The Variety of Polar Simplices

2011/04/14 by Kristian Ranestad, Frank-Olaf Schreyer, Frank–Olaf Schreyer +2
Engineering · Mathematics · #14J45 #14M #Advanced Theoretical and Applied Studies in Material Sciences and Geometry #Algebraic Geometry (math.AG) #FOS: Mathematics #Graph theory and applications #graph theory and CDMA systems #math.AG #msc:14J45 #msc:14M

paper · pdf · doi:10.48550/arxiv.1104.2728

33 pages, a revised presentation

openalex publication_date 2011/04/14 · arxiv created 2012/04/25 · arxiv updated 2012/04/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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= l₁²+...+l_n² 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.

Related

• Polar varieties, Bertini's theorems and number of points of singular complete intersections over a finite field
• Local polar varieties in the geometric study of singularities
• The variety of polar simplices II
• On the geometry of polar varieties
• Solvable points on intersections of quadrics, cubics, and quartics