Open X-ranks with respect to Segre and Veronese varieties


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).

DOI Code: 10.1285/i15900932v41n1p19

Keywords: open rank; open $X$-rank; Segre variety; Veronese variety; secant variety; border rank

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