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

TL;DR

This paper investigates the stratification of moduli spaces of symplectic and orthogonal bundles over a curve using an invariant related to isotropic subbundles, establishing bounds, dimensions, and geometric interpretations.

Contribution

It introduces a new stratification related to secant varieties, provides a sharp upper bound on the invariant, and analyzes the structure and dimensions of the strata, including geometric insights.

Findings

Established a sharp upper bound on the invariant t(V).

Computed the dimension of each stratum in the stratification.

Provided a geometric interpretation of maximal Lagrangian subbundles.

Abstract

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.