Bogomolov-Sommese Vanishing on log canonical pairs
Freitag, 2.11.12, 10:00-11:00, Raum 404, Eckerstr. 1
Gromov Witten Invariants for the Hilbert scheme of points of a K3 surface
Freitag, 9.11.12, 10:00-11:00, Raum 404, Eckerstr. 1
The Yau-Zaslow formula gives an expression of the number of nodal rational curves on a K3 surface in terms of a modular form. In this talk we explain how to extend their result to the Hilbert scheme of 2 points of a K3 surface. In particular, we will present the generating series for the reduced genus 0 GW Invariants which will be given by a weak Jacobi Form.\n
Drinfeld modules and their application to factoring polynomials
Freitag, 16.11.12, 10:00-11:00, Raum 404, Eckerstr. 1
Major works done in Function Field Arithmetic show a strong analogy between the ring of integers Z and the ring of polynomials over a finite field Fq[T]. While an algorithm has been discovered to factor integers using elliptic curves, the discovery of Drinfeld modules, which are analogous to elliptic curves, made it possible to exhibit an algorithm for factorising polynomials in the ring Fq[T]. \nIn this talk, we introduce the notion of Drinfeld modules, then we demonstrate the analogy between Drinfeld modules and Elliptic curves. Finally, we present an algorithm for factoring polynomials over a finite field using Drinfeld modules.\n
Motives of Deligne-Mumford Stacks
Freitag, 23.11.12, 10:00-11:00, Raum 404, Eckerstr. 1
For every smooth and separated Deligne-Mumford stack F,\nwe will associate a motive M(F) in Voevodsky's category of mixed motives with rational coecients DM^eff(k; Q). For F proper over a field of characteristic 0, we will compare M(F) with the Chow motive associated to F by Toen. Without the properness condition we will show that M(F) is a direct summand of the motive of a smooth quasi-projective\nvariety. Then we will generalize a motivic decomposition theorem due to Karpenko to relative geometrically cellular Deligne-Mumford stacks.\nThis will depend on a vanishing result of Voevodsky. Even in the classical case, our method yields a simpler and more conceptual proof of Karpenko's result.
Rational volume of varieties over complete local fields and Galois extensions
Freitag, 30.11.12, 10:00-11:00, Raum 404, Eckerstr. 1
We are interested in the following question: When does a given variety over a complete local field K have a rational point? The rational volume is a motivic invariant of a K-variety X vanishing if X has no K-rational point. For a tame Galois extension L over K, we will compare the rational volume of a K-variety X and of its base change XL to L. To do so, we construct out of a given weak Néron model of XL with an action of the Galois group of L over K a weak Néron model of X with some universal property. As an application, we will show that some varieties over K with potential good reduction have K-rational points.