Homogeneous projective varieties with semi-continuous rank function

TL;DR

This paper classifies homogeneous projective varieties with tame secant varieties, identifying all cases where the secant variety equals the union of lower-rank strata, focusing on equivariant embeddings of reductive group orbits.

Contribution

It provides a complete classification of equivariantly embedded homogeneous varieties with tame secant varieties, extending classical examples to a broader algebraic setting.

Findings

Classified all homogeneous varieties with tame secant varieties.

Identified classical and new examples of such varieties.

Connected secant variety properties to representation theory of reductive groups.

Abstract

Let $\mathbb X\subset\mathbb P(V)$ be a projective variety, which is not contained in a hyperplane. Then every vector $v$ in $V$ can be written as a sum of vectors from the affine cone $X$ over $\mathbb X$. The minimal number of summands in such a sum is called the rank of $v$. The set of vectors of rank $r$ is denoted by $X_r$ and its projective image by $\mathbb X_r$. The r-th secant variety of $X$ is defined $\sigma_r(\mathbb X):=\bar{\sqcup_{s\le r}\mathbb X_s}$; it is called tame if $\sigma_r(\mathbb X)=\sqcup_{s\le r} \mathbb X_s$ and wild if the closure contains elements of higher rank. In this paper, we classify all equivariantly embedded homogeneous projective varieties $\mathbb X\subset\mathbb P(V)$ with tame secant varieties. Classical examples are: the variety of rank one matrices (Segre variety with two factors) and the variety of rank one quadratic forms (quadratic Veronese variety). In the general setting, $\mathbb X$ is the orbit in $\mathbb P(V)$ of a highest weight line in an irreducible representation $V$ of a reductive algebraic group $G$. Thus, our result is a list of all irreducible representations of reductive groups, where the resulting $\mathbb X$ has tame secant varieties.