Fano schemes of symmetric matrices of bounded rank

Abstract

We study the geometry of the Fano schemes $\mathrm{\textbf{F}}_{k}(\mathrm{SD}_n^r)$ of the projective variety $\mathrm{SD}_n^r$ defined by the $r\times r$ minors of a symmetric $n\times n$ matrix filled with indeterminates. These schemes are fine moduli spaces parameterizing $(k+1)$-dimensional linear spaces of $n\times n$ symmetric matrices of rank less than $r$. We prove that the schemes $\mathrm{\textbf{F}}_{k}(\mathrm{SD}_n^r)$ can have generically non-reduced components, and characterize their irreducibility, connectedness, and smoothness. Our approach to connectedness also applies to Fano schemes of rectangular matrices as well as alternating matrices and answers a question of Ilten and Chan. Furthermore, we give a complete description of $\mathrm{\textbf{F}}_{1}(\mathrm{SD}_n^r)$ and show that when $r=n$, the Fano schemes of lines have the expected dimension. As an application, we provide geometric arguments for several previous results concerning spaces of symmetric matrices of bounded rank.