The Variety of Polar Simplices
Algebraic Geometry
2012-04-26 v2
Abstract
A collection of n distinct hyperplanes L_i ={l_i=0} in 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 quadric Q={q=0}, if each L_i is the polar hyperplane of the point p_i, the intersection point of the L_j with j different from i, equivalently, if q= l_1^2+...+l_n^2 for suitable choices of the linear forms l_i. In this paper we study the closure VPS(Q,n) in Hilb_n(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>= 6.
Cite
@article{arxiv.1104.2728,
title = {The Variety of Polar Simplices},
author = {Kristian Ranestad and Frank-Olaf Schreyer},
journal= {arXiv preprint arXiv:1104.2728},
year = {2012}
}
Comments
33 pages, a revised presentation