Course overview -- Sommersemester 2012 (Archive)

I will discuss recent advances in the study of rational points on algebraic surfaces over nonclosed fields.

Let P_1,...P_r be points in the projective space \P^n and let m_1,...,m_r be positive integer. If n=2, then the conjecture of Segre, Harbourne, Gimigliano, Hirschowitz predicts that if the points are general, then the scheme Z=m_1P_1+...+m_rP_r either imposes independent conditions on linear systems of curves of degree d, or this system has a non-reduced base curve. For higher n, even the conjectural picture is less clear. On the other hand, a celebrated result of Alexander and Hirschowitz says, that if m_1,...,m_r are fixed, then the conditions imposed by Z on hypersurfaces of degree d are independent, provided d is sufficiently large. There is no reason to restrict imposing conditions only to points. Hartshorne and Hirschowitz studied the postulation problem for a set of general lines in \P^n. They showed that lines behave as points, i.e. general lines impose independent conditions on hypersurfaces. The proof of this result is pretty involved. We study the problem more generally, asking for conditions imposed by general configurations of linear subspaces and allowing multiplicities.

This is work in progress, joint with Brian Harbourne, Marcin Dumnicki, Joaquim Roe and Halszka Tutaj-Gasinska.

We will investigate the idea introduced by de Fernex and Hacon for studying singularities of normal varieties, via a pull-back for Weil divisors. We will show some pathologies of the new definition and we will explain the main properties, with highlighting how the approach relates to properties of finite generation and to singularities in positive characteristic.

Separation coordinates are coordinates in which the classical Hamilton-Jacobi equation can be solved by a separation of variables. We establish a new and purely algebraic approach to the classification of separation coordinates under isometries. This will be made explicit for the least non-trivial example: the 3-sphere. In particular, we show that the moduli space of separation coordinates on the 3-sphere is naturally isomorphic to a certain moduli space of stable curves with marked points. Several generalisations of this result will be proposed.

A remarkable theorem of Mukai is the existence of symplectic structures on the smooth moduli spaces of semistable sheaves on K3 surfaces, which identifies the cotangent bundle of a moduli space with its dual. In this talk I will describe the cotangent complex of a possibly singular moduli space, viewed as an Artin stack, and show that it shares a similar duality property as in the case of a smooth moduli space.