Back to Algebraic Geometry Seminar: Fall 2009
Algebraic Geometry Seminar
Date: Tuesday, Oct 27, 2009
Time: 3:30PM - 4:30PM
Location: LCB 225
Luca Scala
University of Chicago
Title |
Cohomology of the Hilbert scheme of points on a surface with values in representations of tautological bundles |
Abstract |
We will motivate the work by an important example of the Strange Duality conjecture for moduli spaces of sheaves on the projective plane, in close relation with Barth morphism. The technique used to attack the conjecture in this case, due to Le Potier, He and Danila, uses moduli spaces of coherent systems in order to interpolate the moduli space M_n (of semistable sheaves of rank 2, degree 0 and second Chern class n) with a Hilbert scheme of points in P_2; this procedure allows to reduce the computation of the space of global sections of the determinant line bundle on M_n to the understanding of the cohomology H^*(P_2^[m] , S^q L^[m] ) of the Hilbert scheme of points on P_2 with values in symmetric powers of a tautological bundle L^[m], associated to a certain line bundle L on P_2 . Danila’s results on the cohomology of S^q L_[m] yield the strange duality for 0 < n < 19. We will show how to improve and generalize Danila’s results on the cohomology of the Hilbert scheme of points on a smooth surface X with values in representations of tautological bundles. By making use of the derived McKay correspondence of Bridgeland-King-Reid, adapted by Haiman in the case of the Hilbert scheme X^[m], we prove general formulas for the cohomology of X^[m] with values in the double tensor power and general exterior powers of tautological bundles. Finally, we will sketch the work in progress on higher symmetric powers of tautological bundles, directly useful for the strange duality conjecture on the projective plane. (arxiv: 0710.3072) |
Algebraic Geometry