The $b$-secant variety of a smooth curve has a codimension $1$ locally closed subset whose points have rank at least $b+1$

TL;DR

This paper proves that within the $b$-secant variety of a smooth projective curve, there exists a codimension 1 subset where points have rank at least $b+1$, revealing a nuanced structure of tensor ranks.

Contribution

It establishes the existence of a codimension 1 subset in the $b$-secant variety where points have rank exceeding $b$, providing new insights into the geometry of secant varieties.

Findings

Existence of a codimension 1 subset with higher rank

Points in this subset have rank at least $b+1$

Results apply to smooth, non-degenerate projective curves

Abstract

Take a smooth, connected and non-degenerate projective curve $X\subset \mathbb {P}^r$, $r\ge 2b+2\ge 6$, defined over an algebraically closed field with characteristic $0$ and let $\sigma _b(X)$ be the $b$-secant variety of $X$. We prove that the $X$-rank of $q$ is at least $b+1$ for a non-empty codimension $1$ locally closed subset of $\sigma _b(X)$.