Irreducibility of Lagrangian Quot schemes over an algebraic curve

TL;DR

This paper proves that Lagrangian Quot schemes over a smooth algebraic curve are irreducible and smooth for large degrees, with generic elements being saturated and stable, advancing understanding of their geometric structure.

Contribution

It establishes the irreducibility and generic smoothness of Lagrangian Quot schemes over curves for large degrees, a significant step in their geometric analysis.

Findings

Lagrangian Quot schemes are irreducible for large e.

They are generically smooth of the expected dimension.

A generic element in these schemes is saturated and stable.

Abstract

Let $C$ be a complex projective smooth curve and $W$ a symplectic vector bundle of rank $2n$ over $C$. The Lagrangian Quot scheme $LQ_{-e}(W)$ parameterizes subsheaves of rank $n$ and degree $-e$ which are isotropic with respect to the symplectic form. We prove that $LQ_{-e}(W)$ is irreducible and generically smooth of the expected dimension for all large $e$, and that a generic element is saturated and stable.