Schedule

Senior Talks:

Carolina Araujo

Title: Blowups, Gale duality and moduli spaces

Abstract: The central topic of this mini-course is the geometry of blowups of complex projective spaces at very general points. In dimension 2, it is a classical result that blowups of P² at at most 8 general points are del Pezzo surfaces, and their geometry can be encoded in finite combinatorial data. On the other hand, if we blow up at least 9 very general points, the resulting surface contains infinitely many (-1)-curves, and there are several unsolved conjectures about the geometry of these rational surfaces. After revising the theory for surfaces, we will consider the problem in higher dimensions. In particular, we will introduce the important class of Mori Dream Spaces (MDS), and show that if we blow up few points with respect to the dimension, the resulting variety is a MDS. We will also investigate the border case when the blowup is no longer a MDS. For that, we will make use of Gale correspondence, which provides a duality between sets of n points in projective spaces P^s and P^r when n=r+s+2. For small values of s, this duality has a remarkable geometric manifestation: the blowup of P^r at n points can be realized as a moduli space of vector bundles on the blowup of P^s at the Gale dual points. This realization can be explored to describe the birational geometry of blowups in higher dimensions.

Emanuele Macrì

Title: Non commutative K3 surfaces, with application to Hyperkähler and Fano manifolds

Abstract: The aim of this series of lectures is to explain the relation between polarized Hyperkähler manifolds, non-commutative K3 surfaces, and certain Fano manifolds.

In the first talk I will introduce non-commutative K3 surfaces and moduli spaces of objects in them. The second talk will be centered on examples, including non-commutative K3 surfaces associated to cubic fourfolds and Gushel-Mukai manifolds. In the final talk we will discuss possible ways to associate a non-commutative K3 surface to a polarized Hyperkähler fourfold, and discuss possible applications to Chow groups.

John Christian Ottem

Title: Integral Hodge classes and algebraic cycles

Abstract: The Hodge conjecture predicts which rational cohomology classes on a smooth complex projective variety can be represented by linear combinations of complex subvarieties. The integral Hodge conjecture, the analogous conjecture for integral homology classes, is known to be false in general (the first counterexamples were constructed by Atiyah and Hirzebruch). In this lecture series I will survey the known results on this conjecture, including some of the constructions of counterexamples; positive results; and how algebraicity of integral classes relate to other central questions in algebraic geometry.

Lecture notes: link