Stable sheaves on elliptic fibrations

Abstract

Let X→ B be an elliptic surface and M(a,b) the moduli space of torsion-free sheaves on X which are stable of relative degree zero with respect to a polarization of type aH+ bμ, H being the section and μ the elliptic fibre ( b≫0). We characterize the open subscheme of M(a,b) which is isomorphic, via the relative Fourier–Mukai transform, with the relative compactified Simpson–Jacobian of the family of those curves D↪ X which are flat over B. This generalizes and completes earlier constructions due to Friedman, Morgan and Witten. We also study the relative moduli scheme of torsion-free and semistable sheaves of rank n and degree zero on the fibres. The relative Fourier–Mukai transform induces an isomorphic between this relative moduli space and the relative nth symmetric product of the fibration. These results are relevant in the study of the conjectural duality between F-theory and the heterotic string.