Birational aspects of the Geometry of Varieties of Sum of Powers

TL;DR

This paper investigates the birational geometry of Varieties of Sums of Powers, establishing their rationality, unirationality, or rational connectedness across various degrees and variables by embedding into Grassmannians.

Contribution

It introduces a birational embedding of VSP into Grassmannians and systematically analyzes their birational properties for general degrees and variables.

Findings

Many VSP varieties are shown to be rational, unirational, or rationally connected.

The paper provides a unified approach to understanding the birational geometry of VSP.

Results apply to a broad class of degrees and numbers of variables.

Abstract

Varieties of Sums of Powers describe the additive decompositions of an homogeneous polynomial into powers of linear forms. Despite their long history, going back to Sylvester and Hilbert, few of them are known for special degrees and number of variables. In this paper we aim to understand a general birational behaviour of VSP, if any. To do this we birationally embed these varieties into Grassmannians and prove the rationality, unirationality or rational connectedness of many of those in arbitrary degrees and number of variables.