22 apr 2016 -- 11:30
Aula 103, Dip di Ingegneria dell'Informazione e Sc Mat, Siena
Abstract.
Varieties of sums of powers, $VSP (F, h) $ for short, describe the additive decompositions of a general homogeneous polynomial $F$ into $ h $ powers of linear forms. Studies on these varieties date back to Sylvester and Hilbert, but only a few of them, for certain degrees and numbers of variables, have been concretely identified. Here we investigate the general birational behavior of $V SP $.To do this, we birationally embed the varieties in Grassmannians and prove the rational connectedness of many $V SP $ of arbitrary degrees and number of variables. Now, let $X ⊂ P^N$ be an irreducible, non-degenerate variety. The generalized variety of sums of powers $VSPHX(h) $ of $X $ is the closure in the Hilbert scheme $Hilb^h(X) $ of the locus parametrizing collections of points $\{x_1, \ldots , x_h \}$ such that the $(h−1)$-plane $<x_1, \ldots,x_h >$ passes through a fixed general point $p \in P^N$ . When $X = V_{d}^{n}$ is a Veronese variety we recover the classical variety of sums of powers $V SP (F, h)$ parametrizing additive decompositions of a homogeneous polynomial as powers of linear forms. In this paper we study the birational behavior of $V SPHX (h)$. In particular, we show how some birational properties, such as rationality, unirationality and rational connectedness, of $V SPHX (h) $ are inherited from the birational geometry of variety $X $ itself.