#### DOCUMENTA MATHEMATICA, Vol. 18 (2013), 469-505

Kristian Ranestad and Frank-Olaf Schreyer

The Variety of Polar Simplices

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\$.

2010 Mathematics Subject Classification: 14J45, 14M

Keywords and Phrases: Fano n-folds, Quadric, polar simplex, syzygies

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