Brill--Noether loci on moduli spaces of symplectic bundles over curves

Abstract

The symplectic Brill--Noether locus ${\mathcal{S}}_{2n,K}^k$ associated to a curve $C$ parametrises stable rank $2n$ bundles over $C$ with at least $k$ sections and which carry a nondegenerate skewsymmetric bilinear form with values in the canonical bundle. This is a symmetric determinantal variety whose tangent spaces are defined by a symmetrised Petri map. We obtain upper bounds on the dimensions of various components of ${\mathcal{S}}_{2n,K}^k$. We show the nonemptiness of several ${\mathcal{S}}_{2n,K}^k$, and in most of these cases also the existence of a component which is generically smooth and of the expected dimension. As an application, for certain values of $n$ and $k$ we exhibit components of excess dimension of the standard Brill--Noether locus $B_{2n,2n(g-1)}^k$ over any curve of genus $g\ge 122$. We obtain similar results for moduli spaces of coherent systems.

Several references and acknowledgement added. 30 pp