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Abstract
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The symplectic Brill–Noether locus S2kn,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 S2kn,K . We show the nonemptiness of several S2kn,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 B2kn,2n(g−1) over any
curve of genus g ≥ 122. We obtain similar results for moduli spaces of coherent systems.
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