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 $$B^k_{2n, 2n(g-1)}$$ over any curve of genus $$g \ge 122$$ . We obtain similar results for moduli spaces of coherent systems.