The variety of polar simplices

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+\ldots+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\geq 6 .