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 $\IQeV$ 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 $\IQeV$ 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 \cite{Ho}, and of the symplectic case studied in \cite{CCH1}.