George Hitching, Brill-Noether Loci on Moduli Space of Symplectic Bundles over a Curve

Let C be a smooth projective curve of genus g. The symplectic Brill-Noether locus S(k, 2n, K) parametrises stable bundles of rank 2n over C with at least k independent sections, and which admit a nondegenerate skewsymmetric bilinear form with values in the canonical bundle K. This is a symmetric determinantal variety whose tangent spaces are described by a symmetrised Petri map. We show for many values of g, n and k that S(k, 2n, K) is nonempty and has a component which is generically smooth and of the expected dimension. As an application, we show that for certain n and k, for any curve of large genus the usual Brill-Noether locus B(k, 2n, 2n(g-1)) has a component of excess dimension. This is joint work with Ali Bajravani (Tabriz/Berlin).