Counting maximal isotropic subbundles of orthogonal bundles over a curve

Abstract

Let C be a smooth projective curve and V an orthogonal bundle over C. Let ${\mathrm{IQ}}_e\left(V\right)$ be the isotropic Quot scheme parameterizing degree e isotropic subsheaves of maximal rank in V. We give a closed formula for intersection numbers on components of ${\mathrm{IQ}}_e\left(V\right)$ whose generic element is saturated. As a special case, for $g\ge 2$, we compute the number of isotropic subbundles of maximal rank and degree of a general stable orthogonal bundle in most cases when this is finite. This is an orthogonal analogue of Holla's enumeration of maximal subbundles in [16], and of the symplectic case studied in [7].