Workshop of Algebraic Geometry

   Seminario di Natale 2017 - December 18-19 2017


Program - December 19, 2017 (Aula C, second floor)

You can download the poster and the program.

10:00-10:50

Andrea Maiorana

Moduli of semistable sheaves as quiver moduli

Abstract: In the 1980s Drézet and Le Potier realized moduli spaces of Gieseker semistable sheaves on P2 as what are now called quiver moduli spaces. I will discuss how this can be understood using t-structures and exceptional collections, and how it can be extended to a similar result on P1xP1. This construction can be used to prove easily some of the geometric properties of the moduli space of sheaves, and to do some explicit computations.


11:00-11:50

Dino Festi

The Picard lattice of another family of double covers of the projective plane

Abstract: Double covers of the projective plane ramified along a smooth sextic curve are some of the best known examples of K3 surfaces. The Picard lattice of a K3 surface encodes much information about both the geometry and the arithmetic of the surface. Even though a vast literature about K3 surfaces and their Picard lattice has been produced, a practical algorithm computing the Picard lattice of a given K3 surface has not yet been found. In this talk we are going to explicitly compute the geometric Picard lattice of the generic member of a certain one-dimensional family of K3 surfaces. This specific family comes up in the study of high energy physics.


13:30-14:20

Federico Lo Bianco

On the dynamics of automorphisms preserving a fibration

Abstract: When studying the dynamics of an automorphism (or, more generally, of a birational transformation) $f$ of a (complex) projective manifold $X$, an important property is the existence of a preserved fibration $X \to B$; in this case, the dynamical system can be decomposed into smaller dimension ones: the action on the base $B$ and the action on the fibres. In this talk, I will give some conditions which ensure that the action on the base has finite order, so that all interesting dynamics arise from the action on fibres; this can be applied in particular to the dynamics of birational transformations of irreducible holomorphic symplectic manifolds. The proof is inspired to a strategy used by Tits to show the Tits alternative for finitely generated subgroups of $PG_n(\mathbb \C)$; the main idea consists in embedding such a subgroup in $PGL(\mathbb Q_p)$ (for a well-chosen prime $p$) in order to obtain informations about the subgroup itself. The proof of my result requires the use of basic concepts of $p$-adic integration, which I am going to recall during my talk.


14:30-15:20

Laura Pertusi

Twisted cubics on cubic fourfolds and stability conditions

Abstract: A famous result of Beauville and Donagi states that the Fano variety of lines on a cubic fourfold is a smooth projective irreducible holomorphic symplectic (IHS) variety of dimension four, equivalent by deformation to the Hilbert square on a K3 surface. More recently, Lehn, Lehn, Sorger and van Straten studied curves of degree three on a cubic fourfold Y not containing a plane. In particular, they constructed an IHS variety Z(Y) from the Hilbert scheme of twisted cubics on Y. It turns out that Z(Y) is smooth and projective of dimension eight, deformation equivalent to the Hilbert scheme of points of length four on a K3 surface. In the same spirit of a recent work of Lahoz, Lehn, Macrž and Stellari, we prove that if Y is a cubic fourfold not containing a plane, then Z(Y) is isomorphic to a moduli space of Bridgeland stable objects in the Kuznetsov component of the derived category of Y. This is a joint work with X. Zhao.