Geometry of the Lagrangian Grassmannian Sp(3)/U(3) with applications to Brill-Noether loci

TL;DR

This paper investigates the geometry of the Lagrangian Grassmannian Sp(3)/U(3), linking it to Brill-Noether loci and providing new examples and insights into their structure and relationships with other varieties.

Contribution

It offers a detailed geometric analysis of Sp(3)/U(3), connecting Brill-Noether loci to linear sections and introducing a key technical result about nodal hyperplane sections.

Findings

Brill-Noether loci of vector bundles relate to orthogonal linear sections.

Nodal hyperplane sections of Sp(3)/U(3) are linear sections of Gr(2,6).

New examples of non-abelian Brill-Noether loci are provided.

Abstract

The geometry of Sp(3)/U(3) as a subvariety of Gr(3,6) is explored to explain several examples given by Mukai of non-abelian Brill-Noether loci, and to give some new examples. These examples identify Brill-Noether loci of vector bundles on linear sections of the Lagrangian Grassmannian Sp(3)/U(3) with orthogonal linear sections of the dual variety and vice versa. A main technical result of independent interest is the fact that any nodal hyperplane section of the Lagrangian Grassmannian projected from the node is a linear section of the Grassmannian Gr(2,6).