Geometry of some moduli of bundles over a very general sextic surface for small second Chern classes and Mestrano-Simpson conjecture

Abstract

Let S⊂P3 be a very general sextic surface over complex numbers. Let M(H,c2) be the moduli space of rank 2 stable bundles on S with fixed first Chern class H and second Chern class c2. In this article we study the configuration of certain locally complete intersection zero-dimensional subschemes on S satisfying Cayley-Bacharach property, which leads to the existence of non-trivial sections of a general member of each component of the moduli space for small c2. Using this study we will make an attempt to prove Mestrano-Simpson conjecture on the number of irreducible components of M(H,11) and prove the conjecture partially. We will also show that M(H,c2) is irreducible for c2≤10.