Generalized varieties of sums of powers

2014/01/09 by Alex Massarenti, Massarenti, Alex
Mathematics · #14D06 #14E08 #14M20 #14N05 #14N25 #Algebraic Geometry (math.AG) #FOS: Mathematics #Primary 14M22 #Secondary 14D23 #math.AG #msc:14D06 #msc:14D23 #msc:14E08 #msc:14M20 #msc:14M22 #msc:14N05 #msc:14N25

paper · pdf · doi:10.48550/arxiv.1401.2059

25 pages

arxiv created 2014/01/09 · arxiv updated 2014/01/10

Abstract

Let X⊂ℙ^N be an irreducible, non-degenerate variety. The generalized variety of sums of powers VSP_H^X(h) of X is the closure in the Hilbert scheme Hilb_h(X) of the locus parametrizing collections of points \x₁,...,x_h\ such that the (h-1)-plane ⟨ x₁,...,x_h⟩ passes trough a fixed general point p∈ℙ^N. When X = V_dⁿ is a Veronese variety we recover the classical variety of sums of powers VSP(F,h) parametrizing additive decompositions of a homogeneous polynomial as powers of linear forms. In this paper we study the birational behavior of VSP_H^X(h). In particular we will show how some birational properties, such as rationality, unirationality and rational connectedness, of VSP_H^X(h) are inherited from the birational geometry of variety X itself.

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