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 $\det (x_0{A}_0+\dots +x_r{A}_r)$ for some $(r+1)$-tuple of $n\times n$ matrices $A=(A_0,\dots ,A_r)$. We will refer to $A$ as a presentation of $H$. Another presentation $B=(B_0,B_1,\dots ,B_r)$ of $H$ can be obtained by choosing $g_1,g_2\in {\mathrm{GL}}_n$ and setting $B_i=g_1{A}_i{g}_2$ for every $i=0,1,\dots ,r$. In this case $A$ and $B$ are called equivalent. The second author and A. Vistoli have shown that for $r\ge 3$ 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 $r\ge 2$. Here the matrices $A_0,\dots ,A_r$ are required to be symmetric, and two $(r+1)$-tuples of $n\times n$ symmetric matrices $A=(A_0,A_1,\dots ,A_r)$ and $B=(B_0,B_1,\dots ,B_r)$ are considered equivalent if there exists a $g\in {\mathrm{GL}}_n$ such that $B_i=g^{\mathrm{transpose}}{A}_{i}g$ for every $i=0,\dots ,r$.

8 pages