Some new methods in the theory of gradings of simple Lie algebras
Dienstag, 3.4.12, 10:15-11:15, Raum 404, Eckerstr. 1
On the arithmetic of surfaces
Freitag, 27.4.12, 10:00-11:00, Raum 404, Eckerstr. 1
I will discuss recent advances in the study of rational points on algebraic surfaces over nonclosed fields.
Interpolation on subspaces
Freitag, 25.5.12, 10:00-11:00, Raum 404, Eckerstr. 1
Let P1,...Pr be points in the projective space \bP^n and let m1,...,mr 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=m1P1+...+mrPr 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 m1,...,mr are fixed, then the conditions imposed\nby 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 \bP^n. They showed that lines behave as\npoints, i.e. general lines impose independent conditions on\nhypersurfaces. The proof of this result is pretty involved.\nWe study the problem more generally, asking for conditions\nimposed by general configurations of linear subspaces and allowing multiplicities.\n\nThis is work in progress, joint with Brian Harbourne, Marcin Dumnicki, Joaquim Roe and Halszka Tutaj-Gasinska.\n
Discrepancies on normal varieties
Freitag, 1.6.12, 10:00-11:00, Raum 404, Eckerstr. 1
We will investigate the idea introduced by de Fernex and Hacon for studying singularities of normal varieties, via a pull-back for Weil divisors.\nWe 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.
Klassifikation von Vektorbündeln
Freitag, 15.6.12, 10:00-11:00, Raum 404, Eckerstr. 1
Separation coordinates and moduli spaces of stable curves
Freitag, 29.6.12, 10:00-11:00, Raum 404, Eckerstr. 1
Separation coordinates are coordinates in which the classical Hamilton-Jacobi\nequation can be solved by a separation of variables. We establish a new and\npurely algebraic approach to the classification of separation coordinates\nunder isometries. This will be made explicit for the least non-trivial\nexample: the 3-sphere. In particular, we show that the moduli space of\nseparation coordinates on the 3-sphere is naturally isomorphic to a certain\nmoduli space of stable curves with marked points. Several generalisations of\nthis result will be proposed.
Cotangent Complex of Moduli Spaces and Symplectic Structures
Freitag, 20.7.12, 10:00-11:00, Raum 404, Eckerstr. 1
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.