**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): ****Recursive properties of moduli spaces **

**Abstract: **A connection between algebro-geometric moduli spaces and moduli spaces of polyhedral objects has been under investigation in recent years. The connection, based on the combinatorial properties of the algebro-geometric moduli spaces, expresses the skeleton of an algebro-geometric moduli space as the moduli space of the skeleta of the objects it parametrizes. This talk describes some recent results in the area.

**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):** **Canonical surfaces of high degree and small codimension**

**Abstract:** Given a surface S of general type, we say that it is canonical if the canonical image Y of S is a surface. We denote by d the canonical degree (d is the degree of Y, d=0 if Y is a curve). d is bounded by the canonical volume K^2, and by the BMY inequality we have d <= K^2 <= 9 \chi = 9 (1-q+p_g). Question I: what is the maximum value of d for p_g=4,5,6? Can we find surfaces realizing large canonical degree ? I will recall several known results. For instance, for p_g=4, each value d is achieved in the range [5, 28] by some canonical surfaces. If a surface realizes maximal canonical volume, then K is ample and S is a ball quotient: can we get large d? In joint work with Ingrid Bauer we constructed ball quotients with p_g=4, K^2 = 45, but here d is not maximal. Question II : what is the maximum value of d for a canonically embedded surface S ? (this means: Y is isomorphic to S, or to the canonical model Z of S). This question is interesting for p_g=6, since for p_g=4, Z must be a 5-ic, and, for p_g=5, the canonical model Z must be a complete intersection of type (2,4) or (3,3) (hence d=8 or 9). For p_g = 5, this is a consequence of Severi’s double point formula, and of its extension done in joint work with Keiji Oguiso: it holds for the case of surfaces with RDP’s as singularities. For p_g=6, if S is canonically embedded, there are interesting ties with methods and questions of homological algebra (Walter’s bundle Pfaffians), which led to the question whether 18 would be the upper bound for d (the range [11,17] can be easily filled by bundle methods). Degree d = 24 was achieved by myself with some regular surfaces (q=0), and by Cesarano with a family of surfaces having q=3, polarizations of type (1,2,2) in an Abelian 3-fold. Returning to question I, I recently constructed several connected components of the moduli space, of surfaces S of general type with pg = 5,6, whose canonical map has image Σ of very high degree, d=48 for pg =5, d=56 for pg =6. These surfaces are surfaces isogenous to a product of curves, S = (C1 × C2)/G, with G abelian. Ball quotients S with p_g=6, K^2 = 63, are constructed as unramified Z/7 covers of some fake projective planes X, and in work in progress with Jong Hae Keum we are studying their canonical map. As a preliminary result, we showed that the bicanonical map of these fake projective planes is an embedding.

**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 U.): 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 (Stony Brook U.):** **A theorem of Ran and measures of irrationality **

**Abstract:** I’ll start by presenting a simplified proof of a theorem of Ran’s to the effect that the (n+2)-secant lines to a smooth variety of dimension n sweep out a variety of dimension at most (n+1). Then I’ll explain how this comes up in connection with bounding the “degree of irrationality” of a very general smooth hypersurface of large degree in projective space. This is joint work with Bastianelli, De Poi, Ein and Ullery.

**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): ****Stably irrational hypersurfaces of small slopes**

**Abstract: **We show that a very general complex projective hypersurface in P^{N+1} of degree at least log_2N +2 is not stably rational. The same statement holds over any uncountable field of characteristic p>>N. This significantly improves earlier results of Kollar and Totaro.

**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.