Generic forms of low Chow rank

TL;DR

This paper investigates the Chow rank of generic forms, establishing that for low secant orders, the secant variety of the Chow variety generally has the expected dimension, except for known special cases.

Contribution

It proves that, aside from known exceptions, the secant variety of the Chow variety attains the expected dimension for low secant orders, advancing understanding of Chow rank.

Findings

Secant variety has expected dimension for low s values

Identifies known exceptions where the dimension differs

Provides insights into the structure of generic forms' Chow rank

Abstract

The least number of products of linear forms that may be added together to obtain a given form is the Chow rank of this form. The Chow rank of a generic form corresponds to the smallest s for which the sth secant variety of the Chow variety fills the ambient space. We show that, except for certain known exceptions, this secant variety has the expected dimension for low values of s.