Complete symplectic quadrics and Kontsevich spaces of conics in Lagrangian Grassmannians

TL;DR

This paper constructs a smooth compactification of the space of symmetric and symplectic matrices and explores its geometric properties, applying these results to understand the birational geometry of Kontsevich spaces of conics in Lagrangian Grassmannians.

Contribution

It introduces a new wonderful compactification for symmetric and symplectic matrix spaces and analyzes its geometric structure, with applications to Kontsevich spaces in symplectic geometry.

Findings

Constructed a smooth projective compactification with normal crossing boundary.

Described the birational geometry of Kontsevich spaces of conics in Lagrangian Grassmannians.

Provided geometric insights into the structure of symplectic and symmetric matrix spaces.

Abstract

A wonderful compactification of an orbit under the action of a semi-simple and simply connected group is a smooth projective variety containing the orbit as a dense open subset, and where the added boundary divisor is simple normal crossing. We construct the wonderful compactification of the space of symmetric and symplectic matrices, and investigate its geometry. As an application, we describe the birational geometry of the Kontsevich spaces parametrizing conics in Lagrangian Grassmannians.