On the locus of points of high rank
Abstract
Given a closed subvariety X in a projective space, the rank with respect to X of a point p in this projective space is the least integer r such that p lies in the linear span of some r points of X. Let W_k be the closure of the set of points of rank with respect to X equal to k. For small values of k such loci are called secant varieties. This article studies the loci W_k for values of k larger than the generic rank. We show they are nested, we bound their dimensions, and we estimate the maximal possible rank with respect to X in special cases, including when X is a homogeneous space or a curve. The theory is illustrated by numerous examples, including Veronese varieties, the Segre product of dimensions (1,3,3), and curves. An intermediate result provides a lower bound on the dimension of any GL_n orbit of a homogeneous form.
Keywords
Cite
@article{arxiv.1703.02829,
title = {On the locus of points of high rank},
author = {Jarosław Buczyński and Kangjin Han and Massimiliano Mella and Zach Teitler},
journal= {arXiv preprint arXiv:1703.02829},
year = {2017}
}
Comments
28 pages