The Variety of Polar Simplices

TL;DR

This paper investigates the geometric structure and properties of the variety of polar simplices associated with a quadric in projective space, revealing its smoothness, singularities, and classification for different dimensions.

Contribution

It provides a detailed study of the closure of the variety of sums of powers representing a quadric, including its smoothness, Fano property, and singularities for various dimensions.

Findings

VPS(Q,n) is a smooth Fano variety of index 2 for n<6.

VPS(Q,n) is singular when n≥6.

The geometric properties depend on the dimension n.

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.