Wintersemester 2010/11

 Montag 18.10.2010 Nicola Pagani (KTH Stockholm) The Inertia stack of moduli of hyperelliptic curves Abstract: If X is an orbifold, its Inertia is the orbifold that parametrizes couples (x, g) where x is a point of X and g is an automorphism of it. In general the Inertia of X appears as the disjoint union of distinct connected components: one of them is X itself (with the identical automorphism), the other connected components were called twisted sectors by physicists. Determining the Inertia of an orbifold X is important in the study of many aspects of the geometry of X. For instance, the cohomology of the Inertia orbifold is the degree 0 small quantum cohomology of the orbifold, therefore studying the cohomology of the Inertia orbifold is the first step in the study of the Gromov-Witten theory of X. In this talk we study the twisted sectors of some moduli spaces of curves focusing in particular on the moduli spaces of smooth hyperelliptic curves H_g. We describe the twisted sectors as quotients of moduli spaces of genus 0, n-pointed curves by the action of certain subgroups of the symmetric group S_n. Beginn: 16.00 Uhr 27.10.2010 kein Seminar 03.11.2010 Angela Ortega (HU Berlin) Rank two Brill-Noether Theory and the Maximal Rank Conjecture Abstract: The rank two Brill-Noether theory deals with the linear series of rank 2 on curve C, more precisely with the cycles BN_C(d,k) in the moduli space of semistable rank 2 vector bundles on C of degree d, defined by the condition of admitting at least k sections. Unlikely the classical Brill-Noether theory,the dimension of BN_C(d,k) on a general curve, is not governed by the Brill-Noether number. Related to the non-emptiness problem of BN_C(d,k), the Mercat's conjecture gives an uniform bound for the number of independent sections on a rank 2 vector bundle. In this talk we will explain the link between rank 2 Brill-Noether theory and the Koszul geometry of the curve C. We will show how the Maximal Rank Conjecture implies, in some cases, Mercat's conjecture. Using these ideas we are able to prove Mercat's conjecture for a bounded genus. We also show that, for k=4, there exist Brill-Noether general curves in any genus > 11, for which Mercat's conjecture fails. This is a join work with G. Farkas. 10.11.2010 Angela Gibney (Univ. of Georgia) Morphism on the moduli space of curves from conformal blocks Abstract: Very recently, due to the work of Fakhruddin, we have learned of an enormous supply of semi-ample divisor on the moduli space of curves which come from the theory of conformal blocks. After studying only a few families of them, it seems all known morphisms defined on $\overline{\operatorname{M}}_{0,n}$ can be given using these conformal blocks divisors. Moreover, using conformal blocks divisors, new birational models of the moduli space have been discovered. 17.11.2010 Albrecht Klemm (Univ. Bonn) A proof of the Yau-Zaslow conjecture using mirror symmetry Abstract: We review the relation between mirror symmetry and modular forms. After recapitulating some ingredients of Noether Lefshetz theory we proof the Yau-Zaslow conjecture for non-primitive classes. In particular the proof uses the modular properties of the period integrals on a three parameter family of Calabi-Yau 3-folds, whose mirror exhibits a K3 fibration. 24.11.2010 kein Seminar 01.12.2010 Margherita Lelli Chiesa (HU Berlin) The Gieseker-Petri locus inside the moduli space of curves Abstract: The Gieseker-Petri locus GP_g is defined as the locus inside M_g consisting of curves which violate the Gieseker-Petri Theorem. It is known that GP_g has always some divisorial components and it is conjectured that it is of pure codimension 1 inside M_g. We prove this conjecture for genus up to 13 and we explain why our method is insufficient for the case of genus 14. 08.12.2010 Emanuele Macri (Univ. Bonn) Bogomolov-type inequalities in higher dimension Abstract: In this seminar (based on joint work in progress with A. Bayer, Y. Toda, and A. Bertram), we will present a conjectural approach to the construction of stability conditions on the derived category of a higher dimensional variety. The main ingredient is a generalization to complexes of the classical Bogomolov-Gieseker inequality for sheaves. We will also discuss geometric applications of this result. 15.12.2010 kein Seminar 05.01.2011 Gavril Farkas (HU Berlin) Green's Conjecture on syzygies of canonical curves. Abstract: Mark Green's Conjecture on syzygies of canonical curves, has been one of the most studied questions in the theory of Riemann surfaces in the last few decades. Formulated in 1984 and still open in its full generality, it is a deceptively simple statement which predicts that the intrinsic geometry of the curve (in the form of linear series) can be recovered in a precise way from the extrinsic geometry of the canonical embedding (in the form of syzygies). I will discuss how one can use Voisin's solution to Green's Conjecture for GENERAL curves together with the geometry of the moduli space of curves, in order to prove Green's Conjecture for ALL curves lying on K3 surfaces. This is joint work with M. Aprodu. 12.01.2011 kein Seminar 19.01.2011 Herbert Lange (Univ. Erlangen-Nürnberg) Vector bundles of rank 2 computing Clifford indices Abstract: This is a report on a recent joint paper with Peter Newstead. After defining Clifford indices of vector bundles on algebraic curves and recalling what is known about their possible values, I will outline details of the paper in which we study bundles of rank 2 which compute these Clifford indices. This is of particular interest in the light of recently discovered counterexamples to a conjecture of Mercat. 26.01.2011 Marian Aprodu (IMAR Bucharest) Syzygies of curves 02.02.2011 Yongnam Lee (Sogang Univ. Seoul) Simply connected surfaces of general type with vanishing geometric genus in positive characteristic via deformation theory Abstract: Algebraically simply connected surfaces of general type with p_g = q = 0 and 1 <= K^2 <= 4 in positive characteristic (with one exception in K^2 = 4) are presented by using a Q-Gorenstein smoothing of two dimensional toric singularities, previousconstruction over the field of complex numbers, and Grothendieck's specialization theorem for the fundamental group. It is jointed work with Noboru Nakayama. 09.02.2011 kein Seminar 16.02.2011 Nicola Tarasca (HU Berlin) An extremal ray of the effective cone of M_{6,1} Abstract: Recently Jensen has shown that the divisor in M_{6,1} given by pointed curves [C,p] with a sextic plane model mapping p to a double point, generates an extremal ray in the pseudoeffective cone of M_{6,1}. We compute the class of such a divisor. This will serve as a pretext to discuss classic techniques and new results in enumerative geometry.