# Counting maximal Lagrangian subbundles over an algebraic curve

## TL;DR

This paper derives a formula for intersection numbers on a Lagrangian Quot scheme over a curve and computes the number of maximal degree Lagrangian subbundles for general stable symplectic bundles, extending previous enumerations.

## Contribution

It provides a closed formula for intersection numbers on Lagrangian Quot schemes and calculates the count of maximal degree Lagrangian subbundles in a symplectic setting.

## Key findings

- Closed formula for intersection numbers on Lagrangian Quot schemes
- Number of maximal degree Lagrangian subbundles for general stable symplectic bundles
- Extension of Holla's enumeration to symplectic bundles

## Abstract

Let $C$ be a smooth projective curve and $W$ a symplectic bundle over $C$. Let $LQ_e (W)$ be the Lagrangian Quot scheme parametrizing Lagrangian subsheaves $E \subset W$ of degree $e$. We give a closed formula for intersection numbers on $LQ_e (W)$. As a special case, for $g \ge 2$, we compute the number of Lagrangian subbundles of maximal degree of a general stable symplectic bundle, when this is finite. This is a symplectic analogue of Holla's enumeration of maximal subbundles in [13].