Generalized varieties of sums of powers

Let $X\subset {\mathbb{P}}^N$ be an irreducible, non-degenerate variety. The generalized variety of sums of powers $VS{P}_H^X(h)$ of $X$ is the closure in the Hilbert scheme $Hil{b}_h(X)$ of the locus parametrizing collections of points $\{x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_h\}$ such that the $(h-1)$-plane $\left\langle x_{1},…,x_{h}\right\rangle$passestroughafixedgeneralpoint$p\in$\mathbb{P}$^{N}$\mathrm{.}When$X = V_{d}^{n}$ 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 $VS{P}_H^X(h)$. In particular we will show how some birational properties, such as rationality, unirationality and rational connectedness, of $VS{P}_H^X(h)$ are inherited from the birational geometry of variety $X$ itself.