Title: 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.
I will talk about the loci W_k for values of k larger than the generic rank.
They are nested, I will show a bound on their dimensions, and an estimate
of the maximal possible rank with respect to X in special cases,
including when X is a homogeneous space (particularly, Veronese variety)
or a curve. The theory will be illustrated by 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.
Based on a joint work with Kangjin Han, Massimiliano Mella, and Zach Teitler:
https://arxiv.org/abs/1703.02829 and perhaps also
https://arxiv.org/abs/1503.08253 .