2009 Algebraic Geometry Seminar - 01

Let X be a (smooth, projective) rational surface, with an ample line bundle H. We consider the moduli spaces M_n of H-stable rank 2 torsion- free sheaves on X with second Chern class n. For a line bundle L on X there is an associated line bundle L_n on M_n, and we study the generating function of the holomorphic Euler charcteristics of these line bundles. We prove that it is always a rational function in, which can be determined explicitly in many cases, and has some nice symmetry properties. The rationality and the symmetry properties find their natural explanation in Le Potiers strange duality conjecture, which relates sections of L_n on M_n to sections of line bundles on a moduli space of pure sheaves supported on curves in the linear system associated to L. In some cases we prove the strange duality conjecture.