Geometric conditions for strict submultiplicativity of rank and border rank

TL;DR

This paper investigates geometric conditions under which the rank and border rank of points in projective space exhibit strict submultiplicativity, with specific results for points of rank two and embedded curves.

Contribution

It characterizes strict submultiplicativity for rank two points via trisecant lines and shows that for curves in even-dimensional space, strict submultiplicativity generally occurs except for rational normal curves.

Findings

Strict submultiplicativity is characterized by trisecant lines for rank two points.

For curves in even-dimensional space, strict submultiplicativity occurs except for rational normal curves.

The study provides geometric criteria for understanding rank behavior in projective varieties.

Abstract

The $X$-rank of a point $p$ in projective space is the minimal number of points of an algebraic variety $X$ whose linear span contains $p$. This notion is naturally submultiplicative under tensor product. We study geometric conditions that guarantee strict submultiplicativity. We prove that in the case of points of rank two, strict submultiplicativity is entirely characterized in terms of the trisecant lines to the variety. Moreover, we focus on the case of curves: we prove that for curves embedded in an even-dimensional projective space, there are always points for which strict submultiplicativity occurs, with the only exception of rational normal curves.