Equations for secant varieties of Chow varieties

The Chow variety of polynomials that decompose as a product of linear forms has been studied for more than 100 years. Finding equations in the ideal of secant varieties of Chow varieties would enable one to prove Valiant's conjecture [Formula: see text]. In this paper, I use the method of prolongation to obtain equations for secant varieties of Chow varieties as [Formula: see text]-modules.

Tropical Chow Hypersurfaces

Given a projective variety $X\subset\mathbb{P}^n$ of codimension $k+1$, the Chow hypersurface $Z_X$ is the hypersurface of the Grassmannian $\operatorname{Gr}(k, n)$ parametrizing projective linear spaces that intersect $X$. We introduce the tropical Chow hypersurface $\operatorname{Trop}(Z_X)$. This object only depends on the tropical variety $\operatorname{Trop}(X)$ and we provide an explicit way to obtain $\operatorname{Trop}(Z_X)$ from $\operatorname{Trop}(X)$. We also give a geometric description of $\operatorname{Trop}(Z_X)$. We conjecture that, as in the classical case, $\operatorname{Trop}(X)$ can be reconstructed from $\operatorname{Trop}(Z_X)$ and prove it for the case when $X$ is a curve in $\mathbb{P}^3$. This suggests that tropical Chow hypersurfaces could be the key to construct a tropical Chow variety.

Generic forms of low Chow rank

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 [Formula: see text] for which the [Formula: see text]th 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 [Formula: see text].