Abstracts
Abstract: Associated to a vector bundle over a curve $B$ we can define its positive degree as a natural associated invariant that can be extended to a continuous, convex function. This provides a link to relate, in a simple and compact way, several geographical inequalities on varieties fibred over $B$ such as Clifford-Severi, Slope or Castelnuovo inequalities.
Beauville: Quotients of Jacobians and the Ceresa cycle
Abstract: Let C be a curve of genus g, and G a finite group of automorphisms of C . I will prove that for g > 20 the quotient JC/G has canonical singularities, hence Kodaira dimension 0. On the other hand I'll give examples with g < 5 for which JC/G is uniruled. For g = 3 , I'll show how this implies that the Ceresa cycle [C] -- [(-1)*C] in JC is torsion modulo algebraic equivalence.
Catanese: Manifolds with vanishing Chern classes and some questions by Severi/baldassarri
Abstract: We give a negative answer to a question posed by Severi in 1951, whether the Abelian Varieties are the only manifolds with vanishing Chern classes. We exhibit Hyperelliptic Manifolds which are not Abelian varieties (nor complex tori) and whose Chern classes are zero not only in integral homology, but also in the Chow ring. We prove moreover the surprising result that Bagnera de Franchis manifolds ( quotients T/G whereT is a torus and G is cyclic) have topologically trivial tangent bundle. Motivated by a more general question addressed by Mario Baldassarri in 1956, we discuss the Hyperelliptic Manifolds, the Pseudo- Abelian Varieties introduced by Roth, and we introduce a new notion, of Manifolds Isogenous to a k-Torus Product: the latter have the last k Chern classes trivial in rational homology and vanishing Chern numbers. We show that the latter class is the correct substitute for some incorrect assertions by Enriques, Dantoni, Roth and Baldassarri: in dimension 2 these are the surfaces with KX nef and c2(X) = 0. A similar picture does not hold in higher dimension, unless we consider manifolds (isogenous to manifolds) whose tangent (resp. cotangent bundle) has a trivial summand. We survey old and new results on Kaehler manifolds whose tangent (resp. cotangent bundle) has a trivdrive.google.com/file/d/1p0UcosG_sRQ1fUtQo2dZ0TVUNaKkJEZr/view?usp=sharing ial summand, and pose some open problems.
Abstract: In this talk I will show that on a general blow--up of the projective plane there are slope stable Ulrich bundles of any rank and I will compute the number of moduli of such bundles. This is joint work with F. Flamini and A. L. Knutsen.
Debarre: Complete curves in the moduli space of polarized K3 surfaces and hyper-Kähler manifolds.
Abstract:Building on an idea of Borcherds, Katzarkov, Pantev, and Shepherd-Barron, we prove that the moduli space of polarized K3 surfaces of degree 2e contains complete curves for all e >= 62 and for some sporadic lower values of e (starting at 14). We also construct complete curves in the moduli spaces of polarized hyper-Kähler manifolds of K3^[n]-type or Kum_n-type for all n>=1 and polarizations of various degrees and divisibilities. This is joint work with Emanuele Macrì.
Katzarkov: New Birational Invariants.
Abstract: In this talk we will introduce new birational invariants defined by combining the A and B sides of HMS. Many examples will be considered.
Kawamata: Non-commutative deformations of coherent sheaves.
Abstract: We consider deformations of a coherent sheaf on an algebraic variety. The deformations are usually considered over commutative rings, and the parameter ring of the universal deformation is the local ring of the moduli space of the sheaf. When we allow non-commutativity of the parameter ring, then we obtain similar theory but with more deformations. For example, if we consider the struture sheaf of a line in a projective space, the moduli space of the usual deformations is the Grassmann variety. But there are more non-commutative deformations than the commutative deformations which are obstructed. We will explain how to describe the universal non-commutative deformations.
Abstract: The Cox ring of a variety is the total coordinate ring, i.e., the direct sum of all spaces of global sections of all divisors. When this ring is finitely generated, the variety is called Mori dream (MD). A necessary condition for being MD is the finite generatedness of Pic(X), i.e., the vanishing of the irregularity. Smooth rational surfaces with big anticanonical divisor are MD. Thus all del Pezzo surfaces of any degree are. A K3 surface or an Enriques surface with Picard number at least 3 is MD iff its automorphism group is finite. In this talk I will consider the case of surfaces of general type with p_g=0, and provide several examples that are MD. I will also provide non-minimal examples that are not MD. This is a joint work with Kyoung-Seog Lee.
Abstract: In this talk, we show that there is a natural Lagrangian fibration structure on the map $f$ from the cotangent bundle of a del Pezzo surface $X$ of degree 4 to ${\mathbb C}^2$. Moreover, we describe explicitly all level surfaces of the above natural map $f$. This work is motivated by the work of Hitchin on the moduli spaces $M$ of stable vector bundles over a smooth projective curve from the viewpoint of symplectic geometry of its cotangent bundle, and by the work of Hwang and Ramanan on the Hitchin system and the Hitchin discriminant associated to the Hitchin map on the cotangent bundle of M. This is joint work with Hosung Kim.
Lönne: Fundamental groups of spaces of trigonal curves
Abstract: Some results on trigonal curves and linear systems on Segre-Hirzebruch surfaces are exploited to give presentations
of fundamental groups of various spaces associated to trigonal curves. In particular we address the space PH_4(6), the projectivisation of the space H_4(6) of abelian differentials on non-hyperelliptic genus 4 curves with a single zero of multiplicity 6 providing an even spin structure.
Oguiso: Real forms of a smooth projective surface and cone conjecture
Abstract: Real form problem asks how many different ways one can describe a given complex projective variety by a system of equations with real coefficients, up to isomorphisms over the real number field. It is quite recent that examples with infinitely many real forms are found (Lesieutre, Dinh-Oguiso, Dinh-Oguiso-Yu). In this talk, after explaining basic notions with some concrete examples, we first recall and clarify two important finiteness results, one in terms of Galois cohomology set and one in connection with the cone conjecture. Then, we systematically study the finiteness of real froms for smooth projective surfaces. In particular, we show that a smooth projective surface admits only finitely many real forms unless the surface is either rational or non-minimal and birational to either K3 or Enriques and that there are indeed a rational surface, a one point blow up of some K3 surface and a one point blow up of some Enriques surface, admitting infinitely many real forms. This answers questions by Kharlamov (on rational case), and also by Mukai (on K3 case) and Kondo (on Enrqiues case) to us. This is a joint work in progress with Tien-Cuong Dinh, C\'ecile Gachet, Hsueh-Yung Lin, Long Wang and Xun Yu
Pardini: Smoothing semi-smooth Godeaux surfaces
Abstract: A complex quasi-projective variety is semi-smooth if locally in analytic coordinates it is either smooth, or isomorphic to {uv=0}, or isomorphic to {u^2-v^2w=0}. We compute explicitly the sheaves T_X and T^1_X for a semi-smooth variety and apply these computations to show the smoothability of the semi-smooth Godeaux surfaces classified in a paper by Franciosi-Pardini-Rollenske. This is joint work with Barbara Fantechi and Marco Franciosi.
Abstract: I will report on a work in progress aimed at determining the Betti numbers of the moduli space of admissible G-covers of pointed genus 0 curves, where G is a finite group. We show that the class, in the Grothendieck group, of the stack of admissible G-covers is determined by the class of the open sub-stack corresponding to coverings of the projective line, this reduces our problem to determine the Betti numbers of moduli spaces of G-coverings of P^1. This is a joint work with Massimo Bagnarol.
Abstract: We discuss holomorphic forms on the moduli space M_g of smooth projective complex curves of genus g and on its unramified coverings. Under some hypothesis on the monodromy, we prove the vanishing of holomorphic 1-forms on the preimage of the smooth locus of M_g. This applies to several moduli spaces, as the moduli space of curves with level structures, of spin curves and of Prym curves. It is a joint work with F.Favale ,J.C. Naranjo and S. Torelli.
Rollenske: Stable surfaces with K^2 = 1 and p_g = 2
Abstract: Surfaces of general type with particular invariants and their moduli spaces have been studied for over a century and the case of K^2 = 1 and p_g = 2 is one of the simplest. Nowadays the Gieseker moduli space of surfaces of general type is known to admit a natural compactification in the moduli space of stable surfaces. I will illustrate how even in this special case the stable compactification hides some suprises, so far only partially explored in joint work with Marco Franciosi and Rita Pardini and Julie Rana and Stephen Coughlan.
Abstract: The talk deals with the multiple and historical relations between Enriques Surfaces and Rationality Problems. Artin-Mumford's counterexample X to Lueroth problem is revisited: the role of Enriques surfaces, the family of Reye congruences, is emphasized and the 2-torsion cohomology of X is geometrically reconstructed from that of these surfaces. The same construction extends to higher dimensions.