Generalized varieties of sums of powers

Let X ⊂ PNbe an irreducible, non-degenerate variety. The generalized variety of sums of powers V S PHX(h) of X is the closure in the Hilbert scheme Hilbh(X) of the locus parametrizing collections of points x1,.., xh such that the (h -1)-plane >x1,.., xh> passes through a fixed general point p ∈ PN. When X = Vdnis a Veronese variety we recover the classical variety of sums of powers V S P(F, h) parametrizing additive decompositions of a homogeneous polynomial as powers of linear forms. In this paper we study the birational behavior of V S PHX(h). In particular, we show how some birational properties, such as rationality, unirationalityand rational connectedness, of V S PHX(h) are inherited from the birational geometry of variety X itself.