Abstracts

A theorem of Esnault states that smooth Fano varieties over finite fields always have rational points. A natural question then arises: what happens if we relax the positivity conditions on the anticanonical class? In this seminar, I will discuss the case of 3-folds with nef anticanonical class over finite fields. Specifically, we demonstrate that in the case of negative Kodaira dimension, rational points exist if the cardinality of the field is greater than 19. In the K-trivial case, we prove a similar result, provided that the Albanese morphism is non-trivial. This result draws on a combination of techniques from the Minimal Model Program, semipositivity theorems, and arithmetic considerations. 

This is a joint work with S. Filipazzi. 

In this talk, we devote our attention to infinitesimal deformations of morphisms of locally free sheaves over a smooth projective variety. In particular, we describe a differential graded Lie algebra controlling these deformations over any algebraically closed field of characteristic zero. As an application, we analyse deformations of a locally free sheaf together with a subspace of its global sections. 

This is based on a joint work with Elena Martinengo. 

We present sufficient conditions under which a pair of hyper-Kähler fourfolds of K3^[2]-type of Picard rank 1 is derived equivalent; we describe the Fourier Mukai partners of a given example. 

This is a joint work with Michał Kapustka.

We will introduce the notion of even nodal surfaces of K3 type, special types of nodal surfaces defined by the existence of  smooth double covers with special Hodge theoretic behavior. We will then discuss through examples possible constructions and relations between such surfaces, Fano fourfolds of K3 type and hyperkaehler manifolds.

The talk will include results from various joint works partially in progress involving M. Bernardara, E. Fatighenti, G. Kapustka, A. Kuznetsov, L. Manivel, G. Mongardi, and F. Tanturri.

In this talk, we focus on some singular moduli spaces of sheaves on a K3 surface. More precisely, for any integer n > 1, we consider the moduli space M(n) associated with the Mukai vector 2(1,0,1-n). Looking for new deformation classes of hyper-Kähler manifolds, O’Grady constructed an explicit resolution of every M(n). O’Grady’s resolution is crepant and does give a hyper-Kähler manifold only if n=2. If n>2, it turns out that no crepant resolution exists for M(n), but one may still look for a categorical crepant resolution.

We will report on the preliminary step in this direction, which consists in a geometric analysis of O’Grady’s resolution and of its exceptional locus.

Clifford’s inequality gives an upper bound for the rank of a line bundle of given degree on a smooth curve of given genus.

While this bound still holds for irreducible curves, it can not exist for curves with several components unless we restrict to special multidegrees. In the talk I will discuss joint work with M. Barbosa and K. Christ where we use combinatorial techniques to establish the existence of such special multidegrees.

We present results on existence of moduli spaces of stable vector bundles on general polarized HK varieties of type K3^[2] of arbitrarily large dimension, and we give more precise results in some cases.

We will prove some new slope inequalities for fibred surfaces involving the unitary rank u_f. In particular we can prove, using these results:

1. a new Xiao-type bound on u_f with respect to g for non-isotrivial fibrations – in particular this imples that if u_f=g-1 is maximal, then g is less or equal to 6;

2. a new constrain on the rank of the (−1,0) part of the maximal unitary Higgs subbundle of a curve generically contained in the Torelli locus – a result going in the direction of the Coleman-Oort conjecture.

If time permits, I will describe some constrains to the case g=6, u_f=5, which conjecturally should not exist.