Ranks on the boundaries of secant varieties

TL;DR

This paper investigates the rank of points on the boundaries of secant varieties, establishing conditions under which the X-rank exceeds a certain threshold for specific embeddings like Segre varieties.

Contribution

It proves that for many embeddings, hypersurfaces of secant varieties have X-rank greater than the secant order, using join and tangential variety properties.

Findings

X-rank of general points on joins exceeds secant order

Hypersurfaces of secant varieties have higher X-rank in key cases

Results apply to Segre and multiprojective space embeddings

Abstract

In many cases (e.g. for many Segre or Segre embeddings of multiprojective spaces) we prove that a hypersurface of the $b$-secant variety of $X\subset \mathbb {P}^r$ has $X$-rank $>b$. We prove it proving that the $X$-rank of a general point of the join of $b-2$ copies of $X$ and the tangential variety of $X$ is $>b$.