Isotropic Quot schemes of orthogonal bundles over a curve

Abstract

We study the isotropic Quot schemes IQe(V ) parameterizing degree e isotropic subsheaves of maximal rank of an orthogonal bundle V over a curve. The scheme IQe(V ) contains a compactification of the space IQe◦(V ) of degree e maximal isotropic subbundles, but behaves quite differently from the classical Quot scheme, and the Lagrangian Quot scheme in [6]. We observe that for certain topological types of V, the scheme IQe(V ) is empty for all e. In the remaining cases, for infinitely many e there are irreducible components of IQe(V ) consisting entirely of nonsaturated subsheaves, and so IQe(V ) is strictly larger than the closure of IQe◦(V ). As our main result, we prove that for any orthogonal bundle V and for e << 0, the closure IQe◦(V ) of IQe◦(V ) is either empty or consists of one or two irreducible connected components, depending on deg(V ) and e. In so doing, we also characterize the nonsaturated part of IQe◦(V ) when V has even rank.