On differences between the border rank and the smoothable rank of a polynomial

TL;DR

This paper investigates the difference between border rank and smoothable rank of polynomials, providing explicit examples and proving equality for certain cases, which enhances understanding of secant varieties to Veronese varieties.

Contribution

It presents the first explicit example of a polynomial where smoothable rank exceeds border rank and proves equality for all cubics in up to four variables.

Findings

Explicit example of a cubic with border rank 5 and smoothable rank 6

All cubics in at most four variables have equal border and smoothable ranks

Insights into the structure of secant varieties to Veronese varieties

Abstract

We consider higher secant varieties to Veronese varieties. Most points on the r-th secant variety are represented by a finite scheme of length r contained in the Veronese variety --- in fact, for generic point, it is just a union of r distinct points. A modern way to phrase it is: the smoothable rank is equal to the border rank for most polynomials. This property is very useful for studying secant varieties, especially, whenever the smoothable rank is equal to the border rank for all points of the secant variety in question. In this note we investigate those special points for which the smoothable rank is not equal to the border rank. In particular, we show an explicit example of a cubic in five variables with border rank 5 and smoothable rank 6. We also prove that all cubics in at most four variables have the smoothable rank equal to the border rank.