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Michela Artebani (U. Concepción): Families of Calabi-Yau hypersurfaces of toric varieties 

Abstract: Let X be a Q-Fano toric variety with canonical singularities and let P be its anticanonical polytope. In this talk we provide a sufficient 
condition on a polytope P’ contained in P such that an anticanonical 
hypersurface of X with Newton polytope P’ is a Calabi-Yau variety.
Moreover, we show that there is a natural involution in the set of 
such pairs (P’,P) which gives rise to a duality between families of Calabi-Yau varieties. This construction generalizes Batyrev mirror construction (the case when P=P’) and Berglund-Hübsch-Krawitz construction (the case when both polytopes are simplices). This is joint work with Paola Comparin and Robin Guilbot. 

Francesco Bastianelli (U. Bari): Large theta-characteristics and applications

Abstract: In the moduli space M_{g,n} of n-pointed curves of genus g≥3, I will discuss certain loci described by curves (C,p_1,...,p_n) admitting a theta-characteristic with large number r+1 of independent global sections, and possessing a global section vanishing with prescribed multiplicities at the marked points. I will establish a general upper bound governing the codimension of these loci in M_{g,n}, and when g is large enough with respect to r, I will present irreducible components of these loci attaining the maximal codimension, hence showing the sharpness of the bound. Finally, I will describe some applications to periodic surfaces, i.e.  compact connected Riemann surfaces of genus g admitting a conformal minimal immersion into a flat-3 dimensional real torus, concerning the locus of periodic surfaces in the moduli space M_g and the existence into any flat real 3-torus of periodic surfaces of genus g. The talk reports on joint work with pirola, and with Ballico and Benzo.

Lucia Caporaso (U. Roma tre): TBA

Sebastian Casalaina-Martin (U. Colorado): Algebraic representatives and intermediate Jacobians over perfect fields 

Abstract: Intermediate Jacobians and Abel--Jacobi maps provide a powerful tool for the study of complex projective manifolds. In positive characteristic, over algebraically closed fields, algebraic representatives and regular homomorphisms provide a replacement for the intermediate Jacobian and Abel--Jacobi map. I will discuss recent progress, with Jeff Achter and Charles Vial, extending this theory to the case of perfect fields, as well as some applications to a question of Barry Mazur on weight one Galois representations arising from geometry, and to integral decompositions of the diagonal.

Fabrizio Catanese (U, Bayreuth): TBA

Ciro Ciliberto (U. Roma 2, Tor Vergata): Curves on general hypersurfaces in projective space

Abstract: This talk concerns the existence of curves with low gonality on smooth hypersurfaces $X\subset \mathbb{P}^{n+1}$. After reviewing a series of results on this topic, I will report on a recent progress on the subject in collaboration with F. Bastianelli, F. Flamini and P. Supino. In particular, we obtained that if $X\subset \mathbb{P}^{n+1}$ is a very general hypersurface of degree $d\geqslant 2n+2$, the least gonality of a curve $C\subset X$ passing through a general point of $X$ is $\gon(C)=d-\left\lfloor\frac{\sqrt{16n+1}-1}{2}\right\rfloor$, apart from some exceptions we list in which the gonality could be one less. 

Gavril Farkas (Humboldt University, Berlin): Fundamental groups,  
Alexander invariants and syzygies of canonical curves

Abstract: I will discuss an algebraic statement concerning the vanishing of the Koszul modules associated to any subspace K inside the second exterior product of a complex vector space. This statement, which turns out to be equivalent to Mark Green's Conjecture on syzygies of canonical curves (proven by Claire Voisin), has many interesting topological applications of which I will discuss (1) a universal upper bound on the nilpotence index of the fundamental group of any compact Kaehler manifold and (2) a bound on the length of the nilpotence index on the Torelli groups associated to the moduli space of curves and (3) an explicit description of the Cayely-Chow form of the Grassmannian G(2,n). 
This is joint work with Aprodu, Papadima, Raicu and Weyman.

Gerard van der Geer (U. Amsterdam): Algebraic curves and modular forms of low degree

Abstract: For genus 2 and 3 modular forms are intimately connected with the moduli of curves of genus 2 and 3. We give an explicit way to describe such modular forms for genus 2 and 3 using invariant theory and give some applications.
This is based on joint work with Fabien Clery and Carel Faber.

Alessandro Ghigi (U. Pavia): Some differential-geometric aspects of the Torelli map

Abstract: In the first part of the talk I will briefly describe some results obtained by various people in the last years about the second fundamental form of the Torelli map and the relation between totally geodesic submanifolds of A_g and the Jacobian locus. Next I will discuss a recent result relating divisors in the Jacobian locus to totally geodesic submanifolds.  Finally I will mention a connection between Fujita decomposition and Hodge loci.

Víctor González Alonso (U. Hannover): On the unitary rank of fibred surfaces

Abstract: The unitary rank of a fibred surface is defined as the rank of the flat unitary summand in the so-called second Fujita of the Hodge bundle of the fibration (the direct image of the dualizing sheaf). Understanding this flat summand, and in particular its rank, amounts to understand the locally constant part of the Variation of Hodge Structures associated to the fibration. In this talk I will present an upper bound on the unitary rank, depending on the genus and the Clifford index of a general fibre. Actually, it is the same bound we obtained in a previous work with Barja and Naranjo for the relative irregularity, although the unitary rank can very well be much bigger than the relative irregularity. This is joint work with Lidia Stoppino and Sara Torelli.

Zhi Jiang (Fudan U.): The decomposition formula in generic vanishing and its geometric applications

Abstract: We will explain a decomposition formula in generic vanishing theory, which  was proved by Chen and myself in a special case and was later proved by Pareschi, Popa, and Schnell. Then we will explain that how to use it to study the eventual paracanonical maps and numerically trivial automorphisms.

Martí Lahoz (U. Barcelona): Hyperkaehler manifolds as moduli spaces on non-commutative K3 surfaces. 

Abstract: The derived category of coherent sheaves on a smooth cubic fourfold has a subcategory, recently studied by Kuznetsov, Addington-Thomas and Huybrechts among others, that can be thought as the derived category of a non-commutative K3 surface. In this talk, I will present this category and joint work in progress together with Bayer, Macrì, Nuer, Perry and Stellari about the construction of hyperkaehler manifolds as moduli spaces of objects in it. The starting point is a method to induce Bridgeland stability conditions on semiorthogonal decompositions. 

Robert Lazarsfeld (U. Stony Brook): TBA

Yongnam Lee (KAIST): On function fields of a very general hypersurface sections of Fano 3-folds and their double covers

Abstract: I will talk on dominant rational maps from a very general hypersurface sections of Fano 3-folds. Also vanishing cohomology on a double cover of a very general hypersurface will be discussed. This is a joint work with Gian Pietro Pirola.

Christian Liedtke (Tecnische U. München): A Néron-Ogg-Shafarevich criterion for K3 Surfaces

Abstract: Let R be complete local ring with field of fractions K and residue field k, let X be a K3 surface over K, and assume that X has potential semi-stable reduction (this is automatic if char(k)=0). If char(K)=0, then we show that the following are equivalent:
1) the l-adic Galois representation on H^2(\bar{X},Q_l) is unramified for one l different from p 
2) the l-adic Galois representation on H^2(\bar{X},Q_l) is unramified for all l different from p 
3) the p-adic Galois representation on H^2(\bar{X},Q_p) is crystalline 
4) the surface has good reduction after an unramified extension of K
For example, if R is a power series over the complex numbers, then this says that a family of K3 surfaces over a pointed disk can be filled in smoothly over the origin (that is, has good reduction) if and only the monodromy representation on H^2 is trivial. However, in the arithmetic situation, where the residue field k might not be algebraically closed, then an unramified base change might be needed. We show by example that sometimes, a non-trivial base-change is necessary. In any case, we show that if 1) to 3) hold, then there always exists a model over R whose special fiber X_0 has at worst canonical singularities. Then, good reduction of X is equivalent to having an isomorphism between H^2 of X and the minimal resolution of singularities of X_0, such that this isomorphism is compatible with the natural Galois-actions (or F-isocrystal structures). In my talk, I will introduce all the above notions, which will not give me much time to explain proofs. Part of this is joint with Matsumoto, part of this is joint with Chiarellotto and Lazda.

Rita Pardini (U. Pisa): Fundamental groups of Gorenstein stable Godeaux surfaces

Abstract: A stable Godeaux surface $X$ is a stable surface with $K_X^2=1$ and $\chi(X)=1$. I will describe the classification of Gorenstein stable Godeaux surfaces and explain how their fundamental groups can be computed. This is joint work with M. Franciosi and S. Rollenske.

Stefan Schreieder (LMU, Munich): The rationality problem for quadric bundles 

Abstract: We study the rationality problem for quadric bundles X over rational bases S. By a theorem of Lang, such bundles are rational if r>2^n-2, where r denotes the fibre dimension and n=dim(S) denotes the dimension of the base. We show that this result is sharp. In fact, for any r at most 2^n-2, we show that many smooth r-fold quadric bundles over rational n-folds are not even stably rational. Our result is based on a generalization of the specialization method of Voisin and Colliot-Thélène—Pirutka. 

Alessandro Verra (U. Roma Tre): K3 surfaces and moduli of étale cyclic covers of curves

Abstract: The talk deals with complex K3 surfaces of level d, that is triples (S, L, p) such that p: \tilde S' \to S is a cyclic cover of order d, \tilde S' is birational to a K3 surface and L is orthogonal to the branch divisor B of p. In particular B is contractible to a set of rational singularities and p induces an étale cyclic d:1 cover
 \tilde C \to C of a general C in the linear system of L. The moduli space \mathcal F^{[d]}_g of triples (S, L, p) is clearly related to the moduli space of genus g curves of level d, that is of pairs (C, \pi) such that C is a curve of genus g and \pi: \tilde C \to C is an étale cyclic cover of degree d. Let \mathcal P^{[n]}_g be the moduli of 4-tuples (S, L, p, C) such that C belongs to \vert L \vert, then the assignement (S, L, p, C) \longrightarrow (C, p\vert \tilde C) defines a map 

m^{[d]}_g: \mathcal P^{[d]}_g \to \mathcal R_{g,d}.

This is a variation in the above situation of the Mukai map 
m_g: \mathcal P_g \to \mathcal M_g, 
from the moduli of all triples (S, L, C) such that C belongs to \vert L \vert to the moduli space \mathcal M_g. In the talk some unexpected analogies of the two maps are described, concerning in particular the maximal rank property for m^{[d]}_g. This is then used to study \mathcal R_{g,d} for d = 2, 3.
The talk reports on joint works with A. Knutsen - M. Lelli Chiesa and with A. Garbagnati.

Claire Voisin (College de France):  Segre numbers of tautological bundles on Hilbert schemes 

Abstract:   We establish geometric vanishings in certain ranges for the top Segre classes of tautological bundles of punctual Hilbert schemes of K3 surfaces (which had been first obtained by Marian-Oprea-Pandharipande  by different methods) and also K3 surfaces blown-up at one point. We show how all the Segre numbers for any surface and any polarization are formally determined by these vanishings and we thus reduce the Lehn conjecture to showing that Lehn's function also has these vanishing properties. Marian-Oprea-Pandharipande in turn used the present results to complete recently the proof of Lehn's conjecture.