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

Abstract

The symplectic Brill–Noether locus Sk 2n,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 Sk . We show the nonemptiness of several Sk , 2n,K 2n,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 Bk over any 2n,2n(g−1) curve of genus g ≥ 122. We obtain similar results for moduli spaces of coherent systems.