The variety of polar simplices

Abstract

A collection of n distinct hyperplanes $L_i = {l_i=0} \subset \PP^{n-1}$, 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 $q= l_1^2+\ldots+l_n^2$ for suitable choices of the linear forms $l_i$. In this paper we study the closure $VPS(Q,n) \subset \Hilb_{n}(\check \PP^{n-1})$ 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 $n\geq 6$.