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 IQe (V ) 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 IQe (V ) whose generic element is saturated. As a special case, for g ≥ 2, we compute the number of isotropic subbundles of maximal rank and degree of a general stable orthog onal bundle in most cases when this is finite. This is an orthogonal analogue of Holla’s enumeration of maximal subbundles in [20], and of the symplectic case studied in [8].