Generalized varieties of sums of powers

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Let Xⁿ be an irreducible, non-degenerate variety. The generalized variety of sums of powers VSPₕˣ(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 trough a fixed general point pⁿ. When X = Vdⁿ 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 VSPₕˣ(h). In particular we will show how some birational properties, such as rationality, unirationality and rational connectedness, of VSPₕˣ(h) are inherited from the birational geometry of variety X itself.