The variety of polar simplices

  • Kristian Ranestad

  • Frank-Olaf Schreyer

    053 Campus E2 4 Blindern D-66123 Saarbrücken NO-0316 Oslo Germany Norway
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A collection of n distinct hyperplanes Li=li=0\PPn1L_i = {l_i=0} \subset \PP^{n-1}, the (n1)(n-1)-dimensional projective space over an algebraically closed field of characteristic not equal to 2, is a polar simplex of a smooth quadric Qn2=q=0Q^{n-2}={q=0}, if each LiL_i is the polar hyperplane of the point pi=jiLjp_i = \bigcap_{j \ne i} L_j, equivalently, if q=l12++ln2q= l_1^2+\ldots+l_n^2 for suitable choices of the linear forms lil_i. In this paper we study the closure VPS(Q,n)\Hilbn(\PPˇn1)VPS(Q,n) \subset \Hilb_{n}(\check \PP^{n-1}) of the variety of sums of powers presenting QQ from a global viewpoint: VPS(Q,n)VPS(Q,n) is a smooth Fano variety of index 2 and Picard number 1 when n<6n<6, and VPS(Q,n)VPS(Q,n) is singular when n6n\geq 6.

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Kristian Ranestad, Frank-Olaf Schreyer, The variety of polar simplices. Doc. Math. 18 (2013), pp. 469–505

DOI 10.4171/DM/406