Isotropic Quot schemes of orthogonal bundles over a curve

Abstract

We study the isotropic Quot schemes [Formula: see text] parametrizing degree [Formula: see text] isotropic subsheaves of maximal rank of an orthogonal bundle [Formula: see text] over a curve. The scheme [Formula: see text] contains a compactification of the space [Formula: see text] of degree [Formula: see text] maximal isotropic subbundles, but behaves quite differently from the classical Quot scheme, and the Lagrangian Quot scheme in [D. Cheong, I. Choe and G. H. Hitching, Irreducibility of Lagrangian Quot schemes over an algebraic curve, preprint (2019), arXiv:1804.00052, v2]. We observe that for certain topological types of [Formula: see text], the scheme [Formula: see text] is empty for all [Formula: see text]. In the remaining cases, for infinitely many [Formula: see text] there are irreducible components of [Formula: see text] consisting entirely of nonsaturated subsheaves, and so [Formula: see text] is strictly larger than the closure of [Formula: see text]. As our main result, we prove that for any orthogonal bundle [Formula: see text] and for [Formula: see text], the closure [Formula: see text] of [Formula: see text] is either empty or consists of one or two irreducible connected components, depending on [Formula: see text] and [Formula: see text]. In so doing, we also characterize the nonsaturated part of [Formula: see text] when [Formula: see text] has even rank.