Determinant line bundles on moduli spaces of pure sheaves on rational surfaces and strange duality

摘要

Let M ~H _X (u) be the moduli space of semi-stable pure sheaves of class u on a smooth complex projective surface X. We specify u = (0, L, χ(u) = 0), i.e. sheaves in u are of dimension 1. There is a natural morphism π from the moduli space M ~H _X (u) to the linear system |L|. We study a series of determinant line bundles λ _c ~r _n on M ~H _X (u) via π. Denote gL the arithmetic genus of curves in |L|. For any X and gL ≤ 0, we compute the generating function Z ~r(t) = ∑ _n h ~0(M ~H _X (u), λ _c ~r _n)t ~n. For X being ? ~2 or ?(O _(?1) ?O?1 (-e)) with e = 0, 1, we compute Z ~1(t) for gL > 0 and Z r(t) for all r and gL = 1, 2. Our results provide a numerical check to Strange Duality in these specified situations, together with G?ottsche's computation. And in addition, we get an interesting corollary (Corollary 4.2.13) in the theory of compactified Jacobian of integral curves.