Contents - Home | Mathkollar/book/chap3.pdf · Poincar´e residue map at log canonical centers of codimension ≥2. Then we turn to other ways of studying semi log canonical pairs

Contents

Chapter 3. Semi Log Canonical Pairs 31. Semi-log-canonical singuarities 42. Quotients by finite equivalence relations 153. Descending line bundles to geometric quotients 284. Semi log resolutions 365. Ramified covers 486. Canonical rings of normal crossing surfaces 53

Bibliography 57

1

2 CONTENTS

May 31, 2010

CHAPTER 3

Semi Log Canonical Pairs

We have seen in Section ?? that in order to compactify the moduli theory ofhigher dimensional varieties we need stable pairs (??). That is, pairs (X,∆) withsemi log canonical singularities and ample log canonical class KX + ∆. The aim ofthis Chapter is to study these stable pairs, especially their singularities.

In general X is neither normal nor irreducible. Such varieties can be studiedeither using semi log resolutions or by focusing on their normalization. Both ofthese approaches have difficulties.

A stable curve C has ordinary nodes, and we can encode C by giving a triple(C, D, τ) where C is the normalization of C, D ⊂ C is the preimage of the nodesand τ : D → D is an involution which tells us which point pairs of D are identifiedin C.

Correspondingly, a higher dimensional stable variety has ordinary self-intersectionin codimension 1, and we will encode (X,∆) by a quadruple

(X, D, ∆, τ)

where π : X → X is the normalization, D the preimage of the double normal cross-ing locus ofX , ∆ the preimage of ∆ and the involution τ tells us which point pairs inD are identified in X . (Since X can have rather more complicated self-intersectionsin higher codimensions, τ is an actual involution only on the normalization of D.)

It is easy to see that (X, D, ∆, τ) uniquely determines (X,∆). Our principalaim is to understand which quadruples come from an slc pair (X,∆).

Section 1 gives the precise definitions and works out the complete theory forsurfaces.

Section 2 studies the existence question for X . The main result (23) says thatan easy to state finiteness condition is necessary and sufficient for the existence ofX if (X, D + ∆) is lc. As a corolary of our methods, we also prove that if (X,∆)is slc then X is Du Bois (44).

Even if (X,∆) exists, KX + ∆ may not be Q-Cartier. It is easy to see thatif KX + ∆ is Q-Cartier then the Poincare residue map ωX(D) → ωD has to becompatible with the involution τ (14). If (X, D + ∆) is lc, we show in Section 3that the converse also holds (54). The key ingredient is a new definition of thePoincare residue map at log canonical centers of codimension ≥ 2.

Then we turn to other ways of studying semi log canonical pairs. Section 4contains various resolution theorems that are useful for non-normal schemes andSection 5 investigates finite ramified covers.

Section 6 gives an example of a surface with normal crossing singularities whosecanonical ring is not finitely generated. Thus the minimal model program fails forsemi log canonical varieties. This makes it very hard to use the subtler techniques

3

4 3. SEMI LOG CANONICAL PAIRS

of Chapter 2. For moduli theory it would be especially useful to establish thenon-normal analog of the existence of dlt models (???).

1. Semi-log-canonical singuarities

In this section we define semi-log-canonical pairs (X,∆) and prove some oftheir basic properties. If X is normal, then semi-log-canonical is equivalent to log-canonical, hence we concentrate entirely on the case when X is not normal. We firststudy the normalization (X, ∆) of (X,∆). The key difficulty is then to reconstruct(X,∆) from (X, ∆) and to show that various good properties of (X, ∆) descend to(X,∆). Most of these will be accomplished only in subsequent sections.

Demi-normal schemes.

Definition 1. Recall that, by Serre’s criterion, a schemeX is normal iff it is S2

and regular at all codimension 1 points. As a weakening of normality, it is naturalto consider schemes that are S2 and whose codimension 1 points are either regularor ordinary nodes (??). Such schemes will be called demi-normal. The initial “d”is supposed to remind us that they have double normal crossings in codimension 1.(I really would like to call these schemes “semi-normal,” but that name is alreadytaken.)

The demi normalization of a scheme is usually not defined. (What should thedemi normalization of (xn = yn) ⊂ A2 be for n ≥ 3?) However, if j : U → X is anopen subscheme with only regular points and ordinary nodes such that X \ U hascodimension ≥ 2, then SpecX j∗OU is the smallest demi normal scheme dominatingX . It is called the demi normalization of X .

Roughly speaking, the concept of semi-log-canonical is obtained by replacing“normal” with “demi-normal” in the definition of log canonical (??), but some basicdefinitions and foundational results need to be in place first.

2 (Normalization of demi-normal schemes). Let X be a demi-normal schemeand π : X → X its normalization. The conductor ideal

condX := HomX

(

π∗OX ,OX

)

⊂ OX (2.1)

is the largest ideal sheaf on X that is also an ideal sheaf on X . We write it ascondX when we view condX as an ideal sheaf on X. The conductor subschemes aredefined as

D := SpecX

(

OX/ condX

)

and D := SpecX

(

OX/ condX

)

. (2.2)

Since X is S2, D ⊂ X and D ⊂ X are both of pure codimension 1. Since X hasonly nodes at its codimension 1 points, D and D are generically reduced. Thus Dand D are both reduced divisors.

Let xi ∈ D be a generic point. Then OX,xiis an ordinary node, thus, if

chark(xi) 6= 2, π : D → D is an etale double cover in a neighborhood of xi.In general, D → D is not everywhere etale and not even flat, but the map

between the normalizations πn : Dn → Dn has degree 2 over every irreduciblecomponent. Thus it defines a Galois involution τ : Dn → Dn.

Note that in general, τ does not define an involution of D, not even set-theoretically. As a simple example, consider X := (xyz = 0) ⊂ A3. Here D ⊂ A3 isthe 3 coordinate axes with a triple point at the origin. X is the disjoint union of 3planes each containing a pair of intersecting lines Di and D is their disjoint union.

1. SEMI-LOG-CANONICAL SINGUARITIES 5

The origin 0 ∈ D has 3 preimages in D and they would all have to be in the sameτ -orbit.

Proposition 3. Let X be demi-normal. The triple (X, Dn, τ) defined in (2)uniquely determines X.

Proof. Note that π : X → X is a finite surjection and π n : Dn → Xis τ -ivariant. Assume that π′ : X → X ′ is another finite surjection such thatπ′ n : Dn → X is τ -ivariant. We prove that there is a unique g : X → X ′ suchthat π′ = g π; giving a characterization of X .

Let X∗ ⊂ X × X ′ be the image of (π, π′). Let x ∈ X be a codimension 1point. Then either X is smooth at x, hence X → X∗ → X are isomorphisms nearx, or X has a node at x with preimage x ∈ X. By assumption x → X∗ factorsthrough x→ X∗, hence again X∗ → X is an isomorphisms near x. Since X is S2,this implies that the first projection X∗ → X is an isomorphism. Thus the secondprojection X∗ → X ′ gives the required g.

Note that in the general framework of Section 2, the proof of (3) is equivalentto saying that the relation (n, n τ) : Dn → X × X generates a set-theoreticequivalence relation R X and X is the geometric quotient of X by R.

4 (Main problems). In proving that the moduli problem of stable varietiessatisfies the valuative criterion of properness, we need to construct degenerations.That is, given a flat family of stable varieties Xc : c ∈ C0 over an open curveC0 ⊂ C, we would like to extend the family across the points p ∈ C \ C0 (atleast after a finite base change). In Section ??, our method is to first constructessentially

(

Xp, Dp, τp)

and then recover from it Xp. This, however, turns out tobe quite difficult, and we have to deal with 2 main problems.

Question 5. Let X be a normal variety, D ⊂ X a reduced divisor with nor-malization n : Dn → D and τ : Dn → Dn an involution. Under what conditionsdoes there exist a demi-normal variety X with normalization

(

X, D, τ)

as in (2)

such that(

X, D, τ)

=(

X, D, τ)

?

Question 6. Assume that X is demi-normal and KX + D is Q-Cartier. Underwhat conditions is KX also Q-Cartier?

In order to answer (5), first we may aim to describe the closed fibers of the

putative π : X → X . Since Dn → X is τ -invariant, we see that for any closed pointq ∈ Dn, the points n(q) ∈ X and n(τ (q)) ∈ X must be in the same fiber of π. Therelation n(q) ∼ n(τ (q)) generates an equivalence relation on the closed points of

X. A necessary condition for the existence of X is that this equivalence relation befinite, that is, it should have finite equivalence classes.

Even assuming finiteness, the first question seems rather intractable in general,as shown by the examples in [Kol08, Sec.2]. Thus we consider the case when(

X, D)

is assumed lc. The main result of Section 2 gives a positive answer to (5)when τ is compatible with the lc structure in a weak sense (23).

The second question (6) may, at first, seem puzzling in view of the formulaπ∗KX ∼Q KX + D (8.5). However, as the examples (15) show, in general KX neednot be Q-Cartier, not even if KX + D is Cartier. We show in (14) that a necessarycondition is that the different (13) be τ -invariant. We prove in Section 3 that the

converse holds if(

X, D)

is lc (54), but not in general (16). An explicit study of thesurface case is in (17).


7 (Divisors and divisorial sheaves on demi-normal schemes). Let X be demi-normal. Z-divisors whose support does not contain any irreducible component ofthe conductor DX ⊂ X (2.2) form a subgroup

Weil∗(X) ⊂Weil(X). (7.1)

A rank 1 reflexive sheaf which is locally free at the generic points of DX iscalled a divisorial sheaf on X . Divisorial sheaves form a subgroup

Cl∗(X) ⊂ Cl(X). (7.2)

As usual, the product of two divisorial sheaves L1, L2 is given by

L1⊗L2 :=(

L1 ⊗ L2)∗∗,

the double dual or reflexive hull of the usual tensor product. For powers we usethe notation L[m] :=

(

L⊗m)∗∗

.Let B be a Z-divisor whose support does not contain any irreducible component

of the conductor. Then there is a closed subset Z ⊂ X of codimension ≥ 2 suchthat X0 := X \Z has only smoth and double nc points and B0 := B|X0 has smoothsupport. Thus B0 is a Cartier divisor on X0 and OX0(B0) is an invertible sheaf.Let j : X0 → X denote the natural injection and set

OX(B) := j∗OX0(B0). (7.3)

This establishes a surjective homomorphism Weil∗(X)→ Cl∗(X).Similarly, KX0 is a Cartier divisor on X0 and ωX0

∼= OX0(KX0) an invertiblesheaves. For every m ∈ Z, we get the rank 1 reflexive sheaves

ωX := j∗ωX0 and ω[m]X (B) := j∗

(

ωmX0(B0)

)

. (7.4)

Thus it makes sense to talk about KX or B being Cartier or Q-Cartier, even if B isa Q-divisor. (Even on a nodal curve C one has to be rather careful about viewing anode p ∈ C as a Weil divisor such that 2[p] is Cartier. Fortunately, in the slc case,we only need to deal with divisors in Weil∗(X).)

Let π : X → X be the normalization. For any B in Weil∗(X), let B denotethe divisorial part of π−1(B), as a divisor on X . This establishes a one-to-onecorrespondence between Z-divisors (resp. Q-divisors) on X whose support does notcontain any irreducible component of the conductor DX ⊂ X and Z-divisors (resp.Q-divisors) on X whose support does not contain any irreducible component ofDX ⊂ X.

8. Let Y be a scheme with only double nc points and π : Y → Y its normal-ization. Then Y and the conductors D ⊂ Y and D ⊂ Y are smooth. The naturalmap π : D → D is an etale double cover with Galois involution τ . From

π∗ωY = HomY

(

π∗OY , ωY

)

we conclude that π∗ωY = ωY (−D). (Note that D is not a Cartier divisor on X .)Since the conductor ideals OY (−D) = OY

(

−D)

agree, the latter is equivalent to

ωY = π∗ωY

(

−D)

. Since D is a Cartier divisor, we can take it to the other side toobtain the natural isomorphism

π∗ωY = ωY

(

D)

. (8.1)


If X is an arbitrary demi-normal scheme, we can apply the above considerationto an open subset X0 ⊂ X such that X \ X0 has codimension ≥ 2. By pushingforward from X0 (resp. X0) to X (resp. X) we obtain that

π∗ωX = ωX(−D) and(

π∗ωX

)∗∗= ωX

(

D)

, (8.2)

where the double dual is necessary in general since the pull back of an S2 sheafneed not be S2. Similarly, for any Z-divisor B and integer m we obtain a naturalisomorphism

(

π∗ω[m]X (B)

)∗∗ ∼= ω[m]

X

(

mD + B)

. (8.3)

If ∆ is a Q-divisor, m∆ is integral and m(KX + ∆) is Cartier, this simplifies to

π∗(

ω[m]X (m∆)

) ∼= ω[m]

X

(

mD +m∆)

, (8.4)

which we frequently abbreviate as

π∗(

KX + ∆) ∼Q KX + D + ∆. (8.5)

It is a little more interesting to study which sections of ω[m]

X

(

mD +m∆)

descend

to a section of ω[m]X (m∆). The only question is at the generic points of D, hence

we can work on X0 and ignore ∆.We give an answer in terms of the Poincare residue map (??)

R : ωX0(D0)→(

ωX0(D0))

|D0 = ωD0 .

By taking tensor powers, we get

R⊗m :(

ωX0(D0))⊗m → ωm

D0 .

As a local model, we can take X := (xy = 0) ⊂ A2. A generator of ωX is given byσ := R

(

(xy)−1d(xy))

. Note that

R(d(xy)

xy

)∣

∣

∣

(y=0)=dx

xand R

(d(xy)

xy

)∣

∣

∣

(x=0)= −dy

y.

The two residues differ by a minus sign, thus we obtain the following:

Proposition 9. A section φ of ω[m]

X

(

mD + m∆)

descends to a section of

ω[m]X (m∆) iff R⊗m(φ) is τ-invariant if m is even and τ-anti-invariant if m is

odd.

Remark 10. While it is not necessary, it is instructive to compute the dual-izing sheaf of the curve singularity Cn given by the n coordinate axes in An

k . Itsnormalization Cn is the disjoint union of n lines. Let P ⊂ Cn be the preimageof the origin of Cn; it is n points, each with multiplicity 1. By taking a genericprojection, we see that there is an exact sequence

0→ ωCn→ ωCn

(

P)

P

R−→ k → 0

where the map∑R sends a 1-form η to the sum of its residues at the points P .

From this we see that, for n ≥ 3, the sheaves ωCnare not locally free.

11 (Semi-resolutions). Let X be a demi-normal scheme over a field of char-acteristic 0 and X0 ⊂ X an open subset that has only smooth points (x1 = 0),double nc points (x2

1−ux22 = 0) and pinch points (x2

1 = x22x3) such that X \X0 has

codimension ≥ 2. We show in (87) that there is a projective birational morphismf : X ′ → X such that


(1) X ′ has only smooth points, double nc points and pinch points,(2) f is an isomorphism over X0, and(3) SingX ′ maps birationally onto SingX .

We call any such f : X ′ → X a semi-resolution of X .Moreover, given (X,∆), we can choose f : X ′ → X such that

(4) the normalization of X ′ is a log resolution (???) of (X, D + ∆).

We call any such f : X ′ → X a log semi-resolution of (X,∆). See (90) for details.Note that by (3), X ′ is smooth at the generic point of any f -exceptional divisor.

Semi-log-canonical.

Definition–Lemma 12. Let X be a demi-normal scheme over a field of char-acteristic 0 and ∆ an effective Q-divisor whose support does not contain any irre-ducible component of the conductor D ⊂ X (2.2).

The pair (X,∆) is called semi-log-canonical or slc if

(1) KX + ∆ is Q-Cartier, and(2) one of the following equivalent conditions holds

(a)(

X, D+ ∆)

is lc where D ⊂ X is the conductor (2.2) on X and ∆ is

the divisorial part of π−1(∆), or(b) a(E,X,∆) ≥ −1 for every exceptional divisor E for every semi-

resolution of X (11).

Note that (2.b) is the exact analog of the definition of log canonical given in (??).In order to see that the conditions (2.a) and (2.b) are equivalent, let f : Y → X

be any semi-resolution and Y → Y the normalization. Then we have a commutativediagram

YπY−→ Y

f ↓ ↓ fX

π−→ X

and π∗(

KX + ∆) ∼Q KX + D + ∆ by (8.5).Since Y is smooth at the generic points of Ex(f), we see that πY is an isomor-

phism over the generic points of Ex(f). Thus

a(E,X,∆) = a(E, X, D + ∆) (12.3)

for every exceptional divisor E. Thus (2.a) ⇒ (2.b) and, using (??), the conversealso follows from (11.4); see also (90).

The discrepancy a(E,X,∆) is not defined if KX + ∆ is not Q-Cartier, thus(12.2.b) does not make sense unless (12.1) holds. By contrast, (12.2.a) makes senseif KX + D+ ∆ is Q-Cartier, even if KX + ∆ is not. The point of Question (6) is tounderstand the difference between these two. The answer is given in terms of thedifferent (??), which we recall next.

13. Let (Y,D + ∆) be a pair where Y is normal, D a reduced divisor and ∆a Q-divisor whose support does not contain any irreducible component of D. Letσ : Dn → D be the normalization. Assume that m∆ is an integral divisor andm(KY +D+ ∆) is a Cartier divisor. By (??) there is a unique Q-divisor DiffDn ∆on Dn such that

(1) m ·DiffDn ∆ is integral and m(KDn + DiffDn ∆) is Cartier, and


(2) the mth tensor power of the Poincare residue map (??) extends to anatural isomorphism

σ∗(

ω[m]Y (mD +m∆)

) ∼= ω[m]Dn (m ·DiffDn ∆).

Note that the Poincare residue isomorphism is defined over the snc locus of (Y,D+∆) and the different is then chosen as the unique Q-divisor for which the extensionis an isomorphism.

Let us now apply the above to (X, D, ∆) obtained as the normalization of apair (X,∆). Using (8.5), for m sufficiently divisible, we have isomorphisms

σ∗π∗(

ω[m]X (m∆)

) ∼= σ∗(

ω[m]

X(mD +m∆)

) ∼= ω[m]

Dn (m ·DiffDn ∆). (13.3)

Note that the composite Dn → X → X is τ -invariant. hence the composite iso-morphism in (13.3) is also τ -invariant. As noted above, the isomorphism

σ∗π∗(

ω[m]X (m∆)

) ∼= ω[m]

Dn (m ·DiffDn ∆).

uniquely determines the different DiffDn ∆. Thus we have proved the following:

Proposition 14. Let X be demi-normal and ∆ a Q-divisor whose support doesnot contain any irreducible component of the conductor D ⊂ X. Let (X, D, ∆) andτ : D → D be as in (2) and (7). If KX + ∆ is Q-Cartier then DiffDn ∆ is τ-invariant.

Note that the τ -invariance of DiffDn ∆ depends only on the codimension 2points of X and we prove in (17) that the converse of (14) holds outside a codimen-sion ≥ 3 subset of X . Thus the key question is whether there are futher conditionsat higher codimension points or not. We settle this if (X, D, ∆) is dlt (19), but thegeneral case, treated in Section 3, seems more subtle. As the examples (16) show,there are further conditions if (X, D, ∆) is not lc.

Semi-log-canonical surfaces.

Let us start with a series of examples of non-slc surfaces which seem quite closeto being slc.

Example 15. In A4 consider the surface S that consists of 3 planes, Pxy :=(z = t = 0), Pyz := (x = t = 0), Pzt := (x = y = 0). Its normalization is thedisjoint union S = Pxy ∐ Pyz ∐ Pzt and, correspondingly, the conductor D has 3pieces L1 := (x = 0) ⊂ Pxy, L′

1 + L′2 := (yz = 0) ⊂ Pyz and L2 := (t = 0) ⊂ Pzt.

Its normalization Dn is the disjoint union of the 4 lines Li, L′i. Thus (S, D) is dlt

and both KS and D are Cartier.We see that the origin appears with coefficient 0 in the different on L1 and L2

but with coefficient 1 on L′1 and L′

2. The involution τ interchanges L1 with L′1 and

L2 with L′2. Thus DiffD 0 is not τ -invariant, hence ωS is not Cartier and not even

Q-Cartier.Note that S is a cone over a curve C ⊂ P3 which is a chain of 3 lines, ωC has

degree −1 on the two ends and 0 on the middle line. Thus ωC is not Q-linearlyequivalent to a rational multiple of the hyperplane class and (??) also implies thatωS is not Q-Cartier.

The next example shows that in the non-lc case there is no numerical conditionthat decides whether a demi-normal surface has Q-Cartier canonical class or not.


Example 16. We describe a flat family of demi-normal surfaces parametrizedby C∗ × C∗ such that the canonical class of the fibers is Q-Cartier for a Zariskidense set of pairs (λ, µ) ∈ C∗ ×C∗ and not Q-Cartier for another Zariski dense setof pairs.

Start with a cone S over a hyperelliptic curve and two rulings Cx, Cy ⊂ S.Take two copies of S and glue them together by the isomorphisms C1

x → C2x and

C1y → C2

y which are multiplication by λ ∈ C∗ (resp. µ ∈ C∗) to get a non-normalsurface T (λ, µ). We show that its canonical class is Q-Cartier iff λ/µ is a root ofunity.

To get concrete examples, fix an integer a ≥ 0 and set

S :=(

z2 = xy(x2a + y2a))

⊂ A3 and C := Cx + Cy

where Cx = (y = z = 0) and Cy = (x = z = 0). Note that C is not Cartier but2C = (xy = 0) is. Furthermore, ωS is locally free with generator z−1dx ∧ dy andso ω2

S(2C) is locally free with generator

1

xyz2

(

dx ∧ dy)⊗2

=1

x2y2(x2a + y2a)

(

dx ∧ dy)⊗2

.

The restriction of ω2S(2C) to Cx is thus locally free with generator

1

x2(x2a + y2a)

(

dx ∧ dyy

)⊗2∣∣

∣

Cx

=1

x2+2a

(

dx)⊗2

.

Hence the different on Cx is the origin with coefficient 1+a. Similarly, the restriction

of ω2S(2C) to Cy is locally free with generator y−2−2a

(

dy)⊗2

.Take now 2 copies Si with coordinates (xi, yi, zi) for i ∈ 1, 2. Let τ(λ, µ) :

C1 → C2 be an isomorphism such that τ(λ, µ)∗x2 = λx1 and τ(λ, µ)∗y2 = µy1. LetT (λ, µ) be obtained by gluing C1 ⊂ S1 to C2 ⊂ S2 using τ(λ, µ).

Assume that ω2mT (λ,µ) is locally free with generator σ. Then the restriction of σ

to Si is of the form

σ|Si=

1

x2mi y2m

i (x2ai + y2a

i )m

(

dxi ∧ dyi

)⊗2m · fi(xi, yi, zi)

for some fi such that fi(0, 0, 0) 6= 0. Furthermore,

τ∗(

σ|S2

)

|C2=

(

σ|S1

)

|C1.

Further restricting to the x-axis, this gives

1

(λx1)2m+2am

(

λdx1

)⊗2mf2(λx1, 0, 0) =

1

x2m+2am1

(

dx1

)⊗2mf1(x1, 0, 0).

which implies that

f2(0, 0, 0) = λ2amf1(0, 0, 0).

Similarly, computing on the y-axis we obtain that

f2(0, 0, 0) = µ2amf1(0, 0, 0).

If λ2am 6= µ2am, these imply that f1(0, 0, 0) = f2(0, 0, 0) = 0, hence ω[2m]T (λ,µ) is not

locally free. If λ2am = µ2am then f1(x1, y1, z1) ≡ 1 and f2(x2, y2, z2) ≡ λ2am give

a global generator of ω[2m]T (λ,µ).

For a ≥ 1, we have our required examples. As λ, µ vary in C∗ × C∗, we get aflat family of demi-normal surfaces T (λ, µ). The set of pairs (λ, µ) such that λ/µ


is a root of unity is a Zariski dense subset of C∗ × C∗ whose complement is alsoZariski dense.

Note, however, that for a = 0, ω[2]T (λ,µ) is locally free for every λ, µ. In this

case, S :=(

z2 = xy)

⊂ A3 is a quadric cone and T (λ, µ) is slc. (In fact T (λ, µ)

is isomorphic to the reducible quartic cone (x2 + y2 + z2 + t2 = xy = 0) ⊂ A4 forevery λ, µ.)

We are now ready to prove the converse of (14) for surfaces.

Theorem 17. Let X be demi-normal and ∆ a Q-divisor whose support doesnot contain any irreducible component of the conductor D ⊂ X. Let (X, D, ∆) andτ : D → D be as in (2) and (7). The following are equivalent.

(1) DiffDn ∆ is τ-invariant.(2) There is a codimension 3 set W ⊂ X such that (X \W,∆|X\W ) is slc.

Proof. We have already seen in (14) that (2) ⇒ (1).The converse is etale local near codimension 2 points ofX . We can thus localize

at such a point p ∈ X and assume from now on that X is an affine surface.The conductor D ⊂ X is thus a curve and by passing to a suitable etale neigh-

borhood of p we may assume that the irreducible components of D are analyticallyirreducible at p. (This will make book-keeping easier.)

It is easiest to use case analysis, relying on some of the classification results in(??), but we use only (??).

(17.3) Plt normalization case. Assume that there is an irreducible componentX1 ⊂ X such that (X1, D1 + ∆1) is plt. By (??), D1 is a smooth curve, ⌊∆1⌋ = 0,and and (D1,DiffD1

∆1) is klt by adjunction (??).There are 2 cases:(i) If τ is an involution of D1 then X = X1 is the only component and X = X1

is not normal.(ii) If τ maps D1 to another double curve D2, then, by the τ -invariance of the

different (17.1), DiffD1∆1 = DiffD2

∆2 and so (X2, D2+∆2) is also plt by inversion

of adjunction (??). Thus X = X1 + X2 has 2 irreducible components, both plt andthe Xi are also irreducible components of X .

In the first case, chosoe m ∈ N such that mDiffD1∆1 is a Z-divisor and let σ

be a τ -invariant generator of ω2mD1

(2m ·DiffD1∆1). Since

H0(

X, ω[2m]

X(2mD1 + 2m∆)

)

։ H0(

D1, ω2mD1

(2m ·DiffD1∆1)

)

is surjective, we can lift σ to a generator φ ∈ H0(

X, ω[2m]

X(2mD1 + 2m∆)

)

, and by(9), φ descends to a nowhere zero section

Φ0 ∈ H0(

X \ p, ω[2m]X (2m∆)|X\p

)

which then extends to a local generator Φ ∈ H0(

X,ω[2m]X (2m∆)

)

.In the second case, for i = 1, 2, pick local generators

σi ∈ H0(

Di, ω2mDi

(2mDiffDi∆)

)

that are interchanged by τ and lift them back to sections

φi ∈ H0(

Xi, ω[2m]

Xi(2mDi + 2m∆i)

)

.


As before, the pair (φ1, φ2) descends to a section

Φ0 ∈ H0(

X \ p, ω[2m]X (2m∆)|X\p

)

which then extends to a section Φ ∈ H0(

X,ω[2m]X (2m∆)

)

. Then Φ is a local gener-

ator of ω[2m]X (2m∆), thus 2m(KX + ∆) is Cartier at P .

(17.4) Non-plt normalization case.Let (pj ∈ Dj) be the irreducible components of Dn. Since (X, D+ ∆) is lc but

not plt at any of the preimages of p, we see that

DiffDj

(

D − Dj + ∆)

= 1 · [pj ] for every j.

Thus we can pick local generators

σj ∈ H0(

Dj , ω2mDj

(2m[pj ]))

= H0(

Dj , ω2mDj

(

2mDiffDj(D − Dj + ∆)

))

that have residue 1 at pj and such that together they give a τ -invariant section ofω2m

Dn(2mDiffDn ∆). By (??), D is a curve with only nodes. Since the σj all havethe same residue, by (9), they descend to a section

σ ∈ H0(

D, ω2mD (2mDiffD ∆)

)

.

As before, σ lifts back to φ ∈ H0(

X, ω[2m]

X(2mD + 2m∆)

)

and then descends to

Φ ∈ H0(

X,ω[2m]X (2m∆)

)

.

(17.5) Note the key point of the proof: on a smooth pointed curve p ∈ C, thefiber of ωC(p) over p is not just a 1-dimensional vector space, but the residue givesa canonical isomorphism ωC(p)|p ∼= C. A difficulty in higher dimensions is thatthere is no similar canonical isomorphism.

For instance, if X is a cone with vertex p over an Abelian variety A then thereis a natural isomorphism

ω[m]X |x = ωm

X |x ∼= H0(A,ωA)⊗m

and the latter does not have a canonical isomorphism with C. (Indeed, as A movesin the moduli space of Abelian varieties, the H0(A,ωA)⊗m are fibers of an ampleline bundle on the moduli space.)

We return to this in Section 3.

Divisorial semi-log-terminal.

Definition 18. An slc pair (X,∆) is divisorial semi-log-terminal or dslt ifa(E,X,∆) > −1 for every exceptional divisor E over X such that (X,∆) is notsemi-snc (??) at the generic point of centerX E.

By (12.3), this implies that(

X, D + ∆)

is dlt. The converse is not quite true,

for instance S := (xy = zt = 0) ⊂ A4 is not dslt but its normalization is dlt. Wesee, however, in (19) that this difference appears only in codimension 2.

If (Y,B + ∆) is dlt and B is reduced, then (B,DiffB ∆) is dslt, and this is themain reason for our definition. However, if (B,DiffB ∆) is dslt then (Y,B + ∆)need not be dlt. For instance, take Y := (x1 · · ·xn + xm

n+1 = 0) ⊂ An+1 andB := (xn+1 = 0). Then B is simple normal crossing but (Y,B) is not, hence (Y,B)is not dlt.


It may be useful to develop a variant of dlt/dslt that is compatible both withadjunction and inversion of adjunction. There are, however, enough flavors of “logterminal” floating around, so we will not do this.

In the dlt case, we have the following positive answer to (6).

Proposition 19. Let (X,∆) be a demi-normal pair. Assume that

(1) there is a codimension 3 set W ⊂ X such that (X \W,∆|X\W ) is dslt and

(2) the normalization (X, D, ∆) of (X,∆) is dlt.

Then

(3) the irreducible components of X are normal,(4) KX + ∆ is Q-Cartier and(5) (X,∆) is dslt.

Proof. We may assume that X is affine. At a codimension 2 point p ∈ X , thepair (X,∆) is either snc, and hence the irreducible components of X are normalnear p, or (X, D, ∆) is plt above p and we are in case (17.3.ii). Thus again theirreducible components of X are normal.

Let X1, . . . , Xn be the irreducible components of X with normalization Xj →Xj. Let W ⊂ X be the preimage of W .

Set Bj := Xj ∩(

X1 ∪ · · · ∪Xj−1

)

, as a divisor on Xj . By (??), OXj(−Bj) is a

CM sheaf, hence depthXj∩W OXj(−Bj) ≥ 3. By (???), this implies that

H1(

Xj \W,OXj(−Bj)|Xj\W

)

= H1(

Xj \ W ,OXj(−Bj)|Xj\W

)

= 0.

Hence, by (20.2), each Xj is S2 and hence normal. Thus X and X have the sameirreducible components.

There is an m > 0 such that m(KX + ∆)|Xjis locally free for every j. We can

now apply (20.3) to L := ωmX (m∆)|X\W to conclude that KX + ∆ is Q-Cartier.

Let E be an exceptional divisor over X such that a(E,X,∆) = −1. We needto prove that (X,∆) is snc at the generic point of centerX E. By localizing, we mayassume that centerX E =: p ∈ X is a closed point and (X,∆) is dlt outside p. Byassumption, we are done if codimX p ≤ 2.

Thus assume that dimX ≥ 3 and let (Xi, Di + ∆i) denote the irreduciblecomponents of (X, D + ∆). By permuting the indices, we may assume that E isan exceptional divisor over X1. Then (X1, D1 + ∆1) is snc at p. If Xi is any otherirreducible component such that dimp(X1 ∩Xi) ≥ dimX − 1, then adjunction andinversion of adjunction shows that there is an exceptional divisor Ei over Xi withdiscrepancy −1 whose center is p ∈ Xi. Thus (Xi, Di + ∆i) is also snc at p. SinceX is S2, the complement of any codimension ≥ 2 subset is still connected [Har62],thus every (Xj , Dj + ∆j) is snc at p.

We claim that dim TpX = dimX+1. Note that dimTpX1 = dimX and for anyi 6= 1, dimTp(X1+Xi) = dimX+1. Thus we are done if Tp(X1+Xi) = Tp(X1+Xj)for every i 6= 1 6= j. For this it is enough to find a tangent vector

v ∈(

Tp(X1 +Xi) ∩ Tp(X1 +Xj))

\ TpX1.

Note that (X1 ∩Xi) and (X1 ∩Xj) are divisors in X1, hence their intersection hasdimension ≥ dimX − 2. Since dimX ≥ 3, we conclude that Xi ∩ Xj is strictlylarger than p. Thus dimp(Xi ∩ Xj) ≥ dimX − 1. In particular, Xi ∩ Xj has atangent vector v which is not a tangent vector to X1 ∩Xi ∩Xj.

Thus X has embedding dimension dimX + 1 and so it is snc.


Proposition 20. Let X be affine, pure dimensional and X1, . . . , Xm the irre-ducible components of X. Let W ⊂ X be a closed subset of codimension ≥ 3. LetF be a coherent sheaf on X and set

Ij := ker[

F |X1∪···∪Xj→ F |X1∪···∪Xj−1

]

.

Assume that H1(

Xj \W, Ij |Xj\W

)

= 0 for j ≥ 2. Then

(1) The restriction maps

H0(

X \W,F |X\W

)

→ H0(

X1 ∪ · · · ∪Xj \W,F |X1∪···∪Xj\W

)

are surjective.(2) If depthW F ≥ 2 then depthW F |X1∪···∪Xj

≥ 2 for every j.(3) If F |Xj\W

∼= OXj\W for every j then F ∼= OX .

Proof. The first claim follows from the cohomology sequence of

0→ Ij → F |X1∪···∪Xj→ F |X1∪···∪Xj−1

→ 0

and induction on j. If depthW F |X1∪···∪Xj< 2 then F |X1∪···∪Xj\W has a section φ

which does no extend to a section of F |X1∪···∪Xj. By lifting φ back to a section of

F |X\W , we would get a contradiction. This proves (2).Finally, we prove by induction on j that, under the assumptions of (3), F |X1∪···∪Xj\W

has a nowhere zero section. For j = 1 we have assumed this. Next we lift the sec-tion, going from j − 1 to j.

Since H1(

Xj \W, Ij |Xj\W

)

= 0, we have a surjection

H0(

X1∪· · ·∪Xj \W,F |X1∪···∪Xj\W

)

։ H0(

X1∪· · ·∪Xj−1 \W,F |X1∪···∪Xj−1\W

)

.

Thus F |X1∪···∪Xj\W has a section σj which is nowhere zero on X1∪· · ·∪Xj−1 \W .Note that σj |Xj\W is the section of a trivial line bundle. Thus, if it vanishes at all,then it vanishes along a Cartier divisor Dj on Xj .

Since F |X1∪···∪Xjis S2, Xj ∩

(

X1 ∪ · · · ∪Xj−1

)

has pure codimension 1 in Xj.

Thus, if Dj 6= 0 then Dj ∩(

X1 ∪ · · · ∪Xj−1

)

is a nonempty codimension 2 set ofX1 ∪ · · · ∪Xj−1. On the other hand, W has codimension 3 and σj does not vanishon X1 ∪ · · · ∪Xj−1 \W .

This implies that Dj = 0 and so σj is nowhere zero on X1 ∪ · · · ∪Xj \W .

21 (Dslt models). In the study of lc pairs (X,∆) it is very useful that there isa dlt model, that is, a projective, birational morphism f : (X ′,∆′)→ (X,∆) suchthat (X ′,∆′) is dlt, KX′ + ∆′ ∼Q f∗(KX + ∆) and every f -exceptional divisor hasdiscrepancy −1 (???).

It would be convenient to have a similar result for slc pairs. An obvious ob-struction is given by codimension 1 self-intersections of the irreducible componentsof X . Indeed, this is not allowed on a dslt pair but a semi resolution can not removesuch self-intersections.

Every demi-normal scheme has a natural double cover that removes such codi-mension 1 self-intersections (22), thus it is of interest to ask for dslt models assumingthat every irreducible component of X is normal in codimension 1.

22 (A natural double cover). Every demi-normal scheme has a natural doublecover, constructed as follows.

Let X0 be a scheme whose singularties are double nc points only.

2. QUOTIENTS BY FINITE EQUIVALENCE RELATIONS 15

First let us work over C. Let γ : S1 → X0(C) be a path that intersects thesingular locus only finitely many times. Let c(γ) ∈ Z/2Z be the number of theseintersection points where γ moves from one local component to another. It is easyto see that c : π1(X

0, p) → Z/2Z is a well defined group homomorphism. Let

π0 : X0 → X0 be the corresponding double cover.For a general scheme, we can construct X0 as follows. Let π0 : X0 → X0

denote its normalization with conductors D0 ⊂ X0, D0 ⊂ X0 and Galois involutionτ : D0 → D0. Take two copies X0

1 ∐ X02 and on D0

1 ∐ D02 consider the involution

ρ(p, q) =(

τ(q), τ(p))

.

Note that(

D01∐D0

2

)

/ρ ∼= D0 but the isomorphism is non-canonical. Let X0 be theuniversal pushout (46) of

(

D01 ∐ D0

2

)

/ρ←(

D01 ∐ D0

2

)

→(

X01 ∐ X0

2

)

.

Then π0 : X0 → X0 is an etale double cover and the irreducible components ofX0 are smooth. The normalization of X0 is a disjoint union of two copies of thenormalization of X0.

Another way to construct π0 : X0 → X0 is the following. There is a naturalquotient map q : π∗OX0 → π∗OD0 and τ decomposes the latter as the τ -invariantpart OD and the τ -anti-invariant part, call it LD. Then q−1OD ⊂ π∗OX0 is natu-rally OX and LX := q−1(LD) ⊂ π∗OX0 is also an invertible sheaf. Its tensor squareis OX , since LD · LD = OD (multiplication as in π∗OD0). Thus LX is 2-torsion inPic(X0) and

X0 = SpecX0

(

OX + LX

)

.

Let now X be a demi-normal scheme and j : X0 → X an open subset withdouble nc points only and such that X \X0 has codimenson ≥ 2. Let π0 : X0 → X0

be as above. Then j∗π0∗OX0 is a coherent sheaf of algebras on X . Set

X := SpecX j∗π0∗OX0

with projection π : X → X .By construction, X is S2, π is etale in codimension 1 and the normalization of

X is a disjoint union of two copies of the normalization of X . Furthermore, theirreducible components of X are smooth in codimension 1.

However, as shown by the examples (48), (49) and (50), in general the irre-

ducible components of X need not be normal.

2. Quotients by finite equivalence relations

In this Section we answer question (5) for slc pairs.

Theorem 23. Let X be a normal variety, D ⊂ X a reduced divisor, ∆ a Q-divisor on X and τ : Dn → Dn an involution on the normalization n : Dn → D.Assume that

(1)(

X, D + ∆)

is lc,

(2) τ maps log canonical centers of(

Dn,DiffDn ∆)

to log canonical centers,and

(3) (n, n τ ) : Dn → X× X generates a finite equivalence relation R(τ ) X(26).


Then there is a demi-normal pair(

X := X/R(τ ),∆)

whose normalization is(

X, D + ∆, τ)

(cf. (2)).

As noted after (6), the assumption (23.3) is obviously necessary. The theorem

can fail without (23.1); in the examples of [Hol63, p.342] and [Kol08, 10], X is a

smooth 3-fold, D has cusps along a curve, (23.2–3) both hold yet X does not exist.

Here(

X, D)

is not lc but it is not far from it;(

X, 56D

)

is lc.In order to prove (23), we develop a general theory of geometric quotients by

finite set-theoretic equivalence relations. There are many cases when geometricquotients do not exist; see [Kol08, Sec.2] for a discussion of several such examples.On the positive side, we show in (43) that if a set-theoretic equivalence relationR X satisfies a series of rather restrictive conditions, then the geometric quotientX/R exists.

The proof of (23) then boils down to showing that the relation (n, n τ) :Dn → X × X generates a set-theoretic equivalence relation R X which satisfiesthe assumptions of (43).

Note that if(

Dn,DiffDn ∆)

is τ -invariant, then assumption (23.2) holds.

Finite equivalence relations.

Definition 24. Let X and R be S-schemes. A pair of morphisms σ1, σ2 :R X , or equivalently a morphism σ : R → X ×S X is called a pre-relation. Apre-relation is called finite if the σi are both finite and a relation if σ is a closedembedding.

To any finite pre-relation σ : R → X ×S X one can associate a finite relationi : σ(R) → X ×S X . For the purposes in this section, there is no substantialdifference between σ : R → X ×S X and i : σ(R) → X ×S X . (By contrast, a keyidea of Section ??? is to exploit this difference using stacks.)

Definition 25 (Set theoretic equivalence relations). Let X and R be reducedS-schemes. We say that a morphism σ : R→ X ×S X is a set theoretic equivalencerelation on X if, for every geometric point SpecK → S, we get an equivalencerelation on K-points

σ(K) : MorS(SpecK,R) → MorS(SpecK,X)×MorS(SpecK,X).

Equivalently,

(1) σ is geometrically injective.(2) (reflexive) R contains the diagonal ∆X .(3) (symmetric) There is an involution τR on R such that τX×X σ τR = σ

where τX×X denotes the involution which interchanges the two factors ofX ×X .

(4) (transitive) For 1 ≤ i < j ≤ 3 set Xi := X and let Rij := R when itmaps to Xi ×S Xj . Then the coordinate projection of red

(

R12 ×X2R23

)

to X1 ×S X3 factors through R13:

red(

R12 ×X2R23

)

→ R13π13−→ X1 ×S X3.

Note that the fiber product need not be reduced, and taking the reduced structureis essential.


26 (Equivalence closure). Let R → Y × Y be a finite pre-relation, R reduced.There is a smallest set theoretic equivalence relation generated by R which is con-structed as follows.

First we have to add the diagonal of Y × Y to R and make R symmetric withrespect to the interchange of the two factors. Then we have R1 → Y × Y which isreflexive and symmetric.

Achieving transitivity may be an infinite process. Assume that we have alreadyconstructed Ri → Y × Y with projections σi

1, σi2 : Ri → Y . Ri+1 is obtained by

replacing Ri by the image

Ri+1 := (σi1 τ i

1, σi2 τ i

2)(

Ri ×Y Ri) ⊂ Y × Y, (26.1)

where the maps are defined by the following diagram.

Ri ×Y Ri

ւ τ i1 τ i

2 ցRi Ri

σi1 ւ ց σi

2 σi1 ւ ց σi

2

Y Y = Y Y

(26.2)

At the end we obtain a countable union of reduced subschemes

R ⊂ R1 ⊂ R2 ⊂ · · · ⊂ Y × Yand finite projections σj

1, σj2 : Rj Y . In general, instead of an algebraic relation,

we obtain a pro-finite set theoretic equivalence relations.

Definition 27 (Geometric quotients). Let σ1, σ2 : R X be a set theoreticequivalence relation. We say that q : X → Y is a categorical quotient of X by R if

(1) q σ1 = q σ2, and(2) q : X → Y is universal with this property. That is, given any q′ : X → Y ′

such that q′σ1 = q′σ2, there is a unique π : Y → Y ′ such that q′ = πq.If σ1, σ2 : R X is finite, we say that q : X → Y is a geometric quotient of X byR if, in addition,

(3) q : X → Y is finite and(4) for every geometric point SpecK → S, the fibers of qK : XK(K) →

YK(K) are the σ(

RK(K))

-equivalence classes of XK(K).

Somewhat sloppily, we refer to the last property by saying that “the geometricfibers of q are the R-equivalence classes.”

It is not hard to see [Kol08, 17] that the assumptions (1–3) imply (4), but inour applications we will check (4) directly.

The geometric quotient is denoted by X/R.

There are three cases when the construction of the geometric quotient is easy.

Lemma 28. Let R X be a finite, set theoretic equivalence relation and assumethat there is a finite morphism q′ : X → Y ′ such that q′ σ1 = q′ σ2. Set

OY := ker[

q′∗OXσ∗

1−σ∗

2−→ (q′ σi)∗OR

]

.

Then Y = X/R.

Proof. Y clearly satisfies the assumptions (27.1–3) and the geometric fibers ofX → Y are finite unions of R-equivalence classes. As we noted above, by [Kol08,17], Y also satisfies the assumption (27.4).


Lemma 29. Let R X be a finite, set theoretic equivalence relation with X,Rreduced and over a field of characteristic 0. Let π : X ′ → X and q′ : X ′ → Z befinite surjections. Assume that one of the following holds:

(1) X,Z are semi normal and the geometric fibers of q′ are exactly the preim-ages of R-equivalence classes, or

(2) Z,X are normal, the σi : R→ X are open and, over a dense open subsetof Z, the geometric fibers of q′ are exactly the preimages of R-equivalenceclasses

Then Z = X/R.

Proof. Let X∗ ⊂ Z ×X be the image of X ′ under the diagonal map (q′, π).In the first case, every geometric fiber of π is contained in a geometric fiber of

q′, thus we see that the projectionX∗ → X is one-to-one on geometric points. SinceX is semi normal, this implies that X∗ ∼= X . Thus we get a morphism q : X → Zwhose geometric fibers are exactly the R-equivalence classes.

Therefore, q σ1 agrees with q σ2 on geometric points. Since R is reduced,this implies that q σ1 = q σ2. Define p : Y → Z as in (28). Since X is reduced,so is Y . The geometric fibers of X → Y are finite unions of R-equivalence classes.On the other hand, every geometric fiber of X → Y is contained in a geometricfiber of X → Z which is a single R-equivalence classe. Thus X → Z and X → Yhave the same fibers, hence Y → Z is an isomorphism on geometric points. SinceZ is semi normal, this implies that Y ∼= Z.

In the second case, the same argument gives that X∗ → X is birational. Sinceit is also finite, X∗ ∼= X since the latter is normal. We know that q σ1 = q σ2

holds over a dense open subset of X , hence over a dense open subset of R. Thusq σ1 = q σ2 everywhere. Construct p : Y → Z as before. Here p is birational andfinite, hence an isomorphism since X is normal.

Lemma 30. Let X be an excellent scheme over a field of characteristic 0 that isnormal and of pure dimension d. Let R X be a finite, set theoretic equivalencerelation. Let Rd ⊂ R denote the d-dimensional part of R. Then

(1) Rd X is a finite, set theoretic equivalence relation [BB04, 2.7],(2) the geometric quotient X/Rd exists, and(3) X/Rd is normal.

Proof. Let us prove first that Rd X is a set theoretic equivalence relation.The only question is transitivity. (Easy examples show that transitivity can failif X is not normal [Kol08, 29].) Note that σi : Rd → X is finite with normaltarget. Hence, by (32), Rd×X Rd → Rd is open. In particular, Rd×X Rd has puredimension d. Thus the image of the finite morphism Rd×XR

d → R in (25.3) lies inRd. Therefore Rd X is a set theoretic equivalence relation. It is then necessarilyfinite.

Next assume that X/Rd exists and let Y → X/Rd be the normalization. SinceX is normal, the quotient morphism X → X/Rd lifts to τ : X → Y . Thusτ σ1 = τ σ2 on a dense open set, hence equality holds everywhere. By theuniversal property of geometric quotients (27.2), X/Rd = Y is normal.

It is sufficient to construct a geometric quotient one irreducible component ata time. Thus assume that X is irreducible and let m = deg σi.

Consider the m-fold product X × · · · ×X with coordinate projections πi. LetRij (resp. ∆ij) denote the preimage of R (resp. of the diagonal) under (πi, πj).


A geometric point of ∩ijRij is a sequence of geometric points (x1, . . . , xm) suchthat any 2 are R-equivalent and a geometric point of ∩ijRij \ ∪ij∆ij is a sequence(x1, . . . , xm) that constitutes a whole R-equivalence class. Let X ′ be the normal-ization of the closure of ∩ijRij \ ∪ij∆ij . Note that every πℓ : ∩ijRij → X is finite,hence the projections π′

ℓ : X ′ → X are finite.The symmetric group Sm acts on X × · · · × X by permuting the factors and

this lifts to an Sm-action on X ′. Over a dense open subset of X , the Sm-orbits onthe geometric points of X ′ are exactly the R-equivalence classes. Thus, by (29),X ′/Sm

∼= X/R. Hence the construction of X/Rd is reduced to the construction ofX ′/Sm. This is discussed in (31).

31 (Quotients by finite group actions). Quotients by finite group actions arediscussed at many places. The quasi projective case is quite elementary; see, forinstance [Sha94, Sec.I.2.3] or the more advanced [Mum70, Sec.12]. For generalschemes and algebraic spaces, the quotients are constructed in some unpublishednotes of Deligne. See [Knu71, IV.1.8] for a detailed discussion of this method. Inall cases, the geometric quotient X/G exists.

32 (Chevalley’s criterion). (cf. [Gro67, IV.14.4.4]) Let X,Y be schemes ofpure dimension d and Y normal. Then every quasi finite morphism f : X → Y isuniversally open. That is, for every Z → Y , the induced morphism X ×Y Z → Zis open.

Note also that if fi : Xi → Y are open then so is X1 ×Y X2 → Y .

Definition 33. Let R X be a finite relation and g : Y → X a finitemorphism. Then

g∗R := R×(X×X) (Y × Y ) Y

defines a finite relation on Y . It is called the pull-back of R X . (Strictly speaking,it should be denoted by (g × g)∗R.)

Note that if R is a set theoretic equivalence relation then so is g∗R and theg∗R-equivalence classes on the geometric points of Y map injectively to the R-equivalence classes on the geometric points of X .

If X/R exists then, by (28), Y/g∗R also exists and the natural morphismY/g∗R → X/R is injective on geometric points. If, in addition, g is surjectivethen Y/g∗R→ X/R is finite and an isomorphism on geometric points. Thus, if Xis seminormal and the characteristic is 0, then Y/g∗R ∼= X/R.

Let h : X → Z be a finite morphism andR a finite relation. Then the compositeR X → Z defines a finite pre-relation. If, in addition, R is a set theoreticequivalence relation and the geometric fibers of h are subsets of R-equivalenceclasses, then R X → Z corresponds to a set theoretic equivalence relation

h∗R := (h× h)(R) ⊂ Z × Z,called the push forward of R X . If Z/h∗R exists, then, by (28), X/R alsoexists and the natural morphism X/R → Z/h∗R is finite and an isomorphism ongeometric points.

Stratified equivalence relations.

We saw in (30) that pure dimensional equivalence relations behave well onnormal schemes. In our intended applications, for instance in (23), we start with

a normal scheme X but the inductive nature of the proof leads to equivalence


relations on schemes that are neither normal nor pure dimensional. Furthermore,in (23), the equivalence relation generated by

(n, n τ ) : Dn → X × Xis not pure dimensional, since we always have to add the diagonal of X × X anddim Dn = dim X − 1.

Our aim is to show that R X is still well behaved if X and R can be de-composed into normal and pure dimensional pieces and some strong semi-normalityassumptions hold about the closures of the strata. To do these, we need the conceptof a stratification.

Definition 34. Let X be a scheme. A stratification of X is a decompositionof X into a finite disjoint union of reduced and locally closed subschemes. Wewill deal with stratifications where the strata are pure dimensional and indexed bythe dimension. Then we write X = ∪iSiX where SiX ⊂ X is the i-dimensionalstratum. Such a stratified scheme is denoted by (X,S∗). We also assume that∪i≤jSiX is closed for every j.

The boundary of (X,S∗) is the closed subscheme

BX := ∪i<dim XSiX = X \ Sdim XX.

Let (X,S∗) and (Y, S∗) be stratified schemes. We say that f : X → Y is astratified morphism if f

(

SiX) ⊂ SiY for every i. Equivalently, if SiX = f−1(

SiY)

for every i.Let (Y, S∗) be a stratified scheme and f : X → Y a quasi-finite morphism such

that f−1(

SiY)

has pure dimension i for every i. Then SiX := f−1(

SiY)

defines

a stratification of X , denoted by(

X, f−1S∗

)

. We say that f : X → (Y, S∗) isstratifiable.

Let (X,S∗) be a stratified scheme and f : X → Y a quasi-finite morphism suchthat f−1

(

f(SiX))

= SiX for every i. Then SiY := f(

SiY)

defines a stratification

of Y , denoted by(

Y, f∗(

S∗

))

. We say that f : (X,S∗)→ Y is stratifiable.

Definition 35. Let (X,S∗) be stratified. A relation σi : R (X,S∗) is calledstratified if each σi is stratifiable and σ−1

1 S∗ = σ−12 S∗. Equivalently, there is a

stratification (R, σ−1S∗) such that r ∈ σ−1SiR iff σ1(r) ∈ SiX iff σ2(r) ∈ SiX .Let σi : R (X,S∗) be a stratified set theoretic equivalence relation and

f : (X,S∗)→ Y a stratifiable morphism. If the geometric fibers of f are subsets ofR-equivalence classes then the push forward (33)

f∗R (

Y, f∗(

S∗

))

is also a stratified set theoretic equivalence relation.By contrast, the pull-back of a stratified relation by a stratified morphism is

not always stratified. (Sufficient conditions are given in (36).) As an example, letX be a nodal curve with S1 = X , R X the identity relation and g : Z → Xthe normalization. Then (g−1S)1 = Z but g∗R has 3 components. Besides theidentity, it has 0 dimensional components showing that the 2 preimages of the nodeare equivalent.

Lemma 36. Assume that the strata of (Y, S∗) are all normal.

(1) Let fi : Xi → (Y, S∗) be stratifiable quasi-finite morphisms. Then theinduced maps X1 ×Y X2 Xi Y are all stratifiable.


(2) Let R (Y, S∗) be a stratified relation. Then the pull-back g∗R

(X, g−1S∗) by a stratified morphism g is also a stratified relation.(3) Let R (Y, S∗) be a stratified relation. Then its equivalence closure (26)

is a stratified pro-finite relation.

Proof. The conditions need to be checked one stratum at a time, hence we mayassume that Y is normal and of pure dimension d.

By (32), the fi are universally open and the Xi also have pure dimension d.Thus Xi ×Y X2 → Y is also open hence Xi ×Y X2 has pure dimenion d, proving(1).

Similarly, g∗R := R ×(Y ×Y ) (X × X) → R is also open. Thus g∗R has pure

dimension d and so g∗R (X, g−1S∗) is stratifiable.To see (3), we need to show that all the pre-relations Ri constructed in (26)

are stratified. By induction on i, assume that the maps σij : Ri Y are stratified.

By (1), the fiber products τ ij : Ri ×Y Ri Ri are also stratified. Hence all arrows

in the diagram (26.2) are stratified, and so the composites along the outer edges ofthe triangle are also stratified. Thus all the σi+1

1 : Ri+1 Y are also stratified.

Definition 37. Let X be an excellent scheme. We consider 4 normality con-ditions on stratifications.

(N) We say that (X,S∗) has normal strata, or that it satisfies condition (N),if each SiX is normal.

(SN) We say that (X,S∗) has seminormal boundary, or that it satisfies condition(SN), if X and the boundary BX = ∪i<dim XSiX are both seminormal.

(HN) We say that (X,S∗) has hereditarily normal strata, or that it satisfiescondition (HN), if(a) X satisfies (N),(b) the normalization π : Xn → X is stratifiable, and(c) its boundary B

(

Xn)

satisfies (HN).(HSN) We say that (X,S∗) has hereditarily seminormal boundary, or that it sat-

isfies condition (HSN), if(a) X satisfies (SN),(b) the normalization π : Xn → X is stratifiable, and(c) its boundary B

(

Xn)

satisfies (HSN).

(In order to get a correct inductive definition, we should add that the empty schemesatisfies all these conditions.)

Note that if (X,S∗) satisfies (HN) or (HSN) then(

Xn, π∗S∗

)

also satisfies (HN)or (HSN).

Remark 38. Condition (N) is quite reasonable and usually easy to satisfy butcondition (HN) is more subtle. As an example, take

X =(

x2 = y2(y + z2))

⊂ A3

with S1X = (x = y = 0). Then S1X and S2X are both smooth. The normalizationof X is

Xn =(

x21 = y + z2

)

⊂ A3

where x1 = x/y and the preimage of S1X is (y = x21− z2 = 0) which is not normal.

Actually, one of the trickiest parts of (HN) is to know when the normalizationπ : Xn → X is stratifiable. For example, let

X := (x = y = 0) ∪ (z = t = 0) ⊂ A4


with S1X = (x = y = z = 0). As before, S1X and S2X are both smooth butπ : Xn → X is not stratifiable since the preimage of S1X has a 0-dimensionalirreducible component.

Note also that while every scheme has a stratification satisfying (N) (and prob-ably even (HN)), the conditions (SN) and (HSN) are usually impossible to satisfysince they pose restrictions on the closures of strata.

The conditions (SN) and (HSN) may seem less natural, and indeed they maynot be the best conditions to consider. It would have been possible to requiresemi normality for the closure of every SiX or even for the closure of any union ofirreducible components of strata. The main objective in chosing (SN) and (HSN)was to find the weakest assumptions that make the proof of (43) work.

By contrast, the next conditions are chosen to yield the strongest conclusionsin (44).

Definition 39. Let (X,S∗) be a stratified scheme. Following (34) a subschemej : Z → X is called stratified if j is a stratifiable morphisms. Equivalently, if Z∩SiXis the union of some irreducible components of SiX for every i. We say that (X,S∗)satisfies the stratified closure property if for every irreducible component W ⊂ SiX ,the injection of its closure j : W → X is stratified.

(DB) We say that (X,S∗) is Du Bois, or that it satisfies condition (DB), if ithas the stratified closure property and every stratified subscheme Z → Xis Du Bois (???).

(HDB) We say that (X,S∗) is hereditarily Du Bois, or that it satisfies condition(HDB), if(a) (X,S∗) satisfies (HN),(b) the normalization π : Xn → X is stratifiable, and(c) the boundary of the normalization B

(

Xn)

satisfies (HDB).

Note that by (???) a Du Bois scheme is semi normal, thus (HDB) implies (HSN).Moreover, every stratified subscheme j : Z → X is semi normal.

The main excuse for all these definitions is that they are satisfied in one signif-icant case:

Example 40. Let (X,∆) be lc. Let S∗i (X,∆) ⊂ X be the union of all ≤ i-

dimensional lc centers (???) of (X,∆) and

SiX := S∗i (X,∆) \ S∗

i−1(X,∆).

We call this the log canonical stratification or lc stratification of (X,∆).By (???) [KK09, 1.4] the lc stratification (X,S∗) satisfies all of the conditions

(N), (SN), (HN), (HSN), (DB), (HDB). Furthermore, if D ⊂ ⌊∆⌋ is a divisorwith normalization D then, by (??), D → X is a stratified morphism from the lcstratification of

(

D,Diff∗D ∆

)

to the lc stratification of (X,∆).As a consequence of (43) and (44), we will obtain that the conditions (N), (SN),

(HN), (HSN), (DB), (HDB) also hold if (X,∆) is slc.

Lemma 41. Let (X,S∗) and (Y, S∗) be normal stratified spaces over a field ofcharacteristic zero and f : X → Y a finite stratified morphism. If (X,S∗) satisfiesone of the conditions (N), (SN), (HN), (HSN), (DB), (HDB) then so does (Y, S∗).

Proof. The questions are local on Y . Let us check first (N).


Pick y ∈ SiY . In order to check that SiY is normal at y, we can replace Yby any affine neighborhood of y. Thus we may assume that X,Y are irreducible,affine, SiY is closed in Y and SiX is closed in X . Let φ be a regular functionon the normalization

(

SiY)n

. Since SiX is normal, the pull back f∗φ is a regularfunction on SiX . We can lift it to a regular function ΦX on X . Since Y is normal,

ΦY := 1deg X/Y trX/Y ΦX

is a regular function on Y whose restriction to SiY is φ. Thus SiY is normal.A similar argument with BX instead of SiX shows the (SN) case. Again we

may assume that X,Y are irreducible and affine. Let ψ be a regular function onthe semi normalization

(

BY)sn

. Since BX = red f−1(BY ) is semi normal, the pullback f∗ψ is a regular function on BX , thus it lifts to a regular function ΨX on X .As before, ΨY := 1

deg X/Y trX/Y ΨX is a regular function on Y whose restriction to

BY is ψ. Thus BY is semi normal.The hereditary cases follow by induction using the maps (BX)n → (BY )n.The Du Bois cases follow from (???).

Example 42. Similar results do not hold in positive characteristic, not evenif f is separable. For instance, let f : A3

xyz → A3uvw be given by f(x, y, z) =

(x, y, z2 + zy). Then f is finite of degree 2 and separable. The preimage of thecuspidal curve (v = u2 + w3 = 0) is the curve (y = x2 + z6 = 0), which has 2branches tangent to each other if the characteristic is not 2. In characteristic 2, the(reduced) preimage is the smooth curve (y = x+ z3 = 0).

The following is the main result of this section.

Theorem 43. Let (X,S∗) be an excellent scheme or algebraic space over a fieldof characteristic 0 with a stratification as in (34). Assume that (X,S∗) satisfiesthe conditions (HN) and (HSN). Let R X be a finite, set theoretic, stratifiedequivalence relation. Then

(1) the geometric quotient X/R exists,(2) π : X → X/R is stratifiable and(3)

(

X/R, π∗S∗

)

also satisfies the conditions (HN) and (HSN).

Complement 44. Notation and assumptions as in (43). If (X,S∗) satisfiesthe condition (HDB) then

(

X/R, π∗S∗

)

also satisfies (HDB).In particular, if (X,∆) is slc then X is Du Bois.

Proof. The proof is by induction on d := dimX . We follow the inductive planin [Kol08, 30].

Let (Xn, Sn∗ )→ (X,S∗) be the normalization of X and Rn Xn the pull-back

of R. By (36), Rn is also a finite, set theoretic, stratified equivalence relation.Let Xnd ⊂ Xn (resp. Rnd ⊂ Rn) be the union of all d-dimensional irreducible

components. By (30) Rnd is a finite, set theoretic, stratified equivalence relationon Xnd, the geometric quotient Xnd/Rnd is normal and the quotient map Xnd →Xnd/Rnd is stratifiable. By (41) the push forward of Rn|Xnd to Xnd/Rnd satisfiesthe conditions (HN) and (HSN).

Let Xnl be the union of all lower dimensional irreducible components of Xn.By a slight abuse of notation, we can view Rnd as an equivalence relation on Xn

which is the identity on Xnl. Thus

Xn/Rnd =(

Xnd/Rnd)

∐ Xnl.


Let q : Xn → Xn/Rnd denote the quotient map. Then q∗Sn∗ is a stratification

which agrees with the push forward of Rn|Xnd on Xnd/Rnd and with Rn|Xnl onXnl. Thus

(

Xn/Rnd, q∗Sn∗

)

also satisfies the conditions (HN) and (HSN).

Furthermore, Rn descends to a stratified equivalence relation q∗Rn on Xn/Rnd

which is the identity outside the boundary

B(

Xn/Rnd)

= B(

Xnd/Rnd)

∐ Xnl.

By induction on the dimension, the geometric quotient of B(

Xn/Rnd)

by the

restriction of q∗Rn exists. Let us denote it by B

(

Xn/Rnd)

/q∗Rn.

By (46) we get a universal push-out diagram

B(

Xnd/Rnd)

∐ Xnl = B(

Xn/Rnd)

→ Xn/Rnd

↓ ↓B

(

Xn/Rnd)

/q∗Rn → Y.

We claim that Y = Xn/Rn. To see this note first that the geometric fibers ofXn → Y are exactly the Rn equivalence classes. On the boundary this holds by in-duction and on the open part this follows from (30). Second, Xn/Rnd is normal andB

(

Xn/Rnd)

/q∗Rn is semi normal. Thus Xn/Rnd → Y and B

(

Xn/Rnd)

/q∗Rn →

Y both lift to the semi-normalization of Y . By the universality of the push-out,this implies that Y is semi-normal. Thus Y = Xn/Rn by (29). As we noted in(33), X/R = Xn/Rn.

The open stratum of X/R is also the open stratum of Xn/Rnd which is normal.The lower dimensional strata of X/R are also strata of B

(

Xn/Rnd)

/q∗Rn. These

are normal by induction. Thus(

X/R, π∗S∗

)

satisfies condition (N). We have seen

that both Y = X/R and its bundary B(

Xn/Rnd)

/q∗Rn are semi normal. Thus

(

X/R, π∗S∗

)

also satisfies condition (SN).

Note that Xn/Rnd is normal and Xn/Rnd → Y = X/R is an isomorphism atall d-dimensional generic poins. Hence the normalization of X/R is an open andclosed subscheme of Xn/Rnd. We have seen during the proof that

(

Xn/Rnd, q∗Sn∗

)

satisfies the conditions (HN) and (HSN), hence the same holds for the normalizationof X/R. Together with the previous comments, these show that X/R satisfies theconditions (HN) and (HSN).

Assume finally that (X,S∗) is Du Bois. Then(

Xn/Rnd, q∗Sn∗

)

is Du Bois by

(???) [KK09, 2.3] and then(

X/R, π∗S∗

)

is Du Bois by (???) [KK09, 1.5].

45 (Proof of (23)). Assume that(

X, D+∆)

is lc. Let S∗ be the lc-stratification

constructed in (40). We saw in (40) that (X, S∗) satisfies all of the conditions (N),(SN), (HN), (HSN), (DB), (HDB).

As we noted in (40), n : Dn → X is stratified and τ is stratified by assumption

(23.2). Thus (n, n τ ) : Dn → X × X is a stratified relation. Then by (36), its

equivalence closure R X is a stratified equivalence relation.Thus the assumptions of (43) are satisfied and the geometric quotient X/R

exists. Set X := X/R and let ∆ be the image of ∆. We claim that X is demi

normal and(

X, D + ∆)

is its normalization.

Let W ⊂ X be the union of lc centers of codimension ≥ 2 and W ⊂ X itsimage in X . Then τ is an involution on D \ W , and the universal push out of

(

D \ W)

/τ ←(

D \ W)

→(

X \ W)


is isomorphic to X \W . Thus X \W has duble nc points only.

Let Xd → X be the demi normalization of X (1). Since X is normal, the

quotient map π : X → X lifts to πd : X → Xd. The involution τ is πd-equivariantoutside W , hence it is πd-equivariant. By (28.1), these imply that Xd = X/R = X ,

thus X is demi normal. Moreover,(

X, D, τ)

=(

X, D, τ)

holds over X \W , henceeverywhere.

During the proof of (43) we have used the following theorem of [Art70, Thm.3.1].For elementary proofs, see [Fer03, Rao74] or [Kol08, Sec.6].

Theorem 46. Let X be a Noetherian algebraic space over a Noetherian basescheme S. Let Z ⊂ X be a closed subspace and g : Z → V a finite surjection. Thenthere is a universal push-out diagram of algebraic spaces

Z → Xg ↓ ↓ πV → Y := X/(Z → V )

Furthermore, π is finite, V → Y is a closed embedding, Z = π−1(V ) and the naturalmap ker

[

OY → OV

]

→ ker[

OX → OZ

]

is an isomorphism.

Pro-finite equivalence relations.

In general it is quite hard to see when a finite pre-relation R Y generates afinite set theoretic equivalence relations. Here are some examples which show thatproblems can occur in high codimension, even for dlt pairs.

Example 47. Fix two points a, b ∈ A1 and consider two involutions τ1 : x 7→a− x and τ2 : x 7→ b− x. They correspond to a finite pre-relation

σ : A1 ∐A1 → A1 × A1,

where σ(x) = (x, a − x) and σ(x) = (x, b − x). Note that the composite τ1 τ2 istraslation by b− a. Thus it has infinite order if a 6= b and the characteristic is 0.

Example 48. Let X := A3 with coordinates (x, y, t) and D1 := (y = 0), D2 :=(x = 0) two hyperplanes. Let L := (x = y = 0) be the t-axis. For a, b ∈ C defineinvolutions on Di by

τ1(x, 0, t) 7→ (x, 0, a− t) and τ2(0, y, t) 7→ (0, y, b− t).Note that D1/(τ1) = Spec C[x, t(a− t)] and D2/(τ2) = Spec C[y, t(b− t)]. Thus onX \L the τi generate a finite equivalence relation and we obtain a finite morphism

π0 :(

X \ L)

→(

X \ L)

/(τ1, τ2).

Both involutions act on L. Note that τ2|L τ1|L is translation by b− a, hence hasinfinite order if a 6= b and the characteristic is 0.

This shows that π0 can not be extended to a finite morphism on X .

Example 49. Pick involutions r1, r2, r3 ∈ PGL(2,C) such that any 2 of themgenerate a finite subgroup but the 3 together generate an infinite subgroup.

ConsiderX = A3×P1. Let xi be the coordinates on A3 and Di := (xi = 0)×P1.On Di consider the involution τi which is the identity on Di and ri on the P1-factor.Let R X be the pro-finite set theoretic equivalence relation generated by theτi : i = 1, 2, 3.


Note that

π1 :(

X \D1

)

× P1 →(

X \D1

)

×(

P1/〈r2r3〉)

is finite, thus R|X\D1is a finite set theoretic equivalence relation. Similarly,

(

X \Di

)

/(

R|X\D1

)

exists for i = 2, 3. Set P10 := 0× P1. Then the geometric quotient

(

X \ P10

)

/(

R|X\P1

0

)

exists.Note, however, that the restriction of R to P1

0 is not a finite equivalence relationsince the subgroup generated by r1, r2, r3 is infinite. Thus R is not a finite relationand there is no geometric quotient of X by R.

In order to find such r1, r2, r3, its is easier to work with SO(3,R) ∼= SU(2,C).Let Li ⊂ R3 be 3 lines such that the angles between them are rational multiplesof π. Let ri denote the reflections determined by the lines Li. By assumption, theangle between any 2 lines is a rational multiple of π, hence any 2 rotations generatea finite dihedral group.

The finite subgroups of G ⊂ SO(3,R) are all known. If G is not cyclic ordihedral, then any rotation in G has order ≤ 6. Thus, as soon as the denominatorof the angle between Li, Lj is large enough, the subgroup generated by r1, r2, r3 isinfinite.

Example 50. In Rn consider the hyperplanes

H0 := (x1 = 1), Hn := (xn = 0) and Hi := (xi = xi+1) for i = 1, . . . , n− 1.

Note that any n of these hyperplanes have a common point but the intersection ofall n+ 1 of them is empty.

Let ri denote the reflection on Hi. Each ri is defined over Z and maps Zn toitself. Thus any n of the ri generate a reflection group which has a fixed point andpreserves a lattice. These are thus finite groups. By contrast, all n + 1 of themgenerate a reflection group with no fixed point. It is thus an infinite group.

As in (49) consider X = An+1×An. Let x0, . . . , xn be the coordinates on An+1

and Di := (xi = 0)×An. On Di consider the involution τi which is the identity onDi and ri on the An-factor. Let R X be the pro-finite set theoretic equivalencerelation generated by the τi : i = 0, . . . , n. We see that R is not a finite equivalencerelation on X but it restricts to a finite equivalence relation on

(

An+1 \ 0)

×An.

The next examples have normal irreducible components but they are only lc.

Example 51. Let X be an affine variety, p ∈ X a point and D1, D2 ⊂ Xdivisors such that D1 ∩D2 = p. Take two copies (X i, Di

1 +Di2) for i = 1, 2.

Choose an isomorphism φ(λ, µ)(

(

D11\p1

)

×A1)

∐(

(

D12\p1

)

×A1)

→(

(

D21\p2

)

×A1)

∐(

(

D22\p2

)

×A1)

where φ(λ, µ) =(

1D1×λ

)

∐(

1D2×µ

)

is the identity on the Di and multiplication

by λ (resp. by µ) on the A1-factor of D1 (resp. D2).The corresponding geometric quotient Y ∗(λ, µ) is a non-normal variety whose

irreducible components(

X i \ pi)

× A1 intersect along(

(Di1 +Di

2) \ pi)

× A1.When can we extend this to a non-normal variety Y (λ, µ) ⊃ Y ∗(λ, µ) whose

irreducible components are X1 × A1 and X2 × A1?


Assume that f i =∑

f ijt

j is a function on X i×A1 such that f1, f2 glue together

to a regular fuction on Y ∗(λ, µ). The compatibility conditions are

f1j |D1

1

= λjf2j |D2

1

and f1j |D1

2

= µjf2j |D2

2

.

In particular, we get that

f1j (p1) = λjf2

j (p2) and f1j (p1) = µjf2

j (p2).

If λ/µ is not a root of unity, this implies that the f i are constant on pi × A1.Thus there is no scheme or algebraic space Y (λ, µ) ⊃ Y ∗(λ, µ) whose irreduciblecomponents are X1 × A1 and X2 × A1.

Let us see now some log canonical examples satsfying the above assumptions.

(51.1) Set X = A2, D1 = (x = 0) and D2 = (y = 0). Then (X,D1 +D2) is dlt.Thus we see that gluing in codimension 2 is not automatic for dlt pairs.

(51.2) Set X = (xy − uv = 0), D1 = (x = u = 0), D2 = (y = v = 0) and∆ = (x = v = 0) + (y = u = 0). Here (X,D1 +D2 + ∆) is lc but not dlt. We canreplace ∆ by some other divisor whose coefficients are < 1, but (X,D1 +D2 + ∆)can never be dlt. Thus we see that gluing in codimension 3 is not automatic for lcpairs.

(51.3) Similar examples exists in any dimension. LetX be the cone over P1×Pn,Di the cone over (i : 1)×Pn and ∆ the cone over some P1×B where B ∼Q (n+1)Hon Pn. These examples can be lc but not dlt.

Thus we see that gluing in any codimension is not automatic for lc pairs.

52 (Polarization questions). The above examples were all local, but they canbe easily compactified. However, none of them can be realized on lc pairs (X,D)

such that KX +D is ample. That is, I do not know if there are(

X, D + ∆, τ)

as

in (23) such that KX + D + ∆ is ample yet (n, n τ) : Dn → X × X generates anon-finite set-theoretic equivalence relation.

Note, however, that a pre-relation that is compatible with an ample line bundledoes not always generate a finite set-theoretic equivalence relation.

Indeed, [BT09] gives examples of etale pre-relations RC C on smooth curvesC of genus ≥ 2 that generate non-finite equivalence relations. These are evencompatible with the ample canonical line bundle.

The following obvious finiteness condition turns out to be quite useful. Notethat its assumptions are satisfied if (X,∆) is lc, S∗ is the stratification by lc centersas in (40) and Z does not contain any lc center.

Lemma 53. Let (X,S∗) be a stratified space satisfying (N) and Z ⊂ X a closedsubspace which does not contain any of the irreducible components of the SiX. Letσi : R (X,S∗) be a pro-finite, stratified set theoretic equivalence relation. Assumethat R|X\Z is a finite set theoretic equivalence relation. Then R is also a finite settheoretic equivalence relation.

Proof. Since R is a union of finite relations, it is enough to check that R hasfinitely many irreducible components. The latter can be checked one stratum at atime, hence we may assume that X is normal. Since every irreducible componentof R dominates an irreducible component of X , finitness over the dense open setX \ Z implies finiteness.


3. Descending line bundles to geometric quotients

In this Section we answer question (6) for slc pairs.

Theorem 54. Let X be a demi-normal scheme and ∆ a Q-divisor on X. Asin (2), let (X, D, ∆) be the normalization of (X,∆) and τ : Dn → Dn the corre-sponding involution. The following are equivalent:

(1) (X,∆) is slc.(2) (X, D + ∆) is lc and DiffDn ∆ is τ-invariant.

55 (Plan of the proof). We have seen in (14) that (1) ⇒ (2).By (12), for the converse we only need to prove that KX +∆ is Q-Cartier. The

set of points where KX +∆ is Q-Cartier is open. After localizing at a generic pointof the locus where KX + ∆ is not Q-Cartier, we may assume that X is local withclosed point x ∈ X , k = k(x) is algebraically closed and KX + ∆ is Q-Cartier onX0 := X \ x.

Choose m > 0 such that m∆ is a Z-divisor, m(KX + D + ∆) is Cartier andm(KX + ∆) is Cartier on X0. Let p : XL → X denote the total space of the linebundle OX

(

m(KX + D + ∆))

; that is, p∗OXL=

∑

r≥0OX

(

−rm(KX + D + ∆))

.

Set DL := p−1D and ∆L := p−1∆. Then (XL, DL + ∆L) is lc. The nor-malization Dn

L of DL can be obtained either as the fiber product Dn ×D DL oras the total space of the line bundle ODn

(

m(KDn + DiffDn ∆))

. In particular,

DiffDnL

∆L = p∗ DiffDn ∆ and the lc centers of(

DnL,DiffDn

L∆L

)

are the preimages

of the lc centers of(

Dn,DiffDn ∆)

.

The τ -invariance of DiffDn ∆ is equivalent to saying that τ lifts to an involutionτL : Dn

L → DnL and DiffDn

L∆L is τL-invariant.

(Since we are working locally, (XL, DL + ∆L) is isomorphic to (X, D, ∆)×A1,but τL may not be the product of τ with the identity on A1.)

Thus we have an lc pair (XL, DL + ∆L) and an involution τL : DnL → Dn

L thatmaps log canonical centers of

(

DnL,DiffDn

L∆L

)

to log canonical centers. We are in

a situation considered in (23). We have just established that (23.1–2) both hold.(55.1) In order to apply (23), we need to check assumption (23.3). That is, we

need to prove that (nL, τL nL) : DnL → XL × XL generates a finite set theoretic

equivalence relation RL XL.Note that finiteness holds over X0. Indeed, we assumed that m(KX + ∆) is

Cartier on X0; let X0L → X0 denote the total space of OX

(

m(KX + ∆)|X0

)

. Set

X0L := p−1(X0) ⊂ XL. There is a natural finite morphism X0

L → X0L and in fact

X0L = X0

L/R0L where R0

L denotes the restriction of RL to X0L. Thus R0

L : X0L is

finite and therefore RL : XL is finite iff it is finite over XL \ X0L.

In order to study the latter, let x1, . . . , xr ∈ X be the preimages of x. If none ofthe xi are lc centers of (X, D+∆) then every lc center of (XL, DL +∆L) intersectsX0

L, hence RL XL is finite by (53).Thus we are left with the case when at least one of the xi is an lc center. Then,

by (54.2) and adjunction (??), all the xi are lc centers of (X, D + ∆).The fiber of p : XL → X over xi is the 1-dimensional k-vectorspace

Vi := OX

(

m(KX + D + ∆))

⊗X k(xi)

Thus XL\X0L = V1∪· · ·∪Vr and τL gives a collection of isomorphisms τijk : Vi → Vj .

(A given xi can have several preimages in Dn and each of these gives an isomorphism

3. DESCENDING LINE BUNDLES TO GEOMETRIC QUOTIENTS 29

of Vi to some Vj . Thus there could be several isomorphisms from Vi to Vj for fixedi, j.) The τijk generate a groupoid. All possible composites

τij2k2 τj2j3k3

· · · τjnikn: Vi → Vi

generate a subgroup of Aut(Vi), called the stabilizer stab(Vi). Note that RL XL

is a finite set theoretic equivalence relation iff stab(Vi) ⊂ Aut(Vi) = k∗ is a finitesubgroup for every i.

In the surface case (17), the key step was to observe that, for m even, thePoincare residue map gives a canonical isomorphism

Rm : Vi∼= H0

(

xi, ωmxi

)

and the right hand side is canonically isomorphic to k(xi). With these choices, theτijk become isomorphisms of fields

τijk : k(xi) ∼= k(xj)

and stab(Vi) is the identity. (The last assertion does not quite follow from what wesaid before, but at least we see that stab(Vi) is a subgroup of the Galois group ofk(xi)/k, hence finite.)

As noted in (17.5), in the higher dimensional case the Vi are not canonicallyisomorphic to k(xi). Instead, in (67), we extract from a dlt model of (XL, DL+∆L)(???) klt pairs (Zi,∆Zi

) with m(KZi+ ∆Zi

) ∼ 0 such that, for every i, j, k(55.2) the Poincare residue map gives a canonical isomorphism

Rm : Vi∼= H0

(

Zi, ω[m]Zi

(m∆Zi))

, and

(55.3) τijk : Vi → Vj becomes the pull-back by a birational map φjik : Zj 99K Zi.From these we conclude that

stab(Vi) ⊂ im[

Bir(

Zi,∆Zi

)

→ Autk H0(

Zi, ω[m]Zi

(m∆Zi))

]

. (55.4)

(There does not seem to be an obvious definition of birational self maps of anarbitrary pair (Z,∆Z); see (56) for our approach.) A variant (57) of a result of[NU73] allows us to conclude that the right hand side of (55.4) is finite, hence thestabilizers stab(Vi) ⊂ Aut(Vi) = k∗ are finite.

Thus RL XL is a finite set theoretic equivalence relation and the geometricquotient XL/RL exists by (23).

(55.5) For technical reasons it is more convenient to continue with the comple-ment of the zero section XS ⊂ XL. Note that p : XS → X is a Gm-torsor. Weview p : XS → X as a Seifert bundle over X (69).

Next we use that, by construction, DS + ∆S is Gm-invariant and τS is Gm-equivariant. It is easy to check that the Gm-action descends to a proper Gm-actionon XS/RS and the fibers of XS/RS → X are homogeneous under Gm. By (72)XS/RS → X is a Seifert Gm-bundle. The general theory of Seifert bundles (70)then shows that there are unique torsion free, rank 1, coherent sheaves Li on Xwith multiplication maps Li ⊗ Lj → Li+j such that LM is locally free for someM > 0, and

XS/RS = SpecX

∑

i∈ZLi.

As we noted earlier, the restriction of XL/RL → X to X0 is the total space of theline bundle OX0

(

m(KX0 + ∆|X0))

. Since x ∈ X has codimension ≥ 2, this impliesthat

L−r = j∗OX0

(

rm(KX0 + ∆|X0))

= OX

(

rm(KX + ∆))

,


where j : X0 → X is the natural injection. Thus Mm(KX + ∆) is Cartier.

In the rest of the section we prove the auxiliary results that we used above.

Homotheties.

Definition 56. Let X be an integral k-variety and denote by K(X,ω[m]X ) the

k(X)-vectorspace of rational sections of ω[m]X .

Given a nonzero rational section η of ω[m]X , for some m > 0, let kη ⊂ K(X,ω

[m]X )

denote the 1-dimensional k-subspace generated by η.Examples of such pairs (X, kη) are obtained as follows. Let X be a proper,

normal, geometrically integral k-variety and ∆ a (not necessarily effective) Q-divisoron X such that KX + ∆ ∼Q 0. Then m(KX + ∆) ∼ 0 for some m > 0 and

H0(

X,ω[m]X (m∆)

)

⊂ K(

X,ω[m]X

)

is a 1-dimensional k-subspace. This gives a pair (X, kη) but there is no sensibleway to pick an actual form η ∈ kη.

Given a birational map φ : X ′ 99K X , we get another pair (X ′, kη′) :=(

X ′, kφ∗η)

. Any such (X ′, kη′) is called a birational model of (X, kη).A pair (X, kη) is called lc (resp. klt) if for every birational model (X ′, η′), every

pole of η′ has order ≤ m (resp. < m). As usual (cf. (??)), it is sufficient to checkthis on one model where X ′ is smooth and the support of (η′) is a snc divisor.

A birational map φ : X 99K X is called a homothety of the pair (X, kη) ifφ∗η = λη for some λ = λ(φ) ∈ k, called the scale. The scale does not depend onthe choice of η ∈ kη. All homotheties form a group Homothety(X, kη) and thescale gives a representation

Λ : Homothety(X, kη)→ k∗.

The following theorem is the log version of the key finiteness result of [NU73];see also [Uen75, Sec.14].

Theorem 57. If (X, kη) is klt then the scale representation Homothety(X, kη)→k∗ has finite image.

Remark 58. The method of [K+92, Sec.12] shows that (57) should also holdif (X, η) is lc, but the proof may need MMP for lc pairs.

On the other hand, (57) fails if (X, kη) is not lc. For instance, the image of thescale representation for (A1

t , kdt) is k∗ since d(λt) = λdt. Here dt has a double poleat infinity hence (A1

t , kdt) is not lc.

The first step in the proof is reduction to the case when X is smooth, projectiveand η is a rational section of ωX . Since the poles then are assumed to have order< 1, there are no poles and η is an actual section of ωX . In the latter case, (63)proves that the image of the scale representation consists of roots of unity whosedegree is bounded by the middle Betti number of X . Since there are only finitelymany such roots of unity, this will complete the proof of (57).

59 (Reduction to η ∈ H0(X,ωX)). Pick a birational model (X, kη) such that Xis smooth and (η) is a snc divisor. Since (X, kη) is klt, we can write (η) = mD−m∆where D is an effective Z-divisor and ⌊∆⌋ = 0. Then mKX ∼ −m∆ +mD and wecan view η as an isomorphism

η : OX(−m∆) ∼=(

ωX(−D))⊗m

.


Thus η defines an algebra structure and a cyclic cover

X := X[

m√η]

:= SpecX

m−1∑

i=0

ωX(−D)⊗i(

⌊i∆⌋)

with projection p : X → X . Since p ramifies only along the snc divisor ∆, X is klt(???). Note that ωX has a section η, given by the i = 1 summand in

1 ∈ H0(X,OX(D)) ⊂ p∗ωX =

m−1∑

i=0

HomX

(

ωX(−D)⊗i(

⌊i∆⌋)

, ωX

)

.

Assume now that φ : X 99K X is a homothety with scale λ. Fix an mth root m√λ.

Then φ lifts to a rational algebra map

φ′ : φ∗m−1∑

i=0

ωX(−D)⊗i(

⌊i∆⌋)

99K

m−1∑

i=0

ωX(−D)⊗i(

⌊i∆⌋)

which is the natural isomorphis φ∗ωX → ωX multiplied by(

m√λ)i

on the ith

summand. Thus we get a homothety φ of (X, η) whith scale m√λ. Therefore, if

the scale representation Homothety(X, kη)→ k∗ has finite image then so does thescale representation Homothety(X, kη)→ k∗.

Next we compare the pull-back of holomorphic forms with the pull-back mapon integral cohomology. Note that one can pull back holomorphic forms by rationalmaps, but one has to be careful when pulling back integral cohomology classes byrational maps.

60. Let f : M → N be a map between oriented compact manifolds of dimensionm. Then one can define a push forward map

f∗ : Hi(M,Z)/(torsion)→ Hi(N,Z)/(torsion)

as follows. Cup product with α ∈ Hi(M,Z)/(torsion) gives

α∪ : Hm−i(N,Z)→ Hm(M,Z) = Z given by β 7→ α ∪ f∗β.

Since the cup product Hi(N,Z) × Hm−i(N,Z) → Hm(N,Z) = Z is unimodular,there is a unique class γ ∈ Hi(N,Z)/(torsion) such that γ ∪ β = α ∪ f∗β for everyβ. Set f∗α := γ.

Note that if α = f∗γ for some γ ∈ Hi(N,Z) then

f∗γ ∪ f∗β = f∗(γ ∪ β) = deg f · (γ ∪ β)

shows that f∗(f∗γ) = deg f · γ. Thus f∗ f∗ : H∗(N,Z)→ H∗(N,Z) is multiplica-

tion by deg f . In particular, if deg f = 1 then f∗ f∗ is the identity.

Lemma 61. Let X,X ′, Y be smooth proper varieties over C and g : X 99K Ya map. Let f : X ′ → X be a birational morphism such that (g f) : X ′ → Y is amorphism. Then the following diagram is commutative

H0(Y,ΩiY ) → Hi(Y (C),C)

g∗ ↓ ↓ f∗ (g f)∗

H0(X,ΩiX) → Hi(X(C),C)


Proof. For a holomorphic i form φ, let [φ] denotes its cohomology class inHi( ,C). Pull-back by a morphism commutes with taking cohomology class, thus[

(g f)∗φ]

= (g f)∗[φ]. On the other hand, (g f)∗φ = f∗(

g∗φ)

. Thus (g f)∗[φ] = f∗

[

g∗φ]

. As noted in (60), f∗ f∗[

g∗φ]

=[

g∗φ]

. Thus f∗ (g f)∗[φ] =

f∗ f∗[

g∗φ]

=[

g∗φ]

.

Corollary 62. Let X be a smooth proper variety over C and g : X 99K Xa birational map. Then every eigenvalue of g∗ : H0(X,Ωi

X) → H0(X,ΩiX) is an

algebraic integer of degree ≤ dimHi(X(C),C).

Proof. Let f : X ′ → X be a birational morphism such that (g f) : X ′ → Y isa morphism. By (61), every eigenvalue of g∗ : H0(X,Ωi

X)→ H0(X,ΩiX) is also an

eigenvalue of f∗ (g f)∗ : Hi(X(C),Z)→ Hi(X(C),Z). The latter is given by anintegral matrix, hence its eigenvalues are algebraic integers.

Warning 62.1. Although f∗ (g f)∗ : Hi(X(C),Z) → Hi(X(C),Z) does notdepend on the choice of f , the correspondence g 7→ f∗ (g f)∗ is not a grouphomomorphism. In fact, usually f∗ (g f)∗ is not invertible; it need not even havemaximal rank.

Corollary 63. Let X be a smooth proper variety over C and g : X 99K X abirational map. Then every eigenvalue of g∗ : H0(X,ωX) → H0(X,ωX) is a rootof unity of degree ≤ dimHdim X(X(C),C).

Proof. Assume that η is an eigenform and g∗η = λη. Since η ∧ η is a (singular)volume form,

∫

X(C)

η ∧ η =

∫

X(C)

g∗(η ∧ η) = (λλ)

∫

X(C)

η ∧ η.

Thus |λ| = 1 and, by (62), it is an algebraic integer.Let σ ∈ Aut(C/Q) be any field automorphism. By conjugating everything by

σ, we get gσ : Xσ 99K Xσ such that(

gσ)∗ησ = λσησ. Thus λσ also has absolute

value 1. We complete the proof by noting that an algebraic integer is a root ofunity iff all of its conjugates have absolute value 1 [?].

Poincare residue map.

64. Let (X,∆) be dlt with ∆ either effective or snc and f : X → Y a morphism

with connected fibers such that ω[m]X (m∆) ∼f 0. Let Z ⊂ X be a lc center of (X,∆)

such that s := f(Z) ⊂ Y is a closed point. For m > 0 and even, we have the generalPoincare residue map (??)

RmX→Z : ω

[m]X (m∆)|Z

∼=−→ ω[m]Z (m ·Diff∗

Z ∆). (64.1)

Note that H0(Z,OZ) is a finite field extension of k(s) and by our assumptions thereis a natural map

f∗(

ω[m]X (m∆)

)

⊗Y k(s)→ H0(

Z, ω[m]X (m∆)|Z

)

.

Taking global sections in (64.1) gives a nonzero map

RmX→Z : f∗

(

ω[m]X (m∆)

)

⊗Y k(s)→ K(Z, ω[m]Z ). (64.2)

and the resulting (X, kη) is lc (cf. (??.1)). Furthermore, if Z is a minimal lc centerthen (X, kη) is klt.


The following result shows, that, for minimal lc centers, (64.2) is essentiallyindependent of the choice of Z.

Proposition 65. Let (X,∆) be dlt (with ∆ effective) over a field k and f :

X → Y a proper morphism with connected fibers such that ω[m]X (m∆) ∼f 0 for some

m > 0 even. Let Z1, Z2 be minimal lc centers of (X,∆) such that f(Z1) = f(Z2) =s ∈ S is a closed point. Then there is a birational map φ : Z2 99K Z1 such that thefollowing diagram commutes

f∗(

ω[m]X (m∆)

)

⊗Y k(s) = f∗(

ω[m]X (m∆)

)

⊗Y k(s)

RmX→Z1

↓ ↓ RmX→Z2

K(Z1, ω[m]Z1

)φ∗

−→ K(Z2, ω[m]Z2

)

Proof. By (??) it is sufficient to prove this in case there is an lc center W of(X,∆) that dominates s, contains Z1, Z2 as divisors and such that W is birationalto a P1-bundle P1 × U with Z1, Z2 as sections. Thus projection to U provides abirational isomorphism φ : Z2 99K Z1.

Since RX→Zi= RW→Zi

RX→W (??.2), it is sufficient to check the commu-tativity of the diagram

f∗(

ω[m]W (mDiff∗

W ∆))

⊗Y k(s) = f∗(

ω[m]W (mDiff∗

W ∆))

⊗Y k(s)

RmW→Z1

↓ ↓ RmW→Z2

K(Z1, ω[m]Z1

)φ∗

−→ K(Z2, ω[m]Z2

)

(65.1)

Note that L := H0(U,PU ) is a finite field extension of k(s) and (65.1) is equivalentto the commutativity of the following diagram of isomorphisms of 1-dimensionalL-vector spaces.

H0(

W,ω[m]W (mDiff∗

W ∆))

= H0(

W,ω[m]W (mDiff∗

W ∆))

RmW→Z1

↓ ↓ RmW→Z2

H0(

Z1, ω[m]Z1

(mDiff∗Z1

∆)) φ∗

−→ H0(

Z2, ω[m]Z2

(mDiff∗Z2

∆))

(65.2)

This in turn can be checked over the generic point of U . This reduces us to thecase when W = P1

L with coordinates (x:y), Z1 = (0:1) and Z2 = (1:0). A generatorof H0

(

P1, ωP1(Z1 + Z2))

is dx/x which has residue 1 at Z1 and −1 at Z2. Thus(65.2) commutes for m even and anticommutes for m odd.

Definition 66. Let (X,∆) be lc and W ⊂ X an lc center. Let f : (X ′,∆′)→(X,∆) be a log resolution and Z ⊂ X ′ a minimal among the lc centers of (X ′,∆′)that dominate W . By (64), we obtain a Poincare residue map

RmX→Z : ω

[m]X (m∆)⊗ k(W )→ K

(

Zk(W ), ω[m]Z

)

,

defining a pair(

Zk(W ), k(W )η)

.Note that this depends on the choice of f and Z.Let D ⊂ ⌊∆⌋ be a divisor with normalization π : Dn → D. If W ⊂ D then

every irreducible component of π−1(W ) is an lc center of(

Dn,Diff∗Dn ∆

)

(??); let

WD ⊂ D be any one of them. Let ZD denote a corresponding choice of Z as above.

Theorem 67. Notation and assumptions as in (66). Then

(1) (Zk(W ), k(W )η) is independent of f and Z up to homothety.


(2) The construction of(

Zk(W ), k(W )η)

is local in the Zariski (and even inthe strict etale) topology.

(3) If W ⊂ D then there is a birational map φ : ZD 99K Z such that for msufficiently divisible, the following diagram commutes

ω[m]X (m∆)⊗ k(W )

RX→D−→ ω[m]

D(mDiff∗

D ∆)⊗ k(WD)

RmX→Z ↓ ↓ Rm

D→ZD

K(Z, ω[m]Z )

φ∗

−→ K(ZD, ω[m]ZD

).

Proof. Let (Xm,∆m) → (X,∆) be a dlt model of (X,∆). Then (65) proves(1) if we choose Z on Xm and (??) takes care of every other choice. Part (2) isclear form the construction.

In order to see (3), let Dm ⊂ Xm denote the birational tarnsform of D. ThenDm → D is a dlt model. By (1) we may choose ZD on Dm and take Z = ZD. Then(3) is just the composition property (??.2) of the Poincare residue map.

Seifert bundles.

Seifert bundles were introduced to algebraic geometry in the works of [OW75,

Dol75, Pin77]. The main emphasis has been on the case of smooth varietiesand orbifolds. We need to study Seifert bundles over non-normal bases, so we gothrough the basic definitions.

Notation 68. Gm denotes the multiplicative group scheme GL(1). As ascheme over SpecA, it is SpecAA[t, t−1]. For any natural number r > 0, therth roots of unity form the subgroup scheme

µr := SpecAA[t, t−1]/(tr − 1).

These are all the subgroup schemes of Gm. (Note that µr is nonreduced when thecharacteristic divides r.)

Every linear representation ρ : Gm → GL(W ) is completely reducible, and thesame holds for µr ⊂ Gm (see, for instance, [SGA70, I.4.7.3]). This implies thatevery quasi coherent sheaf with a Gm-action is a direct sum of eigensubsheaves.The set of vectors v : ρ(λ)(v) = λiv is called the λi-eigenspace. We use thisterminology also for µM -actions. In this case i is determined modulo M .

If a group G acts on a scheme X via ρ : G → Aut(X), we get an action onrational functions on X given by f 7→ f ρ(g−1). (The inverse is needed mostly fornoncommutative groups only.)

Thus ifGm acts on itself by multiplication, we get an induced action on A[t, t−1]where λ ∈ Gm(k) acts as ti 7→ λ−iti. Thus ti spans the λ−i-eigenspace.

A Gm-action on an A-algebra R is equivalent to a Z grading R =∑

i∈Z Ri

where Ri is the λ−i-eigenspace.The natural Gm-action on Gm/µM corresponds to the algebra

∑

i∈MZ A∼=

A[tM , t−M ].

Definition 69. Let X be a semi-normal scheme (or algebraic space) over S.A Seifert bundle (or a Seifert Gm-bundle) over X is a reduced scheme (or algebraicspace) Y together with a morphism f : Y → X and a Gm-action on Y satisfyingthe following conditions.

(1) f is affine and Gm-equivariant (with the trivial Gm-action on X).

(2) The natural map OX →(

f∗OY

)Gmis an isomorphism,


(3) For every point x ∈ X , the Gm-action on the reduced fiber redYx isisomorphic to the natural Gm-action on Gm/µm(x) for some m(x) ∈ N,called the multiplicity of the fiber over x.

(If X is normal, one usually assumes that m(x) = 1 at the generic point, but for Xreducible this is not a natural condition.)

One can thus view the theory of Seifert bundles as a special chapter of thestudy of algebraic Gm-actions. The emphasis is, however, quite different.

Theorem 70. Let X be a pure dimensional semi-normal scheme (or algebraicspace). There is a one-to-one correspondence between

(1) Seifert Gm-bundles f : Y → X, and(2) graded OX-algebras

∑

i∈Z Li such that(a) each Li is a torsion free, coherent sheaf on X whose rank is 1 or 0

at the generic points,(b) Li ⊗ Lj → Li+j are isomorphisms at the generic points,(c) LM is locally free for some M > 0, and(d) Li ⊗ LM → Li+M is an isomorphism for every i.

Proof. Let f : Y → X be a Seifert bundle. Since f : Y → X is affine,f∗OY is a quasicoherent sheaf with a Gm-action. Thus it decomposes as a sum ofquasicoherent Gm-eigensubsheaves

f∗OY =∑

j∈Z Lj , (70.3)

where Lj is the λ−j eigensubsheaf, with multiplication maps mij : Li⊗Lj → Li+j .Note that L0 = OX by (69.2).

Pick any point x ∈ X . By assumption redYx∼= Gm/µm(x), thus t−m(x) on

Gm descends to an invertible function hx on Yx which is a Gm-eigenfunction witheigencharacter m(x). There is an affine neighborhood x ∈ U ⊂ X such that hx

lifts to an invertible function hU on f−1(U) which is a Gm-eigenfunction witheigencharacter m(x). This hU is a generator of Lm(x) on U and the multiplicationmaps Li ⊗ Lm(x) → Li+m(x) are isomorphisms over U for every i.

Setting M = m(X) := lcmm(x) : x ∈ X, we see that LM is locally free onX and the multiplication maps Li ⊗ LM → Li+M are isomorphisms for every i.

If x is a generic point, then Li⊗k(x) ∼= k(x) if m(x) divides i and Li⊗k(x) = 0otherwise.

We still need to prove that the Li are torsion free and coherent. Any torsionsection of Li is killed by L⊗M

i → LMi, hence it would give a nilpotent section ofOY , a contradiction. Thus every Li is torison free. Coherence is a local question,thus assume that X is affine. For a generic point xg ∈ X , Li⊗k(xg) 6= 0 iff m(xg)|iiff L−i ⊗ k(xg) 6= 0. Thus there is a section s ∈ H0(X,L−i) that is a generator atall generic points xg such that Li ⊗ k(xg) 6= 0. Then the composite

Li∼= Li ⊗OX

(1,s)−→ Li ⊗ L−i → L0∼= OX

is an isomorphism at the generic points, hence an injection. Thus every Li is acoherent sheaf on X .

Conversely, assume that∑

i∈Z Li satisfies the conditions of (70.2.a–d). Then∑

i∈Z Li is generated by the coherent submodule∑

−m≤i≤m Li. Thus Y := SpecX

∑

i∈Z Li

is affine over X . The grading gives a Gm-action.


Pick any x ∈ X , then the fiber Yx over x is Specx

(∑

i∈Z Li ⊗ k(x))

. By (71),

Li⊗k(x) is nilpotent unless Li is locally free at x and L⊗ri → Lri is an isomorphism

near x for every r. Hence the reduced fiber is Specx

∑

i∈m(x)Z k(x)∼= k(x)[t, t−1]

for some m(x) ∈ N.

Lemma 71. Let L,M be rank 1 torsion free sheaves and assume that there isa surjective map h : L⊗M ։ OX . Then L,M are both locally free.

Proof. Pick x ∈ X . By assumption there is an affine nieghborhood x ∈ U andsections α ∈ H0(U,L), β ∈ H0(U,M) such that h(α⊗ β) is invertible.

Let γ ∈ H0(U,L) be arbitrary. Then h(γ ⊗ β) = f · h(α⊗ β) for some f ∈ OU ,thus h((γ − fα)⊗ β) = 0. Thus γ − fα is zero on the open set where M is locallyfree, hence it is zero since L is torsion free. Thus α generates L|U and so L is locallyfree.

Proposition 72. Let f : Y → X be a Seifert Gm-bundle. Let RX X bea finite, set theoretic equivalence relation and RY Y a Gm-equivariant finite,set theoretic equivalence relation. Assume that the geometric quotients X/RX andY/RY both exist. Then

(1) f descends to f/R : Y/RY → X/RX iff f(RY ) ⊂ RX , and(2) f/R : Y/RY → X/RX is a Seifert Gm-bundle iff RY → RX is surjective.

Proof. The first part follows from the universal property of geometric quotients(27.2).

Next assume that f/R : Y/RY → X/RX exists. Since Y → X is affine andX → X/RX is finite, Y → X/RX is also affine. Since Y → Y/RY is finite,Y/RY → X/RX is affine by Chevalley’s theorem. Since RY Y is Gm-equivariant,the Gm action descends to Y/RY , again by the universal property of geometricquotients.

The only remaining question is about the fibers of Y/RY → X/RX . Pick apoint x ∈ X/RX , let xi ∈ X be its preimages and Yi ⊂ Y the reduced Seifert fiberover xi. Then

red(f/R)−1(x) =(

∐iYi

)

/(

RY |∐iYi

)

.

Thus red(f/R)−1(x) is a union of Gm-orbits and it is irreducible iff for every i, j,every point of Yi is RY -equivalent to some point of Yj . Using the Gm-action, thisholds iff for every i, j, some point of Yi is RY -equivalent to some point of Yj . Thelatter holds iff RY → RX is surjective.

4. Semi log resolutions

The aim of this section is to discuss resolution theorems that are useful in thestudy of semi log canonical varieties.

Definition 73 (Simple normal crossing). Let k be a field, X a k-scheme andD =

∑

aiDi a Weil divisor on X with the Di irreducible.We say that (X,D) has simple normal crossing or snc at a point p ∈ X if

X is smooth at p and there are local coordinates x1, . . . , xn such that SuppD ⊂(x1 · · ·xn = 0) near p. Alternatively, if for each Di there is a c(i) such that Di =(xc(i) = 0) near p.

We say that (X,D) has normal crossing or nc at a point p ∈ X if (XK , D|XK)

is snc at p where XK denotes the completion at p and K is an algebraic closure ofk(p).

4. SEMI LOG RESOLUTIONS 37

Let p ∈ D be a nc point of multiplicity 2. If the characteristic is different from2, then, in suitable local coordinates, D can be given by an equation x2

1 − ux22 = 0

where u ∈ Op,X is a unit. D is snc at p iff u is a square.For example, (y2 = x2 +x3) ⊂ A2 is nc but it is not snc at the origin. Similarly,

(x2 + y2 = 0) ⊂ A2 is nc but it is snc only if√−1 is in the base field k.

We say that (X,D) is snc (resp. nc) if it is snc (resp. nc) for every p ∈ X .Given (X,D), there is a largest open set U ⊂ X such that (U,D|U ) is snc (resp.

nc). This open set is called that snc (resp. nc) locus of (X,D).

Definition 74 (Log resolution). Let k be a perfect field, X a reduced k-scheme and D a Weil divisor on X . A log resolution of (X,D) is a proper birationalmorphism f : X ′ → X such that

(

X ′, D′ := Supp(

f−1(D) + Ex(f)))

has snc. (Inparticular, all of the irreducible components of D′ have codimension 1.) Here Ex(f)denotes the exceptional set of f , that is, the set of points where f is not a localisomorphism. We also say that f : (X ′, D′)→ (X,D) is a log resolution.

The basic existence result on resolutions was established by [Hir64]. We alsoneed a strengthening of it, due to [Sza94], see also [BM97, Sec.12].

Theorem 75 (Existence of log resolutions). Let X be an algebraic space offinite type over a field of characteristic 0 and D a Weil divisor on X.

(1) [Hir64] (X,D) has a log resolution.(2) [Sza94, BM97] (X,D) has a log resolution f : X ′ → X such that f is

an isomorphism over the snc locus of (X,D).

Corollary 76 (Resolution in families). Let C be a smooth curve over a fieldof characteristic 0, f : X → C a flat morphism and D a divisor on X. Then thereis a log resolution g : Y → X such that g−1

∗ D+Ex(g)+Yc is a snc divisor for everyc ∈ C where Yc denotes the fiber over a point c.

Proof. Let p : X ′ → X be any log resolution of (X,D). There are only finitelymany fibers f p of such that p−1D+ Ex(p)+X ′

c is not a snc divisor. Let these beX ′

ci: i ∈ I. Let p′ : Y → X ′ be a log resolution of

(

X ′, p−1D+ Ex(p) +∑

iX′ci

)

that is an isomorphism over X ′ \∑

iX′ci

and set g = p′ p. Then g−1∗ D+ Ex(g) +

∑

i Yciis a snc divisor. Thus, if c = ci for some i then g−1

∗ D + Ex(g) + Yc is a sncdivisor. For other c ∈ C, g−1

∗ D + Ex(g) + Yc is a snc divisor, except possibly nearYc. By construction, the map p′ is an isomorphism near Yc, and p−1D+Ex(p)+X ′

c

is snc divisor, hence so is g−1∗ D + Ex(g) + Yc.

Next we show how (75.2) can be reduced to the Hironaka-type resolution theo-rems presented in [Kol07]. The complication is that the Hironaka method and itsvariants proceed by induction on the multiplicity. Thus, for instance, the methodwould normally blow up every triple point of D before dealing with the non-sncdouble points. In the present situation, however, we want to keep the snc triplepoints untouched.

We can start by resolving the singularities of X , thus it is no restriction toassume from the beginning that X is smooth. To facilitate induction, we work witha more general resolution problem.

Definition 77. Consider the object (X, I1, . . . , Im, E) where X is a smoothvariety, the Ij are ideal sheaves of Cartier divisors and E a snc divisor. We saythat (X, I1, . . . , Im, E) has simple normal crossing or snc at a point p ∈ X if X is


smooth at p and there are local coordinates x1, . . . , xr, xr+1, . . . , xn and an injectionσ : 1, . . . , r → 1, . . . ,m such that

(1) Iσ(i) = (xi) near p for 1 ≤ i ≤ r and p /∈ cosupp Ij for every other Ij ;(2) SuppE ⊂ (

∏

i>r xi = 0) near p.

Thus E +∑

j cosupp Ij has snc support near p, but we also assume that no two ofE, cosupp I1, . . . , cosupp Im have a common irreducible component near p. Further-more, the Ij are assumed to vanish with multiplicity 1, but we do not care aboutthe multiplicities in E. The definition is chosen mainly to satisfy the followingrestriction property:

(3) Assume that I1 is the ideal sheaf of a smooth divisor S ⊂ X , E + S is asnc divisor and that none of the irreducible components of S is containedin E or in cosupp Ij for j > 1. Then (X, I1, . . . , Im, E) is snc near S iff(S, I2|S , . . . , Im|S , E|S) is snc.

The set of all points where (X, I1, . . . , Im, E) is snc is open. It is denoted bysnc(X, I1, . . . , Im, E).

Definition 78. Let Z ⊂ X be a smooth, irreducible subvariety that has sim-ple normal crossing with E (cf. [Kol07, 3.25]). Let π : BZX → X denote theblow-up with exceptional divisor F ⊂ BZX . Define the birational transform of(X, I1, . . . , Im, E) as

(X ′ := BZX, I′1, . . . , I

′m, E

′ := π−1totE) (78.1)

where I ′j = g∗Ij(−F ) if Z ⊂ cosupp Ij and I ′j = g∗Ij if Z 6⊂ cosupp Ij . Note that if

Z has codimension 1, then X ′ = X but I ′j = Ij(−Z) whenever Z ⊂ cosupp Ij .By an elementary computation, the birational transform commutes with re-

striction to a smooth subvariety (cf. [Kol07, 3.62]). As in [Kol07, 3.29] we candefine blow-up sequences.

The assertion (75.2) will be a special case of the following result.

Proposition 79. Let X be a smooth variety, E an snc divisor on X and Ijideal sheaves of Cartier divisors. Then there is a smooth blow-up sequence

Π : (Xr, I(r)1 , . . . , I(r)

m , E(r))→ · · · → (X1, I(1)1 , . . . , I(1)

m , E(1)) = (X, I1, . . . , Im, E)

such that

(1) (Xr, I(r)1 , . . . , I

(r)m , E(r)) has snc everywhere,

(2) for every j, cosupp I(r)j is the birational transform of (the closure of)

cosupp Ij ∩ snc(X, I1, . . . , Im, E), and(3) Π is an isomorphism over snc(X, I1, . . . , Im, E).

Proof. The proof is by induction on dimX and on m.

Step 79.i. Reduction to the case where I1 is the ideal sheaf of a smooth divisor.

Apply order reduction [Kol07, 3.107] to I1. (Technically, to the marked ideal(I1, 2); see [Kol07, Sec.3.5].) In this process, we only blow up a center Z if the(birational transform of) I1 has order ≥ 2 along Z. These are contained in thenon-snc locus. A slight problem is that in [Kol07, 3.107] the transformation ruleused is I1 7→ π∗I1(−2F ) instead of I1 7→ π∗I1(−F ) as in (78.1). Thus each blow-up


for (I1, 2) corresponds to two blow ups in the sequence for Π: first we blow upZ ⊂ X and then we blow up F ⊂ BZX .

At the end the maximal order of I(r)1 becomes 1. Since I

(r)1 is the ideal sheaf

of a Cartier divisor, cosupp I(r)1 is a disjoint union of smooth divisors.

Step 79.ii. Reduction to the case when (X, I1, E) is snc.

The first part is an easier version of Step (79.iii), and should be read afterit. Let S be an irreducible component of E. Write E = S + E′ and considerthe restriction (S, I1|S , E′|S). By induction on the dimension, there is a blow-upsequence ΠS : Sr → · · · → S1 = S such that

(

Sr, (I1|S)(r), (E′|S)(r))

is snc andΠS is an isomorphism over snc(S, I1|S , E′|S). The “same” blow-ups give a blow-up

sequence Π : Xr → · · · → X1 = X such that(

Xr, I(r)1 , E(r)

)

is snc near Sr and Πis an isomorphism over snc(X, I1, E).

We can repeat the procedure for any other irreducible component of E. Notethat as we blow up, the new exceptional divisors are added to E, thus E(s) hasmore and more irreducible components as s increases. However, we only add newirreducible components to E that are exceptional divisors obtained by blowing upa smooth center that is contained in (the birational transform of) cosupp I1. Thusthese automatically have snc with I1. Therefore the procedure needs to be repeatedonly for the original irreducible components of E.

After finitely many steps, (X, I1, E) is snc near E and X and cosupp I1 aresmooth. Thus (X, I1, E) is snc everywhere.

(If we want to resolve just one (X, Ij , E), we can do these steps in any order,but for a functorial resolution one needs an ordering of the index set of E andproceed systematically.)

Step 79.iii. Reduction to the case when (X, I1, . . . , Im, E) is snc near cosupp I1.

Assume that (X, I1, E) is snc. Set S := cosupp(I1). If an irreducible compo-nent Si ⊂ S is contained in cosupp Ij for some j > 1 then we blow up Si. Thisreduces multSi

I1 and multSiIj by 1. Thus eventually none of the irreducible com-

ponents of S are contained in cosupp Ij for j > 1. Thus we may assume that theIj |S are ideal sheaves of Cartier divisors for j > 1 and consider the restriction(S, I2|S , . . . , Im|S , E|S).

By induction there is a blow-up sequence ΠS : Sr → · · · → S1 = S such that(

Sr, (I2|S)(r), . . . , (Im|S)(r), (E|S)(r))

is snc

and ΠS is an isomorphism over snc(S, I2|S , . . . , Im|S , E|S). The “same” blow-upsgive a blow-up sequence Π : Xr → · · · → X1 = X such that the restriction

(

Sr, I(r)2 |Sr

, . . . , I(r)m |Sr

, E(r)|Sr

)

is snc

and Π is an isomorphism over snc(X, I1, . . . , Im, E). (Since we use only order 1blow-ups, this is obvious. For higher orders, one would need the Going-up theorem[Kol07, 3.84], which holds only for D-balanced ideals. Every ideal of order 1 isD-balanced [Kol07, 3.83], that is why we do not need to worry about subtletieshere.)

As noted in (77.3), this implies that(

Xr,OXr(−Sr), I

(r)2 , . . . , I(r)

m , E(r))

is snc near Sr.


Note, furthermore, that Sr = cosupp I(r)1 , hence

(

Xr, I(r)1 , . . . , I

(r)m , E(r)

)

is snc near

cosupp I(r)1 .

Step 79.iv. Induction on m.

By Step 3, we can assume that (X, I1, . . . , Im, E) is snc near cosupp I1. Apply(79) to (X, I2, . . . , Im, E). The resulting Π : Xr → X is an isomorphism oversnc(X, I2, . . . , Im, E). Since cosupp I1 is contained in snc(X, I2, . . . , Im, E), all the

blow up centers are disjoint from cosupp I1. Thus(

Xr, I(r)1 , . . . , I

(r)m , E(r)

)

is alsosnc.

Finally, we may blow up any irreducible component of cosupp I(r)j that is not

the birational transform of an irreducible component of cosupp Ij which intersectssnc(X, I1, . . . , Im, E).

80 (Proof of (75)). Let Dj be the irreducible components of D. Set Ij :=OX(−Dj) and E := ∅. Note that (X,D) is snc at p ∈ X iff (X, I1, . . . , Im, E) issnc at p ∈ X .

If X is a variety, we can apply (79) to (X, I1, . . . , Im, E) to get Π : Xr → X

and (Xr, I(r)1 , . . . , I

(r)m , E(r)). Note that E(r) contains the whole exceptional set of

Π, thus the support of D′ = Π−1∗ D + Ex(Π) is contained in E(r) +

∑

j cosupp I(r)j .

Thus D′ is snc. By (79.3), Π is an isomorphism over the snc locus of (X,D).The resolution constructed in (79) commutes with smooth morphisms and with

change of fields [Kol07, 3.34.1–2], at least if in (78) we allow reducible blow-upcenters.

As in [Kol07, 3.42–45], we conclude that (75) and (79) also hold for algebraicand analytic spaces over a field of characteristic 0.

Starting with (X,D), the above proof depends on an ordering of the irreduciblecomponents of D. This is an artificial device, but I don’t know how to avoid it. Thisis very much connected with the difficulies of dealing with general nc divisors.

81. It should be noted that (75.2) fails for nc instead of snc. The simplestexample is given by the pinch point D := (x2 = y2z) ⊂ A3 =: X . Here (X,D) hasnc outside the origin. At a point along the z-axis, save at the origin, D has 2 localanalytic branches. As we go around the origin, these 2 branches are interchanged.We can never get rid of the pinch point without blowing up the z-axis.

Note that (X,D) is not snc along the z-axis, thus in constructing a log resolutionas in (75.2), we are allowed to blow up the z-axis.

This leads to the following general problem:

Problem 82. For each n, describe the smallest class of singularities Sn suchthat for every (X,D) of dimension n there is a proper birational morphism f :X ′ → X such that

(1) (X ′, D′) has only singularities in Sn, and(2) f is an isomorphism over the nc locus of (X,D).

In dimension 2 we can take, up to etale equivalence, S2 = (xy = 0) ⊂ A2and in dimension 3 we can almost certainly take

S3 = (xy = 0), (xyz = 0), (x2 = y2z) ⊂ A3.(Bierstone and Milman informed me that this can be proved using their method ofresolution [BM97].) In higher dimensions, there is not even a clear conjecture onwhat Sn should be.


It is natural to deal with the problem inductively. For this it is better to viewSn as a set of polynomials in n-variables, and allowing any number of dummyvariables.

One could then hope that Sn consists of Sn−1 plus a few other polynomialsfi(x1, . . . , xn) such that, for every i the singularities of (fi = 0) \ 0 are in Sn−1.That is, for every (c1, . . . , cn) 6= (0, . . . , 0), fi(x1 − c1, . . . , xn − cn) is in Sn−1 (upto an analytic change of coordinates).

Unfortunately, already in dimension 4, the situation is more complicated, asillustrated by the following example.

Example 83. In A4xyzu consider the hypersurface

H :=(

(x+ uy + u2z)(x+ ǫuy + ǫ2u2z)(x+ ǫ2uy + ǫu2z) = 0)

where ǫ is a primite 3rd root of unity. Note that H ∩ (u = c) consists of 3 planesintersecting transversally if c 6= 0 while H ∩ (u = 0) is a triple plane. The Z3-action (x, y, z, u) 7→ (x, y, z, ǫu) permutes the 3 irreducible components of H andA4

xyzu/Z3 = A4xyzt where t = u3. By explicit computation, the image of H in the

quotient is the irreducible hypersurface

D =(

x3 + ty3 + t2z3 − 3txyz = 0) ⊂ A4xyzt.

The singular locus of D can be parametrized as

SingD = im[

(u, z) 7→ (ǫu2z,−(1 + ǫ)uz, z, u3)]

.

Since the quotient map H → D is etale away from (t = 0), we conclude that(

A4xyzt, D

)

is nc outside (t = 0). By explicit computation,

(t = 0) ∩ SingD = (x = y = t = 0).

The singularity D ⊂ A4 can not be improved by futher smooth blow-ups whosecenters are disjoint from the nc locus. Indeed, in our case the complement of the nclocus is the z-axis, hence there are only 2 choices for such a smooth blow-up center.

(1) Blow up (x = y = z = t = 0). In the chart x = x1t1, y = y1t1, z =z1t1, t = t1 we get the birational transform

D′ =(

x31 + t1y

31 + t21z

31 − 3t1x1y1z1 = 0).

(2) Blow up (x = y = t = 0). In the chart x = x1t1, y = y1t1, z = z1, t = t1we get the birational transform

D′′ =(

t1x31 + t21y

31 + z3

1 − 3t1x1y1z1 = 0).

In both cases we get a hypersurface which is, up to a coordinate change, iso-morphic to D.

Note, however, that not every birational morphism between smooth 4-folds isa composite of smooth blow-ups, and I do not know if the singularity of D can beimproved by some other birational morphism.

The transversal singularity type along the z-axis is a degenerate cusp of multi-plicity 2. Indeed, in the coordinates x2 = x/z−(y/z)2, y2 = y/z, t2 = t− 1

2 (y3−3xy)the equation of D ∩ (z = c) (where c 6= 0) becomes

t22 + x22(x2 + 3

4y22) = 0.

Thus S4 also contains the 3-variable polynomial x2 + y2(y+ z2), which is, however,not in S3.


Definition 84 (Semi snc). The ideal local model of an snc Q-divisor is givenby D =

∑ni=1 ai(xi = 0) on X = An. We can also view this as sitting on An+1,

where X = (xn+1 = 0) and D is defined using the other coordinates.Following this example, we can define a non-normal version of snc where X ⊂

An+1 is defined by the product of some of the coordinates and D is defined usingthe remaining coordinates.

For n = 2 we get three possible local models.

(1) S = (z = 0) ⊂ A3 and D = ax(x|S = 0) + ay(y|S = 0). This is the usualnormal case.

(2) S = (yz = 0) ⊂ A3 and D = ax(x|S = 0). Note that as a Weil divisor,D has two irreducible components, namely D1 := (x = y = 0) and D2 :=(x = z = 0). The support of the Weil R-divisor a1D1 + a2D2 is alwayssnc, but the pair (S, a1D1 + a2D2) is semi-snc only if a1 = a2. It is easyto see that a1D1 + a2D2 is R-Cartier only if a1 = a2.

(3) S = (xyz = 0) ⊂ A3 and D = 0.

Let Y be a smooth variety and∑

i∈I Bi a snc divisor. Let IX , ID ⊂ I bedisjoint subsets and c : ID → R a function. Then

X :=∑

i∈IX

Bi

is an snc divisor on Y , which we view now as a subscheme, and

D :=∑

i∈ID

c(i)Bi|X

is a Weil (even Cartier) R-divisor on X . We call any such (X,D) an embeddedsemi-snc pair.

Let X be a reduced variety and D a Weil Q-divisor on X . We say that (X,D) issemi-snc if every point x ∈ X has an open neighborhood x ∈ U such that (U,D|U )is isomorphic to an embedded semi-snc pair.

Note that, by our definition, neither of the following examples are semi-snc:(

(xy = 0), (x = y = 0))

⊂ A3 or(

(xy = 0), (x = z = 0))

⊂ A3.

As in (73), one can also define semi-nc.

85 (Semi log resolutions). What is the right notion of resolution or log resolu-tion for non-normal varieties?

The simplest choice is to make no changes and work with resolutions. In par-ticular, if X = ∪iXi is a reducible scheme and f : X ′ → X is a resolution thenX ′ = ∪iX

′i such that each X ′

i → Xi is a resolution. Note that we have not com-pletely forgotten the gluing data determining X since f−1(Xi ∩Xj) is part of theexceptional set, and so we keep track of it.

There are, however, several inconvenient aspects. For instance, f∗OX′ 6= OX ,and this makes it difficult to study the Picard group of X or the cohomology of linebundles on X using X ′. Another problem is that although Ex(f) tells us whichpart of Xi intersects the other components, it does not tell us anything about whatthe actual isomorphism is between (Xi ∩Xj) ⊂ Xi and (Xi ∩Xj) ⊂ Xj .

It is not clear how to remedy these problems for an arbitrary reducible scheme,but we are dealing with with schemes that have only double normal crossing incodimension 1.


We can thus look for f : X ′ → X such that X ′ has only double normal crossingsingularities and f is an isomorphism over codimension 1 points of X .

As in (75), this works for simple nc but not in general. We need to allow atleast pinch points.

Definition 86 (Pinch points). Let X be a smooth variety over a field ofcharacteristic 6= 2 and D ⊂ X a divisor. We say that D has a pinch point at p ∈ Dif, in suitable local coordinates, D can be defined by the equation x2

1 − x22x3 = 0.

Note that this notion is invariant under field extensions and even completion.Indeed, if the singular set of D is a codimension 2 smooth subvariety, then D canbe locally given by an equation ax2

1 + bx1x2 + cx22 = 0 where a, b, c are regular

functions. If the quadratic part of the equation is a square times a unit, then, aftera coordinate change, we can write the equation as x2

1 + cx22 = 0. This gives a pinch

point after a field extension and completion iff the linear term of c is independentof x1, x2. Thus we can take x3 = −c to get the equation x2

1 − x22x3 = 0.

Let us blow up Z := (x1 = x2 = 0). The normalization of D is contained inthe affine charts with coordinates x′1 := x1/x2, x2, . . . , xn. If we introduce x′3 :=

x3 − x′12then the normalization of D is given by (x′3 = 0). The preimage of Z

is the smooth divisor x2 = 0 and the involution on it is (x′1, 0, 0, x4, . . . , xn) 7→(−x′1, 0, 0, x4, . . . , xn).

A function f defines a τ -invariant divisor iff

f(x′1, x2, x4, . . . , xn) =

g(x′12, x4, . . . , xn) + x2h(x

′1, x2, x4, . . . , xn), or

x′1g(x′12, x4, . . . , xn) + x2h(x

′1, x2, x4, . . . , xn).

In the first case f is τ -invariant and descends to a regular function on D. In thesecond case f is not τ -invariant, but f2 descends to a regular function on D.

In particular, (x1 = x3 = 0) ⊂ (x21 = x2

2x3) is not a Cartier divisor but it isQ-Cartier since 2(x1 = x3 = 0) = (x3 = 0) is Cartier.

Conversely, let Y be a smooth variety, B ⊂ Y a smooth divisor and τ aninvolution on B whose fixed point set F ⊂ B has pure codimension 1 in B. LetZ := B/τ and X the universal push-out of Z ← B → Y , cf. [Art70, Thm.3.1].Then X has only nc and pinch points.

To see this, pick a point p ∈ F and local coordinates y1, . . . , yn such thatB = (y1 = 0), τ∗y2|B = −y2|B and τ∗yi|B = yi|B for i > 2. Then

x1 := y1y2, x2 := y1, x3 := y22 and xi := yi−1 for i > 3

give local coordinates on X with the obvious equation x21 − x2

2x3 = 0.

Theorem 87. Let X be a reduced scheme over a field of characteristic 0. LetXncp ⊂ X be an open subset such that Xncp has only smooth points (x1 = 0),double nc points (x2

1 − ux22 = 0) and pinch points (x2

1 − x22x3 = 0). Then there is a

projective birational morphism f : X ′ → X such that

(1) X ′ has only smooth points, double nc points and pinch points,(2) f is an isomorphism over Xncp,(3) SingX ′ maps birationally onto the closure of SingXncp.

If X ′ has any pinch points then they are on an irreducible component of B ⊂SingX ′ along which X ′ is nc but not snc. Then, by (87.3), X is nc but not sncalong f(B). Thus we obtain the following simple nc version.


Corollary 88. Let X be a reduced scheme over a field of characteristic 0. LetXsnc2 ⊂ X be an open subset which has only smooth points (x1 = 0) and simplenc points of multiplicity ≤ 2 (x1x2 = 0). Then there is a projective birationalmorphism f : X ′ → X such that

(1) X ′ has only smooth points and simple nc points of multiplicity ≤ 2,(2) f is an isomorphism over Xsnc2,(3) SingX ′ maps birationally onto the closure of SingXsnc2.

89 (Proof of (87)). The method of [Hir64] reduces the multiplicity of a schemestarting with the highest multiplicity locus. We can use it to find a proper birationalmorphism g1 : X1 → X such that every point of X1 has multiplicity ≤ 2 and g2 isan isomorphism over Xncp. Thus by replacing X by X1 we may assume to startwith that every point of X has multiplicity ≤ 2.

The next steps of the Hironaka method would not distinguish the nc locus(that we want to keep intact) from the other multiplicity 2 points (that we want toeliminate). Thus we proceed somewhat differently.

Let n : X → X be the normalization with reduced conductor B ⊂ X.Near any point of X , in local analytic or etale coordinates we can write X as

X =(

y2 = g(x)h(x)2)

⊂ An+1

where (x) := (x1, . . . , xn) and g has no multiple factors. (We allow g and h to havecommon factors.) The normalization is then given by

X =(

z2 = g(x))

where z = y/h(x).

Here B = (h(x) = 0) and the involution τ : (z,x) 7→ (−z,x) is well defined onB. (By contrast, the τ action on X depends on the choice of the local coordinatesystem.)

Thus we have a pair (Y2, B2) := (X, B) plus an involution τ2 : B2 → B2 suchthat for every b ∈ B2 there is an etale neighborhood Ub of b, τ2(b) such that τ2extends (nonuniquely) to an involution τ2b of (Ub, B2|Ub

).Let us apply an etale local resolution procedure (as in [W lo05] or [Kol07])

to (Y2, B2). Let the first blow up center be Z2 ⊂ Y2. Since the procedure is etalelocal, we see that Ub ∩ Z2 is τ2b-invariant for every b ∈ B2. Let Y3 → Y2 be theblow up of Z2 and let B3 ⊂ Y3 be the birational transform of B2. Then τ2 liftsto an involution τ3 of B3 and the τ2b lift to extensions on suitable neighborhoods.Moreover, the exceptional divisor of Y3 → Y2 intersected with B3 is τ3-invariant. Inparticular, there is an ample line bundle L3 on Y3 such that L3|B3

is τ3-invariant.At the end we obtain g : Yr → Y2 = X such that

(1) Yr is smooth and Ex(g) +Br is an snc divisor,(2) Br is smooth and τ lifts to an involution τr on Br, and(3) there is a g-ample line bundle L such that L|Br

is τr-invariant.

The fixed point set of τr is a disjoint union of smooth subvarieties of Br. Byblowing up those components whose dimension is < dimBr − 1, we also achieve(after replacing r + 1 by r) that

(4) the fixed point set of τr has pure codimension 1 in Br.

Let Zr := Br/τr and Xr the universal push-out of Zr ← Br → Yr. As wenoted in (86), Xr has only nc and pinch points.


Further, let D be a divisor on Yr such that D|Bris τr-invariant. As noted in

(86), 2D is the pull back of a Cartier divisor on Xr. In particular, if D is amplethen Xr is projective.

We would like not just a semi resolution of X but a log resolution of the pair(X,D). Thus we need to take into account the singularities of D as well. As wenoted in (81), this is not obvious even when X is a smooth 3-fold. The followingweaker version, which gives the expected result only for the codimension 1 part ofthe singular set of (X,D), will be sufficient for us.

Theorem 90. Let X be a reduced scheme over a field of characteristic 0 andD a Weil divisor on X. Let Xnc2 ⊂ X be an open subset which has only nc pointsof multiplicity ≤ 2 and D|Xnc2 is smooth and disjoint from SingXnc2. Then thereis a projective birational morphism f : X ′ → X such that

(1) the local models for(

X ′, D′ := f−1∗ (D) + Ex(f)

)

are(a) (Smooth) X ′ = (x1 = 0) and D′ = (

∏

i∈I xi = 0) for some I ⊂2, . . . , n+ 1,

(b) (Double nc) X ′ = (x21 − ux2

2 = 0) and D′ = (∏

i∈I xi = 0) for someI ⊂ 3, . . . , n+ 1, or

(c) (Pinched) X ′ = (x21 = x2

2x3) and D′ = (∏

i∈I xi = 0) +D2 for someI ⊂ 4, . . . , n+ 1 where either D2 = 0 or D2 = (x1 = x3 = 0).

(2) f is an isomorphism over Xnc2.(3) SingX ′ maps birationally onto the closure of SingXnc2.(4) Let X → X be the normalization, B ⊂ X the closure of the conductor of

Xnc2 → Xnc2 and D ⊂ X the preimage of D. Then f is a log resolutionof

(

X, B + D)

.

As before, (90) implies the simple nc version:

Corollary 91. Let X be a reduced scheme over a field of characteristic 0 andD a Weil divisor on X. Let Xsnc2 ⊂ X be an open subset which has only snc pointsof multiplicity ≤ 2 and D|Xsnc2 is smooth and disjoint from SingXsnc2. Then thereis a projective birational morphism f : X ′ → X such that

(1) the local models for(

X ′, D′ := f−1∗ (D) + Ex(f)

)

are(a) (Smooth) X ′ = (x1 = 0) and D′ = (

∏

i∈I xi = 0) for some I ⊂2, . . . , n+ 1, or

(b) (Double snc) X ′ = (x1x2 = 0) and D′ = (∏

i∈I xi = 0) for someI ⊂ 3, . . . , n+ 1.

(2) f is an isomorphism over Xsnc2.(3) SingX ′ maps birationally onto the closure of SingXsnc2.

92 (Proof of (90)). First we use (87) to reduce to the case when X has onlydouble nc and pinch points. Let X → X be the normalization and B ⊂ X theconductor. Here X and B are both smooth.

Next we want to apply embedded resolution to (X, B + D). One has to be alittle careful with D since the preimage D ⊂ X need not be τ -invariant.

As a first step, we move the support of D away from B. As in [Kol07, 3.102]this is equivalent to multiplicity reduction for a suitable ideal ID ⊂ OB. Let usnow apply multiplicity reduction for the ideal ID + τ∗ID. All the steps are nowτ -invariant, so at the end we obtain g : Yr → X such that Br + Dr + Ex(g) hasonly snc along Br and τ lifts to an involution τr.


As in the proof of (87), we can also assume that the fixed locus of τr has purecodimension 1 in Br and that there is a g-ample line bundle L such that L|Br

isτr-invariant.

As in the end of (89), let Xr be the universal push-out of Br/τr ← Br →Yr. Then (Xr, D

′r) has the required normal form along SingXr. The remaining

singularities of D′r can now be resolved as in (75).

The following analog of (75) is still open:

Problem 93. Let X be a reduced scheme over a field of characteristic 0 andD a Weil divisor on X . Let Xsnc ⊂ X be the largest open subset such that(Xsnc, D|Xsnc) is semi snc. Is there a projective birational morphism f : X ′ → Xsuch that

(1) (X ′, D′) is semi snc and(2) f is an isomorphism over Xsnc?

The following weaker version is sufficient for many applications. We do notguarantee that f : X ′ → X is an isomorphism over Xsnc, only that f is an iso-morphism over an open subset X0 ⊂ Xsnc that intersects every semi log canonicalcenter of (Xsnc, D|Xsnc). (One can see easily that the latter are exactly the ir-reducible components of intersections of irreducible components of Xsnc and ofD|Xsnc .) This implies that we do not introduce any “unnecessary” f -exceptionaldivisors with discrepancy −1. The latter is usually the key property that one needs.

Unfortunately, the proof only works in the quasi projective case.

Proposition 94. Let X be a reduced quasi projective scheme over a field ofcharacteristic 0 and D a Weil divisor on X. Let X0 ⊂ X be an open subset suchthat (X0, D|X0) is semi snc. There is a projective birational morphism f : X ′ → Xsuch that

(1) (X ′, D′) is an embedded semi snc pair and(2) f is an isomorphism over the generic point of every semi log canonical

center of (X0, D|X0).

Proof. In applications it frequently happens that X+B is a divisor on a varietyY and D = B|X . Applying (75.2) to (Y,X + B) gives (94). In general, not every(X,D) can be obtained this way, but one can achieve something similar at the priceof introducing other singularities.

Take an embedding X ⊂ PN . Pick a finite set W ⊂ X such that each semi logcanonical center of (X0, D|X0) contains a point of W .

Choose d ≫ 1 such that the scheme theoretic base locus of OPN (d)(−X) is Xnear every point of W . Taking a complete intersection of (N − dimX − 1) generalmembers in |OPN (d)(−X)|, we obtain Y ⊃ X such that Y is smooth at every pointof W . (Here we use that X has only hypersurface singularities near W .)

For every Di choose di ≫ 1 such that the scheme theoretic base locus ofOPN (di)(−Di) is Di near every point of W . For each i, let DY

i ∈ |OPN (di)(−Di)|be a general member.

We have thus constructed a pair (Y,X +∑

DYi ) such that

(1) (Y,X +∑

DYi ) is snc near W , and

(2) (X,∑

DYi |X) is isomorphic to (X,

∑

Di) in a suitable neighborhood ofW .


By (75.2) there is a semi log resolution of

f : (Y ′, X ′ +∑

Bi)→ (Y,X +∑

DYi )

such that f is an isomorphism over an open neighborhood of W . Then f |X′ : X ′ →X is the log resolution we want.

Definition 95 (Total transform). Let X be a smooth variety and D ⊂ X a ncdivisor. An irreducible subvariety Z ⊂ D is called a closed stratum if, at a generalpoint z ∈ Z, the intersection of the local analytic branches of D that pass throughz is Z. If D is snc, then Z is an irreducible component of the intersection of someof the irreducible components of D.

In general, Z can be singular. For smooth Z, let π : BZX → X denote theblow-up of Z with exceptional divisor EZ ⊂ BZX . Let D′ ⊂ BZX denote thebirational transform of D.

Then EZ +D′ ⊂ BZX is a nc divisor, called the total transform of D in BZX .Let JZ ⊂ OX denote the ideal sheaf of Z ⊂ X and IZ ⊂ OD denote the ideal

sheaf of Z ⊂ D. Then

D′ = ProjD∑

m≥0

ImZ and EZ = ProjZ

∑

m≥0

Sm(

JZ/J2Z

)

.

Note that a priori the total transform also depends onX , that is, the embeddingof D into a smooth variety. We claim, however, that any two total transforms arecanonically isomorphic. In fact, we construct the total transfrom for non-embeddednc schemes as well.

Thus let D be a nc scheme and Z ⊂ D an irreducible, smooth, closed stratum.In the trivial case, when Z is an irreducible component of D, the total transform isD itself. Thus assume from now on that Z has codimension at least 2. This impliesthat D is singular along Z.

Let IZ ⊂ OD denote the ideal sheaf of Z ⊂ D. As in the embedded case, thebirational transform of D is given by the blow-up of D along Z:

D′ = BZD = ProjD∑

m≥0

ImZ .

The preimage of Z in D′ is thus

Z ′ := ProjD∑

m≥0

ImZ /I

m+1Z .

Since D is singular along Z, we know that JZ/J2Z = IZ/I

2Z . Thus

EZ = ProjZ∑

m≥0

Sm(

IZ/I2Z

)

,

and there is a natural injection Z ′ → EZ coming from the surjections Sm(

IZ/I2Z

)

։

ImZ /I

m+1Z . Thus we can glue D′ and EZ along Z ′ to obtain the total transform

π : EZ +D′ → D.The total transform commutes with etale maps D∗ → D.


5. Ramified covers

In this section we study finite ramified morphisms between demi normal schemes.We consider only morphisms that are unramified over the generic points of the con-ductor. This restriction is satisfied in our applications, but the general case isneeded in some other contexts [?].

Definition 96 (Ramified covers).

A finite morphism of demi normal schemes π : X → X is called a ramified coverof degree m if there is a dense open subset U ⊂ X which contains the generic pointsof the conductorDX such that π is etale and has degreem over πU : U → U . In thiscase there is an open subscheme j : X0 → X whose complement has codimension≥ 2 such that π is finite and flat of degree m over X0. Indeed, we can takeX0 = U ∪Xns. Set X0 = π−1(X0) and π0 : X0 → X0 the induced map. Since Xis S2,

j∗(

π0∗OX0

)

= π∗OX .

In particular, π : X → X is uniquely determined by the finite, flat morphismπ0 : X0 → X0.

If πU : U → U is Galois with Galois group G then the G action on U extendsto a proper G-action on X. The action is free on U , hence it is free on an open setthat contains the generic points of the conductor DX .

Conversely, let X be a demi normal scheme with a proper action of a finitegroup G. The geometric quotient X := X/G exists by (31). Let U ⊂ X be the

largest open set on which the G-action is free. Then U → U/G is etale. (We could

take this as the definition of a free action.) Thus if U contains the generic points

of the conductor DX then X → X = X/G is a ramified cover of X .

Let g : Y → X be a morphism with Y demi normal. The pull back X ×X Y →Y defines a finite, flat, ramified cover over g−1(X0). Thus, if Y \ g−1(X0) has

codimension ≥ 2, then there is a unique ramified cover Y → Y that agrees withX ×X Y → Y over g−1(X0). There is always a morphism Y → X ×X Y which is

an isomorphism iff X ×X Y is S2.

97 (Pull-back and push-forward of divisors). The pull back of a Weil divisorby π can be defined as follows.

Take any Weil divisor B on X , restrict it to X0 as in (96), pull it back and

then extend uniquely to a Weil divisor B =: π∗B on X.If B is Cartier, then π∗B is Cartier on X and agrees with the usual pull back.

Conversely, if π∗B is Cartier then m ·B is also Cartier. Thus the pull back and thenorm (???) take Q-Cartier Q-divisors to Q-Cartier Q-divisors.

98 (Hurwitz formula). (cf. [Har77, Sec.IV.2]) Let g : X ′ → X be a ramifiedcover of an n-dimensional demi normal scheme defined over a field k.

The ramification divisor of g is defined as

R(g) =∑

F⊂X′

R(F )[F ] :=∑

F⊂X′

lengthk(F )

(

ΩX′/X

)

F[F ], (98.1)

where the summation is over all prime divisors of X ′ and ΩX′/X denotes the sheafof relative differentials. If r(F ) denotes the ramification index of g along F thenR(F ) ≥ r(F )−1. The ramification is called tame along F if R(F ) = r(F )−1. Thisholds iff char k(F ) does not divide r(F ).

5. RAMIFIED COVERS 49

The support of g(

R(g))

is called the branch divisor of g.The Hurwitz formula says that KX′ = g∗KX +R. More generally, if ∆ is any

Q-divisor on X , then the Q-divisor g∗∆−R makes sense and then

KX′ + g∗∆−R = g∗(KX + ∆). (98.2)

Thus KX + ∆ is Q-Cartier iff KX′ + g∗∆−R is.In general g∗∆−R need not be effective, but there are three important special

cases when it is effective and the pull back formula is very simple.The first is when R = 0, that is, when g is unramified in codimension 1. Then

KX′ + g∗∆ = g∗(KX + ∆). (98.3)

The second is when g is tamely ramified and ∆ = B+∆1 where B is an integraldivisor whose support contains the branch divisor of g. Then g∗(B) contains thesupport of R and g∗B = R+ red g∗B. Thus we obtain the pull back formula

KX′ + red g∗B + g∗∆1 = g∗(KX +B + ∆1). (98.4)

More generally, assume that for every Di ⊂ X , the coeffcient of Di in ∆ isat least 1 − 1

rifor some ri ≥ supjeij where D′

ij are the irreducible components

of g∗Di and eij denotes the ramification index along D′ij . We can then write

∆ =∑

i

(

1− 1ri

)

Di + ∆′ where ∆′ ≥ 0. This gives as the formula

KX′ +∑

ij

(

1− eij

ri

)

D′ij + g∗∆′ = g∗

(

KX +∑

i

(

1− 1ri

)

Di + ∆′)

. (98.5)

In particular, if ∆′ = 0 and eij |ri for every i, j then every coefficient of the pull-backalso has the form 1− 1

rijfor some integer rij .

The following general principle compares discrepancies under finite morphisms.A result of this type first appeared in [Rei80].

Proposition 99. Let g : X ′ → X be a finite, separable morphism betweenn-dimensional normal varieties defined over a field k. Let ∆ be a Q-divisor on Xand ∆′ a Q-divisor on X ′ such that KX′ + ∆′ = g∗(KX + ∆). Then

(1) discrep(X ′,∆′) ≥ discrep(X,∆);(2) (deg g)(discrep(X,∆) + 1) ≥ (discrep(X ′,∆′) + 1) if one of the following

conditions holds(a) char k = 0,(b) degX ′/X < char k, or(c) X ′/X is Galois and char k 6 | degX ′/X.

Proof: Consider the fiber product diagram with exceptional divisors given be-low:

F ⊂ Y ′ f ′

→ X ′

↓ ↓ h ↓ gE ⊂ Y

f→ X.

(99.4)


Near the generic point of F we compute that

KY ′ = f ′∗(KX′ + ∆′) + a(F,X ′,∆′)F

= f ′∗g∗(KX + ∆) + a(F,X ′,∆′)F

= h∗f∗(KX + ∆) + a(F,X ′,∆′)F, and

KY ′ = h∗KY +R(F )F

= h∗f∗(KX + ∆) + a(E,X,∆)h∗E +R(F )F

= h∗f∗(KX + ∆) +(

r(F )a(E,X,∆) +R(F ))

F.

This shows that

a(F,X ′,∆′) + 1 = r(F )(

a(E,X,∆) + 1)

+(

R(F ) + 1− r(F ))

.

Since R(F ) + 1 ≥ r(F ) ≥ 1 this implies (1) if discrep(X,∆) ≥ −1. Otherwisediscrep(X,∆) = −∞ and there is nothing to prove.

Conversely, if one of the conditions of (99.2.a–c) hold then R(F ) + 1 = r(F )and so

(

a(E,X,∆) + 1)

=1

r(F )

(

a(F,X ′,∆′) + 1)

≥ 1

deg g

(

a(F,X ′,∆′) + 1)

.

We are done if these considerations apply to all possible divisors E and F . Givenany divisor E over X , we get a divisor F over X ′ from the diagram (99.4). Theconverse is proved in [KM98, 2.45].

Corollary 100. Notation and assumptions as above. If (X,∆) is klt (resp.lc) then so is (X ′,∆′). Conversely, if (X ′,∆′) is klt (resp. lc) then so is (X,∆)provided one of the conditions of (99.2.a–c) hold.

Next we consider ramified covers with cyclic Galois group. These are easy toconstruct and especially useful in the study of slc pairs.

101 (µm-covers). Let π : X → X be a ramified cover with Galois group µm.Since µm is reductive, its action decomposes π∗OX into a sum of eigensheaves Li,

each of rank 1. Multiplication gives maps L⊗i1 → Li, hence there are divisors Di

such that

π∗OX =m−1∑

i=0

L[i]1 (Di). (101.1)

The i ≡ 0 mod m eigensubsheaf is isomorphic to OX , hence we get an isomorphism

γ : L[m]1 (Dm) ∼= OX .

Since L1 tends to be negative, we usually choose L := L[−1]1 as our basic sheaf

and D := Dm as the key divisor. Then γ corresponds to a section s of L[m] whosezero divisor is D.

Conversely, let X be a demi normal scheme, L a divisorial sheaf on X ands a section of L[m] for some m > 0 that does not vanish along any irreduciblecomponent of the conductor DX ⊂ X . Set D := (s) and ∆ := 1

mD. The section scan be identified with an isomorphism

γs : L[−m](

⌊m∆⌋)

= L[−m](

D) ∼= OX . (101.2)

This in turn defines an algebra structure on

OX + L[−1] + · · ·+ L[−(m−1)],

5. RAMIFIED COVERS 51

where, for i + j < m the multiplication L[−i] × L[−j] → L[−(i+j)] is the tensorproduct and for i+ j ≥ m we compose the tensor product with the isomorphisms

L[−(i+j)] = L[−(i+j−m)] ⊗ L[−m] 1⊗γs−→ L[−(i+j−m)] ⊗OX = L[−(i+j−m)].

The spectrum of this algebra gives X overX\D, but it is usually quite singular overD since we have not yet found the correct divisors Di. By the universal property of

the normalization, Di is the largest divisor such that(

L[−i](Di))[m] ⊂ OX . That

is, Di = ⌊ imD⌋ = ⌊i∆⌋ and so

X = SpecX

(

OX + L[−1](

⌊∆⌋)

+ · · ·+ L[−(m−1)](

⌊(m− 1)∆⌋)

)

. (101.3)

We frequently write X =: X[

L, m√s] to emphasize its dependence on L and s.

Alternatively, let Is be the ideal sheaf of∑∞

i=0 L[−i]

(

⌊i∆⌋)

generated by φ− γs(φ)

where φ is any local section of L[−m](

⌊m∆⌋)

. Then

X[

L, m√s] = SpecX

(

∞∑

i=0

L[−i](

⌊i∆⌋)

)

/Is. (101.4)

Duality for finite morphisms now gives that

π∗ωX = ωX + ωX⊗L(

−⌊∆⌋)

+ · · ·+ ωX⊗L[m−1](

−⌊(m− 1)∆⌋)

. (101.5)

Note also that

π∗π[∗]

(

L(−⌊∆⌋))

= L(

−⌊∆⌋)

+OX + L[−1](

⌊∆⌋)

+ · · ·+ L[−(m−2)](

⌊(m− 2)∆⌋)

,

which shows that

π[∗](

L(−⌊∆⌋)) ∼= OX . (101.6)

102 (Normal forms of µm-covers). There are several ways to change L and swithout changing the corresponding µm-cover.

First of all, if (i,m) = 1 then the same cover is constructed if we think ofthe ith summand as the basic divisorial sheaf. That is using Li

(

⌊−i∆⌋)

and theisomorphism

(

L[i](

⌊−i∆⌋))[m]

= L[mi](

m⌊−i∆⌋) ∼= OX

(

mi∆ +m⌊−i∆⌋)

. (102.1)

Second, if D0 is any divisor then by replacing L by L(−D0) and ∆ by ∆−D0 givesthe same µm-cover. Thus we can always assume that ⌊∆⌋ = 0.

Finally, there is the choice of the isomorphism s : OX∼= L[m](−m∆). Given

two such isomorphisms si, their quotient u := s1/s2 is a unit in OX . If u = vm isan mth power, then acting by v on L shows that the two µm-covers are isomorphic.Thus we should think of s as an element

s ∈ H0(X,OX)⊗m\ IsomX

(

OX(m∆), L[m])

. (102.2)

Different choices of s can result in quite different covers. For instance, if C = A1\0with coordinate x then C

[

OC ,m√

1]

is the reducible plane curve ym = 1 while

C[

OC , m√x]

is the irreducible plane curve ym = x.

If the residue characteristics do not divide m, then X[

OX , m√u]

→ X is etale

and the two µm-coversX[

L, m√s1

]

and X[

L, m√s2

]

become isomorphic after pulling

back to X[

OX , m√u]

. In particular, they have isomorphic etale covers.


However, they can be quite different in positive characteristic. For m = p theabove example gives C

[

OC ,p√

1]

which is the nonreduced plane curve yp = 1 while

C[

OC , p√x]

is the smooth plane curve yp = x.We can summarize these discussions as follows.

Corollary 103. Let X be a demi normal scheme over a field k and U ⊂ Xan open subset which contains every generic point of the conductor DX . Assumethat char k does not divide m. Then there is a natural one-to-one correspondancebetween the following 3 sets.

(1) Etale Galois covers U → U plus an isomorphism Gal(U/U) ∼= µm.

(2) Ramified Galois covers X → X whose branch divisor is in X \ U plus an

isomorphism Gal(X/X) ∼= µm.(3) Triples (L,∆, s) where

(a) L is a divisorial sheaf on X,(b) ∆ is a Q-divisor whose support is in X \ U such that ⌊∆⌋ = 0 and

m∆ is a Z-divisor and(c) s ∈ H0(X,OX)⊗m\ IsomX

(

OX(m∆), L[m])

.

104 (Local properties of µm-covers). Given X and (L,∆, s) as in (103.3), let π :

X → X be the corresponding µm-cover. Write ∆ =∑

(mi/ri)Di where (mi, ri) = 1and assume that chark does not divide m.

By (71), for x ∈ X the evaluation of the product

L[−i](

⌊i∆⌋)

× L[i−m](

⌊(m− i)∆⌋)

→ L[−m](

⌊m∆⌋)

⊗ k(x) ∼= k(x)

is zero, unless i∆ is a Z-divisor near x and L[−i](

⌊i∆⌋)

is locally free.This implies the following:

(1) The number of preimages of x equals the number of indices 0 ≤ j < msuch that j∆ is a Z-divisor near x and L[−j]

(

j∆)

is locally free at x.(2) π is etale at x ∈ X iff L is locally free at x and x 6∈ Supp ∆.(3) The ramification index of π over Di is ri.

Definition 105 (Index 1 covers). Let (X,∆) be a demi normal pair. Write∆ = B + ∆′ where B is a Z-divisor and ⌊∆′⌋ = 0.

The index of (X,∆) at a point x ∈ X , denoted by indexx(X,∆) is the smallest

positive integer m such that m∆ is a Z-divisor and ω[m]X (m∆) is locally free at x.

(If there is no such m, set indexx(X,∆) =∞.)For a subset Z ⊂ X , let indexZ(X,∆) be the least common multiple of

indexx(X,∆) for all x ∈ Z. We write index(X,∆) := indexX(X,∆)Thus KX + ∆ is Q-Cartier iff index(X,∆) <∞.Pick a point x ∈ X and set m = indexx(X,∆). After replacing X with an

open neighborhood of x, we may assume that there is an isomorphism s : OX∼=

ω[m]X (m∆). Thus L := ω

[−1]X (−B), ∆′ and s determine a µm-cover π : X → X .

Since m is the smallest, (104.1) implies that π−1(x) consists of a single point x,

hence we get a pointed scheme (x ∈ X). Note that (x ∈ X) depends on s if wework Zariski locally, but it does not depend on s if we work etale locally. Thus, inthe latter case, we can talk about the index 1 cover of (x ∈ X,∆).

π∗OX =

m−1∑

i=0

ω[i]X

(

iB + ⌊i∆′⌋)

and π∗ωX

(

B)

=

m−1∑

i=0

ω[1−i]X

(

(1− i)B − ⌊i∆′⌋)

.

6. CANONICAL RINGS OF NORMAL CROSSING SURFACES 53

As in (101.6), the i = 1 summand shows that ωX

(

B) ∼= OX . Furthermore, the

µm-action on ωX

(

B)

⊗ k(

x)

is the standard representation, hence it is faitful.

Theorem 106. In each of the following 4 cases, taking the index 1 cover givesa natural one-to-one correspondence between the sets described in (a) and (b). Localis understood in the etale topology and char k(x) 6 |m is always assumed.

(1) (a) Local demi normal schemes (x ∈ X) such that indexxX = m.

(b) Local demi normal schemes (x ∈ X) such that ωX is localy free witha proper µm-action that is free outside a codimension ≥ 2 subset andthe induced action on ωX ⊗ k(x) is faithful.

(2) (a) Local demi normal pairs (x ∈ X,B) such that indexx(X,B) = m.

(b) Local demi normal pairs (x ∈ X, B) such that ωX(B) is localy freewith a proper µm-action that is free outside a codimension ≥ 2 subsetand the induced action on ωX(B)⊗ k(x) is faithful.

(3) (a) Local demi normal pairs(

x ∈ X,∆)

where ∆ =∑

i(1 − 1ri

)Di with

ri ∈ N such that indexx(X,∆) = m.

(b) Local demi normal schemes (x ∈ X) such that ωX is localy free witha proper µm-action that is free on a dense open subset that containsall generic points of the conductor DX and the induced action onωX ⊗ k(x) is faithful.

(4) (a) Local demi normal pairs(

x ∈ X,B + ∆)

where ∆ =∑

i(1 − 1ri

)Di

with ri ∈ N such that indexx(X,B + ∆) = m.

(b) Local demi normal pairs (x ∈ X, B) such that ωX(B) is localy freewith a proper µm-action that is free on a dense open subset thatcontains all generic points of B + DX and the induced action on

ωX(B)⊗ k(x) is faithful.

Moreover, in all cases the pair (X, 0) (resp. (X,B), (X,∆), (X,B + ∆)) is klt (or

lc or slc) iff the index 1 cover (X, 0) (resp. (X, B), (X, 0), (X, B)) is klt (or lc orslc).

Proof. Strating with (X,B+ ∆), the conctruction of (X, B) was done in (105)

and we also saw that ωX(B) is localy free and induced µm-action on ωX(B)⊗ k(x)is faithful.

The pull back of the canonical class is computed in (98) and (99) shows thelast claim about the properties klt, lc or slc.

The following two special cases are especially important.

Corollary 107. (1) A singularity (x ∈ X) is lt iff it is a quotient of an

index 1 canonical singularity(

x ∈ X)

by a proper µm-action that is freeoutside a codimension ≥ 2 subset.

(2) A singularity(

x ∈ X,∆)

where ∆ =∑

i(1 − 1ri

)Di with ri ∈ N is klt iff

it is a quotient of an index 1 canonical singularity (x ∈ X) by a properµm-action.

6. Canonical rings of normal crossing surfaces

In this section we show, following [?], that the minimal model program doesnot work for varieties with semi log canonical singularities. Problems arise even forsurfaces with normal crossing singularities.


Proposition 108. There are irreducible, projective surfaces of general typewith only normal crossing singularities whose canonical ring is not finitely gener-ated.

Proof. Let S → P1×P1 be a double cover ramified along a curve B of bidegree(6, 6) and πi : S → P1 the coordinate projections. The canonical class of S is thepull-back of OP1×P1(1, 1), hence ample.

Pick distinct points p, q, r1, . . . , r6 ∈ P1 and choose B such that

B∩p×P1 = p×r1, r2, r3, r4, 2r5 and B∩q×P1 = q×r1, r2, r3, r4, 2r6.Set Fp := π−1

1 (p) and Fq := π−11 (q) with normalizations of Fp, Fq. Note that Fp is

singular at p0 = (p, r5) and Fq is singular at q0 = (q, r6). Furthermore, π2 : Fp → P1

and π2 : Fq → P1 both ramify over the points r1, r2, r3, r4, hence there are twoisomorphisms τF , τ

′F : Fp

∼= Fq that commute with π2. Let p1, p2 ∈ Fp ⊂ S be the2 preimages of (p, r6) and q1, q2 ∈ Fq ⊂ S be the 2 preimages of (p, r5).

As in (23), the triple (S, Fp + Fq, τF ) defines a demi-normal surface T :=S/R(τF ). Note that T has 2 triple points, P with preimages p0, q1, q2 ∈ S andQ with preimages q0, p1, p2 ∈ S. Set Z := p0, q1, q2, q0, p1, p2. Then τF is aninvolution on (Fp +Fq) \Z and T \ P,Q has only double nc points. Note that Tis a double cover of

(

P1/(p ∼ q))

× P1, hence projective. By (9),

H0(

T, ω[m]T

)

=

s ∈ H0(

S\Z, ωmS\Z(mFp+mFq)

)

: s|Fp+Fqis (−1)mτF -invariant

.

We show next that the canonical ring∑

m≥0H0(

T, ω[m]T

)

is not finitely generated.This is, however, caused by the singularities of T ; in fact, KT is not even Q-Cartier.So at the end we construct a surface T1 with only normal crossing singularities whosecanonical ring is isomorphic to the canonical ring of T .

Near the two triple points P,Q ∈ T , we are in the situation described in (108.1).

In particular, we know that the OT -algebra∑

m≥0 ω[m]T is not finitely generated,

not even locally near P or Q.Local computation (108.1) Let C1 := (xy = 0) ⊂ C2

x,y =: S1. Let C21 := (u1 =

0) ⊂ C2u1,v1

=: S21 and C22 := (v2 = 0) ⊂ C2u2,v2

=: S22. Set S2 := S21 ∐ S22 andC2 := C21 ∐ C22

The gluing is defined by σ : C1 \ (0, 0)→ C2 sending (0, y) 7→ (0, y) ∈ C21 and(x, 0) 7→ (x, 0) ∈ C22.

Note that T := (S1 ∐ S2)/σ is not a nc surface. Rather, it has a triple pointwith embedding dimension 4. A local model is given by

(t1 = t2 = 0) ∪ (t2 = t3 = 0) ∪ (t3 = t4 = 0) ⊂ C4.

The isomorphism is given by (x, y) 7→ (0, x, y, 0), (u1, v1) 7→ (v1, u1, 0, 0) and(u2, v2) 7→ (0, 0, v2, u2).

A local generator of ωS21(C21) is u−1

1 du1∧dv1, and the restriction ωS21(C21)|C21

=ωC21

is given by the Poincare residue map

dff ∧ dg|(f=0) 7→ dg|(f=0).

Thus ωmS21

(mC21)|C21= (dv1)

m · OC21. The situation on C22 is similar.

On the other hand, a local generator of ωS1(C1) is (xy)−1dx∧dy. Its restriction

to C1 gives a local generator η of ωC1. Note that

η|(y=0) = − dxx and η|(x=0) = dy

y . (108.2)

6. CANONICAL RINGS OF NORMAL CROSSING SURFACES 55

Thus

ωmS1

(mC1)|C1= ηm · OC1

.

The interesting feature appears when we compute that

σ∗(dv1)m = ym · η|(x=0) and σ∗(du2)

m = (−x)m · η|(y=0).

Thus the image of the restriction map

ωmT → ωm

S1(mC1) is (xy, xm, ym) ·

(

dx∧dyxy

)m. (108.3)

Local finite generation fails since the C[x, y]-algebra∑

m≥0

(xy, xm, ym) · wm ⊂ C[x, y, w] is not finitely generated,

where w = (xy)−1dx ∧ dy is a formal variable taking care of the grading. Indeed,for every m, the element xy · wm needs to be added as a new generator.

To go from the local infinite generation to global infinite generation we considerthe natural map

ρ :∑

m≥0

H0(T, ω[m]T )→

∑

m≥0

ω[m]T .

Assume that for all m ≫ 1 there are global sections tm ∈ H0(T, ω[m]T ) such that

ρ(tm) is not contained in the subsheaf of ω[m]T generated by the ω

[i]T for i < m. Then

tm is not contained in the subalgebra generated by the H0(T, ω[i]T ) for i < m, hence

∑

m≥0H0(T, ω

[m]T ) is not finitely generated.

Since ωS is ample and Fp, Fq are nef, we see that ωmS (mFp +mFq)(−Fp − Fq)

is globally generated for m ≫ 1. Sections of ωmS (mFp + mFq)(−Fp − Fq) vanish

along Fp + Fq, hence they automatically glue and descend to sections of ω[m]T .

Thus if sm ∈ H0(S, ωmS (mFp +mFq)) vanishes along Fp + Fq with multiplicity

1, then we obtain a corresponding tm ∈ H0(T, ω[m]T ) which, up to a unit, equals

xy ·(

(xy)−1dx∧ dy)m

in (108.3). Thus∑

m≥0H0(T, ωm

T ) is not finitely generated.Finally we construct T1. Let S1 → S be obtained by blowing up p1, p2, q1, q2

with the corresponding exceptional curves are Ep1, Ep2

, Eq1, Eq2

⊂ S1. The normal-ization of our surface will be S1 with conductorD1 := Fp+Fq+Ep1

+Ep2+Eq1

+Eq2.

Fix isomorpisms τp : Ep1

∼= Ep2and τq : Eq1

∼= Eq2that map Fp ∩Ep1

to Fp ∩Ep2

and Fq∩Eq1to Fq∩Eq2

. Let τ1 be the involution on the normalization of D1 whichis τF on Fp + Fq, τp on Ep1

+ Ep2and τq on Eq1

+ Eq2.

Set T1 := S1/R(τ1) with normalization map n : S1 → T1. T1 has only normalcrossing singularities, 2 of them triple points. By (8),

n∗ωT1= ωS1

(D1) = ωS1(Fp + Fq + Ep1

+ Ep2+ Eq1

+ Eq2)

and this line bundle has negative degree along the 4 curves Eq1, Eq2

, Ep1, Ep2

.Therefore, every section of ωm

S1(mD1) is the pull-back of a section of ωm

S

(

mFp +

mFq

)

. Therefore∑

m≥0

H0(

T1, ωmT1

)

=∑

m≥0

H0(

T, ω[m]T

)

.

Note that T1 is projective since T is projective that ω−1T1

is relatively ample onT1 → T .


Note 109. The explicit computation in (??) is a special case of the followinggeneral result:

Let X be a reduced, S2 surface and F a rank 1 sheaf on X . Then the OX -algebra

∑

m≥0 F[m] is finitely generated iff F [m] is locally free for some m > 0.

It seems that the minimal model of a typical nc surface has such singularitiesand its canonical ring is not finitely generated.

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Contents - Home | Mathkollar/book/chap3.pdf · Poincar´e residue map at log canonical centers of codimension ≥2. Then we turn to other ways of studying semi log canonical pairs