The variety of polar simplices

Abstract

A collection of n distinct hyperplanes $L_i=l_i=0\subset {\mathbf{P}}^{n-1}$, the $(n-1)$-dimensional projective space over an algebraically closed field of characteristic not equal to 2, is a polar simplex of a smooth quadric $Q^{n-2}=q=0$, if each $L_i$ is the polar hyperplane of the point $p_i={\bigcap }_{j\ne i}{L}_j$, equivalently, if $q=l_1^2+\dots +l_n^2$ for suitable choices of the linear forms $l_i$. In this paper we study the closure $VPS(Q,n)\subset {\mathrm{Hilb}}_n({\check{\mathbf{P}}}^{n-1})$ of the variety of sums of powers presenting $Q$ from a global viewpoint: $VPS(Q,n)$ is a smooth Fano variety of index 2 and Picard number 1 when $n<6$, and $VPS(Q,n)$ is singular when $n\ge 6$.

A correction to this paper is available.