Schedule and Abstracts

TITLES AND ABSTRACTS:

Arend Bayer: Non-commutative abelian surfaces, Kummer-type Hyperkaehler varieties, and Weil-type abelian fourfolds

The derived category D(A) of an abelian surface A has a six-dimensional space of deformations, but A has only a 3-dimensional space of deformations as an algebraic variety.  I will explain a construction that reduces this gap, by constructing a family of categories over a four-dimensional space that we call "non-commutative abelian surfaces".

This gives rise to an interpretation of every general Kummer-type Hyperkaehler variety via a moduli space. Finally, I will explain how this also leads to a different approach to and proof of O'Grady's and Markman's results concerning the relation to Weil-type abelian fourfolds.

This is based on joint work with Laura Pertusi, Alex Perry and Xiaolei Zhao.

Samir Canning: Cycles on moduli spaces of curves and abelian varieties

I will show how the study of non-tautological classes on the moduli space of abelian varieties helps explain the structure of the tautological ring of the moduli space of curves of compact type. On the curves side, this is joint work with Hannah Larson and Johannes Schmitt, and on the abelian varieties side, it is joint with Dragos Oprea and Rahul Pandharipande. 

Paola Frediani: Higher Gaussian maps for special classes of curves

I will report on some results obtained in collaboration with Dario Faro and Antonio Lacopo on the rank of higher Gaussian maps of the canonical bundle for hyperelliptic curves, for plane curves and for curves contained in Enriques surfaces. I will show how these maps are related to asymptotic directions in the tangent bundle of the moduli space of curves and in the tangent bundle of the hyperelliptic locus with respect to the second fundamental form of the Torelli map and of the hyperelliptic Torelli map.

Víctor González Alonso: 527 elliptic fibrations on any Enriques surface

Barth and Peters showed that a generic complex Enriques surface admits 527 isomorphism classes of elliptic fibrations. In this talk I will present a joint work with Simon Brandhorst, showing that this result also holds for any Enriques surface over an algebraically closed field of characteristic different from two, if the elliptic fibrations are counted with the appropriate multiplicity. The proof involves constructing a moduli space of elliptically fibred Enriques surfaces and its natural map to the space of unpolarized Enriques surfaces. Moreover, in the complex case the multiplicities can be naturally interpreted as the ramification indices of the forgetful map.

Frank Gounelas: Some new results on curves on K3s

There are numerous questions and open conjectures about the (non-)existence of curves on K3 surfaces which vary maximally in moduli, or, respectively, do not vary at all (the isotrivial case) - a classical open problem in the area, which goes back to Deligne/Schoen/Serre is whether a very general K3 surface can be dominated by the product of two smooth curves. In this talk I will summarise some recent results and in particular focus on recent work with Chen and Dutta proving that if the Picard rank is one, then there can be no isotrivial family of curves whose generic member is smooth.

Bruno Klingler: Recent progress on Hodge loci

Given a quasi projective family S of complex algebraic varieties, its Hodge locus is the locus of points of S where the corresponding fiber admits exceptional Hodge classes (conjecturally: exceptional algebraic cycles). In this talk I will survey the many recent advances in our understanding of such loci, both geometrically and arithmetically.

Margherita Lelli Chiesa: Prym-Gauss maps and curves on abelian surfaces

Wahl's conjecture, finally proved by Arbarello, Bruno and Sernesi, predicted that Brill-Noether general curves lying on K3 surfaces are characterized by the non-surjectivity of their Gaussian map. Wahl himself had related the conjecture to the deformation theory of the affine cone over a canonical curve C in P^g, thus reducing part of it to the vanishing of the higher cohomology groups of the square of the ideal of C twisted by hypersurfaces of degree >=3.

I will show that the Prym-Gauss map of a Prym curve on an abelian surface is never surjective. I will then talk about a joint work in progress with Arbarello and Bruno aimed to show that a general Prym canonical curve with non-surjective Prym-Gauss map always lies on a surface in P^{g-2}. Since there is no affine cone entering this picture, during the seminar I will provide an alternative proof of Wahl's result avoiding the deformation theory of cones.

Emanuele Macrì: Mukai models for Fano varieties

Mukai's classification of Fano manifolds in dimension 3 and index 1 is one of the fundamental results in algebraic geometry; it was completed by Iskovskikh and Mukai more than 30 years ago.

In this talk, based on joint work with Arend Bayer and Alexander Kuznetsov, I will present a new proof of this result, still inspired by Mukai's ideas, which also extends to the singular case and in higher dimension.

Rita Pardini: The classification of stable I-surfaces of index 2

A  stable I-surface (also called a (1,2)-surface) is a complex projective surface with K^2=1, h^2(O)=2, slc singularities and ample canonical class. Gorenstein stable I-surfaces are hypersurfaces of degree 10 in P(1,1,2,5).

We give a complete classification of stable  I-surfaces of Cartier index 2, using a twofold approach.

If the canonical curves are non-reduced, we show that the surface has two irreducible components and we classify the possible "half" surfaces and the ways in  which they can be glued.

If the surface has a reduced canonical curve C, then restricting the canonical bundle of the surface to C and modding out the torsion, one obtains a "generalized spin curve" of genus 2. This is a pair (C,L), where C is a reduced Gorenstein curve  with ample canonical class, and L is a torsion free rank 1 sheaf on C with vanishing Euler characteristic and with a generically injective map from L⊗L to the dualizing sheaf of C. We obtain a complete classification  of such pairs with C of genus 2 and derive from it the classification of stable I-surfaces of index 2 with a reduced canonical curve.

This is joint work with S.Coughlan, M.Franciosi and S.Rollenske.

Laura Pertusi: Irreducible symplectic varieties via relative Prym fibrations

Hyperkahler manifolds are one of the building blocks of manifolds with trivial canonical bundle, and their study has deep connections to differential geometry and theoretical physics. One of the most intriguing problems is their classification. A recent important development is the proof of the decomposition theorem in the singular setting, where hyperkahler manifolds are replaced by irreducible symplectic varieties.

The goal of this talk is to explain the construction of new infinite series of irreducible symplectic varieties starting from a K3 surface with an anti-symplectic involution and the choice of a linear system on the quotient surface. This is a joint work with Emma Brakkee, Chiara Camere, Annalisa Grossi, Giulia Saccà and Sasha Viktorova.

Gian Pietro Pirola: Asymptotic directions on the moduli space of curves

We study the second fundamental form associated with the image of the period map of curves started many years ago and present some computational improvements that allow us to study asymptotic lines in the tangent of the moduli space M_g of the curves of genus g. The asymptotic directions are those tangent directions that are annihilated by the second fundamental form induced by the Torelli map. We give examples of asymptotic lines for any g> 3 and we study their rank. The rank r(v) of a tangent direction at M_g is defined to be the rank of the cup product map associated to the infinitesimal deformation map, that is the infinitesimal variation of Hodge structure in that direction. 

We show that if v is not zero and r(v)< (cliff(C) +1) where cliff(C) is the Clifford index of C, then v is not asymptotic and we study the case when r(v)= cliff(C). Finally all asymptotic directions of rank 1 are determined and a description of the rank 2 case is given. 

It is a joint work with Elisabetta Colombo and Paola Frediani.

Tong Zhang: Moduli spaces of threefolds on the Noether line

I will talk about the classification of canonical threefolds on the Noether line by describing their moduli spaces, which includes an explicit stratification, an estimate of the number of irreducible components and the dimension formula for every such moduli space. I will also discuss the key idea behind the proof, which is to relate canonical threefolds on the Noether line to simple fibrations in (1,2)-surfaces. This is a joint work with S. Coughlan, Y. Hu and R. Pignatelli.

Xiaolei Zhao: Moduli of stable objects on Fano threefolds

The derived category of a Fano variety sometimes contains an interesting admissible subcategory, known as the Kuznetsov component. Moduli spaces of Bridgeland semistable objects on the Kuznetsov component provide many examples to study, often generalizing constructions from classical algebraic geometry. In this talk, I will review several cases of Fano threefolds, and explain how moduli spaces on their Kuznetsov components behave similarly to moduli of sheaves on curves. Based on joint work with Chunyi Li, Yinbang Lin and Laura Pertusi.