Compactifications of spaces of symmetric matrices and pointed Kontsevich spaces of isotropic Grassmannians

TL;DR

This paper constructs and analyzes compactifications of spaces of symmetric matrices and pointed Kontsevich spaces of isotropic Grassmannians, revealing their birational geometry and modular interpretations.

Contribution

It introduces a Kausz--type compactification of symmetric matrices and links it to Kontsevich spaces, providing new insights into their birational geometry.

Findings

Explicit description of the birational geometry of the compactification

Realization of the compactification as a fiber in a Kontsevich space

Applications to the geometry of pointed conics in Lagrangian Grassmannians

Abstract

We study two closely related families of varieties arising from genus $0$ stable maps to the Lagrangian Grassmannian $\operatorname{LG}(n,2n)$. First, we construct the Kausz--type compactification $\mathcal {TL}_n$ of the space of symmetric matrices and give an explicit description of its birational geometry. Second, we realize $\mathcal {TL}_n$ as a general evaluation fiber in a Kontsevich space, and then exploit this modular interpretation to derive consequences for the birational geometry of the space of pointed conics $\overline{M}_{0,1}(\operatorname{LG}(n,2n),2)$. Analogous compactifications related to orthogonal Grassmannians are also presented.