Stable Gorenstein surfaces with K2=1

Stable surfaces and compact moduliStable surfaces from log-canonical pairs

Stable pairs with K + ∆ Cartier and (K + ∆)2 = 1Stable Gorenstein surfaces with K 2 = 1

Stable Gorenstein surfaces with K 2 = 1

Rita Pardini(joint work with M. Franciosi and S. Rollenske)

“Moduli and Automorphic Forms”Berlin

May 22-24, 2014

Rita Pardini (joint work with M. Franciosi and S. Rollenske) Stable Gorenstein surfaces with K 2 = 1



Outline of the talk

1 Stable surfaces and compact moduli

2 Stable surfaces from log-canonical pairs

3 Stable pairs with K + ∆ Cartier and (K + ∆)2 = 1

4 Stable Gorenstein surfaces with K 2 = 1




I work over C, varieties are projective.Prototypical situation:

take g > 1 an integer: there exists a quasi-projective modulispaceMg parametrizing isomorphism classes of curves ofgenus g.

to get a geometrically significant compactificationMg ⊂Mg

one introduces stable curves :

a stable curve of genus g is a connected nodal curve X suchthat ωX is ample and h1(OX ) = g.

Generalizations: higher dimension, relative case (“stablepairs”).




I’m going to focus on surfaces. Here’s how the analogy works:

smooth curve of genus g > 1 =⇒ (canonical model of)

surface of general type , i.e., surface X with mild singularities(RDP’s) and ωX = OX (KX ) an ample line bundle

genus =⇒ numerical invariants : K 2X , χ(OX ), pg(X ) = h2(OX )

(= h0(ωX )) geometric genus, q(X ) = h1(OX ) (= h0(Ω1X ))

irregularity). Observe: χ = pg − q + 1

If we fix integers a,b, there exists a quasi-projective modulispaceMa,b parametrizing isomorphism classes of canonicalmodels of surfaces of general type with K 2 = a, χ = b.

To get a compact moduli space, one has to considerstable surfaces .




What is a stable surface X?

in codimension 1, X is either smooth or analyticallyisomorphic to xy = 0 (“double crossings”);X may be non-normal, but it S2 (a function that is regularoutside finitely many points is regular). The dualizing sheafωX = OX (KX ) is reflexive (KX is a Weil divisor).some “multiple” of the dualizing sheaf ωX is a line bundle(eq., some multiple of KX is Cartier), and it is ampleX has semi-log-canonical (“slc”) singularities.




Theorem (Kollár-Shepherd-Barron, Kollár, Alexeev, 1988–1993)

There exists a projective moduli spaceMa,b parametrizingstable surfaces with K 2

X = a, χ(X ) = b.

Taking the closure ofMa,b ⊆Ma,b one obtains a geometricallymeaningful compactification ofMa,b.

Warning: the situation is more complicated than in the case ofcurves.

1 Ma,b may be reducible and disconnected2 in generalMa,b ⊆Ma,b is not dense. For instance, it may

happen thatMa,b = ∅, whileMa,b 6= ∅.




A stable surface is Gorenstein if ωX is a line bundle.

Singularities of stable Gorenstein surfaces:Rational double points (e.g., z2 − x2 + y4 = 0)




Elliptic Gorenstein singularity (e.g., zy2 + (x − z)(x + z)x = 0)




Normal crossings xy = 0




Pinch point z2 − yx2 = 0




Degenerate cusp (e.g., xyz + x3 + y3 = 0 )




X a (non-normal) stable surface, D the double locus . Takenormalization π : X → X and set ∆ = π−1(D). One has:

π∗KX = KX + ∆.

hence K 2X = (KX + ∆)2.

(X ,∆) is a stable pair , i.e.:1 KX + ∆ is Q-Cartier and ample2 (X ,∆) has log-canonical singularities *

The 2-to-1 map ∆→ D induces an involution τ of thenormalization ∆ν of ∆ such that Diff∆ν (0) is τ -invariant.




Definition

Let η : Y → X be a resolution such that η−1∗ (∆) +

∑Ei has

normal crossing support, where Ei are the irreducibleexceptional curves.Then (X ,∆) is a log-canonical pair iff in the pull back formula

KY + η−1∗ (∆) = η∗(KX + ∆) +

∑aiEi

one has ai ≥ −1 for every i .




Theorem (Kollár)

Associating to a stable surface X the triple (X ,∆, τ : ∆ν → ∆ν)induces a one-to-one correspondence

stablesur-

faces

↔

(X ,∆, τ)

∣∣∣∣∣∣∣(X ,∆) stable pair,τ : ∆ν → ∆ν involu-tion s.th. Diff∆ν (0) is τ -invariant.

.

AddendumIn the above correspondence X is Gorenstein if and only ifKX + ∆ is Cartier and τ induces a fixed-point free involution onthe preimages of the nodes of ∆.




An example with K 2 = 1

X = P2, ∆ = L1 + L2 + L3 + L4, Li lines in general position.Pij ∈ Li = preimage of Li ∩ Lj

An involution τ of ∆ν acting freely on the Pij is the same as apair of isomorphisms: φ12 : L1 → L2, φ34 : L3 → L4.An example:

φ12 =

(P12 P13 P14P23 P21 P24

), φ34 =

(P31 P32 P34P41 P43 P42

).

Remark: in this example the points Pij are all mapped to thesame point in X .




P2

L1 L2

L3

L4

π normalisation

P2

L1 L2

L3

L4

blow up

nodes

glue L1 ∼ L2

L3 ∼ L4L12

L34

E

Y0.1

contract E

X0.1Rita Pardini (joint work with M. Franciosi and S. Rollenske) Stable Gorenstein surfaces with K 2 = 1






Theorem (A)

Let (X ,∆) be a stable pair with ∆ > 0 such that KX + ∆ isCartier and (KX + ∆)2 = 1. Then (X ,∆) is one of the following:(P) X = P2 and ∆ is a quartic.

(dP) X is a (possibly singular) Del Pezzo surface of degree 1and ∆ ∈ | − 2KX |.

(E−) X is obtained from a P1-bundle over an elliptic curve E bycontracting to a point P a section C0 with C2

0 = −1 and ∆is a bisection not containing P.

(E+) X = S2E, where E is an elliptic curve, and ∆ is a trisectionof the Albanese map X → E.

REMARKS: (1) There is a also a rough classification, according toKodaira dimension, for the case ∆ = 0.(2) Both theorems are proven by “classical” methods: study of linearsystems and their adjoints.




From now on X is a stable Gorenstein surface with K 2X = 1 .

Why assume Gorenstein?

can study the pluricanonical maps as in the classical caseX is irreducible and, if X is non-normal, by Theorem (A) wehave four families, according to the type of thenormalization (X ,∆).

In order to obtain a complete classification, for each type wehave to determine the involutions τ of ∆ν that act freely on thepreimages of the nodes of ∆.




We start by looking at the numerical invariants:

Proposition (B)There exists a non-normal Gorenstein stable surface withnormalization of given type exactly in the following cases:

normalization χ(X ) = 0 χ(X ) = 1 χ(X ) = 2 χ(X ) = 3

(P) X X X X(dP) X X X(E−) X X(E+) X X

REMARK: This improves on results by Liu and Rollenske: ingeneral one has χ ≥ −K 2.




Theorem (C)If X is stable Gorenstein, then the possible invariants of X are:

1 pg = 2, q = 02 pg = 1, q = 03 pg = 0, q = 0 (“stable Godeaux surfaces”)4 pg = 0, q = 1 (these are the surfaces X0.1 and X0.2 of the

Table )

Remarks: (1) all possibilities can be realized starting from(P2,4 lines).(2) main steps in the proof:

proving q = 0 if χ > 0 (classical method, paracanonicalsystems).proving that if χ = 0 then the normalization is (P2,4 lines)+ computation of q for the 2 surfaces




We are mainly interested in the case X smoothable ; this is afirst step in the understanding of the closure of the modulispace of surfaces of general type with K 2 = 1.

Recall: if X has canonical singularities, the possible invariantsare

1 pg = 2, q = 0 (described by Horikawa, 1976; moduli spaceis irreducible of dimension 28)

2 pg = 1, q = 0 (described by Catanese in 1980; modulispace is irreducible of dimension 18)

3 pg = 0, q = 0 (“numerical Godeaux surfaces”; modulispace has several irreducible components, only some havebeen described)




Theorem (D)

Let X be a stable Gorenstein surface with K 2 = 1 and pg = 1 or2.Then X is smoothable .

Idea: since X is Gorenstein, one can study the pluricanonicallinear systems |mKX | and the pluricanonical rings, adapting theclassical arguments to the stable case.




Theorem (D)

Let X be a stable Gorenstein surface with K 2 = 1 and pg = 2.Then:

1 the bicanonical map is a finite degree 2 mapφ2 : X → Q := x0x1 = x2

2 ⊂ P3

2 φ2 is branched on the vertex V = [0,0,0,1] of Q and on aquintic section B of Q such that (Q, 1

2B) is a log canonicalpair and V /∈ B.

3 Conversely, given an lc pair (Q, 12B) as in (2), the double

cover X → Q branched on B and on the vertex is a stableGorenstein surface with K 2 = 1, pg = 2.




Recall that a non-normal Gorenstein surface with K 2 = 1 andpg = 2 has type (P), (dP) or (E−). These are obtained asfollows:

(dP) B = 2H + C3, where H is a plane section and C3 is a cubicsection with at most negligible singularities.

(E−) B = 2H + C3, as in case (dP), but C3 has a (3,3)-point.(P) B = 2C2 + H, where H is a plane section and C2 is a

quadric section.

So:(dP) specializes to (E−)

by letting C3 acquire a double component in (E−) or in(dP) one obtains a surface of type (P).




Some non-Gorenstein degenerations:

1 Letting B pass through V we obtain a stable surfaces Ysuch that 2KY is Cartier but KY is not. If B = C4 + 2F ,where C4 is a quartic section and F is a ruling of Q, thenthe normalization Y is a K 3 surface with two nodes.

2 Degenerating Q to a union of planes one obtains areducible surface, consisting of two nodal K 3 surfacesglued along a rational curve.

Degeneration (2) occurs also by letting B acquire a quintuplepoint and applying semi-stable reduction.




To-do list/open problems:

understand the non-Gorenstein degenerations, at least forpg = 2. (E.g., is the Cartier index at most 2?)understand which non-normal Gorenstein surfaces withK 2 = 1 specialize to whichdecide which Gorenstein Godeaux surfaces aresmoothable (have a classification and some partial results). . .


Documents

Stable Gorenstein surfaces with K2=1