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 Abstracts Ingrid Bauer Perspectives of algebraic surfaces with pg=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 pg=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 3-folds of general type First we give a classification of 3-folds 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 3-folds of general type satisfying K3> 1⁄1767. In the second part, we present some new classes of 3-folds 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 Sg parametrizes theta-characteristics and is a highly interesting cover of the moduli space of curves. I will discuss a complete birational classification of Sg, obtained recently in joint work with A. Verra. The nature of the even spin space Sg+ changes around genus 8. Using Nikulin K3 surfaces we provide a unirational parametrization of the moduli space for g<8, whereas S8+ turns out to be a mysterious Calabi-Yau 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 pg(S)=0 and c1(S)2=3c2(S)=9. Its universal cover is the unit 2-ball in C2 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 non-abelian group of order 21. We also refer to the wonderful list produced by Cartwright and Steger, using a computer-based but very difficult group-theoretic calculation, of all fundamental groups of fake projective planes. Let (X, G) be a pair of a fake projective plane X and a non-trivial 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 KZ-system 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 Kac-Moody 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 2-dimensional 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 3-dimensional 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 Jun-Muk Hwang (KIAS) about complete classification of general singular fibers of proper holomorphic Lagrangian fibrations over local base in terms of the characteristic 1-cycles, 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 one-forms. A consequence is the invariance of all Hodge numbers for derived equivalent threefolds.   Francesco Russo On irreducible projective varieties Xr+1 ⊂ PN 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 quadro-quadric 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 Shepherd-Barron Exceptional loci in Ag Ag is the moduli space of principally polarized abelian g-folds. When g = 1 or 2 much about the projective geometry of compact models of Ag 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.   Yu-jong 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 Yau-Zaslow 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 Yau-Zaslow 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 L-functions, 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 Abel-Jacobi map, integral Hodge classes, and decomposition of the diagonal Given a smooth projective 3-fold Y, with H3,0(Y)=0, the Abel-Jacobi map induces a morphism from each smooth variety parameterizing 1-cycles in Y to the intermediate Jacobian J(Y). We study in this talk the existence of families of 1-cycles 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 Iliev-Markoushevich. 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.
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