The Variety of Polar Simplices
نویسندگان
چکیده
A collection of n distinct hyperplanes Li = {li = 0} ⊂ P , 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 = {q = 0}, if each Li is the polar hyperplane of the point pi = ⋂ j 6=i Lj , equivalently, if q = l 2 1 + . . .+ l 2 n for suitable choices of the linear forms li. In this paper we study the closure V PS(Q,n) ⊂ Hilbn(P̌) of the variety of sums of powers presenting Q from a global viewpoint: V PS(Q,n) is a smooth Fano variety of index 2 and Picard number 1 when n < 6, and V PS(Q,n) is singular when n ≥ 6. 2010 Mathematics Subject Classification: 14J45, 14M
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