Open $X$-ranks with respect to Segre and Veronese varieties

Abstract

Let $X\subset \mathbb{P}^N$ be an integral and non-degenerate variety. Recall (A. Bialynicki-Birula, A. Schinzel, J. Jelisiejew and others) that for any $q\in \mathbb{P}^N$ the open rank $or_X(q)$ is the minimal positive integer such that for each closed set $B\subsetneq X$ there is a set $S\subset X\setminus B$ with $\#S\le or_X(q)$ and $q\in \langle S\rangle$, where $\langle \ \ \rangle$ denotes the linear span. For an arbitrary $X$ we give an upper bound for $or_X(q)$ in terms of the upper bound for $or_X(q')$ when $q'$ is a point in the maximal proper secant variety of $X$ and a similar result using only points $q'$ with submaximal border rank. We study $or_X(q)$ when $X$ is a Segre variety (points with $X$-rank $1$ and $2$) and when $X$ is a Veronese variety (points with $X$-rank $\le 3$ or with border rank $2$).