A stratification on the moduli spaces of symplectic and orthogonal bundles over a curve

A symplectic or orthogonal bundle V of rank 2n over a curve has an invariant t(V) which measures the maximal degree of its isotropic subbundles of rank n. This invariant t defines stratifications on moduli spaces of symplectic and orthogonal bundles. We study this stratification by relating it to another one given by secant varieties in certain extension spaces. We give a sharp upper bound on t(V), which generalizes the classical Nagata bound for ruled surfaces and the Hirschowitz bound for vector bundles, and study the structure of the stratifications on the moduli spaces. In particular, we compute the dimension of each stratum. We give a geometric interpretation of the number of maximal Lagrangian subbundles of a general symplectic bundle, when this is finite. We also observe some interesting features of orthogonal bundles which do not arise for symplectic bundles, essentially due to the richer topological structure of the moduli space in the orthogonal case

Lagrangian subbundles of symplectic bundles over a curve

A symplectic bundle over an algebraic curve has a natural invariant s Lag determined by the maximal degree of its Lagrangian subbundles. This can be viewed as a generalization of the classical Segre invariants of a vector bundle. We give a sharp upper bound on s Lag which is analogous to the Hirschowitz bound on the classical Segre invariants. Furthermore, we study the stratifications induced by s Lag on moduli spaces of symplectic bundles, and get a full picture for the case of rank fou