Fano schemes of symmetric matrices of bounded rank

Abstract

We study the geometry of the Fano schemes ${\text{F}}_k({\mathrm{SD}}_n^r)$ 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 ${\text{F}}_k({\mathrm{SD}}_n^r)$ 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 ${\text{F}}_1({\mathrm{SD}}_n^r)$ 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.

45 pages, 11 figures, 3 tables This version proves the statement on classifying the smoothness of the Fano schemes of symmetric matrices of bounded rank previously stated as a conjecture (Conjecture 7.2 in v1)