Abstracts
Ingrid Bauer
Perspectives of algebraic surfaces with p_{g}=0
In the first part of the talk, I will discuss recent developments, methods and
open problems in the systematic construction and classification of surfaces
of general type with geometric genus zero.
In the second part I will concentrate on Burniat surfaces (= surfaces with
p_{g}=0 constructed by P. Burniat in 1966 as singular bidouble covers of
the projective plane). I will explain the results on their moduli spaces
which are recently obtained in several joint papers with F. Catanese, and in
particular a new pathology concerning deformation spaces of surfaces of
general type with non ample canonical bundle.
Christian Böhning
Affine groups and levels of stable rationality
We provide an overview of recent results on the rationality and stable rationality of group quotients.
The main series of examples considered will be quotients of linear spaces by linear group actions. Obstructions to rationality and stable rationality will also be discussed.
Fabrizio Catanese
Special Galois coverings and the irreducibility of certain spaces
of coverings of curves,
with applications
Special Galois coverings are e.g. cyclic or dihedral coverings, for
which I will describe old and new results,
and new examples, obtained together with Fabio Perroni.
In the case of curves I will show some irreducibility results for
coverings of a fixed topological type:
in the dihedral case this is work in progress with Michael Loenne and
Fabio Perroni.
One application will be the description of an irredundant irreducible
decomposition for
the Singular locus of the compactified Moduli space of curves $\overline{\mathfrak M_g}$,
extending the result of Cornalba for the open set $\mathfrak M_g$.
Meng Chen
Geography on 3folds of general type
First we give a classification of 3folds of general type with small volume up to the level of Reid's baskets of orbifold points. The direct result is that the canonical volume of any minimal 3folds of general type satisfying K^{3}> ^{1}⁄_{1767}. In the second part, we present some new classes of 3folds that are canonically fibred by surfaces or curves with big invariants.
Ciro Ciliberto
On Severi varieties of nodal curves on K3 surfaces
In this talk I consider some irreducibility problems
concerning Severi varieties of nodal curves on general
polarised K3 surfaces (work in collaboration with Th.
Dedieu).
Olivier Debarre
On nodal prime Fano threefolds of degree 10
I will explain joint results with A. Iliev and L. Manivel about the geometry and the period map of nodal complex prime Fano threefolds with index 1 and degree 10. These threefolds are birationally isomorphic to Verra solids, i.e., hypersurfaces of bidegree (2,2) in the product of two projective planes. Using Verra's results on the period map for these solids, we prove that the fiber of the period map for our nodal threefolds is the union of two surfaces, which have various geometric birational descriptions.
Gavril Farkas
Explicit geometry of moduli spaces of even spin curves
The spin moduli space of curves S_{g} parametrizes
thetacharacteristics and is a highly interesting cover of the moduli
space of curves. I will discuss a complete birational classification of
S_{g}, obtained recently in joint work with A. Verra. The nature of the even
spin space S_{g}^{+} changes around genus 8. Using Nikulin K3 surfaces we
provide a unirational parametrization of the moduli space for g<8, whereas
S_{8}^{+} turns out to be a mysterious CalabiYau variety.
JongHae Keum
Toward a geometric construction of fake projective planes
Let S be a fake projective plane, i.e., a surface of
general type with p_{g}(S)=0 and c_{1}(S)^{2}=3c_{2}(S)=9.
Its universal cover is the unit 2ball in C^{2}
and hence its fundamental group is infinite.
Recently, Prasad and Yeung classified all possible
fundamental groups of fake projective planes. According to their
beautiful result, many fake projective planes admit a nontrivial
group of automorphisms, and in that case the group is isomorphic
to Z/3Z, (Z/3Z)^{2},
Z/7Z, or 7:3, where
7:3 is the unique nonabelian group of order 21. We also refer
to the wonderful list produced by Cartwright and Steger,
using a computerbased but very difficult grouptheoretic
calculation, of all fundamental groups of fake projective planes.
Let (X, G) be a pair of a fake projective plane X and a
nontrivial group G of automorphisms. We
classified and described all possible structures of the quotient surface X/G
and its minimal resolution.
In this talk, we will show that the description gives a criterion for a projective surface to become a quotient
of a fake projective plane.
Shigeyuki Kondo
The supersingular K3 surface with Artin invariant 1 in characteristic 2 and the Leech lattice
I shall construct, on the supersingular K3 surface with Artin invariant 1 in characteristic 2, a set of 21 disjoint smooth rational curves and another set of 21 disjoint smooth rational curves such that each curve in one set intersects exactly 5 curves from the other set.
This configuration of curves is similar to the (16)_{6} configuration of 32 smooth rational curves on Jacobian Kummer surfaces.
I also discuss a relation with the Leech lattice. This is a joint work with Igor Dolgachev and Toshiyuki Katsura.
Alexander Kuznetsov
Hochschild homology and cohomology of admissible subcategories
I will explain how one can compute the Hochschild homology and
cohomology of admissible subcategories (i.e. components of semiorthogonal
decompositions) of bounded derived categories of coherent sheaves on
smooth projective varieties. Some interesting examples will be discussed.
Eduard Looijenga
Representations out of polydifferentials and the KZsystem
We show how the KZ system and the WZW systems in genus zero admit a
topological interpretation.
A major tool is a way of realizing the standard highest weight
representations of KacMoody algebra
inside an algebra of polydifferentials.
Massimiliano Mella
The Automorphism group of the moduli space of pointed rational curves
The compactified moduli space of pointed rational curves has natural automorphisms coming from the permutation of the markings. Fulton conjectured that these are the only automorphisms, as soon as there are at least 5 markings. In this talk I will prove this conjecture using Kapranov's description of the moduli space as iterated blow up of the projective space.
Shigeru Mukai
Enriques surfaces of type E7
Horikawa constucted a 2dimensional family of Enriques surfaces in his study
of period map. These Enriques surfaces are characterized by the presence of
the root lattice of type E8 in their twisted Picard lattices. Enlarging this family
we construct a 3dimensional one characterized by E7. An Enriques surface
of type E7 is reconstructed from its period using Richelot isogeny, i.e., Hecke
correspondence between curves of genus 2.
Keiji Oguiso
Classifications and constructions of general singular fibers of proper holomorphic Lagrangian fibrations
I would like to talk on my joint work with Professor JunMuk Hwang (KIAS) about complete classification of general singular fibers of proper holomorphic Lagrangian fibrations over local base in terms of the characteristic 1cycles, together with several concrete examples.
Rita Pardini
Curves on irregular surfaces
I will report on recent joint work with M. Mendes Lopes and G.P. Pirola.
I will describe a new characterization of the symmetric product of a curve of genus q>2.
Then I will describe some work in progress, explaining how some of the ideas of the proof of this result can be generalized and used to obtain restrictions on the existence of curves of low genus on irregular surfaces of general type not fibered onto curves of genus >1.
Gianpietro Pirola
Lagrangian surfaces
Using Galois closure we construct algebraic varieties with non abelian fundamental group. In particular we produce a new lagrangian surface in its albanese variety which is an abelian fourfold. We discuss its topological signature and other geometric invariants, some related conjectures and new results on the classification of irregular surfaces.
It is a joint work with Francesco Bastianelli and Lidia Stoppino.
Mihnea Popa
Derived equivalence and the Picard variety
I will explain joint work with C. Schnell in which we show that
derived equivalent varieties have isogenous Picard varieties. This implies
the derived invariance of the number of independent holomorphic oneforms.
A consequence is the invariance of all Hodge numbers for derived
equivalent threefolds.
Francesco Russo
On irreducible projective varieties X^{r+1}
⊂ P^{N} such that
through n ≥ 2 general points there passes an irreducible curve of
degree δ
For varieties as in the title L. Pirio and J. M. Trépreau
proved a sharp bound N ≤ π(r,n,δ)1 in terms of a suitable Castelnuovo bound
function π(r,n,δ) . The extremal cases for this
bound are particularly relevant, share deep geometrical properties
and for n=δ=3 lead to
interesting connections with quadroquadric Cremona
transformations and with cubic complex Jordan algebras.
I shall present the main results of joint work with L. Pirio on the
geometry of extremal varieties in the above sense, both from a
projective and an abstract point of view, focusing especially on the
case n=δ=3
and on its various applications.
Matthias Schütt
Arithmetic of singular Enriques surfaces
We will discuss joint work with Klaus Hulek on the arithmetic of singular Enriques surfaces,
i.e. Enriques surfaces covered by K3 surfaces with Picard number 20.
Nick ShepherdBarron
Exceptional loci in A_{g}
A_{g} is the moduli space of principally polarized abelian
gfolds. When g = 1 or 2 much about the projective geometry of compact
models of A_{g} has been known for more than a century; this talk will
describe extensions of these results, in terms of extremal rays and
associated contractions, for all values of g.
Yujong Tzeng
Universal Formulas for Counting Nodal Curves on Surfaces
The problem of counting nodal curves on algebraic surfaces has
been studied since the nineteenth century. On the projective surface, it
asks how many curves defined by homogeneous degree d polynomials have only
nodes as singularities and pass through points in general position. On K3
surfaces, the number of rational nodal curves was predicted by the
YauZaslow formula. Goettsche conjectured that for sufficiently ample line
bundles L on algebraic surfaces, the numbers of nodal curves in L are
given by universal polynomials in four topological numbers. Furthermore,
based on the YauZaslow formula he gave a conjectural generating function
in terms of quasi modular forms and two unknown series. In this talk, I
will discuss how degeneration methods can be applied to count nodal curves
and sketch my proof of Goettsche's conjecture.
Bert van Geemen
Explicit moduli spaces of abelian varieties with automorphisms
We will discuss some recent results, obtained in collaboration with
M. Schütt, on two projective models of Shimura varieties. One is a compact
Shimura curve and the other is (somewhat surprisingly) a Hilbert modular
surface. We will explain some of the theta function techniques we used to find
these varieties as well as their Lfunctions, which we determined using the
explicit models.
Alessandro Verra
On the geometry of some moduli spaces related to curves in low genus
The aim of the talk is to review some new or recent results on the moduli of pairs (C,L), where C is a curve of genus g and L is a
line bundle on C with prescribed properties. The talk is mainly concerned with the (uni)rationality problem for very low values of g.
The cases to be considered are when L is a root of the trivial or of the canonical sheaf on C as well as the case of the universal
Picard variety of degree zero line bundles.
Claire Voisin
AbelJacobi map, integral Hodge classes, and decomposition of the diagonal
Given a smooth projective 3fold Y, with H^{3,0}(Y)=0, the AbelJacobi map induces a morphism from each smooth variety parameterizing 1cycles in Y to the intermediate Jacobian J(Y).
We study in this talk the existence of families of 1cycles in Y for which this induced morphism is surjective with rationally connected general fiber, and various applications of this property.
When Y itself is rationally connected, we relate this property to the existence of an integral homological decomposition of the diagonal.
We also study this property for cubic threefolds, completing the work of IlievMarkoushevich. We then conclude that the Hodge conjecture holds for degree 4 integral Hodge classes on fibrations into cubic threefolds over curves, with restriction on singular fibers.
