Generalized varieties of sums of powers

Abstract

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 $\{x_1,...,x_h\}$ such that the $(h-1)$-plane $⟨x_1,...,x_h⟩$ passes trough a fixed general point $p\in {\mathbb{P}}^N$. 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.

arXiv comment25 pages