On the number of symmetric presentations of a determinantal hypersurface

摘要

A hypersurface H in 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) + ... + x(r)A(r)) for some (r + 1)-tuple of n x n matrices A = (A(0), . . . , A(r)). We will refer to A as a presentation of H. Another presentation B = (B-0, B-1, . . . , B-r) of H can be obtained by choosing g(1), g(2) is an element of GL(n), and setting B-i = g(1)A(i)g(2) for every i = 0, 1, . . . , r. In this case A and B are called equivalent. The second author and A. Vistoli have shown that for r >= 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 >= 2. Here the matrices A(0), . . . , A(r) are required to be symmetric, and two (r + 1)-tuples of n x n symmetric matrices A = (A(0), A(1), . . . , A(r)) and B = (B-0, B-1, . . . , B-r) are considered equivalent if there exists a g is an element of GL(n) such that B-i = g(t)A(i)g for every i = 0, . . . , r. (C) 2021 Elsevier Inc. All rights reserved.