Abstract

Let $S$ be a smooth projective surface with p_g>0 and H^{1(S,\Z)=0.  We consider the moduli spaces $M=M_S}H(r,c_1,c_2)$ of $H$-semistable sheaves on $S$ of rank $r$ and with Chern classes $c_1,c_2$. Associated a suitable class $v$ the Grothendieck group of vector bundles on $S$ there is a deteminant line bundle $\lambda(v)\in Pic(M)$, and also a tautological sheaf $\tau(v)$ on M.In this talk we derive a conjectural generating function for the virtual Verlinde numbers, i.e. the virtual holomorphic Euler characteristics of all determinant bundles $\lambda(v)$ on M, and for Segre invariants associated to $\tau(v)$. The argument is based on conjectural blowup formulas and a virtual version of Le Potier’s strange duality. Time permitting we also sketch a common refinement of these two conjectures, and their proof for Hilbert schemes of points.