On the number of symmetric presentations of a determinantal hypersurface

Abstract

A hypersurface $H$ in $\mathbb{P}^r$ of degree $n$ is called determinantal if it is the zero locus of a polynomial of the form $\operatorname{det}(x_0 A_0 + \ldots + x_r A_r)$ for some $(r+1)$-tuple of $n \times n$ matrices $A = (A_0, \ldots, A_r)$. We will refer to $A$ as a presentation of $H$. Another presentation $B = (B_0, B_1, \ldots, B_r)$ of $H$ can be obtained by choosing $g_1, g_2 \in \operatorname{GL}_n$ and setting $B_i = g_1 A_i g_2$ for every $i = 0, 1, \ldots, r$. In this case $A$ and $B$ are called equivalent. The second author and A. Vistoli have shown that for $r \geq 3$ a general determinantal hypersurface admits only finitely many presentations up to equivalence. In this paper we prove a similar result for symmetric presentations for every $r \geq 2$. Here the matrices $A_0, \ldots, A_r$ are required to be symmetric, and two $(r+1)$-tuples of $n \times n$ symmetric matrices $A = (A_0, A_1, \ldots, A_r)$ and $B = (B_0, B_1, \ldots, B_r)$ are considered equivalent if there exists a $g \in \operatorname{GL}_n$ such that $B_i = g^{\rm transpose} A_i g$ for every $i = 0, \ldots, r$.