Università di Pisa - unipi.itpardini/files/MSRI.pdf · Surfaces of general type Geography Irregular surfaces The geography of irregular surfaces Rita Pardini Università di Pisa

Surfaces of general typeGeography

Irregular surfaces

The geography of irregular surfaces

Rita Pardini

Università di Pisa

Classical Algebraic Geometry todayM.S.R.I., 1/26 – 1/30 2008

Rita Pardini The geography of irregular surfaces


Irregular surfaces

Summary

1 Surfaces of general type

2 Geography

3 Irregular surfacesIrregular surfaces and irrational pencilsThe slope inequalityThe Severi inequalityThe Castelnuovo–De Franchis inequality



Irregular surfaces

Surface = smooth projective complex surface

Given a surface S, we denote as usual:

OS , the structure sheaf;

KS, the canonical divisor.

The surface S is of general type iff KS is big , i.e. if the linearsystem |mKS | is birational for m >> 0.



Irregular surfaces

Surfaces not of general type are classified (“Enriques’classification”). They are divided in 7 classes.

Such a fine classification is not possible for surfaces of generaltype!



Irregular surfaces

Surfaces of general type:

Definition: A surface S of general type is minimal iff KS is nef .

Every birational equivalence class of surfaces of generaltype contains precisely one minimal surface, called theminimal model.

two minimal surfaces of general type are birational if andonly if they are isomorphic.

So it is enough to study minimal surfaces up to biregularequivalence.



Irregular surfaces

Numerical invariants:pg(S) := h0(KS) = h2(OS), the geometric genus;q(S) := h1(OS) = h0(Ω1

S), the irregularity (recall:q(S) = 1

2b1(S));χ(S) := χ(OS) = 1 − q(S) + pg(S), the holomorphic Eulercharacteristic.K 2

S , the self intersection number of KS .

All these invariants, excepting K 2S , are birational.

However, K 2S is well defined if we take S minimal.

The numerical invariants are determined by the topology of S.Usually K 2

S , χ(S) are chosen as the main numerical invariants.Both invariants are known to be > 0.



Irregular surfaces

Theorem (Gieseker 1977)

For every pair of positive integers (a, b) there exists a coarsemoduli space Ma,b parametrizing minimal surfaces of generaltype with K 2 = a, χ = b. The space Ma,b is quasi projective.



Irregular surfaces

Question:

for which values of a, b is Ma,b nonempty?

namely, what are the pairs (a, b) for which there exists aminimal surface of general type with K 2 = a, χ = b?



Irregular surfaces

Restrictions on K 2, χ:

K 2 > 0, χ > 0;

K 2 ≥ 2χ − 6 (Noether’s inequality);

K 2 ≤ 9χ (Bogomolov-Miyaoka-Yau inequality).



Irregular surfaces

K 2

χ

K 2 = 9χ

K 2 = 2χ − 6q

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Irregular surfaces

Surfaces on the “borders”:

Surfaces with K 2 = χ = 1 (“Godeaux surfaces”): manyexamples are known and there is a partial classification.

Surfaces on the line K 2 = 2χ − 6 are classified (Horikawa,1976). They exist for every value of χ ≥ 4 and are simplyconnected.

Surfaces on the line K 2 = 9χ have the unit ball in C2 as

universal cover (Yau’s proportionality theorem).



Irregular surfaces

Surfaces in the “interior”:

Persson (1981) has "filled" the region 2χ − 6 ≤ K 2 ≤ 8χ, apartfrom a few exceptions. A great part of the remaning area hasbeen then “filled” by Z.J. Chen (1988, 1991).

Sommese (1984) has shown that the possible ratios K 2/χ area dense subset of the interval [2, 9].

So it seems there are no “desertic areas" in the geography ofsurfaces.



Irregular surfaces

Irregular surfaces and irrational pencilsThe slope inequalityThe Severi inequalityThe Castelnuovo–De Franchis inequality

A surface S is called irregular if the irregularityq(S) = h0(Ω1

S) = h1(OS) is > 0.

Remark: most of the examples by Persson and Chen aresimply connected. In general, a lot of effort has been put intoshowing the existence of simply connected surfaces with all thepossible invariants, while irregular surfaces have hardly beenconsidered.



Irregular surfaces


For an irregular surface, one defines an abelian varietyAlb(S) := H0(Ω1

S)∨/H1(S, Z), the Albanese variety, and amorphism a : S → Alb(S), the Albanese map.The Albanese dimension Albdim(S) of S is by definition thedimension of a(S).

An irrational pencil of genus b ≥ 1 of S is a morphism withconnected fibres f : S → B, where B is a smooth curve ofgenus b.

If a surface has an irrational pencil, then of course it is irregular.



Irregular surfaces


Remark: (Catanese) the existence of an irrational pencil ofgenus b ≥ 2 is a topological property.

A surface has Albanese dimension 1 iff it has an irrationalpencil of genus q. So the Albanese dimension is a topologicalproperty, too.



Irregular surfaces


Question:

How do irregular surfaces fit in the geography of surfaces ofgeneral type?

More precisely, what are the restrictions on the invariants for:

irregular surfaces?

surfaces with an irrational pencil?

surfaces with Albanese dimension 2?

irregular surfaces without irrational pencils?



Irregular surfaces


Irregular surfaces:

Surfaces on the Noether line K 2 = 2χ − 6 are simplyconnected.

There are no irregular surfaces with K 2 < 2χ.The irregular surfaces with K 2 = 2χ have q = 1 and the fibresof the Albanese pencil have genus 2 (Horikawa).



Irregular surfaces


Surfaces with an irrational pencil:

Slope inequality (Xiao Gang, Cornalba-Harris):

Let f : S → B be a relatively minimal fibration onto a smoothcurve of genus b, with general fibre of genus g. If f is notsmooth & isotrivial, then:

K 2S − 8(g − 1)(b − 1)

χ(S) − (g − 1)(b − 1)≥

4(g − 1)

g.

Corollary: surfaces with an irrational pencil of curves of genusg satisfy

K 2 ≥4(g − 1)

gχ.



Irregular surfaces


K 2

χ

K 2 =4χK 2 = 9χ

K 2 =2χ − 6,

K 2=2χ

K 2=3χ

K 2= 83 χ

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As g → ∞, the slope lines K 2 = 4(g−1)g χ converge to the line

K 2 = 4χ.



Irregular surfaces


Surfaces with Albdim = 2:

Theorem (Severi inequality):

Let S be a smooth minimal surface of general type. IfAlbdim(S) ≥ 2, then:

K 2S ≥ 4χ(S).



Irregular surfaces


Some history:

The inequality was first claimed by Severi in 1932, but hisproof was wrong and it does not seem possible to fix it;

it was rediscovered at the end of the ’70’s by Catanese,who realized that Severi’s argument was incorrect, and byReid, and stated as a conjecture;

in 1987, Xiao Gang gave a proof for surfaces with anirrational pencil;

Konno in 1993 proved it for surfaces with KS divisible by 2;

at the end of ’90’s, Manetti gave a proof under theassumption that KS be ample;

proved in full generality in 2004 (–).



Irregular surfaces


Remarks & questions:

The Severi inequality is sharp, but all the known exampleswith K 2 = 4χ have q = 2. This leads to the following

Conjecture (Reid, Manetti):

If q ≥ 4 and Albdim = 2, then K 2 ≥ 4χ + 4q − 12.



Irregular surfaces


If the Albanese map is an immersion, except possibly atfinitely many points, then K 2 ≥ 6χ. Can one get aninequality of the form K 2 ≥ cχ, with c > 4, under lessrestrictive assumptions on the Albanese map?

In order to discuss these questions, it is useful to compare thetwo proofs of the inequality, which are completely different.



Irregular surfaces


Sketch of proof (–):A := Alb(S), µd : A → A multiplication by d . Fix H very ampleon A. Note: µ∗

d (H) ∼num d2H.Have a cartesian diagram:

Sdπd−−−−→ S

ad

y

y

a

Aµd−−−−→ A

Set L := a∗H, Ld := a∗

dH.The line bundle Ld is nef & big and |Ld | is base point free.



Irregular surfaces


Compute:χ(Sd) = d2qχ(S), K 2

Sd= d2qK 2

S ,

KSdLd = d2q−2KSL > 0, L2

d = d2q−4L2 > 0.

As d grows, Ld becomes “small” with respect to KS.



Irregular surfaces


Choose a general pencil in |Ld | and blow up Sd to get arelatively minimal fibration

fd : Xd → P1.

with general fibre of genus

gd = d2q−2KL/2 + o(d2q−2).

Note:K 2

Xd= d2qK 2

S + o(d2q), χ(Xd) = d2qχ(S).



Irregular surfaces


Write down the slope inequality for fd :

K 2Xd

+ 8(gd − 1)

χ(Xd) + (gd − 1)≥

4(gd − 1)

gd.

The left hand side is equal to:

d2qK 2S + o(d2q)

d2qχ(S) + o(d2q)

Take the limit for d → ∞:

K 2S

χ(S)≥ 4.



Irregular surfaces


Remarks:

This proof tells us nothing about surfaces with K 2S = 4χ(S);

it cannot be adapted to give a bound of the formK 2 ≥ 4χ + aq + b;

given a refinement of the slope inequality, it might give abound of the form K 2 ≥ cχ, with c > 4, under suitableassumptions on the Albanese map.



Irregular surfaces


Outline of Manetti’s proof: (needs the extra assumption KSample).

Set P := Proj(Ω1S), π : P → S the projection, L := OP(1).

Note: L2(L + π∗KS) = 3(K 2S − 4χ(S)).

Assume for simplicity that there is no divisor on which every1-form of S vanishes. Then the base locus of |L| has dimension≤ 1 and L2 is represented by an effective cycle Γ.If L + π∗KS is nef, this finishes the proof.

Unfortunately this is not the case in general.



Irregular surfaces


Decompose the cycle Γ representing L2:Γ = Γ0 + Γ1 + Γ2, where the cycles Γi are effective and– π(Γ0) is contracted by a,– π(Γ1) is in the ramification locus of a but is not contracted,– Γ2 is not in the base locus of |L|.Show: (L + π∗KS)Γ0 can be < 0, but(L + π∗KS)(Γ0 + Γ1 + Γ2) ≥ 0.This uses the connectedness of canonical divisors on a surfaceand a very fine analysis of the possible components of Γ0. Theassumption KS ample is used here in an essential way.



Irregular surfaces


Remarks:

It seems very difficult to adapt this method to the casewhen S has −2-curves.

Under the assumption KS ample, this proof yields acharacterization of surfaces with K 2

S = 4χ(S): they aredouble covers of abelian surfaces, branched on an ampledivisor.

it can be adapted to give a bound of the formK 2 ≥ 4χ + aq + b.



Irregular surfaces


Theorem 1 (Mendes Lopes, – 2008):

Let S be a smooth surface of maximal Albanese dimension,with KS ample and irregularity q ≥ 5. Then:

K 2S ≥ 4χ(S) +

103

q − 8.

This is obtained by giving a lower bound for the term(L + π∗KS)Γ2 in Manetti’s proof. We do this by a very carefulstudy of the subsystem of |KS | generated by the divisors of theform α ∧ β = 0, where α and β are 1-forms.It is possible to get better bounds under extra assumptions onthe Albanese map.



Irregular surfaces


Theorem 2: (Mendes Lopes, – 2008):

Let S be a smooth surface of maximal Albanese dimension,with KS ample and irregularity q ≥ 5.

1 If the Albanese map a : S → A := Alb(S) is not birationalonto its image, then

K 2S ≥ 4χ(S) + 4q − 13;

2 if S has no irrational pencil and the Albanese mapa : S → A is unramified in codimension 1, then

K 2S ≥ 6χ(S) + 2q − 8.



Irregular surfaces


Remarks:

We can give better inequalities when the canonical map isnot birational.

It is clear from the proofs that our results are not sharp: forfixed q or for q >> 0 one can give better inequalities, but itis very hard to give unified statements.



Irregular surfaces


Surfaces with no irregular pencil of genus ≥ 2:

Theorem (Castelnuovo– de Franchis):

Let S be an irregular surface of general type. If S has noirrational pencil of genus b ≥ 2, then

pg(S) ≥ 2q(S) − 3.

This can be rewritten as:

χ(ωS) ≥ (q − 2).



Irregular surfaces


In this form the inequality has been recently extended to higherdimension:

Theorem (Pareschi–Popa 2008:)

Let X be a compact Kähler manifold withdim X = Albdim X = n. If there exists no surjective morphismX → Z with Z a normal analytic variety such that0 < dim Z = Albdim Z < minn, q(Z ), then:

χ(ωX ) ≥ (q(X ) − n),

where q(X ) := h0(Ω1X ) = h1(OX ).



Irregular surfaces


Question:

What can one say about minimal surfaces with pg = 2q − 3?

If pg = 2q − 3 and S has an irrational pencil of genus ≥ 2, thenthere is a complete classification (Mendes Lopes – 2008,Barja-Naranjo-Pirola 2007): one gets either the product of twocurves of genus 3 or a free Z2-quotient of a product of curves.

So, from now on, assume:S minimal, pg(S) = 2q(S) − 3 and S has no irrational pencil ofgenus ≥ 2.



Irregular surfaces


Properties:Barja-Naranjo-Pirola have shown that if |KS | has no fixedcomponent, then K 2 ≥ 8χ. Moreover we have:

Theorem 3: (Mendes Lopes – 2008):

K 2 ≥ 7χ − 1;

if q ≥ 7 and K 2S < 8χ(S) − 6, then the canonical map is

birational;

the Albanese map is birational onto its image;

if q ≥ 5, the canonical map does not have degree 2.



Irregular surfaces


What about the examples?

If q = 3, then S is the symmetric product of a curve ofgenus 3 (Hacon –, Pirola-Catanese-Ciliberto-MendesLopes);

There is no such surface with q = 5 (Pirola)

If q = 4, then K 2 = 16 or 17 (Barja-Naranjo-Pirola,Causin-Pirola).

The difficulty in producing the examples is that the standardconstructions either give surfaces with an irrational pencil orsurfaces with K 2 < 8χ but |KS | free, and we have seen that thiscannot happen for pg = 2q − 3.



Irregular surfaces


Conjectural example:

Let S be a minimal surface with q = 4, pg = 5 = 2q − 3 andwithout irrational pencils.If the canonical map ϕ of S has degree 2, then K 2

S = 16, thecanonical image Σ ⊂ P

4 is the complete intersection of aquadric and a quartic. The map ϕ is a morphism, branchedprecisely on 40 nodes, which are the only singularities of Σ.

Conversely, given such a Σ with an even set of 40 nodes, thedouble cover of Σ branched over the nodes is a surface S asabove.

Note: 40 is the maximum possible number of nodes for acomplete intersection of a quadric and a quartic in P

4. The codeV corresponding to the even sets of nodes of Σ has length 40,dimension 8 and weights 16, 20, 24, 40. Such a code exists.



Irregular surfaces


Question 1: do minimal surfaces with pg = 2q − 3 satisfy theinequality K 2 ≥ 8χ?

Question 2: do surfaces with pg = 2q − 3 and no irrationalpencil have bounded invariants?


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Università di Pisa - unipi.itpardini/files/MSRI.pdf · Surfaces of general type Geography Irregular surfaces The geography of irregular surfaces Rita Pardini Università di Pisa