup of principal divisors is

called the divisor class group of Z, and is denoted by ClZ. The correspondence D → [D]

induces a homomorphism Pic(Z)→ Cl(Z) which is an embedding when Z is normal.

The theory of divisors on normal toric varieties is particularly tractable. Let A ⊆ ZL

be as in section 2.0.1, with P := the convex hull of A in RL. Assume that aff(A) = ZL

(by assertion 2 of theorem 2.0.2, this does not change the class of corresponding toric


varieties). Then each facet Q of P admits the ‘smallest inner normal’ ηQ: in other words

ηQ is the unique generator of the semigroup {η ∈ ZL : 〈η, α〉 ≤ 〈η, β〉 for all α ∈ Q and

β ∈ P}, where for all γ ∈ ZL, 〈η, γ〉 :=∑L

i=1 ηiγi is the usual inner product. Note that

by assertion 1 of theorem 2.0.2, dim(XP) = L, and we can (and will) identify (C∗)L with

the torus T of XA via the isomorphism induced by φP .

Theorem 2.0.5 ([9, Chapter 3]). For each facet Q of P, let VQ be the torus invariant

subvariety corresponding to Q (as in assertion 3 of theorm 2.0.2).

1. The principal divisor of xα on XA is [xα] =∑Q〈ηQ, α〉[VQ], where the sum is over

the facets Q of P.

2. {[VQ] : Q is a facet of P} generate Cl(XP) and there is a commutative diagram

with exact rows:

0→ ZL →⊕

Z[VQ]→ Cl(XP)→ 0,

where the map ZL →⊕

Z[VQ] is defined by α 7→∑Q〈ηQ, α〉[VQ].

Example 2.0.6. Let A and P be as in example 2.0.3. Below we reproduce the picture

of P along with the inner normals of its facets.

According to assertion 4 of theorem 2.0.2, variety XP ∼= XP∩Z2 . Since P ∩ Z2 =

{(0, 0), (1, 0), (0, 1), (1, 1), (2, 1)}, map φP sends (x, y) ∈ (C∗)2 to [1 : x : y : xy : x2y] ∈

P4(C). Let the homogeneous coordinates on P4(C) be [W : X : Y : T : Z]. A straight-

forward calculation shows that the ideal of XP ↪→ P4(C) is generated by polynomials

(TW −XY ), (ZW −XT ) and (T 2−Y Z). Let (x, y, z, t) := (X/W, Y/W,Z/W, T/W ) be

the affine coordinates on UW := P4(C) \ V (W ). Then in UW , variety XP ∩ UW =


V (t − xy, z − xt, t2 − yz) = V (t − xy, z − xt) ∼= C2. Similary one can show that

XP \V (G) ∼= C2 for the polynomials G being equal to any of the remaining homogeneous

coordinates. It follows that XP is non-singular; in particular it is normal. The equations

of the subvarieties Vi corresponding to faces Fi are: V1 = V (Y, T, Z), V2 = V (T, Z,X),

V3 = V (X,W, T 2 − Y Z) and V4 = V (W,Y, T ). The restriction to XP of projection

π : [W : X : Y : T : Z] 7→ [W : X : Y : Z] gives a well defined map from XP to

XA. Then π maps the torus of XP isomorphically onto the torus of XA, and maps Vi

onto Vi(XA) where Vi(XA) are the codimension one subvarieties of XA corresponding

to Fi (the subvarieties we denoted by Vi in example 2.0.3). In particular it is a finite

birational map, and therefore it is the normalization map (by the universal property of

normalization).

A straightforward calculation shows that map φP : C2 → P4(C) defined by φP :

(x, y) 7→ [1 : x : y : xy : x2y] embeds C2 isomorphically onto XP \ V (W ). Then V1 and

V2 are the closures of the x and, respectively, of the y-axis of C2. It follows that for each

α := (α1, α2) ∈ Z2, ordV1(xα1yα2) = α2 and ordV2(x

α1yα2) = α1. To calculate ordV3(·)

and ordV4(·), it suffices to examine the affine chart UZ := P4(C) \ V (Z) with coordinates

(ξ1, ξ2, ξ3, ξ4) := (T/Z,X/Z, Y/Z,W/Z). Then in UZ variety XP ∩ UZ coincides with

V (ξ1ξ4−ξ2ξ3 , ξ4−ξ2ξ1 , ξ21−ξ3). It follows that XP is a graph over the (ξ1, ξ2)-plane of the

map f(ξ1, ξ2) := (ξ21 , ξ1ξ2) and projection to (ξ1, ξ2)-plane maps XP ∩ UZ isomorphically

onto (ξ1, ξ2)-plane. Under this isomorphism V3 ∩ UZ and V4 ∩ UZ are mapped to the

ξ1 and, respectively, to the ξ2-axis. Note that x = X/W = ξ2/ξ4 = ξ2/(ξ1ξ2) = 1/ξ1,

and y = Y/W = ξ3/ξ4 = ξ21/(ξ1ξ2) = ξ1/ξ2. Hence xα1yα2 = ξα2−α1

1 ξ−α22 and therefore

ordV3(xα1yα2) = −α2 and ordV4(x

α1yα2) = α2−α1. Finally, observe that the smallest inner

normals associated to the Fi are: η1 = (0, 1), η2 = (1, 0), η3 = (0,−1) and η4 = (−1, 1)

and, indeed, ordVi(xα1yα2) = 〈ηi, α〉 for all α ∈ Z2.


2.1 Semidegree and Subdegree

Throughout the remainder of this chapter let A denote a finitely generated domain over

any field K.

Definition. A degree like function on A is a map δ : A \ {0} → Z ∪ {−∞} such that:

1. δ(K) = 0.

2. δ(f + g) ≤ max{δ(f), δ(g)} for all f, g ∈ A, with < in the preceding equation

implying δ(f) = δ(g).

3. δ(fg) ≤ δ(f) + δ(g) for all f, g ∈ A.

Remark. Even though we allow degree like functions to have values−∞, it will follow from

theorems 2.2.11 and 2.2.16 that we do not need to leave the realm of integer valued degree

like functions by either of the operations of normalizing degree like functions or taking

associated semidegrees of a subdegree (we introduce both notions below). Intermediately

we allow for a theoretical possibility of ending up with a degree like function that takes

the value −∞ on some nonzero f ∈ A.

There is a one-to-one correspondence between degree like functions and filtrations:

Filtrations ←→ Degree like functions

F = {Fd}d∈Z −→ δF : f ∈ A 7→ inf{d : f ∈ Fd}

Fδ := {Fd := {f ∈ A : δ(f) ≤ d}}d∈Z ←− δ

In the remainder of our thesis we identify degree like functions with the corresponding

filtrations. In particular, we refer to a degree like function δ as complete (resp. finitely

generated) iff the corresponding filtration Fδ is complete (resp. finitely generated). More-

over, Aδ and grAδ will be shorthand notations for the rings AFδ and, respectively, grAFδ

and ψδ will denote the natural embedding SpecA ↪→ Xδ := ProjAδ, while for the sake

of convenience we would freely refer to Xδ as the “completion ψδ” (of SpecA).

The protagonists of this chapter are two classes of degree like functions which satisfy

stronger versions of the multiplicative property (i.e. property 3 above).


Definition.

• A degree like function δ on A is a semidegree iff δ(fg) = δ(f) + δ(g) for all

f, g ∈ A \ {0}.

• We say that δ is a subdegree if there are semidegrees δ1, . . . , δN such that

δ(f) = max1≤i≤N

δi(f) for all f ∈ A \ {0}. (2.1)

Given a subdegree δ as in (2.1), we may assume by getting rid of some δi’s, if need be,

that every δi that appears in (2.1) is not redundant in the sense that for every i, there

is an f ∈ A such that δi(f) > δj(f) for all j 6= i (indeed, if there is an i such that for

all f ∈ A, δi(f) ≤ δj(f) for some j 6= i, then δ(f) = maxj 6=i δj(f) for all f ∈ A). In the

latter case we will say that (2.1) is a minimal presentation of δ.

Example 2.1.1 (Weighted Degree). Every weighted degree δ on the polynomial ring

A := K[x1, . . . , xn] is a semidegree. In this case there is of course a unique homomorphism

ι : grAδ → K[x1, . . . , xn] commuting with maps A 3 f 7→ [(f)δ(f)] ∈ grAδ and A 3 f 7→

Lδ(f) ∈ K[x1, . . . , xn], where [(f)δ(f)] is the equivalence class of (f)δ(f) in grAδ and Lδ(f)

is the leading weighted homogeneous form of f corresponding to the weighted degree δ.

Moreover, a straightforward calculation shows that ι is a graded K-algebra isomorphism

of grAδ and K[x1, . . . , xn]. If di := δ(xi) is positive for each i, then δ is a complete

semidegree and the corresponding completion of Kn is the weighted projective space

Pn(K; 1, d1, . . . , dn).

Example 2.1.2 (An iterated semidegree). LetX := K2, ring A := K[x1, x2] and filtration

F := {Fd : d ≥ 0} on A be: F0 := K , F1 := K〈1, x21 − x3

2〉 , F2 := (F1)2 + K〈x2〉 ,

F3 := F1F2 + K〈x1〉 and Fd :=∑d−1

j=1 FjFd−j for d ≥ 4. We claim that function δ := δF

is a semidegree. Indeed,

Aδ = K[(1)1, (x1)3, (x2)2, (x21 − x3

2)1] ∼= K[X1, X2, Y, Z]/〈Y Z5 −X21 +X3

2 〉 ,

where the last isomorphism is induced by a K-algebra homomorphism which sends

X1 7→ (x1)3, X2 7→ (x2)2, Y 7→ (x21 − x3

2)1 and Z 7→ (1)1. The inverse image of the ideal


I := 〈(1)1〉 of Aδ under this isomorphism coincides with ideal 〈Z, Y Z5 − X21 + X3

2 〉 =

〈Z,X21 − X3

2 〉. Since the latter is a prime ideal of K[X1, X2, Y, Z], it follows that I is

a prime ideal of Aδ as well. Then δ is a semidegree according to theorem 2.2.1. The

isomorphism constructed above induces a closed embedding of Xδ onto hypersurface

V (Y Z5 −X21 + X3

2 ) ⊆WP, where WP is the weighted projective space P3(K; 1, 3, 2, 1)

with (weighted homogeneous) coordinates [Z : X1 : X2 : Y ]. Constructed semidegree δ

is an example of an iterated semidegree (introduced in section 3.2).

By definition a polytope P ⊆ Rn is integral provided its vertices are in Zn.

Example 2.1.3 (Subdegrees determined by integral polytopes). Let X be the n-torus

(K∗)n and A := K[x1, x−11 , . . . , xn, x

−1n ] be its coordinate ring. Let P be a convex rational

polytope (i.e. a convex polytope in Rn with vertices in Qn) of dimension n containing

origin in its interior. Define a function δ′ : A \ {0} → Q+ as follows:

δ′(xα) := inf{r ∈ Q+ : α ∈ rP},

δ′(∑

aαxα) := max

aα 6=0δ′(xα)

Claim. There is k ∈ N such that kδ′ is a complete subdegree.

Proof. For each facet Q of P , let ωQ be the smallest ‘outward pointing’ integral vector

normal to Q and let cQ = 〈ωQ, α〉, where α is any element of the hyperplane that contains

Q. Since P is rational, in each Q there is an α with rational coordinates, and therefore

each cQ is a positive rational number. Let δ′Q be the Q-valued weighted degree on A

given by:

δ′Q(∑

aαxα) := max

aα 6=0

〈ωQ, α〉cQ

.

For each r ∈ R+, rP = {β ∈ Rn : 〈ωQ, β〉 ≤ rcQ for every facet Q of P}. It follows that


for each α ∈ Rn,

δ′(xα) := inf{r ∈ Q+ : α ∈ rP},

= inf{r ∈ Q+ : 〈ωQ, α〉 ≤ rcQ for every facet Q of P}

= inf{r ∈ Q+ : δ′Q(xα) ≤ r for every facet Q of P}

= max{δ′Q(xα) : Q is a facet of P}.

Let k ∈ N be such that k/cQ is an integer for each Q. Then kδ′Q is an integer valued

semidegree for each Q and hence kδ′ is a subdegree.

We claim that Akδ′

is finitely generated. Indeed, since P is rational, there is an l ∈ N

such that lP is integral. Identify Rn with the hyperplane xn+1 = 1 in Rn+1. Let C be the

cone in Rn+1 over lP ⊆ Rn (vertex of C being the origin). Then, with d := kl, the d-th

truncated ring of Akδ′

is

(Akδ′)[d] :=

⊕m≥0

{f ∈ A : kδ′(f) ≤ klm}

=⊕m≥0

{f ∈ A : δ′(f) ≤ lm}

=⊕m≥0

K-span{xα : α ∈ lmP}

=⊕m≥0

K-span{xα : (α,m) ∈ C ∩ Zn+1}.

By Gordan’s lemma [9, Proposition 1, Section 1.2] the semigroup C ∩ Zn+1 is finitely

generated. It follows that (Akδ′)[kl] is a finitely generated K-algebra. But ring Akδ

′is

integral over ring (Akδ′)[kl], since for every (f)e ∈ Akδ

′, e ∈ Z+, ((f)e)

kl ∈ (Akδ′)[kl].

Therefore Akδ′

is a (Akδ′)[kl]-submodule of the integral closure of (Akδ

′)[kl]. Since the

latter is a finite (Akδ′)[kl]-module, it follows that Akδ

′is also a finite (Akδ

′)[kl]-module and

hence is a finitely generated algebra over K.

Let δ := kδ′, where k is an integer as in the above claim. Recall from section 1.0.3

that for each d > 0, a set of homogeneous generators of (Aδ)[d] gives rise to the d-uple


embedding of Xδ into a weighted projective space. Let d be a positive integer such that

(Aδ)[d] is generated as a K-algebra by elements in the d-th graded component of Aδ, or

equivalently the K-span of monomials xα such that α ∈ dkP . Then the image of the d-

uple embedding of Xδ is the closure (in the appropriate dimensional projective space) of

the image of (K∗)n under the map whose components are the monomials with exponents

in dkP . For K = C, it follows by theorem 2.0.2 that Xδ is isomorphic via the d-uple

embedding to toric variety XP .

2.2 Properties of Subdegree

2.2.1 Uniqueness of minimal presentations of subdegrees by

semidegrees and a characterization via absence of nilpo-

tents in their graded rings.

Recall that an ideal q of a ring R is called primary if q 6= R and for all f, g ∈ R,

fg ∈ q ⇒ either f ∈ q or gk ∈ q for some k ≥ 1. A primary decomposition of an

ideal a of R is an expression of a as a finite intersection of primary ideals. a is called

decomposable if it has a primary decomposition. Below we use the following commonly

used notation: for any ideal a of R and any element f ∈ R, (a : f) is the ideal of R

defined by: (a : f) := {g ∈ R : fg ∈ a}.

Proposition 2.2.0 (See [8, Chapter 3]). a is a decomposable radical ideal if and only if

it is a finite intersection of unique prime ideals of the form (a : f) for f ∈ R\a. If R is a

graded ring and a is a homogeneous ideal, then the elements f in the preceding statement

are also homogeneous. If R is Noetherian, then every ideal of R is decomposable.

Theorem 2.2.1 (see [25, Theroem 2.1] and [26, Theorem 2.2.1]). Let δ be a degree like

function on A and let I be the ideal of Aδ generated by (1)1.

1. δ is a semidegree if and only if I is a prime ideal.


2. δ is a subdegree if and only if I is a decomposable radical ideal.

3. If Aδ is Noetherian, then δ is a subdegree if and only if I is a radical ideal.

Proof. I is a prime ideal iff (f)d, (g)e 6∈ I implies (f)d(g)e = (fg)d+e 6∈ I. Since (f)d 6∈ I

iff δ(f) = d, the preceding condition is equivalent to the condition that whenever δ(f) = d

and δ(g) = e, then δ(fg) = d + e. The latter is the defining property of a semidegree,

and assertion 1 is proved.

Due to proposition 2.2.0, assertion 3 follows from assertion 2. So we proceed with

proving assertion 2. The following lemma reformulates the property of I being radical:

Lemma 2.2.1.1. I is radical iff δ(fk) = kδ(f) for all f ∈ A and k ≥ 0.

Proof. I is radical iff f ∈ A and (f)d ∈ Aδ \ I imply ((f)d)k = (fk)dk ∈ Aδ \ I for all

k ≥ 0. Since (f)d ∈ Aδ \ I iff δ(f) = d, it follows that I is radical iff δ(fk) = kδ(f) for

all f ∈ A and k ≥ 0.

At first assume δ is a subdegree. Let δ = maxNi=1 δi be a minimal presentation for δ.

Since for all f ∈ A and k ≥ 0, δi(fk) = kδi(f) for each i, it follows that δ(fk) = kδ(f).

Therefore, by lemma 2.2.1.1, I is radical.

It remains to show for the ‘only if’ implication that ideal I is decomposable, i.e. it is

the intersection of finitely many primes. Recall that temporarily (up to Theorem 2.2.11)

we allow degree like functions (including semi- and subdegrees) to have −∞ as a value.

For each i with 0 ≤ i ≤ N , fix an fi ∈ A such that δi(fi) > δj(fi) for all j 6= i. Then, in

particular, δi(fi) ∈ Z. Let di := δ(fi) = δi(fi). That I is decomposable is a consequence

of assertion 2 of the following lemma.

Lemma 2.2.1.2.

(1) For each i, 1 ≤ i ≤ N , (I : (fi)di) is a homogeneous ideal, and the homogeneous

elements of (I : (fi)di) are precisely the elements of Li := {(f)d : f ∈ A, d > δi(f)}.


(2) For each i, (I : (fi)di) is a distinct minimal prime ideal of Aδ containing I. In

particular, (fi)di ∈ (⋂j 6=i(I : (fj)dj)) \ (I : (fi)di) for each i. Moreover,

N⋂i=1

(I : (fi)di) = I. (2.2)

Proof. We first prove assertion 1. Fix an i, 1 ≤ i ≤ N . Since (fi)di is a homogeneous

element in Aδ and I is a homogeneous ideal of Aδ, it follows that (I : (fi)di) is also a

homogeneous ideal of Aδ. Let (f)d be an arbitrary homogeneous element of (I : (fi)di).

If δ(f) < d, then δi(f) ≤ δ(f) < d and (f)d ∈ Li. So assume δ(f) = d. Since (f)d(fi)di =

(ffi)d+di ∈ I, it follows that δ(ffi) < d + di. But then δi(ffi) = δi(f) + δi(fi) < d + di,

which implies that δi(f) < d = δ(f), and thus (f)d ∈ Li. To summarize, all homogeneous

elements of (I : (fi)di) belong to Li.

Now let (f)d be an arbitrary element of Li. If d > δ(f), then (f)d ∈ I ⊆ (I : (fi)di).

So assume d = δ(f). Then δi(f) < d = δ(f), and thus δi(ffi) = δi(f) + δi(fi) < d + di.

Also, for each j 6= i, δj(fi) < di, so that δj(ffi) = δj(f) + δj(fi) < d + di. It follows

that δ(ffi) < d + di and (f)d(fi)di = (ffi)d+di ∈ I, i.e. (f)d ∈ (I : (fi)di), which proves

inclusion Li ⊆ (I : (fi)di) and completes the proof of assertion 1.

We now show that (I : (fi)di) is prime. Let (g1)e1 , (g2)e2 be homogeneous elements of

Aδ such that (g1)e1(g2)e2 ∈ (I : (fi)di). Due to assertion 1, (g1g2)e1+e2 = (g1)e1(g2)e2 ∈ Li,

which means that δi(g1g2) = δi(g1) + δi(g2) < e1 + e2. Since ej ≥ δ(gj) ≥ δi(gj) for each

j, it follows that there is j, j = 1 or 2, such that δi(gj) < ej. Then assertion 1 implies

that (gj)ej ∈ (I : (fi)di), i.e. (I : (fi)di) is a prime ideal. Next we show that (I : (fi)di) is

a minimal prime ideal of Aδ containing ideal I.

Indeed, if I ⊆ p ⊆ (I : (fi)di) for a prime ideal p of Aδ, and some i ≤ N , then

(fi)di 6∈ p (since assertion 1 implies (fi)di 6∈ (I : (fi)di)). But if (g)e ∈ (I : (fi)di), then

(g)e(fi)di ∈ I ⊆ p and it follows that (g)e ∈ p. Hence (I : (fi)di) ⊆ p, and therefore

(I : (fi)di) = p, as required.

To see that the ideals (I : (fi)di) are distinct, note that (fi)di ∈ (⋂j 6=i Lj) \ Li for


1 ≤ i ≤ N , which implies due to assertion 1 that (fi)di ∈ (⋂j 6=i(I : (fj)dj)) \ (I : (fi)di)

for every i.

Finally, pick any homogeneous (g)e ∈⋂

(I : (fi)di). Then by assertion 1, for each

i, δi(g) < e. Therefore δ(g) = maxi δi(g) < e, and hence (g)e ∈ I. It follows that⋂(I : (fi)di) = I, which concludes the proof of the lemma.

We now return to the proof of theorem 2.2.1 and prove the ‘if’ implication of assertion

2. Assume I is a decomposable radical ideal of Aδ. We have to show that δ is a subdegree.

According to proposition 2.2.0 there exist (f1)d1 , . . . , (fN)dN ∈ Aδ \I such that (I : (fi)di)

is a prime ideal containing I for each i and I =⋂Ni=1(I : (fi)di). Since (fi)di 6∈ I, it follows

that δ(fi) = di ∈ Z for each i. For each i = 1, . . . , N , let δi : A→ Z ∪ {−∞} be defined

as follows:

δi(f) := limk→∞

δ((fi)kf)− δ((fi)k). (2.3)

We first show that δi is well defined for each i. Indeed, fix an i, 1 ≤ i ≤ N . Let f ∈ A

and k ≥ 1. Since (fi)di 6∈ I and I is radical, it follows that ((fi)di)k = ((fi)

k)kdi 6∈ I, so

that δ((fi)k) = kdi. Therefore,

δ((fi)k+1f)− δ((fi)k+1) ≤ δ((fi)

kf) + δ(fi)− δ((fi)k+1)

= δ((fi)kf) + di − (k + 1)di

= δ((fi)kf)− kdi

= δ((fi)kf)− δ((fi)k).

(2.4)

It follows that δi(f) is a well defined element in Z ∪ {−∞}.

Multiplicativity: Let f, g ∈ A. We now show that δi(fg) = δi(f) + δi(g), splitting

the proof into the following three cases:


Case 1: δi(f) and δi(g) are integers. Note that δi(f) = d ∈ Z

⇐⇒ ∃kf ∈ N such that δ((fi)k+1f)− δ((fi)k+1) = d for all k ≥ kf ,

⇐⇒ ∃kf ∈ N such that δ((fi)k+1f) = (k + 1)di + d for all k ≥ kf , (2.5)

⇐⇒ ∃kf ∈ N such that ((fi)kf)kdi+d(fi)di = ((fi)

k+1f)(k+1)di+d 6∈ I for all k ≥ kf ,

⇐⇒ ∃kf ∈ N such that ((fi)kf)kdi+d 6∈ (I : (fi)di) for all k ≥ kf .

Therefore, if both δi(f) and δi(g) are integers, then there is a k′ := max{kf , kg} ∈ N such

that for all k ≥ k′, ((fi)kf)kdi+δi(f) and ((fi)

kg)kdi+δi(g) do not lie in (I : (fi)di). Since

(I : (fi)di) is a prime ideal, it follows that for all k ≥ k′, ((fi)kf)kdi+δi(f)((fi)

kg)kdi+δi(g) =

((fi)2kfg)2kdi+δi(f)+δi(g) 6∈ (I : (fi)di). Therefore δi(fg) = δi(f) + δi(g) ∈ Z due to (2.5),

as required.

If case 1 does not take place, we may w.l.o.g. assume that δi(f) = −∞.

Case 2: δi(f) = −∞ and δ(g) ∈ Z. Note that for all h ∈ A, δi(h) = −∞ if and

only if for every n ∈ Z there is a k′ ∈ N such that for all k ≥ k′, δ((fi)kh)− δ((fi)k) < n.

Pick an n ∈ Z. Since δi(f) = −∞, we can choose k′ ∈ N such that for all k ≥ k′,

δ((fi)kf)−δ((fi)k) < n−δ(g). Then δ((fi)

kfg)−δ((fi)k) ≤ δ((fi)kf)+δ(g)−δ((fi)k) < n

for all k ≥ k′. Therefore δ(fg) = −∞ = δ(f) + δ(g).

Case 3: δi(f) = −∞ and δ(g) = −∞. If δ(g) is−∞, then for all h ∈ A, δ(gh) ≤

δ(g) + δ(h) ≤ −∞, and hence δ(gh) = −∞. But then δ(fki fg) = −∞ for all k and

therefore δi(fg) = −∞ = δi(f) + δi(g).

The above 3 cases cover all possible options and therefore we proved that each δi is

multiplicative.

Additivity: Let f, g ∈ A. We first show that δi(f + g) ≤ max{δi(f), δi(g)}. Since δ

is a degree like function, it follows that for all k ∈ N,

δ(fki (f + g))− δ(fki ) ≤ max{δ(fki f)− δ(fki ), δ(fki g)− δ(fki )}.

The required inequality then follows from the following simple observation: if {ak}, {bk}


and {ck} are decreasing sequences of real numbers such that ak ≤ max{bk, ck} for each

k, then lim ak ≤ max{lim bk, lim ck}.

Next assume that δi(f + g) < max{δi(f), δi(g)}. We will show, as required, that

δi(f) = δi(g). Assume otherwise, say δi(f) < δi(g). Then δi(g) is an integer and it follows

by definition of δi that there exists k′ ∈ N such that for all k ≥ k′, δ(fki f) − δ(fki ) <

δi(g) = δ(fki g) − δ(fki ). Consequently, for all k ≥ k′, δ(fki g) > δ(fki f), and therefore

due to property 2 of the definition of degree like functions, δ(fki (f + g)) − δ(fki ) =

δ(fki g) − δ(fki ) = δi(g). It follows that δi(f + g) = δi(g) contrary to our assumption.

Therefore δi(f) = δi(g) and it concludes the proof that δi is a semidegree.

The only remaining assertion to prove is δ = maxNi=1 δi. First of all note that for all

f ∈ A, and 1 ≤ i ≤ N , δ(fif) ≤ δ(fi) + δ(f), so that δ(f) ≥ δ(fif) − δ(fi) ≥ δi(f). It

follows that δ ≥ maxNi=1 δi.

Now pick any f ∈ A. If δ(f) = −∞, then for all i and for all k ≥ 0, δ((fi)kf) = −∞,

and therefore δi(f) = −∞ = δ(f).

It remains to consider the case that δ(f) ∈ Z. Since (f)δ(f) 6∈ I, it follows that there

is an i such that (f)δ(f) 6∈ (I : (fi)di). Note that (fi)di also does not lie in (I : (fi)di), for

otherwise ((fi)di)2 would be an element of I and this would imply that (fi)di ∈ I (since I

is a radical ideal), contradicting our choice of (fi)di . Thus neither of (f)δ(f) and (fi)di is

an element of (I : (fi)di). Since (I : (fi)di) is a prime ideal, it follows that for all k ≥ 0,

((fi)di)k(f)δ(f) = ((fi)

kf)kdi+δ(f) 6∈ (I : (fi)di). Then according to (2.5), δi(f) = δ(f),

which proves δ = maxNi=1 δi and that δ is indeed a subdegree.

Corollary 2.2.2. Assume Aδ is Noetherian. Then δ is a subdegree on A iff δ(fk) = kδ(f)

for all f ∈ A and k ≥ 0.

Proof. This claim is a straightforward consequence of assertion 3 of theorem 2.2.1 and

lemma 2.2.1.1.

Proposition 2.2.3. Let δ be a subdegree on A. Semidegrees δ1, . . . , δN of a minimal

presentation δ = max1≤i≤N δi of δ are unique.


Proof. Assume semidegrees δ1, . . . , δN and δ′1, . . . , δ′N ′ provide two distinct minimal pre-

sentations δ = max1≤i≤N δi and δ = max1≤i≤N ′ δ′i of δ. As in the proof of theorem 2.2.1,

there exist f1, . . . , fN and f ′1, . . . , f′N ′ ∈ A such that

di := δi(fi) > δj(fi) for 1 ≤ j 6= i ≤ N, (2.6)

d′i := δ′i(f′i) > δ′j(f

′i) for 1 ≤ j 6= i ≤ N ′. (2.6′)

Then di and d′i are integers, and by lemma 2.2.1.2,

I =N⋂i=1

(I : (fi)di) and I =N ′⋂i=1

(I : (f ′i)d′i)

are unique minimal primary decompositions of I, with ideal I in Aδ being generated by

(1)1. The latter uniqueness implies that N = N ′ and, after an appropriate re-indexing

of δ′i’s, that ideals (I : (fi)di) and (I : (f ′i)d′i) coincide for all i ≤ N .

Fix an i, 1 ≤ i ≤ N . According to assertion 2 of lemma 2.2.1.2, elements

(f ′i)d′i ∈ (⋂j 6=i

(I : (f ′j)d′j)) \ (I : (f ′i)d′i) = (⋂j 6=i

(I : (fj)dj)) \ (I : (fi)di) .

Therefore assertion 1 of lemma 2.2.1.2 implies that (f ′i)d′i ∈ (⋂j 6=i Lj) \ Li, where Lj’s

are as in lemma 2.2.1.2. It follows that δi(f′i) = δ(f ′i) = δ′i(f

′i), and δj(f

′i) < δi(f

′i) for all

j 6= i. The latter implies property (2.6) with f ′i ’s replacing fi’s. Since fi’s were assumed

to be arbitrary elements in A such that (2.6) is true, we may assume without loss of

generality that fi = f ′i for each i. It follows that δi = δ′i for all i by making use of

Lemma 2.2.4. If f1, . . . , fN ∈ A satisfy (2.6), then δi(f) = limk→∞ δ((fi)kf)− δ((fi)k)

for all f ∈ A.

Proof. Fix an i, 1 ≤ i ≤ N . Let us write δi(f) := limk→∞ δ((fi)kf) − δ((fi)

k) for all

f ∈ A. It follows by the arguments in the proof of theorem 2.2.1 that δi is a well

defined semidegree on A. Moreover, for all k ≥ 0 and all f ∈ A, δ((fi)kf) − δ((fi)k) ≥

δi((fi)kf)−δ((fi)k) = δi(f), so that δi ≥ δi. To see the opposite inequality, let f ∈ A and

d ∈ Z be such that d ≥ δi(f). Then kdi+d ≥ δi((fi)kf) for all k. Moreover, (2.6) implies


that for sufficiently large k, kdi + d > kδj(fi) + δj(f) = δj((fi)kf) for all j 6= i. It follows

that for sufficiently large k, δ((fi)kf) ≤ kdi+d, and hence δ((fi)

kf)−δ((fi)k) ≤ d. Since

{δ((fi)kf) − δ((fi)k)} is a decreasing sequence due to (2.4), it follows that δi(f) ≤ d.

With d = δi(f) when δi(f) ∈ Z, and otherwise letting d converge to δi(f) from above,

we see that δi(f) ≤ δi(f). It follows that δi(f) = δi(f), which completes the proof of

proposition 2.2.3.

Corollary 2.2.5. Let δ be a subdegree on A and δ = maxNi=1 δi be its minimal presenta-

tion. Then X∞ := ProjAδ \ SpecA has exactly N irreducible components.

Proof. By theorem 1.1.4, X∞ = V (I) ⊆ ProjAδ, where I is the ideal in Aδ generated

by (1)1. Let f1, . . . , fN , d1, . . . , dN be as in (2.6). Then pi := (I : (fi)di) is a minimal

prime ideal containing I and I =⋂Ni=1 pi is the prime decomposition of I. It follows that

X∞ = V (I) =⋃Ni=1 V (pi) is the decomposition of X∞ into irreducible components.

Example 2.2.6. Let X = (K∗)n and let P be a convex integral polytope of dimension

n containing the origin in its interior. We saw in example 2.1.3 that the toric variety XP

is isomorphic to Xδ for a subdegree δ on K[x1, x−11 , . . . , xn, x

−1n ]. Moreover, the minimal

presentation of δ is of the form: δ = max{δQ : Q is a facet of P}. It follows by corollary

2.2.5 that the components of XP \ (K∗)n are in a one-to-one correspondence with the

facets of P (cf. theorem 2.0.2).

A subdegree on an integrally closed domain has an additional property:

Proposition 2.2.7. If A is an integrally closed domain and δ is a subdegree on A, then

Aδ is also integrally closed.

Proof. In (1.1) we introduced an isomorphism Aδ ∼=⊕

i∈Z Fiti ⊆ A[t, t−1], with (1)1

being mapped to t. Since A is an integrally closed domain, it follows that A[t, t−1] is also

integrally closed [1, Exercise 5.9]. So it suffices to show that Aδ is integrally closed in


A[t, t−1]. Pick f =∑r

i=q fiti ∈ A[t, t−1] integral over Aδ, where fi ∈ A for each i. Then

f satisfies an equation of the form

Tm +G1Tm−1 + · · ·+Gm = 0 (2.7)

for G1, . . . , Gm ∈ Aδ. Taking the highest degree terms in t, we see that frtr is integral

over f . Replacing f by f − frtr and repeating the procedure, it follows that each fiti

is integral over Aδ. Therefore it suffices to show that if ftk ∈ A[t, t−1] is integral over

Aδ for some f ∈ A, then ftk ∈ Aδ. To that end, let f ∈ A be such that ftk satisfies

an equation of form (2.7). Comparing the coefficients at tkm in that equation, we may

assume that Gi = gitik for some gi ∈ A. Since git

ik ∈ Aδ, it follows that δ(gi) ≤ ik.

Assume contrary to the assertion of the proposition that ftk 6∈ Aδ. Then d := δ(f) >

k. Set T = ftk, Gi = gitik and t = 1 in (2.7). It follows that fm = −

∑mi=1 gif

m−i in

A. For each i ≥ 1, δ(gifm−i) ≤ δ(gi) + δ(fm−i) ≤ ik + (m − i)d < id + (m − i)d = md.

Therefore δ(fm) = δ(−∑m

i=1 gifm−i) ≤ maxmi=1 δ(gif

m−i) < md. On the other hand,

since δ is a subdegree, δ(fm) = mδ(f) = md. The contradiction we arrived at proves

ftk ∈ Aδ, as required.

Corollary 2.2.8. Let X be a normal affine variety with A being the ring of regular

functions on X and δ being a complete subdegree on A. Then SpecAδ and ProjAδ are

normal varieties as well.

Example 2.2.9. In this example rings A := K[x] and Aδ are integrally closed, where

degree like function δ on K[x] is defined by

δ(xk) :=

3k/2 if k is even,

3(k − 1)/2 + 2 if k is odd.

We claim that degree like function δ is not a subdegree.

Indeed, K[x]δ = K[(1)1, (x)2, (x2)3] ∼= K[x, y, z]/〈x2 − yz〉, where the isomorphism

is induced by the K-algebra homomorphism K[x, y, z] → K[x]δ which sends x 7→ (x)2,


y 7→ (x2)3 and z 7→ (1)1. If K is not of characteristic 2, then K[x]δ is integrally closed

[14, Exercise II.6.4]. On the other hand, δ(x2) = 3 < 4 = 2δ(x) and it follows by making

use of corollary 2.2.2 that δ is not a subdegree.

2.2.2 For finitely generated subdegrees, the semidegrees of the

minimal presentation do not take −∞ value.

In the proof of the next two theorems we make an essential use of the theory of Rees

valuation. We first describe following [24, chapter XI] the relevant results of Rees (starting

with a reminder of the notion of a Krull domain).

Definition. A domain B is a Krull domain iff

1. Bp is a discrete valuation ring for all height one prime ideals p of B, and

2. every non-zero principal ideal of B is the intersection of a finite number of primary

ideals of height one.

Every normal Noetherian domain is a Krull domain [23, Section 41]. In particular,

the integral closure of Aδ is a Krull domain provided that Aδ is finitely generated.

For an ideal I of a ring R define νI : R→ N ∪ {∞} and νI : R→ Q+ ∪ {∞} by:

νI(x) := sup{m : x ∈ Im}, and

νI(x) := limm→∞

νI(xm)

m,

for all x ∈ R. Recall that the integral closure J of an ideal J of R is the ideal defined by:

J := {x ∈ R : x satisfies an equation of the form: xs + j1xs−1 + · · ·+ js = 0 with jk ∈ Jk

for all k = 1, . . . , s}. The following is due to Rees [24, Propositions 11.1 – 11.6]:

Theorem (Rees). For any ring R and any ideal I of R, νI is well defined. Assume R is

a Noetherian domain. Then

(1) there is a positive integer e such that for all x ∈ R, νI(x) ∈ 1eN, and


(2) if k ≥ 0 is an integer then νI(x) ≥ k if and only if x ∈ Ik, where Ik is the integral

closure of Ik in R.

(3) Assume in addition that I is a principal ideal generated by u and R is an integral

extension of R which is a Krull domain. Let p1, . . . , pr be the height 1 prime ideals

of R containing u. Then for all x ∈ R, νI(x) = min{νi(x)ei

: i = 1, . . . , r}, where for

each i = 1, . . . , r, νi is the valuation associated with the discrete valuation ring Rpi and

ei := νi(u).

The following theorem gives a characterization of the semidegrees δi associated to a

subdegree δ, provided that Aδ is finitely generated. In particular, it states that each δi

is integer valued (which is not a priori obvious relying only on the limit definition of δi’s

in (2.3)). We start with the following

Lemma 2.2.10. If Aδ is Noetherian, then δ(f) ∈ Z for all f ∈ A, i.e. δ does not take

the value −∞.

Proof. Recall that A is a domain and hence Aδ is also a domain by lemma 1.1.1. Let I

be the ideal generated by (1)1 in Aδ. Since (1)0 6∈ I, ideal I is a proper ideal of Aδ. It

follows that if Aδ is Noetherian, then⋂∞k=1 I

k = ∅ [1, Corollary 10.18]. Assume contrary

to the claim that δ(f) = −∞. Then (f)k ∈ Aδ for all k ∈ Z, and therefore (f)0 ∈⋂∞k=1 I

k

(since for each k ≥ 0, (f)0 = (f)−k · ((1)1)k ∈ Ik). It follows that⋂∞k=1 I

k 6= ∅, which is

in contradiction with the previous conclusion, and therefore δ(f) 6= −∞ for all f ∈ A,

as required.

Theorem 2.2.11 (see [26, Theorem 2.2.7]). Let δ be a finitely generated subdegree on A

with a minimal presentation δ = max1≤i≤N δi. Let B be any Krull domain which is also

an integral extension of Aδ and p1, . . . , pr be the height one primes of B containing (1)1.

For each j, 1 ≤ j ≤ r, define a function δj on A \ {0} by

δj(f) = δ(f)−νj((f)δ(f))

ej


where νj is the discrete valuation of the discrete valuation ring Bpj and ej := νj((1)1).

Then for each i, 1 ≤ i ≤ N , δi ≡ δj for some j, 1 ≤ j ≤ r. In particular, semidegrees δi

are integer valued for all i.

Proof. Let I be the ideal generated by (1)1 in Aδ and νI be the Rees’ valuation corre-

sponding to I. Let f ∈ A. According to lemma 2.2.10, δ(f) ∈ Z. Since δ is a subde-

gree, δ(fk) = kδ(f) for all k ≥ 0 (corollary 2.2.2) and hence ((f)δ(f))k = (fk)kδ(f) 6∈ I

for all k ≥ 0. It follows that νI((f)δ(f)) = 0. Now assertion 3 of Rees’ theorem im-

plies that minrj=1νj((f)δ(f))

ej= 0, where ej := νj((1)1) for every j. Therefore δ(f) =

δ(f) − minrj=1νj((f)δ(f))

ej= maxrj=1 δj(f) for all f ∈ A. Next we show that for each j,

function ej δj is a Z-valued semidegree, which suffices to complete the proof of theorem

2.2.11 due to the uniqueness of the minimal presentation of subdegrees (proposition 2.2.3)

applied to the minimal presentation eδ = maxi eδi and the presentation eδ = maxj eδj

with an appropriate e ∈ Z+ (e.g. e =∏

j ej).

Multiplicativity: Fix j, 1 ≤ j ≤ r. Let f, g ∈ A \ {0}. First we show that

δj(fg) = δj(f) + δj(g). Let ε := δ(f) + δ(g)− δ(fg). Then

δj(fg) = δ(fg)− 1

ejνj((fg)δ(fg)) = δ(f) + δ(g)− 1

ej(ejε+ νj((fg)δ(fg)))

= δ(f) + δ(g)− 1

ej(νj((1)ε) + νj((fg)δ(fg)))

= δ(f) + δ(g)− 1

ejνj((fg)δ(fg)+ε) = δ(f) + δ(g)− 1

ejνj((fg)δ(f)+δ(g))

= δ(f) + δ(g)− 1

ej(νj((f)δ(f)) + νj((g)δ(g))) = δj(f) + δj(g) .

Additivity: Let d := δ(f)− δ(g). Note that

δj(f) ≥ δj(g) ⇐⇒ δ(f)−νj((f)δ(f))

ej≥ δ(g)−

νj((g)δ(g))

ej

⇐⇒ νj((f)δ(f)) ≤ dej + νj((g)δ(g)).

(∗)

Moreover, strict inequality in any one of the expressions of (∗) implies strict inequality

in the other expressions of (∗). The remainder of the proof splits into several cases.


Case 1: d > 0. In this case δ(f + g) = δ(f) and (f + g)δ(f+g) = (f)δ(f) + ((1)1)d(g)δ(g).

Then δj(f+g) = δ(f+g)− 1ejνj((f+g)δ(f+g)) = δ(f)− 1

ejνj((f)δ(f) +((1)1)d(g)δ(g)).

Subcase 1(a): δj(f) > δj(g). In this case due to (∗) it follows that νj((f)δ(f)) <

dej + νj((g)δ(g)) = νj(((1)1)d) + νj((g)δ(g)) = νj(((1)1)d(g)δ(g)). Therefore

νj((f)δ(f)+((1)1)d(g)δ(g)) = νj((f)δ(f)). Hence δj(f+g) = δ(f)− 1ejνj((f)δ(f)) =

δj(f).

Subcase 1(b): δj(f) = δj(g). As in 1(a) it follows by making use of (∗) that

νj((f)δ(f)) = νj(((1)1)d(g)δ(g)), and therefore νj((f)δ(f) + ((1)1)d(g)δ(g)) ≥

νj((f)δ(f)). Hence δj(f + g) ≤ δ(f)− 1ejνj((f)δ(f)) = δj(f).

Case 2: d = 0. Let e := δ(f) = δ(g) and ε := e − δ(f + g) ≥ 0. It follows that

δj(f + g) = δ(f + g) − 1ejνj((f + g)δ(f+g)) = δ(f) − ε − 1

ejνj((f + g)δ(f+g)) =

δ(f)− 1ejνj(((1)1)ε(f + g)δ(f+g)) = e− 1

ejνj((f + g)δ(f+g)+ε) = e− 1

ejνj((f + g)e) =

e− 1ejνj((f)e + (g)e).

Subcase 2(a): δj(f) > δj(g). In this case due to (∗) with d = 0 it follows that

νj((f)e) < νj((g)e). Therefore νj((f)e + (g)e) = νj((f)e). Hence δj(f + g) =

e− 1ejνj((f)e) = δj(f).

Subcase 2(b): δj(f) = δj(g). As in 2(a) (∗) implies that νj((f)e) = νj((g)e).

Consequently νj((f)e + (g)e) ≥ νj((f)e). Hence δj(f + g) ≤ e − 1ejνj((f)e) =

δj(f).

Combining above conclusions it follows that δj is indeed a semidegree. As we remarked

earlier in the proof, conclusions of theorem 2.2.11 now follow from the uniqueness of the

minimal presentation of a subdegree (proposition 2.2.3).

Let X be an affine algebraic variety and δ be a finitely generated subdegree on A :=

K[X] with a minimal presentation δ = max1≤i≤N δi. Fix i, 1 ≤ i ≤ N . Theorem

2.2.11 implies that δi is integer-valued. Let di be the positive generator of the subgroup

of Z generated by {δi(f) : f ∈ A}. Then νi(·) := − 1diδi(·) is a discrete valuation on


K(X). Recall (corollary 2.2.5) that δi corresponds to an irreducible component Vi of

X∞ := Xδ \ X. We next show that the local ring OVi,Xδ of Xδ at Vi is precisely the

valuation ring of νi. In particular, OVi,Xδ is regular, i.e. is a discrete valuation ring.

Proposition 2.2.12. Let X, δ and A be as above. For each i, the local ring OVi,Xδ is

regular and hence is a discrete valuation ring. The valuation associated with OVi,Xδ is

precisely νi(·) := − 1diδi(·) .

Proof. Fix an i, 1 ≤ i ≤ N . Let pi be the prime ideal of Aδ corresponding to δi. Then

OVi,Xδ is the degree zero part of the local ring Aδpi . Let us identify Aδ with∑Fdt

d. Then

OVi,Xδ = { ftkd

(gtd)k: gtd 6∈ pi, d ≥ δ(g), k ≥ 0}

= { fgk

: (g)δ(g) 6∈ pi, δ(f) ≤ kδ(g), k ≥ 0}.(2.8)

Let Ri ⊆ K(X) be the valuation ring of the discrete valuation νi. We have to show

that OVi,Xδ = Ri. Recall that Ri := {g1/g2 : g1, g2 ∈ A, g2 6= 0, νi(g1/g2) ≥ 0}.

Pick g1, g2 ∈ A such that g1/g2 ∈ Ri. Then νi(g1/g2) = νi(g1) − νi(g2) ≥ 0 and hence

δi(g1) ≤ δi(g2). Pick fi ∈ A such that δi(fi) > δj(fi) for all j 6= i. It follows due to (2.5)

that there is k ≥ 1 such that δ(glfki ) = δi(glf

ki ) for l = 1, 2. Then δ(g1f

ki ) = δi(g1f

ki ) =

δi(g1) + δi(fki ) ≤ δi(g2) + δi(f

ki ) = δi(g2f

ki ) = δ(g2f

ki ). Moreover, lemma 2.2.1.2 implies

that (glfki )δ(glfki ) 6∈ pi for l = 1, 2. Then, according to (2.8), g1f

ki /(g2f

ki ) = g1/g2 ∈ OVi,Xδ .

Consequently Ri ⊆ OVi,Xδ . Therefore OVi,Xδ = Ri due to

Lemma 2.2.12.1. Let R be a discrete valuation ring and K be the quotient field of R.

If S is a proper subring of K such that R ⊆ S, then R = S.

Proof. Let ν be the discrete valuation associated to R and h ∈ R be a parameter for

ν, in particular ν(h) = 1. Assume contrary to the claim that R 6= S. Let f ∈ S \ R.

Then f = u/hk for some unit u of R and k > 0. It follows that h−1 = u−1fhk−1 ∈ S.

Let g ∈ K \ {0}. Then ν(gh−ν(g)) = 0 and therefore gh−ν(g) ∈ R ⊆ S and also g =


gh−ν(g) · hν(g) ∈ S. Therefore S = K, contrary to the assumptions, which completes the

proof.

To summarize, we have proved that OVi,Xδ is precisely the valuation ring of νi. Since

valuations are completely determined by their valuation rings [36, Section VI.8], it fol-

lows that νi is the valuation corresponding to OVi,Xδ , which completes the proof of the

proposition.

Example 2.2.13. Let P be a convex integral polytope of dimension n containing the

origin in its interior. As in example 2.1.3, for each facet Q of P , let ωQ be the smallest

‘outward pointing’ integral vector normal to Q and cQ := 〈ωQ, α〉, where α is any element

of the hyperplane spanned by Q. Recall that the toric completion XP of X := (K∗)n is

isomorphic to Xδ for a subdegree δ on K[x1, x−11 , . . . , xn, x

−1n ] and the minimal presenta-

tion of δ is:

δ = maxQ is a facet of P

δQ,

where δQ(xα) := k 〈ωQ,α〉cQ

as in example 2.1.3 (in particular k is such that δQ takes integer

values). Since the greatest common divisor of the components of ωQ is 1, it follows that

the positive greatest common divisor of {δQ(f) : f ∈ A} is dQ := kcQ

. Therefore, due

to proposition 2.2.12, the order of zero of xα along VQ is ordQ(xα) = −δQ(xα)/dQ =

−〈ωQ, α〉. This provides a proof of assertion 1 of theorem 2.0.5.

2.2.3 Finitely generated subdegrees with associated filtrations

preserving intersections at ∞.

The main existence theorem below provides a link between the contents of this chapter

and that of section 1.3. We start with a definition in the spirit of section 1.3.

Definition. Let δ and η be degree like functions on A.

• η preserves the intersections at ∞ for the completion determined by δ (in short, for

δ), provided that for any closed subsets X1, . . . , Xk of SpecA such that the closures


of Xi in ProjAδ have an empty intersection at infinity, also the closures of Xi in

ProjAη have an empty intersection with ProjAη \ SpecA.

• We say η is integral over δ provided η(f) ≤ δ(f) for all f ∈ A, so that there is a

natural inclusion Aδ ↪→ Aη and Aη is integral over Aδ under the above inclusion.

Lemma 2.2.14. Let δ and η be degree like functions on A. Assume there is a commuting

diagram as follows:

SpecA

xxpppppp&&NNNNNN

ProjAη // ProjAδ

where the maps corresponding to the slanted arrows are the natural embeddings associated

with δ and η. Then η preserves the intersections at ∞ for δ.

Proof. Straightforward from the definitions.

Corollary 2.2.15. Let δ and η be non-negative degree like functions on A. If η is integral

over δ or η = nδ for some n > 0, then η preserves the intersections at ∞ for δ.

Proof. If η is integral over δ, then the inclusion Aδ ⊆ Aη induces a morphism ProjAη →

ProjAδ (since the irrelevant ideal Aη+ of Aη is integral over the irrelevant ideal Aδ+ of Aδ)

which is identity on SpecA. Thus by lemma 2.2.14 η preserves the intersections at ∞

for δ.

Now assume η = nδ. Let F := {Fd}d≥0 (resp. G := {Gd}d≥0) be the filtration

corresponding to δ (resp. η). Recall that Aδ ∼=⊕

d≥0 Fdtd and Aη ∼=

⊕d≥0Gds

d, where

s, t are indeterminates over A. Note that for each d ≥ 0, Gd := {f ∈ A : η(f) ≤ d} =

{f ∈ A : δ(f) ≤ d/n} = Fbd/nc. Therefore Aη ∼=⊕

d≥0 Fbd/ncsd. Sending t ∈ Aδ to

sn ∈ Aη induces an inclusion of Aδ into Aη such that Aη is integral over Aδ. At this point

the argument of the previous paragraph completes the proof of the corollary.

Theorem 2.2.16 (Main Existence Theorem, see [25, Theroem 2.3] and [26, Theorem

2.2.9]). Let δ be a finitely generated degree like function on A. Let I be the ideal generated

by (1)1 in Aδ. Then


1. there is a positive integer e and a finitely generated subdegree δ on A such that for

all h ∈ A,

δ(h) := e limm→∞

δ(hm)

m.

2. Subdegree δ is integral over eδ and ring Aδ is integral over Aδ. If δ is non-negative

(resp. complete), then δ is also non-negative (resp. complete), and δ preserves the

intersections at ∞ for δ.

Proof. Fix h ∈ A and m ∈ N. Therefore δ(hm) ≤ mδ(h). Moreover, since ideal I is

generated by (1)1, it follows that k := mδ(h) − δ(hm) is the largest integer such that

(hm)mδ(h) ∈ Ik. The definition of νI implies that νI(((h)δ(h))m) = k = mδ(h) − δ(hm).

Therefore δ(hm)/m = δ(h) − νI(((h)δ(h))m)/m. It follows according to assertion 1 of

Rees’ theorem that δ(h) := limm→∞ δ(hm)/m = δ(h) − νI((h)δ(h)) is well defined and

there exists a positive integer e (independent of h) such that δ(h) ∈ 1eZ, which proves

the displayed formula of assertion 1.

To complete the proof of assertion 1 it remains to show that δ is a finitely generated

subdegree. Let F := {Fd}d∈Z be the filtration on A corresponding to δ. As usual, we

identify Aδ with⊕

i∈Z Fiti. For m ∈ Z, let Fm

e:= {f ∈ A : δ(h) ≤ m

e} and define

Aδ :=⊕

m∈Z Fmetme . Since δ ≤ δ, it follows that Fk ⊆ Fk for each k ∈ Z. Therefore

Aδ ⊆ Aδ. We will make use of the following

Claim 2.2.16.1. Aδ is integral over Aδ.

Proof. It suffices to show that for each h ∈ A, htδ(h) is integral over Aδ. Pick h ∈ A.

Then htδ(h) is integral over Aδ if and only if H := (htδ(h))e is integral over Aδ. Note that

eδ(h) = δ(he) by construction of δ, and therefore H = hetδ(he). Let H := hetδ(h

e) ∈ Aδ and

k := νI(H), where I is the ideal generated by (1)1 in Aδ. Since νI(Hm) = mδ(H)−δ(Hm)

and δ(H) = δ(he), it follows that k = δ(he)− δ(he) = δ(he)−eδ(h). Then k is an integer.

Hence according to assertion 2 of Rees’ theorem, H is in the integral closure of Ik in Aδ,

i.e. H satisfies an equation of the form H l + G1Hl−1 + · · · + Gl = 0, where Gi ∈ I ik


for each i. Comparing the coefficients at tlδ(he) in the above equation, we may assume

w.l.o.g. that each Gi is of the form gitiδ(he) for some gi ∈ A. Since Gi ∈ I ik, it follows

that iδ(he) ≥ δ(gi) + ik, implying that giti(δ(he)−k) is an element of Aδ. Moreover, in the

ring Aδ, (via embedding Aδ ↪→ Aδ) element H = hetδ(he) = hetδ(h

e)+k = hetδ(he)tk = tkH.

Substituting these values of H and Gi into the equation of integral dependence for H

and then canceling a factor of tlk we conclude that (H)l +∑l

i=1 giti(δ(he)−k)(H)l−i = 0.

But then H is integral over Aδ, which completes the proof of the claim.

Let F := {Fd}d∈Z be the filtration corresponding to δ = eδ. Since δ ≤ eδ, Aeδ ⊆ Aδ.

But observe that Fd = {f : eδ(h) ≤ d} = F de, and it follows that the homomorphism

χ : Aδ :=⊕

d∈Z F detde −→

⊕d∈Z Fds

d ∼= Aδ that sends t 7→ se and is the identity map

on the coefficients (i.e. on F de

for d ∈ Z) is an isomorphism of K-algebras. Arguments

similar to those in the conclusion of the proof of corollary 2.2.15 show that the restriction

of χ to Aδ induces a chain of inclusions Aδ ⊆ Aeδ ⊆ Aδ and that Aeδ is integral over Aδ.

Also, due to claim 2.2.16.1 it follows that Aδ is integral over Aδ, as required. Therefore

Aδ is integral over Aeδ, implying that δ is integral over eδ. Moreover, since Aδ is a finitely

generated K-algebra, it follows that Aδ is a finitely generated K-algebra.

From the constructive definition of δ it follows that δ(fm) = mδ(f) for all f and m.

Then corollary 2.2.2 implies that δ is a subdegree. This completes the proof of the first

assertion of the theorem.

If δ is non-negative (resp. complete), then by construction δ is also non-negative

(resp. complete), and applying lemma 2.2.15 the second assertion follows as well, which

completes the proof of the theorem.

Remark 2.2.17. Let X be an affine variety with coordinate ring A and δ be a finitely

generated degree like function on A. Form subdegree δ as in theorem 2.2.16. If in addition

X is normal, then according to corollary 2.2.8 X δ is also normal. Since Aδ is integral

over Aδ (assertion 2 of theorem 2.2.16), it follows that there is a finite map X δ → Xδ


which is identity on X. But then X δ must be the normalization of Xδ. Motivated by

the latter, we will refer to δ as the normalization of δ.

Example 2.2.18. Let A be a finite subset of Zn and P be the convex hull of A in Rn.

Assume that Zn = ZA and that P contains the origin in its interior. Let XA and XP be

as in section 2.0.1 (with arbitrary algebraically closed field K in place of C). Recall from

example 2.1.3 that XP is isomorphic to the completion of X := (K∗)n corresponding to

a subdegree δ on A := K[x1, x−11 , . . . , xn, x

−1n ] defined by:

δ(∑

aαxα) := max

aα 6=0eδ′(xα),

where δ′(xα) := inf{r ∈ Q+ : α ∈ rP} and e is a suitable positive integer chosen to

ensure that δ is integer valued. Let η be the degree like function on A corresponding to

the completion φA : (K∗)n ↪→ XA ⊆ P|A|−1(K) (recall that components of map φA are

the monomials xα with α ∈ A). We claim that δ is the normalization of η. Once verified,

remark 2.2.17 would imply that XP is the normalization of XA (cf. theorem 2.0.2).

Indeed, the definition of η implies that η(xα) = min{k : ∃ α1, . . . , αk ∈ A such that

α =∑k

i=1 αi}, and η(∑aαx

α) = maxaα 6=0 η(xα). Note that if η(xα) ≤ k then α ∈ kP .

Hence δ ≤ eη. Since δ is a subdegree, it follows that δ ≤ eη, where as in theorem 2.2.16,

η(h) := limm→∞ η(hm)/m for all h ∈ A. Let α ∈ Zn and d := δ(xα). Then α ∈ deP

and hence there is an expression of the form: α = de

∑riαi such that αi ∈ A for each

i and ri’s are positive rational numbers such that∑

i ri ≤ 1. Let k ∈ N be such that

kdri is an integer for all i. Then keα =∑

i kdriαi, and the definition of η implies that

η(xkeα) ≤∑

i kdri = kd. Therefore η(xα) ≤ kdke

= de

= 1eδ(xα), so that δ(xα) ≥ eη(xα). It

follows that δ = eη and δ is the normalization of η, as claimed.

We summarize the results of theorems 2.2.16, 1.3.2 and 1.3.7 in the following

Corollary 2.2.19. Let X be an affine variety of dimension n and A be the coordinate

ring of X.


1. Let V1, . . . , Vm be closed subvarieties of X such that⋂i Vi is a finite set. Then there

is a complete subdegree δ on A such that the corresponding completion ψδ of X

preserves the intersection of the Vi’s at ∞.

2. Let f : X → Y be a dominating map of affine varieties. Then there is a complete

subdegree δ on A such that ψδ preserves map f at ∞.

Example 2.2.20. None of the assertions of corollary 2.2.19 would remain valid if we

replace in its conclusion the ‘subdegree’ by a ‘semidegree’. Let X = Y = K2 and

f := ((x21 − x4

2)2 + x1x2, (x21 − x4

2)3 + x1x2) : X → Y . Then f is a quasifinite map.

We show below that there is no complete semidegree δ on A := K[x1, x2] such that ψδ

preserves {f1, f2} at ∞ over any point of Y .

Let δ be any complete semidegree on A with associated filtration F = {Fd}d≥0.

Recall the notation of section 1.3: given an ideal a of A, let aδ :=⊕

d≥0(a ∩ Fd) be the

corresponding homogeneous ideal of Aδ. Let a := (a1, a2) ∈ K2 and pi = 〈fi − ai〉 ⊆ A

for i = 1, 2. If suffices to show that ψδ does not preserve {f1, f2} at ∞ over a, which is

equivalent to showing that√I $ Aδ+ (lemma 1.3.1), where I := 〈pδ1, pδ2, (1)1〉 ⊆ Aδ and

Aδ+ is the irrelevant ideal of Aδ. We will make use of the following two simple lemmas in

the proof of the latter.

Lemma 2.2.21. Let I and Aδ be as above. If I is generated by k < 3 elements, then√I 6= Aδ+.

Proof. Assume to the contrary that√I = Aδ+. Since Aδ+ is a maximal ideal of Aδ, it

follows that I is an Aδ+-primary ideal of Aδ [1, Proposition 4.2] and therefore dimension

of SpecAδ is at most k ([1], Theorems 11.14 and 11.25). But dimension of SpecAδ is

1 + dim(SpecA) = 3 > k and the derived contradiction proves the lemma.

Lemma 2.2.22. Assume δ is a semidegree. Let f 6= 0 ∈ A and a be the ideal generated

by f in A. Then ideal aδ is generated by (f)δ(f).


Proof. If g ∈ a ∩ Fd, then g = fh for some h ∈ A and δ(g) ≤ d. Since δ is a semidegree,

δ(g) = δ(f) + δ(h) and hence (g)δ(g) = (f)δ(f)(h)δ(h). But then (g)d = (g)δ(g)(1)d−δ(g) =

(f)δ(f)(h)δ(h)(1)d−δ(g) ∈ 〈(f)δ(f)〉. It follows that aδ ⊆ 〈(f)δ(f)〉, which completes the proof

(since the inclusion in the opposite direction is obvious).

We return to example 2.2.20. Let di := δ(xi) for i = 1, 2. We show√I $ Aδ+,

splitting the proof into several cases:

Case 1: d1 > 2d2. In this case δ(f1−a1) = δ(x41) = 4d1, and δ(f1−a1−x4

1) < 4d1. It

follows that (f1−a1)4d1− (x41)4d1 ∈ 〈(1)1〉. According to lemma 2.2.22, p1 = 〈(f1−a1)4d1〉

and therefore 〈p1, (1)1〉 = 〈(f1 − a1)4d1 , (1)1〉 = 〈((x1)d1)4, (1)1〉. Similarly 〈p2, (1)1〉 =

〈((x1)d1)6, (1)1〉. Hence I = 〈((x1)d1)

4, (1)1〉. Lemma 2.2.21 then implies that I 6= Aδ+.

Case 2: d1 < 2d2. It follows by an argument similar to that of case 1 that I =

〈((x2)d2)8, (1)1〉. Consequently an application of lemma 2.2.21 yields that

√I 6= Aδ+.

Case 3: d1 = 2d2. Let d := d1 = 2d2. Then δ(x1 ± x22) ≤ d. On the other hand,

since δ((x1 − x22) + (x1 + x2

2)) = d, at least one of δ(x1 + x22) and δ(x1 − x2

2) is precisely

d, whereas the other is ≥ 1 (since δ is complete). It follows that d′ := δ(x21 − x4

2) =

δ(x1 − x22) + δ(x1 + x2

2) ≥ d + 1 . Therefore δ((x1 − x42)2) = 2d′ ≥ 2d + 2 =

4d2 + 2 > 3d2 = δ(x1x2) . The latter implies that δ(f1 − a1) = 2d′ and also equality

(f1 − a1)2d′ = ((x21 − x4

2)2)2d′ + (x1x2)3d2((1)1)2d′−3d2 − a1((1)1)2d′ . Therefore 〈pδ1, (1)1〉 =

〈((x21 − x4

2)d′)2, (1)1〉. It similarly follows that 〈pδ2, (1)1〉 = 〈((x2

1 − x42)d′)

3, (1)1〉. Conse-

quently I = 〈((x21 − x4

2)d′)2, (1)1〉 and lemma 2.2.21 implies that

√I 6= Aδ+.

Summarizing all cases considered, we have proved that√I 6= Aδ+. Therefore ψδ does

not preserve {f1, f2} at ∞ over a.

Example 2.2.23. Let f : X → Y be as in example 2.2.20. For each complete semidegree

δ on A := K[x1, x2], we have shown that ψδ does not preserve {f1, f2} over any point

in Y , and hence ψδ does not preserve map f at ∞. On the other hand, according to

corollary 2.2.19 it follows that there exists a completion of X determined by a complete


subdegree on A which preserves map f at ∞. Below we exhibit an example of such a

subdegree, i.e. a complete subdegree which preserves map f at ∞.

We start with defining a complete filtration F on A by:

F0 := K , F1 := K〈1, x2, x21 − x4

2〉 , F2 := (F1)2 + K〈x1〉 , Fd :=d−1∑j=1

FjFd−j for d ≥ 3 .

We claim δ := δF is a subdegree. Indeed, Aδ ∼= K[W,X1, X2, Z]/〈W 3Z−X21 +X4

2 〉 via the

surjective K-algebra homomorphism φ : K[W,X1, X2, Z]→ Aδ which sends X1 7→ (x1)2,

X2 7→ (x2)1, Z 7→ (x21 − x4

2)1 and W 7→ (1)1. The pull back by φ of ideal I generated by

(1)1 in Aδ is 〈W,X21 −X4

2 〉, which is a radical ideal. It follows that I itself is radical and

therefore δ is a subdegree due to theorem 2.2.1. Moreover, one can show (by a lengthy

calculation which we did not include) that δ = max{δ1, δ2} where δ1 is the weighted

degree on A that assigns weight 1 to x2 and −1 to x1− x22, and δ2 is the weighted degree

on A that assigns weight 1 to x2 and −1 to x1 +x22. Finally, the following claim completes

the proof of the assertions of example 2.2.23.

Claim. Completion ψδ associated to δ preserves map f at ∞.

Proof. Let a := (a1, a2) ∈ K2 and ξ := (ξ1, ξ2) : K2 → K2 be a nondegenerate linear

transformation. Define pi := 〈(ξi ◦ f)(x) − ξi(a)〉 ⊆ K[x, y] for i = 1, 2. Let I :=

〈pδ1, pδ2, (1)1〉 ⊆ Aδ. It suffices to show that for a homogeneous prime ideal P of Aδ, if

P ⊇ I, then P ⊇ {(1)1, (x1)2, (x2)1, (x21 − x4

2)1} (since the latter elements generate the

irrelevant ideal Aδ+).

Let i = 1 or 2. Let ξi(x1, x2) := ξi1x1 + ξi2x2. Then

(ξi ◦ f)(x) = ξi1(x21 − x4

2)2 + ξi2(x21 − x4

2)3 + (ξi1 + ξi2)x1x2 ∈ A,

and therefore, in ring Aδ, elements ((ξi ◦ f)(x)− ξi(a))3 =

ξi1((x21 − x4

2)1)2(1)1 + ξi2((x21 − x4

2)1)3 + (ξi1 + ξi2)(x1)2(x2)1 − ξi(a)((1)1)3 .

Let P be a homogeneous prime ideal of Aδ such that P ⊇ I. Since (1)1 ∈ P , it follows

that ξi2((x21− x4

2)1)3 + (ξi1 + ξi2)(x1)2(x2)1 ∈ P , i = 1, 2. The nondegeneracy of ξ implies


that (ξi2, ξi1 + ξi2), for i = 1, 2 are linearly independent. It follows that both (x1)2(x2)1

and ((x21 − x4

2)1)3 are in P . Consequently (x21 − x4

2)1 and either (x1)2 or (x2)1 are in P .

Note that

((x1)2)2 − ((x2)1)4 = (x21 − x4

2)4 = (x21 − x4

2)1((1)1)3.

Since the element in the right hand side is in P , it follows that if either (x1)2 or

(x2)1 is in P , then the other one is in P as well! Therefore we showed that P ⊇

{(x21 − x4

2)1, (x1)2, (x2)1, (1)1}, which completes the proof of the claim.

Chapter 3

Affine Bezout-type theorems

We introduce a family of iterated semidegrees extending the notion of weighted homo-

geneous degrees and prove an explicit Bezout-type bound on the number of solutions

of a system of n polynomials on Kn, which is sharp whenever the associated projective

completion preserves the intersection of the component hypersurfaces at ∞. We also

establish a Bezout-type bound in the general case of a subdegree.

3.0 Background

3.0.1 Valuations and convex bodies

We start with a brief description of the theory of convex bodies associated to a valuation

on a variety and their relation to intersection theory on the variety. Our general geometric

bound for semidegrees (unlike the explicit bound in the case of iterated semidegrees)

makes use of the theorem we quote below from this theory. We follow the exposition of

[17], where the theory was developed.

Let X be a quasi-projective variety of dimension n over an algebraically closed field K.

A valuation on the field K(X) of rational functions on X with value group Zn (equipped

with an addition preserving total order ≺) is a surjective map ν : K(X) \ {0} → Zn such

76

Chapter 3. Affine Bezout-type theorems 77

that:

1. ν(λf) = ν(f) for all λ 6= 0 ∈ K,

2. ν(f + g) ≥ max{ν(f), ν(g)} for all f, g 6= 0 ∈ K(X),

3. for every pair of elements f, g 6= 0 ∈ K(X) such that ν(f) = ν(g), there is λ 6= 0 ∈ K

such that ν(f − λg) � ν(f), and

4. ν(fg) = ν(f) + ν(g) for all f, g 6= 0 ∈ K(X).

Example 3.0.1 (Monomial Valuation). Let x be any smooth point of X. Choosing a

set of local parameters at x, one can embed K[X] into the completion Ox,X of the local

ring Ox,X of X at x. According to Cohen structure theorem [8, Theorem 7.7] Ox,X is

isomorphic to the ring K[[t1, . . . , tn]] of formal power series in n indeterminates t1, . . . , tn.

Choose any addition preserving well ordering ≺ on (Z+)n (e.g. the ‘lexicographic order-

ing’ of Zn). Define ν(∑aαt

α) := min≺{α : aα 6= 0}.Then map ν extends uniquely to the

field of fractions of K[[t1, . . . , tn]], and the restriction of ν on K(X) is a valuation in the

above sense.

The intersection index [L1, . . . , Ln] of finite dimensional vector spaces L1, . . . , Ln of

rational functions on X is the number of ‘solutions’ of the system f1 = f2 = · · · = fn = 0

for generic fi ∈ Li, where we count solutions only on the maximal Zariski open subset U

of X with all functions in⋃ni=1 Li being regular.

Let ν be a (surjective) valuation onX with values in Zn. Then every finite dimensional

vector subspace L of rational functions on X admits an associated semigroup S(L) :=

{(k, ν(f)) : f ∈ Lk, k ≥ 0} ⊆ N⊕

Zn. Let C(L) be the closure of the convex hull

of S(L) in Rn+1. Then C(L) is a convex cone. Let ∆(L) := C(L) ∩ ({1} × Rn). The

main theorem of [17] gives a connection between ∆(L) and the self intersection number

[L, . . . , L] of L.

Theorem 3.0.2 (see [17, Main theorem]). Let L and ν be as above. Choose any basis

f0, . . . , fl of L. Let YL be the the closure in Pl(K) of the image of the rational map

ΦL : X → Pl(K) defined by f0, . . . , fl.


1. ∆(L) is a compact convex body and the real dimension of ∆(L) is equal to the

dimension of YL as an algebraic variety over K.

2. Assume dimK(YL) = n. Then [L, . . . , L] = n!d(L)s(L)

Voln(∆(L)), where Voln is the

n-dimensional Euclidean volume, d(L) is the mapping degree of ΦL and s(L) is

the index in Zn of the subgroup generated by all of the differences α − β such that

(k, α), (k, β) ∈ S(L) for some k ≥ 1.

Remark. Although in [17] only the case K = C is considered, the arguments used there

to prove theorem 3.0.2 remain valid for all algebraically closed fields.

3.1 Bezout Theorem for Semidegrees

Let X be an n dimensional affine variety. Let δ be a complete degree like function on

the coordinate ring A of X with associated filtration F := {Fd : d ≥ 0}. Recall from

example 1.0.6 that there exists d > 0 such that (Aδ)[d] :=⊕

k≥0 Fkd is generated by Fd as a

K-algebra and then the d-uple embedding embeds Xδ into Pl(K), where l := dimK Fd−1.

Below we will make use of a notion of multiplicity at an isolated point b of fiber f−1(a)

of morphism f : X → Kn, where X is an affine variety and a := (a1, . . . , an) ∈ Kn.

The latter multiplicity we define following the definition in [10, Example 12.4.8] as the

intersection multiplicity at b of the effective Cartier divisors determined by regular (on

X) functions fj − aj, 1 ≤ j ≤ n.

Theorem 3.1.1 (see [25, Theroem 1.3] and [26, Theorem 3.1.1]). Let X, A, δ, d and l

be as above. Denote by D the degree of Xδ in Pl(K). Let f = (f1, . . . , fn) : X → Kn be

any generically finite map. Then for all a ∈ Kn,

|f−1(a)| ≤ D

dn

n∏i=1

δ(fi) (A)

where |f−1(a)| is the number of the isolated points in fiber f−1(a) each counted with the

multiplicity of f−1(a) at the respective point. If in addition δ is a semidegree and ψδ


preserves {f1, . . . , fn} at ∞ over a, then (A) holds with an equality.

Proof. Let a = (a1, . . . , an) ∈ Kn. For each i, let ai be the ideal generated by (fi − ai)d

in A and aδi :=⊕

j≥0(〈(fi − ai)d〉 ∩ Fj) be the corresponding homogeneous ideal in Aδ.

Clearly Gi := ((fi − ai)d)ddi ∈ ai, where di := δ(fi) for each i = 1, . . . , n. It follows that

n⋂i=1

{x ∈ X : (fi(x)− ai)d = 0} ⊆n⋂i=1

{x ∈ Xδ : G(x) = 0 for all G ∈ aδi} (3.1)

⊆n⋂i=1

{x ∈ Xδ : Gi(x) = 0} (3.2)

where the first inclusion is due to lemma 1.3.1, since the closure of the hypersurface V (ai)

of X in Xδ is V (aδi ). Note that the sum of the multiplicities of the intersections of Cartier

divisors determined by (fi − ai)d at the isolated points in the set on the left hand side

of (3.1) is precisely dn times the sum of the multiplicities of fiber f−1(a) at the isolated

points in f−1(a).

Pick a set of homogeneous coordinates [y0 : · · · : yL] of Pl(K). The d-uple embedding

of Xδ into Pl(K) induces a surjective homomorphism φ : K[y0, . . . , yL]→ (Aδ)[d] of graded

K-algebras. For each i, choose an arbitrary homogeneous polynomial Gi ∈ K[y0, . . . , yL]

of degree di such that φ(Gi) = Gi. According to the classical Bezout theorem on Pl(K),

the sum of the multiplicities of isolated points of Xδ ∩ V (G1) ∩ · · · ∩ V (Gn) in Pl(K) is

at most Dd1 · · · dn, and it is equal to Dd1 · · · dn if all the points in the intersection are

isolated.

Since Xδ ∩ V (G1) ∩ · · · ∩ V (Gn) is precisely Xδ ∩ V (G1) ∩ · · · ∩ V (Gn), inequality

(A) follows from (3.2) and the conclusions of the two preceding paragraphs. The last

assertion of theorem 3.1.1 follows from the following observations:

1. if δ is a semidegree, then according to lemma 2.2.22 ideal aδi is generated by Gi and

hence ⊆ in (3.2) can be replaced by =, and

2. completion ψδ preserves {f1, . . . , fn} at ∞ over a iff the ⊆ in (3.1) is in fact an

equality.


Remark 3.1.2. Both assertions of theorem 3.1.1 are valid for δ = maxNj=1 δj being a

subdegree with δ1(fi) = · · · = δN(fi) for all i = 1, . . . , n.

Remark 3.1.3. Let f := (f1, . . . , fn) : X → Kn be a quasifinite map and δ be a

complete semidegree on A := K[X]. We claim that if ψδ preserves {f1, . . . , fn} at ∞

over any point, then ψδ preserves {f1, . . . , fn} at ∞ over all points. Indeed, let a :=

(a1, . . . , an) ∈ Kn. For each i, let pi(a) be the ideal generated by (fi − ai) in A, and

let pδi (a) :=⊕

j≥0(pi(a) ∩ Fj) ⊆ Aδ. Due to lemma 1.3.1, ψδ preserves {f1, . . . , fn}

at ∞ over a iff√I(a) = Aδ+, where I(a) := 〈pδ1(a), . . . , pδn(a), (1)1〉 ⊆ Aδ. According

to lemma 2.2.22, pδi (a) = 〈(fi − ai)di〉, where di := δ(fi) for each i. Then I(a) =

〈(f1 − a1)d1 , . . . , (fn − an)dn , (1)1〉 = 〈(f1)d1 , . . . , (fn)dn , (1)1〉. Since the latter expression

is independent of ai’s, the claim follows. Note that we have shown (in the notation of

remark 1.3.6) that either Sψδ = ∅ or Sψδ = f(X).

Example 3.1.4 (Weighted Bezout theorem, cf. [7], [25, Example 7], [26, Example 3.1.4]).

Let X := Kn and δ be a weighted degree on A := K[x1, . . . , xn] which assigns weights

di > 0 to xi, 1 ≤ i ≤ n. Then (1)1, (x1)d1 , . . . , (xn)dn is a set of K-algebra generators

of Aδ. A straightforward application of lemma 1.3.1 implies that ψδ preserves at ∞ all

components of the identity map 1 of Kn over 0. Therefore theorem 3.1.1 with f = 1

and d as in the preamble to theorem 3.1.1 implies that 1 = Ddn

∏ni=1 di and therefore

D = dn∏ni=1 di

. Consequently, theorem 3.1.1 implies for any f that:

|f−1(a)| ≤n∏i=1

δ(fi)

di. (3.3)

Recall (example 2.1.1) that grAδ ∼= K[x1, . . . , xn] via an identification of [(g)δ(g)] ∈ grAδ

and Lδ(g) ∈ K[x1, . . . , xn], for 0 6= g ∈ A. Let a := (a1, . . . , an) ∈ Kn and ai := 〈fi − ai〉

(cf. the proof of theorem 3.1.1). According to lemma 2.2.22, ideals aδi are generated by


(fi − ai)δ(fi) = (fi)δ(fi) − ai((1)1)δ(fi) for 1 ≤ i ≤ n. Therefore,

(3.3) holds with equality ⇐⇒ ψδ preserves {f1, . . . , fn} at ∞ over a

⇐⇒ V (aδ1, . . . , aδn, (1)1) = ∅ ⊆ ProjAδ

⇐⇒ V ((f1)δ(f1), . . . , (fn)δ(fn), (1)1) = ∅ ⊆ ProjAδ

⇐⇒ V ([(f1)δ(f1)], . . . , [(fn)δ(fn)]) = ∅ ⊆ Proj grAδ

⇐⇒ V (Lδ(f1), . . . ,Lδ(fn)) = {0} ∈ Kn,

where the last relation holds since the maximal ideal of the origin in Kn corresponds to the

irrelevant ideal⊕

k>0 Fk/Fk−1 of grAδ via the isomorphism grAδ ∼= K[x1, . . . , xn]. Finally

note that formula (3.3) in combination with condition V (Lδ(f1), . . . ,Lδ(fn)) = {0} ∈ Kn

for equality in (3.3) constitute the content of the Weighted Bezout theorem stated in

section 0.2.

Applying theorem 3.0.2, number D of theorem 3.1.1 admits a geometric description

as the volume of a convex body associated to δ and a Zn valued (surjective) valuation ν

of K(X) (cf. section 3.0), namely:

Proposition 3.1.5 (see [25, Proposition 1.4] and [26, Theorem 3.1.3]). Let X, A, d, δ,

l and D be as in theorem 3.1.1 and ν be a valuation on A with values in Zn. Let C be

the smallest closed cone in Rn+1 containing

G := {(1

dδ(f), ν(f)) : f ∈ A} ∪ {(1, 0, . . . , 0)}.

Let ∆ be the convex hull of the cross-section of C with the first coordinate having value

1. Then D = n! Voln(∆), where Voln is the n-dimensional Euclidean volume.

Proof. Consider L := Fd = {f ∈ A : δ(f) ≤ d}. As in section 3.0, we introduce

S(L) := {(k, ν(f)) : f ∈ Lk, k ≥ 0} ⊆ N⊕

Zn. Let C(L) be the closure of the convex

hull of S(L) in Rn+1 and ∆(L) := C(L) ∩ ({1} × Rn). Mapping ΦL of theorem 3.0.2

is precisely the embedding X ⊆ Xδ ↪→ Pl(K), so that the mapping degree d(L) of ΦL


is 1. Therefore theorem 3.0.2 implies that [L, . . . , L] = n!s(L)

Voln(∆(L)), where s(L) is

the index in Zn of the subgroup S ′(L) generated by all the differences α − β such that

(k, α), (k, β) ∈ S(L) for some k ≥ 1. Note also that:

1. according to its definition [L, . . . , L] is equal to the degree D of Xδ in PL(C),

2. Fd generates (Aδ)[d], so that for Lk = (Fd)k = Fkd for all k ≥ 1, and therefore for

all f ∈ A, (k, ν(f)) ∈ S(L) for all k ≥ δ(f)d

, and

3. for all k ≥ 1, 1 ∈ Lk and therefore (k, ν(1)) = (k, 0, . . . , 0) ∈ S(L).

Properties 2 and 3 imply that S ′(L) ⊇ {ν(f) : f ∈ A}. Since ν is surjective, it follows that

S ′(L) = Zn and hence s(L) = 1. But then D = n! Voln(∆(L)) due to property 1. Finally,

properties 2 and 3 also imply that C = C(L), and consequently that ∆ = ∆(L).

Example 3.1.6. Let X = Kn and δ be as in example 3.1.4. Let ν be any monomial

valuation on A := K[x1, . . . , xn], e.g. the one that assigns to∑aαx

α ∈ A \ {0} the

lexicographically (coordinatewise) minimal exponent α among all α := (α1, . . . , αn) with

aα 6= 0. With d being any common multiple of d1, . . . , dn, it is straightforward to see that

∆ = {(1, x) ∈ Rn+1+ :

∑ni=1 xidi ≤ d} and therefore Voln(∆) = 1

n!

∏ni=1

ddi

. It follows that

D = n! Voln(∆) =∏n

i=1ddi

, which is, of course, the value we have calculated in example

3.1.4.

3.2 Iterated Semidegree

In this section we describe particularly simple semidegrees generalizing weighted homoge-

neous degrees for which we establish a constructive version of affine Bezout-type theorem.

Our dream is that a stronger version of Main Existence Theorem 2.2.16 would be valid

with subdegrees in whose minimal presentations only constructive semidegrees ‘like’ the

iterated semidegrees of this section would appear, and we expect that a precise construc-

tive version of an affine Bezout-type theorem for any generically finite map f : Kn → Kn

would follow.


Let A be a domain and δ be a degree like function on A. Pick f ∈ A and an integer

w with w < δ(f). Let s be an indeterminate over A and δe be a ‘natural’ extension of δ

to A[s] such that δe(s) = w, namely: δe(∑ais

i) := maxai 6=0(δ(ai) + iw). Of course δ is a

degree like function iff δe is a degree like function.

Lemma 3.2.1. δ is a semidegree iff δe is a semidegree.

Proof. The (⇐) direction is obvious, since δ ≡ δe|A. For the proof of the ‘only if’

implication, let G :=∑gis

i, H :=∑hjs

j ∈ A[s] with d := δe(G), e := δe(H). For

each k ≥ 0, let Gk :=∑

δ(gi)+iw=k gisi and Hk :=

∑δ(hi)+iw=k his

i. It suffices to show

that δe(GdHe) = d + e, and hence we may w.l.o.g. assume G = Gd and H = He. Let

i0 (resp. j0) be the largest integer such that gi0 6= 0 (resp. hj0 6= 0). Then GH =

gi0hj0si0+j0 +

∑m<i0+j0

amsm. Thus δe(GH) ≥ δe(gi0hj0s

i0+j0) = δ(gi0hj0) + (i0 + j0)w =

δ(gi0) + i0w + δ(hj0) + j0w = d + e. Since the inequality δe(GH) ≤ d + e is obviously

true, it follows that δe(GH) = d+ e.

Remark 3.2.2. In fact δ is a subdegree iff δe is a subdegree, which follows by means of

calculations similar to those in the proof of lemma 3.2.1 and of the characterization of

subdegrees as degree like functions η satisfying η(fk) = kη(f) for all f ∈ A and k ≥ 0

(corollary 2.2.2).

Let J denote the ideal generated by s− f in A[s]. Identify A with A[s]/J and define

δ to be the degree like function on A induced by δe, i.e. δ(g) := min{δe(G) : G− g ∈ J}.

Let a be the principal ideal generated by f in A and let aδ be the ideal induced in Aδ by

a (as defined in lemma 1.3.1). Denote by gr a the ideal generated by the image of aδ in

grAδ, i.e. gr a := 〈[(g)δ(g)] : g ∈ a〉, where [(g)δ(g)] denotes the equivalence class of (g)δ(g)

in grAδ.

Remark 3.2.3. A straightforward application of definitions shows that δ is a degree like

function provided that δe is a degree like function and δ ≡ 0 on K. Note that δ is a


meaningful degree like function only if [(f)δ(f)] is not a unit in grAδ. Indeed, if [(f)δ(f)]

is a unit in grAδ, then [(1)0] = [(f)d][(g)−d] ∈ grAδ for some g ∈ A with δ(g) = −d.

It follows that if 1 − fg 6= 0 then δ(1 − fg) < 0. Let G := 1 − fg + gs ∈ A[s].

Then δe(gs) = w − d < 0 and therefore δe(G) < 0. Moreover, 1 ≡ G mod J in A[s].

Consequently δ(1) ≤ δe(G) < 0. Since, for all h ∈ A and n ∈ Z+, δ(h) = δ(h · (1)n) ≤

δ(h) + nδ(1), it follows that δ(h) = −∞ for all h ∈ A. Moreover, (1)1 is a unit in Aδ

(since ((1)1)−1 = (1)−1) and therefore grAδ ∼= Aδ/〈(1)1〉 is the zero ring.

Theorem 3.2.4 (cf. [25, Example 5] and [26, Theorem 2.1.3]).

1. (a) If δ is non-negative and w > 0, then δ is non-negative.

(b) δ(g) ≤ δ(g) for all g ∈ A. If [(g)δ(g)] 6∈ gr a, then δ(g) = δ(g).

(c) δ(f) ≤ w < δ(f). If δ is a semidegree and [(f)δ(f)] is not a unit in grAδ, then

δ(f) = w.

2. Assume δ is a semidegree. Then

(a) The homomorphism of K-algebras A[s]δe ∼= Aδ[s]→ Aδ induced by the inclusion

Aδ ↪→ Aδ (available due to δ ≤ δ) and s 7→ (f)w is surjective with kernel

〈(s− f)δe(s−f)〉.

(b) grAδ is isomorphic to (grAδ/ gr a)[z], where z is an indeterminate of degree w

and the isomorphism maps z to the class of equivalence [(f)w] of (f)w ∈ Aδ.

(c) δ is a semidegree if and only if gr a is a prime ideal of grAδ.

Note that if δ is a semidegree, then gr a is a principal ideal in grAδ generated by

[(f)δ(f)] due to lemma 2.2.22.

Proof. Assertion 1(a) is a straightforward consequence of the definitions. Note that

δ(g) ≤ δe(G) for all g ∈ A and G ∈ A[s] such that G ≡ g mod J . The first assertion

of 1(b) follows by setting G := g in the previous sentence. Similarly, the first assertion

of 1(c) follows by setting g := f and G := s. As for the second assertion of 1(b),

let g ∈ A and G ∈ A[s] be such that G ≡ g mod J . Let a0, . . . , ak ∈ A such that


G = g + (s− f)(a0 + a1s+ · · ·+ aksk) = (g − fa0) +

∑ki=1(ai−1 − fai)si + aks

k+1. Note

that if e := δ(g−fa0) < d := δ(g), then δ(fa0) = d and (g)d = (fa0)d+((1)1)d−e(g−fa0)e,

so that [(g)d] = [(fa0)d] ∈ gr a. In other words, if [(g)δ(g)] 6∈ gr a, then δ(g − fa0) ≥ δ(g)

and therefore δe(G) ≥ δ(g). It follows that δ(g) ≥ δ(g), which concludes the proof of

1(b).

Next we prove the second assertion of 1(c). Assume δ is a semidegree and contrary

to the conclusion of 1(c) that δ(f) < w. Then it suffices to show that [(f)d] is a unit in

grAδ, where d := δ(f). Indeed, δ(f) < w in view of the definition of δ in terms of δe

implies that there is an identity

f = akfk + ak+1f

k+1 + · · ·+ alfl (3.4)

with ak, . . . , al ∈ A such that for all j, 0 ≤ k ≤ j ≤ l, δ(aj) + jw < w. In particular,

δ(a0) < w if a0 6= 0 and δ(a1) < 0 if a1 6= 0.

If k > 1, then dividing both sides of (3.4) by f it follows that fg = 1, where

g :=∑l

j=k ajfj−2. Then δ(g) = −δ(f) = −d and therefore [(f)d] · [(g)−d] = [(1)0] ∈ grAδ.

Since [(1)0] is the identity in grAδ, it follows that [(f)d] is a unit in grAδ, which proves

assertion 1(c) in the case that k > 1.

If k = 1, then (3.4) implies that 1 − a1 = fg1, where g1 := a2 + a3f + · · · + alfl−2.

Since δ(a1) < 0, it follows that δ(1 − a1) = 0 and therefore δ(g1) = −d. Moreover,

[(a1)0] = 0 ∈ grAδ. Hence [(f)d] · [(g1)−d] = [(1)0] ∈ grAδ. Consequently [(f)d] is a unit

in grAδ, as required.

It remains to consider the case of k = 0. In this case a0 = fg2, with element g2 :=

1−a2f−a3f2−· · ·−alf l−1. Then δ(g2) = δ(a0)−δ(f) < w−d < 0 and g2 = 1−fg1 with

g1 := a2 + a3f + · · · + alfl−2. Since δ(g2) < 0 = δ(1), it follows that δ(fg1) = δ(1) = 0,

and therefore δ(g1) = −δ(f) = −d. Consequently [(f)d] · [(g1)−d] = [(1)0] − [(g2)0] =

[(1)0] ∈ grAδ, implying [(f)d] is a unit in grAδ, which completes the proof of 1(c).

Next we prove assertion 2. Assume δ is a semidegree. Due to remark 3.2.3 it suffices

to consider the case when [(f)δ(f)] is not a unit in grAδ. Then δ(f) = w according


to assertion 1(c). We start with introducing two surjective K-algebra homomorphisms

φ : A[s] � A and Φ : A[s]δe � Aδ by means of formulae

φ(k∑i=1

aisi) :=

k∑i=1

aifi for any a1, . . . , ak ∈ A,

Φ((H)d) := (φ(H))d for all H ∈ A[s], d ≥ δe(H) ∈ Z.

Clearly φ is surjective and kerφ = J . It follows that Φ is a surjective homomorphism of

graded rings with ker Φ = Jδe , and consequently, Aδ = A[s]δe/Jδe . Moreover, δ being a

semidegree on A implies that δe is also a semidegree on A[s] (lemma 3.2.1) and therefore

Jδe = 〈(s− f)δe(s−f)〉 (lemma 2.2.22), which completes the proof of 2(a).

Next we prove assertion 2(b). Ring grAδ = Aδ/〈(1)1〉 ∼= A[s]δe/(Jδe + 〈(1)1〉) because

Aδ = A[s]δe/Jδe . The element z in the assertion of 2(b) (which corresponds to [(f)w])

is precisely the equivalence class [(s)w] of (s)w ∈ A[s]δe . Note that the homomorphism

defined by Aδ[s] 3∑

(fi)d−iwsi 7→ (

∑fis

i)d ∈ A[s]δe is an isomorphism. Since δe(s) =

w < δ(f) = δe(f), it follows that (s − f)δe(s−f) = (s)w(1)δ(f)−w + (f)δ(f) and therefore

Jδe + 〈(1)1〉 = 〈(s− f)δe(s−f), (1)1〉 = 〈(f)δ(f), (1)1〉. Hence grAδ ∼= Aδ[s]/〈(f)δ(f), (1)1〉 =

(Aδ/〈(f)δ(f), (1)1〉)[s]. But Aδ/〈(f)δ(f), (1)1〉 ∼= (Aδ/〈(1)1〉)/(〈(f)δ(f), (1)1〉/〈(1)1〉) and

〈(f)δ(f), (1)1〉/〈(1)1〉 is precisely the ideal generated by [(f)δ(f)] in grAδ, which is gr a,

while Aδ/〈(1)1〉 ∼= grAδ. It follows that grAδ ∼= (grAδ/ gr a)[s], and completes the proof

of assertion 2(b).

It remains only to prove assertion 2(c). Due to assertion 2(b), gr a is a prime ideal of

grAδ iff grAδ is a domain and, of course, iff 〈(1)1〉 is a prime ideal of Aδ. But 〈(1)1〉 is a

prime ideal of Aδ iff δ is a semidegree (theorem 2.2.1), which completes the proof of the

theorem.

Theorem 3.2.4 motivates the following:

Definition. Let δ be a semidegree on A. The leading form Lδ(f) of an element f of A

is the equivalence class [(f)δ(f)] of (f)δ(f) in grAδ.


If δ is a semidegree on A and the ideal 〈Lδ(f)〉 of grAδ generated by the leading form

Lδ(f) of f ∈ A is prime, then δ is also a semidegree on A (theorem 3.2.4). Semidegree

δ differs from δ according to assertion 1. of theorem 3.2.4. On the other hand, assertion

1(b) of theorem 3.2.4 shows that δ agrees with δ off gr a. We will say that δ is formed by

the iteration procedure starting with semidegree δ by means of f ∈ A.

Example 3.2.5. Let A := K[x1, x2] and δ be the semidegree on A defined in 2.1.2. Recall

that δ(x1) = 3, δ(x2) = 2 and δ(x21 − x3

2) = 1. Moreover, K-algebra Aδ coincides with

K[(1)1, (x1)3, (x2)2, (x21 − x3

2)1] = K[X1, X2, Y, Z]/〈Y Z5 −X21 + X3

2 〉. We claim that δ is

formed by an iteration procedure by means of f := x21 − x3

2 starting with the weighted

degree η which assigns weight 3 to x1 and 2 to x2. Indeed, grAη ∼= K[x1, x2] via the map

that sends Lη(h) ∈ grAη to the leading weighted homogeneous component of h. Then,

since f = x21 − x3

2 is weighted homogeneous, Lη(f) = f . Since Lη(f) = f ∈ K[x1, x2] =

grAη, it follows that ideal 〈Lη(f)〉 is prime. Therefore according to assertion 2. of theorem

3.2.4, degree like function η formed by the iteration procedure by means of f starting with

η is in fact a semidegree. Also Aη = A[s]ηe/〈(s − f)6〉, where ηe is the weighted degree

on A[s] that extends η and sends s to 1, as defined in the paragraph preceding lemma

3.2.1. Then with t := (1)1, A[s]ηe/〈(s − x21 + x3

2)6 = K[x1, x2, s, t]/〈st5 − x21 + x3

2〉 ∼= Aδ.

To summarize, semidegree δ of example 2.1.2 coincides with the iterated semidegree η.

Remark 3.2.6. Assume a semidegree δ on the coordinate ring A of an affine variety X

is constructed by means of finitely many iterations starting with a semidegree η. Denote

by Xη and Xδ the completions of the d-uple embedding of X into appropriate projective

spaces (valid for appropriate d ∈ Z+ [27, Lemma in section III.8]). Then we can express

the degree of Xδ in terms of the degree of Xη (in a straightforward generalization of

theorem 3.2.7 below). In particular, in the special case of η being a weighted homogeneous

degree on A := K[x1, . . . , xn] with weights 0 < di := η(xi), 1 ≤ i ≤ n, degXη = 1d1·····dn

(example 3.1.4) and an explicit formula for D := degXδ, which appears in the affine

Bezout-type theorem 3.1.1, follows:


Theorem 3.2.7 (see [25, Example 9] and [26, Theorem 3.1.5]). Let δ0 be a weighted

degree on A := K[x1, . . . , xn]. Let k ≥ 1 and for each i = 1, . . . , k, let δi be a semidegree

on A obtained by an iteration procedure starting with δi−1 by means of a polynomial hi

( with 〈Lδi−1(hi)〉 being prime in grAδi−1) by assigning to the polynomial hi a weight wi

with 0 < wi < δi−1(hi). Then

D

dn=

δ0(h1) · · · δk−1(hk)

δ0(x1) · · · δ0(xn)w1 · · ·wk, (B)

where D := degXδ and d ∈ Z+ are as in theorem 3.1.1 for X := Kn, A := K[x1, . . . , xn]

and δ := δk.

Proof. Let ei := δi−1(hi) for 1 ≤ i ≤ k. According to assertion 2. of theorem 3.2.4, rings

Aδi ∼= (Aδi−1 [si])δei−1/〈(si − hi)ei〉, where si are indeterminates and δei−1 extend δi−1 by

assigning weights wi to si. It follows by induction on i with x0 = (1)1 that

Aδi = K[x0, . . . , xn, s1, . . . , si]/Ji ,

where Ji := 〈h1 − xe1−w10 s1, . . . , hi − xei−wi0 si〉, 1 ≤ i ≤ k, and hj ∈ K[x, s] are weighted

homogeneous polynomials in (x, s) := (x0, . . . , xn, s1, . . . , sj−1) whose equivalence classes

in Aδi are (hj)ej , 1 ≤ j ≤ i.

Let δ be the weighted degree on Rk := K[x0, . . . , xn, s1, . . . , sk] which assigns weight

1 to x0, di := δ0(xi) to xi, 1 ≤ i ≤ n, and wj to indeterminates sj, 1 ≤ j ≤ k. Then

homomorphism π : Rk/Jk → Aδ of graded K-algebras is surjective, and, therefore, for

each f ∈ A, there is a polynomial f in Rk with δ(f) = δ(f) and f 7→ (f)δ(f) under

homomorphism π. Moreover, homomorphism π induces an embedding of Xδk into the

weighted projective space WP := Pn+k(K; 1, d1, . . . , dn, w1, . . . , wk). Since Jk is generated

by exactly k polynomials in Rk, it follows that the image of Xδ in WP is a complete

intersection. Identifying Xδ with its image in WP, it follows that Xδ = V (Jk) and,


therefore, that for any f1, . . . , fn ∈ K[x1, . . . , xn]

n⋂i=1

{x ∈ Kn : fi(x) = 0} ⊆ Xδ ∩ V (f1) ∩ · · · ∩ V (fn) = (3.5)

V (h1 − xe1−w10 s1) ∩ · · · ∩ V (hk − xek−wk0 sk) ∩ V (f1) ∩ · · · ∩ V (fn).

Arguing via an embedding of WP ↪→ PN(K) it suffices to choose as fj’s the pull backs

of generic linear polynomials on PN(K) and then the intersection on the right hand side

of the equality in (3.5) would consist of isolated points in X := Kn ↪→ Xδ ↪→ PN(K) (of

multiplicities one and of the total number being the degree of Xδ in PN(K) according to

the commonly used geometric definition of degree of a projective variety [13, Definition

18.1]). Consequently due to weighted homogeneous Bezout theorem (example 3.1.4) on

Kn+k, the sum of the intersection multiplicitiess of Cartier divisors corresponding to

hi − xei−wi0 si, 1 ≤ i ≤ k, and fj, 1 ≤ j ≤ n, at the points in the left hand side of the

equality in (3.5) is δ(f1)···δ(fn)e1···ekd1···dnw1···wk

, and by the Bezout theorem for semidegrees, the sum of

the multiplicities of the fiber f−1(0) at the points in the left hand side of the inclusion in

(3.5) is Ddnδ(f1) · · · δ(fn). Formula (B) follows by comparing these two expressions, which

completes the proof.

Example 3.2.8. Let fk := (x1 + (x21 − x3

2)2, (x21 − x3

2)k) : K2 → K2. We estimate the

size of fibers of fk =: (fk1, fk2) in three different ways. The first one is by means of the

weighted homogeneous Bezout formula (3.3). It is straightforward to see that the smallest

upper bound given by (3.3) for fk is achieved for d1 = 3p and d2 = 2p for some p ≥ 1, in

which case the bound is 12p·6kp3p·2p = 12k. The second approach we take is via Bernstein’s

theorem (see section 0.2). Let a := (a1, a2) ∈ K2 and let P and Qk be the Newton

polygons of x1 + (x21 − x3

2)2 − a1 and, respectively, of (x21 − x3

2)k − a2. The BKK bound

for |f−1k (a)| for non-zero a1, a2 is then 2M(P ,Q) = Vol(P + Q) − Vol(P) − Vol(Q) =

12(2k + 4)(3k + 6) − 12− 3k2 = 12k. (A similar argument implies that the BKK bound

for a1 or a2 being zero is as well 12k.)

Let δ be the iterated semidegree on K[x1, x2] from example 3.2.5, so that δ(x1) = 3,


δ(x2) = 2 and δ(x21 − x3

2) = 1. Then D/dn = 63·2·1 = 1 (theorem 3.2.7). The estimate of

|f−1k (a)| given by (A) with this δ is then δ(x1 + (x2

1− x32)2)δ((x2

1− x32)k) = 3k. Moreover,

ideals generated by (fk1)δ(fk1), (fk2)δ(fk2) and (1)1 in Aδ are primary to the irrelevant ideal

of Aδ and, therefore, completion ψδ preserves fk at ∞ over all points in K2. In other

words our bound with iterated semidegree δ is exact!

3.3 A Formula for Subdegrees

Assume f := (f1, . . . , fn) : X → Kn is a dominating morphism of affine varieties with

generically finite fibers and δ := max{δj : 1 ≤ j ≤ N} is a complete subdegree on

A := K[X], i.e. δ is non-negative, finitely generated and δ−1(0) = K \ {0}. Assume

δj(fi) > 0 for each i, j. In the spirit of theorem 3.1.1 we derive in this section an upper

bound for the number of points in a generic fiber of f in X (counted with multiplicity)

in terms of the degree of a projective completion of X.

Definition.

• Let g ∈ A and divX(g) be the principal Cartier divisor corresponding to g on

X. Assume that the corresponding Weil divisor is [divX(g)] =∑ri[Vi]. Given a

completion X ↪→ Y of X, we write [divY

X(g)] for the Weil divisor on Y given by:

[divY

X(g)] :=∑

ri[V i],

where V i is the closure of Vi on Y . If Y = Xδ for some degree like function δ on

A, then we also make use of notation divδ

X(g) for divY

X(g).

• If g ∈ A is such that δj(g) > 0 for all j = 1, . . . , N , then

δg := eg ·max{ δjδj(g)

: 1 ≤ j ≤ N},

where eg is a suitable integer to ensure that δg is integer valued (e.g. one can take

eg :=∏N

j=1 δj(g)).


Let the filtration corresponding to δ be F := {Fd : d ≥ 0}. Identify Aδ with∑d∈Z Fdt

d. Recall that Xδ := ProjAδ is the union of affine charts of the form SpecAδ(gtd)

,

where d > 0 and Aδ(gtd)

is the subring of elements of degree zero of the localizations Aδgtd

.

Say U0, . . . , Um is an open cover of Xδ with Uj := SpecAδ(gjt

lj )for some lj ≥ 1 and g ∈ Flj

for every j. Moreover, assume that g0 = 1 and l0 = 1, so that U0 = SpecAδ(t) = SpecA.

Let d be a common multiple of l1, . . . , lm. Then for each j, hj := td

(gj)d/lj td

is a regular

function on Uj and hj/hk is a unit on Uj ∩ Uk. Therefore collection {(hj, Uj)}j defines

an effective Cartier divisor Dδd,∞ on Xδ, which we call the d-uple divisor at infinity. Its

associated Weil divisor is

[Dδd,∞] :=

N∑j=1

ordj(Dδd,∞)[Vj],

where V1, . . . , VN are the irreducible components of X∞ and ordj is the shorthand for

ordVj (where ordVj is as defined in section 2.0.2). Support of [Dδd,∞] being X∞ justifies

index ∞ as a subscript of Dδd,∞.

Lemma 3.3.1. Let X, A, δ and Dδd,∞ be as above. Then

1. [Dδd,∞] =

∑Nj=1

ddj

[Vj] where for every j, integer dj is the positive generator of the

subgroup of Z generated by {δj(f) : f ∈ A}.

2. Let g ∈ A be such that δg is finitely generated. Then the principal divisor of gd on

Xδg is [divXδg (gd)] = d[divδgX (g)]− eg[Dδg

d,∞].

Proof. 1. Fix integer j, 1 ≤ j ≤ N . Local ring OVj ,Xδ is a discrete valuation ring

and its associated valuation is νj(·) := − δj(·)dj

(proposition 2.2.12). Pick k, 1 ≤ k ≤ N ,

such that Vj ∩ Uk 6= ∅. Recall that Uk := SpecAδ(gkt

lk )and a local equation for Dδ

d,∞

on Uk is td

(gk)d/lk td. Let pj be the ideal of Aδ corresponding to Vj. Since Vj ∩ Uk 6= ∅,

it follows that gktlk 6∈ pj and therefore δj(gk) = lk according to assertion 1 of lemma

2.2.1.2. Therefore ordj(Dδd,∞) = νj(

td

(gk)d/lk td) = νj(1/g

d/lkk ) = − d

lkνj(gk) = d

lk· lkdj

= ddj

. It

follows that [Dδd,∞] :=

∑Nj=1 ordj(D

δd,∞)[Vj] =

∑Nj=1

ddj

[Vj], which completes the proof of

assertion 1.


2. Reindexing the δj’s if necessary, we may assume that the minimal presentation of

δg is δg = max{ eg ·δjδj(g)

: 1 ≤ j ≤ M} for some M ≤ N . For 1 ≤ j ≤ M , let V ′j be the

irreducible component of Xδg∞ corresponding to δj and d′j be the positive generator of the

subgroup of Z generated by { eg ·δj(h)

δj(g): h ∈ A}. Then, as a straightforward consequence of

proposition 2.2.12 it follows that ordV ′j (gd) = − egd

d′jfor every j and therefore [divXδg (gd)] =

d[divδgX (g)]+

∑Mj=1 ordV ′j (g

d)[V ′j ] = d[divδgX (g)]−

∑Mj=1

egd

d′j[V ′j ]. On the other hand, applying

assertion 1 with δg in place of δ yields that [Dδgd,∞] =

∑Mj=1

dd′j

[V ′j ]. Therefore [divXδg (gd)] =

d[divδgX (g)]− eg[Dδg

d,∞], as required.

Theorem 3.3.2. Assume that f := (f1, . . . , fn) : X → Kn and δ := max{δj : 1 ≤ j ≤ N}

are as in the first paragraph of this section and that δj(fi) > 0, 1 ≤ i, j ≤ n. Assume also

that for i = 1, . . . , n, subdegrees δfi are finitely generated. Let dfi ≥ 1, 1 ≤ i ≤ n, be such

that the dfi-uple embedding of Xδfi is a closed immersion of Xδfi into a projective space

PLi(K). Let L :=∏n

i=1(Li + 1) − 1 and X be the closure of the image of X in PL(K)

under the composition of the following maps:

X ↪→ Xδf1 × · · · ×Xδfn ↪→ PL1(K)× · · · × PLn(K) ↪→ PL(K),

where the first map is the diagonal embedding and the last map is the Segre embedding.

Then for all a ∈ Kn,

|f−1(a)| ≤ ef1 · · · efnnndf1 · · · dfn

deg(X), (C)

where |f−1(a)| is the number of the isolated points in f−1(a) each counted with the mul-

tiplicity of f−1(a) at the respective point.

Question: Is it true that the completeness of δ implies the finite generation property of

every δfi?

Proof of theorem 3.3.2. Denote by [yi,0 : · · · : yi,Li ] the homogeneous coordinates on

PLi(K). Without loss of generality we may assume that Xδfi∞ = Xδfi ∩ V (yi,0).


Denote by Xη the closure of the diagonal embedding of variety X into the product

Xδf1 × · · · ×Xδfn ⊆ PL1(K)× · · · × PLn(K) =: Y . Denote by [y0 : · · · : yL : z1 : · · · : zn]

the homogeneous coordinates on PL′(K), where L′ := L+ n. Let us identify PL(K) with

the subspace V (z1, · · · , zn) of PL′(K) and let s : PL1(K)× · · · × PLn(K)→ PL(K) denote

the Segre embedding. Let s′ : Xη → PL′(K) be the map defined by:

s′ : Xη 3 ([y1,0 : · · · : y1,L1 ], . . . , [yn,0 : · · · : yn,Ln ]) 7→ [s(y) : (y1,0)n : · · · : (yn,0)n] ∈ PL′(K).

Fix an i, 1 ≤ i ≤ n. Let Di := π∗i (Dδfidfi ,∞

), where πi : Xη → Xδfi is the projection

onto the i-th factor of Y . Due to our choice of yi,0, Cartier divisor Dδfidfi ,∞

is precisely the

restriction of the divisor of yi,0 to Xδfi . It follows that s′∗(D′i) = nDi, where D′i is the

restriction of the divisor of zi to s′(Xη).

Then (D1, . . . , Dn) = 1nn

(s′∗(D′1), . . . , s′∗(D′n)) = 1nn

(D′1, . . . , D′n), since intersection

numbers are preserved under the pull backs by proper birational morphisms [10, Exam-

ple 2.4.3]. Since each D′i is the divisor of a linear form on s′(Xη), it follows that the

intersection number (D′1, . . . , D′n) = deg s′(Xη). On the other hand, s|Xη = π′ ◦ s′, where

π′ is the projection onto the first L coordinates. Since the mapping degree of π′|s′(Xη) is 1,

it follows that deg(s(Xη)) = deg s′(Xη) [28, Proposition 5.5]. Combining three equalities

established in this paragraph it follows that (D1, . . . , Dn) = 1nn

deg s(Xη).

Let Ei := π∗i (divδfiX (fi)). According to assertion 2 of lemma 3.3.1 [div

Xδfi

(fdfii )] =

dfi [divδfiX (fi)]− efi [D

δfidfi ,∞

]. Since π∗i (divXδfi

(fi)) = divXη(fi) it follows that

(E1, . . . , En) =ef1 · · · efndf1 · · · dfn

(D1, . . . , Dn) =ef1 · · · efnnndf1 · · · dfn

deg s(Xη) .

Fix an i, 1 ≤ i ≤ n. Note that Dδfidfi ,∞

are very ample, i.e. are the pull backs of

the hyperplane sections under the embeddings of Xδfi into PLi(K). In particular, these

divisors are base point free [14, Section II.7]. Then the pull backsDi ofDδfidfi ,∞

are also base

point free, and so are efiEi (the latter being linearly equivalent to dfiDi). Also, since Ei’s

are effective (defined in section 2.0.2), it follows that the intersection number (E1, . . . , En)

bounds the sum of the intersection multiplicities of Ei’s at the isolated points of the


intersection⋂ni=1 Supp(Ei) [10, Section 12.2]. Of course X ∩ (

⋂ni=1 Supp(Ei)) = f−1(0).

Therefore |f−1(0)| ≤ (E1, . . . , En), which completes the proof of the theorem.

Future plans:

1. Let f := (f1, . . . , fn) : X → Kn be any generically finite map (not necessarily

satisfying the hypotheses of theorem 3.3.2). Replacing f by ξ ◦ f for a generic affine

transformation ξ of Kn (and reordering δj’s if necessary), one may assume that there is

an M ≤ N such that

(a) δj(fi) > 0 for all i and all j = 1, . . . ,M , and

(b) δj(fi) = 0 for all i and all j = M + 1, . . . , N .

We expect that it should be possible to extend theorem 3.3.2 in this setting as a con-

sequence of extending the arguments of the proof of theorem 3.3.2 to the case of X :=

Spec A, where A := {g ∈ A : δj(g) ≤ 0 for M + 1 ≤ j ≤ N}, and for the completion X δ

of X determined by δ := max{δj : 1 ≤ j ≤M}.

2. Moreover, we hope to prove (by means of an extension of our theorem A.1 to the

case of subdegrees) that for completion Xδ that preserves map f at ∞ (in the setting of

theorem 3.3.2), inequality in (C) can be replaced by equality.

Appendix A

Non-degeneracy condition in

Bernstein’s theorem

Starting with a precise statement of Bernstein’s theorem, we provide a proof of our

interpretation of Kushnirenko-Bernstein’s non-degeneracy condition (E3) as

toric completion XP of (C∗)n preserves {f1, . . . , fn} at ∞ over 0, (E)

where XP is the toric variety associated with the Minkowski sum P of the Newton

polytopes of components of f .

Theorem (Bernstein [2]). For each i = 1, . . . , n, let Ai be a finite subset of Zn and fi be

a Laurent polynomial in C[z1, z−11 , . . . , zn, z

−1n ] such that Supp(fi) ⊆ Ai. Then

|V (f1, . . . , fn)| ≤ n!M(A1, . . . , An) (∗)

where |V (f1, . . . , fn)| is the number of isolated points in the set V (f1, . . . , fn) :=

{x ∈ (C∗)n : f1(x) = · · · = fn(x) = 0} each counted with the multiplicity of f−1(0) at

the respective point, and M(A1, . . . , An) is the mixed volume of A1, . . . , An. Let fi =∑β∈Ai ai,βx

β, and for each α ∈ (Zn)∗, let Ai,α := {β ∈ Ai : 〈α, β〉 ≤ 〈α, γ〉 for all γ ∈ Ai}

and fi,α :=∑

β∈Ai,α ai,βxβ. Inequality (∗) holds with an equality iff

for all α ∈ Zn, V (f1,α, . . . , fn,α) = ∅. (∗∗)

95

Appendix A. Non-degeneracy condition in Bernstein’s theorem 96

Let fi’s and Ai’s be as in Bernstein’s theorem. Let P be the convex hull of A1+· · ·+An.

Recall from section 2.0.1 that if dimP = n, then there is an embedding φP : (C∗)n ↪→ XP ,

where XP is the toric variety corresponding to P .

Theorem A.1 (see [26, Appendix]). Assume dimP = n. Then (∗∗) holds iff property

(E) takes place.

Proof. Recall that if ψ : Zn → Zn is an injective affine transformation, then XP ∼= Xψ(P)

(theorem 2.0.2). Moreover, for g :=∑aβx

β, the image of V (g) ∩ (C∗)n under the above

isomorphism is V (gψ)∩(C∗)n, where gψ :=∑aβx

ψ(β). For f := (f1, . . . , fn) : (C∗)n → Cn,

we also define fψ := (f1ψ, . . . , fnψ). It follows that for either one of the properties

(asserted to be equivalent in the statement of the theorem), f satisfies the property if

and only if so does fψ. Therefore we may without loss of generality replace in the course

of this proof f by fψ for any injective affine transformation ψ of Zn.

We first prove the ‘only if’ implication. So, assume φP does not preserve {f1, . . . , fn}

at ∞ over 0. Let z ∈ V (f1) ∩ · · · ∩ V (fn) ∩ X∞. Recall that X∞ = ∪XQ, where the

union is over faces Q of P := P ∩ Zn (i.e. Q = Q ∩ Zn, where Q is a face of P). Let

mP ∩ Zn := {α0, . . . , αN}, for an m ≥ n − 1 as in assertion 4 of theorem 2.0.2. Fix

m. Then XP is isomorphic to the closure of the image of φP : (C∗)n → PN which maps

x 7→ [xα0

: · · · : xαN

]. Under this isomorphism, for a face Q := Q ∩ Zn of P , XQ is the

closure of the image of the map x 7→ [φQ,0(x) : · · · : φQ,N(x)], where

φQ,j(x) =

xαj

if αj ∈ mQ,

0 otherwise.

Let Q be the smallest dimensional face of P such that z ∈ XQ. Replacing f by fψ for a

suitable injective affine transformation ψ of Zn we may assume w.l.o.g. that

1. Ai ⊆ (Z+)n for each i, and

2. 0 ∈ Q ⊆ Ed := Z〈e1, . . . , ed〉, where d := dim(convQ) and ei ∈ Zn, 1 ≤ i ≤ d, have

the j-th coordinate being 1 if j = i and 0 otherwise.


(In particular, P = conv(A1 + · · ·+An)∩Zn ⊆ (Z+)n.) Reindexing the αj’s if necessary,

we may also assume that α0 = 0, αi ∈ Q for 1 ≤ i ≤ d and α1, . . . , αn are linearly

independent. Let matrix U of size n× n be defined by

U :=

α1

1 · · · α1n

......

αn1 · · · αnn

.

The assumptions on αj’s imply that matrix U is ‘block triangular’, namely:

U =

A 0

B C

,

where A is an invertible d× d matrix and C is an invertible (n− d)× (n− d) matrix over

Q . Let V := U−1 and let vi := (vi1, . . . , vin) be the i-th row of V . It follows that

V =

A−1 0

B′ C−1

, (A.1)

for an appropriate (n − d) × d matrix B′. Denote by [z0 : · · · : zN ] the homogeneous

coordinates on PN . Due to our choice of Q, zj 6= 0 iff αj ∈ conv(mQ), in particular

z0 6= 0.

Fix an i ∈ {1, . . . , n}. Since z ∈ V (fi), there is a curve Ci ⊆ V (fi) such that z ∈ Ci.

Let the j-th coordinate (Ci(t))j of the Puiseux expansion of Ci at z be as follows (with

index i suppressed in abuse, but for simplicity of notation):

(zj/z0)(t) = tγj(bj,0 + bj,1tγj,1 + bj,2t

γj,2 + · · · ) =: (Ci(t))j,

where each γj is a non-negative rational number and {γj,k}k is an increasing sequence of

positive rational numbers with a common denominator in Z+. Since (zj/z0)(0) = zj/z0

and zj 6= 0 for αj ∈ conv(mQ), it follows that in fact

γj = 0 and bj,0 = zj/z0 (A.2)


whenever αj ∈ conv(mQ), in particular for 1 ≤ j ≤ d. Note that for all sufficiently small

t ∈ C∗, points Ci(t) ∈ (C∗)n, i.e. (zj/z0)(t) = xαj(t)/xα

0(t) = xα

j(t). Therefore, for

1 ≤ j ≤ n,

xj(t) := (Ci(t))j = x(t)ej

= x(t)vj1α

1+···+vjnαn

=n∏k=1

(xαk

(t))vjk

= tγj(bj,0 + bj,1tγj,1 + bj,2t

γj,2 + · · · ),

(A.3)

where γj :=∑n

k=1 vjkγk =

∑nk=d+1 v

jkγk are (not necessarily positive) rational numbers,

{γj,k}k are increasing sequences of positive rational numbers with common denominators

in Z+, and bj,0 :=∏n

k=1 bvjkk,0. Here for all j, k such that vjk is not an integer, we may

define bvjkk,0 as follows: at first write vjk = pjk/q

jk with pjk and qjk being relatively prime and

qjk > 0. Then fix for all curves Ci the choice of a qjk-th root of the respective bk,0, and set

bvjkk,0 to be the pjk-th power of that root.

Let γ := (γ1, . . . , γn) ∈ Qn and b := (b1,0, . . . , bn,0) ∈ (C∗)n. (Recall that γ = γ(i) and

b = b(i) =: (b1,0(i), . . . , bn,0(i)) depend on the choice of curve Ci.) Since αj ∈ (Z+)n for

every j, it follows (with once again index i in the right hand side of the following formula

being suppressed) that

φP(Ci(t)) = [1 : (bα1

t〈γ,α1〉 + h.o.t.) : · · · : (bα

N

t〈γ,αN 〉 + h.o.t.)],

where h.o.t. stands for the ‘higher order terms’ (in t). Since limt→0 φP(Ci(t)) = z,

and since zj 6= 0 iff αj ∈ conv(mQ), it follows that 〈γ, αj〉 > 0 if αj 6∈ conv(mQ), and

〈γ, αj〉 = 0 if αj ∈ conv(mQ). But then Q is the ‘face’ of P corresponding to γ. Therefore

Q = conv(A1,γ + · · · + An,γ) ∩ Zn, where each Aj,γ is the ‘face’ of Aj corresponding to

γ as defined in the statement of Bernstein’s theorem. With all exponents γj, γj,k and

coefficients bj,k, 1 ≤ j ≤ n and k ≥ 0, from the expression for Ci(t) in (A.3) depending


on i, it follows

fi(Ci(t)) = fi(tγ1∑k≥0

b1,ktγ1,k , . . . , tγn

∑k≥0

bn,ktγn,k)

=∑β

ai,β

n∏j=1

(tγj∑k≥0

bj,ktγj,k)βj

=∑β

ai,β(bβt〈γ,β〉 + h.o.t.)

= tdi(∑β∈Ai,γ

ai,β bβ) + h.o.t.,

for some di ∈ Q. Since fi(Ci(t)) ≡ 0, it follows that fi,γ(i)(b(i)) =∑

β∈Ai,γ(i) ai,β b(i)β

= 0.

Moreover, since Q ⊆ Ed, there exists β(i) ∈ Zn such that Ai,γ(i) ⊆ β(i) + Ed. It

follows that fi := x−β(i)fi,γ(i) depends only on coordinates (x1, . . . , xd). Hence for each

x = (x1, . . . , xn) ∈ (C∗)n, fi,γ(i)(x) = xβ(i)fi(x1, . . . , xd). Note that according to (A.1),

vjk = 0 for 1 ≤ j ≤ d and k > d. Therefore (A.2) implies

bj,0(i) =n∏k=1

bk,0(i)vjk =

d∏k=1

bk,0(i)vjk =

d∏k=1

(zkz0

)vjk ,

with bj,0(i)’s a priori depending on curves Ci, but for 1 ≤ j ≤ d due to the latter

formula de facto not depending on i. Let z′j := bj,0(i), 1 ≤ j ≤ d. Then fi,γ(i)(b(i)) =

b(i)β(i)fi(b1,0(i), . . . , bd,0(i)) = b(i)

β(i)fi(z

′1, . . . , z

′d). It follows that fi(z

′1, . . . , z

′d) = 0.

Repeating the above arguments starting with the choice of a curve Ci ⊆ V (fi)

for every i, 1 ≤ i ≤ n, we construct γ(1), . . . , γ(n) ∈ Qn, β(1), . . . , β(n) ∈ Zn and

b(1), . . . , b(n) ∈ (C∗)n such that for every i,

(i) Q = conv(A1,γ(i) + · · ·+An,γ(i))∩Zn, i.e. Q is the ‘face’ of P corresponding to γ(i)

and Aj,γ(i) are the ‘faces’ of Aj corresponding to γ(i) for every j, 1 ≤ j ≤ n,

(ii) first d-coordinates of b(i) are z′1, . . . , z′d (in particular do not depend on i),

(iii) Laurent polynomials fi := x−β(i)fi,γ(i) depend only on coordinates (x1, . . . , xd) and

fi(z′1, . . . , z

′d) = 0.

Since a decomposition of a face of a Minkowski sum of polytopes as the Minkowski

sum of faces of the summands is unique, it follows that for every choice of i, j, k,


Ai,γ(j) = Ai,γ(k). Let γ := γ(1) and b := b(1). Then for each i, fi,γ(b) =∑

β∈Ai,γ ai,βbβ =∑

β∈Ai,γ(i) ai,βbβ = fi,γ(i)(b) = bβ(i)fi(b1, . . . , bd) = bβ(i)fi(z

′1, . . . , z

′d) = 0. In other words,

assuming that φP does not preserve {f1, . . . , fn} at∞ over 0, we derived that (∗∗) is not

satisfied, as required for the proof of the ‘only if’ implication.

Next we prove the ‘if’ implication. Assume (∗∗) is not satisfied. Replacing f by

fψ for a suitable injective affine transformation ψ of Zn we may assume w.l.o.g. that

Ai ⊆ (Z+)n for every i (similarly to the argument in the beginning of the proof of the

‘only if’ implication). Let a := (a1, . . . , an) ∈ V (f1,α, . . . , fn,α).

Lemma A.2. We claim that for every i, 1 ≤ i ≤ n, there is a curve Ci ⊆ V (fi) with

parametrization

Ci(t) = (a1tα1 + h.o.t. , . . . , ant

αn + h.o.t.) .

Proof. Indeed, fix an i, 1 ≤ i ≤ n, and let

fi(t, w1, . . . , wn) := t−difi(tα1(a1 + w1), . . . , tαn(an + wn)),

where t, w1, . . . , wn are indeterminates and di := 〈α, β〉 for β ∈ Ai,α. Then fi is a

polynomial in t, w1, . . . , wn, with a vanishing constant term, and therefore the origin O

belongs to the hypersurface V (fi) of Cn+1. Since every point of a complex hypersurface

(of a variety of dimension larger than or equal to two) is the origin of a germ of a complex

analytic curve contained in the hypersurface, it follows that there are convergent Puiseux

series wj(t), 1 ≤ j ≤ n, in t with positive rational exponents and common denominators

in Z+ such that

Ci(t) := (t, w1(t), . . . , wn(t))

is the parametrization of a curve contained in V (fi) and centered at O. Then

Ci(t) := (tα1(a1 + w1(t)), . . . , tαn(an + wn(t))) ,

as claimed by the assertion of the lemma.


Returning to the proof of the ‘if’ implication of theorem A.1 with the point a in

the set V (f1,α, . . . , fn,α) picked preceding lemma A.2 and with Ci(t) from lemma A.2 it

follows that φP(Ci(t)) = [(aα0t〈α,α

0〉+ h.o.t.) : · · · : (aαNt〈α,α

N 〉+ h.o.t)]. (The ‘h.o.t.’s

appear as they do because all αj ∈ (Z+)n, 1 ≤ j ≤ N). Multiplying every coordinate by

t−d, where d := min{〈α, αj〉}Nj=0, it follows that limt→0 φP(Ci(t)) = b := [b0 : · · · : bN ],

where

bj =

aαj

if 〈α, αj〉 = d, 0 ≤ j ≤ N,

0 otherwise.

(In particular, point b does not depend on the choices of curves Ci, 1 ≤ i ≤ n, of lemma

A.2.) Then b ∈ Ci ⊆ V (fi) for each i, and therefore φP does not preserve {f1, . . . , fn} at

∞ over 0, which completes the proof of the theorem.

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Towards a Bezout-type theory of affine varieties...Introduction 0.1 Introduction The most prominent example of complete spaces in algebraic geometry is perhaps the complex projective