Secants of Lagrangian Grassmannians

TL;DR

This paper investigates the dimensions of secant varieties of Lagrangian Grassmannians, providing explicit calculations for third and fourth secant varieties through normal forms and tangent space analysis.

Contribution

It introduces a normal form for four general points on Lagrangian Grassmannians and computes tangent spaces to determine secant dimensions.

Findings

Dimensions of third and fourth secant varieties are explicitly calculated.

Normal form for four general points on Lagrangian Grassmannians is established.

Tangent space analysis is used to determine secant dimensions.

Abstract

We study the dimensions of secant varieties of the Grassmannian of Lagrangian subspaces in a symplectic vector space. We calculate these dimensions for third and fourth secant varieties. Our result is obtained by providing a normal form for four general points on such a Grassmannian and by explicitly calculating the tangent spaces at these four points.