[Paper Review] On the locus of points of high rank

Jarosław Buczyński, Kangjin Han+2 more|arXiv (Cornell University)|Mar 8, 2017

Tensor decomposition and applications17 references4 citations

TL;DR

This paper investigates the loci of points of high rank with respect to a nondegenerate irreducible projective variety $X$, focusing on ranks exceeding the generic rank. It establishes a nesting structure for high-rank loci $W_k$ ($k > g$), proves dimension bounds, and applies these to determine maximal ranks for curves and homogeneous varieties, including Veronese and Segre embeddings, showing that high-rank points are more numerous than previously thought.

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.

Motivation & Objective

• To understand the structure and geometry of points of rank higher than the generic rank with respect to a projective variety $X$.
• To analyze the loci $W_k = \overline{\{p \in \mathbb{P}^N : \operatorname{rank}_X(p) = k\}}$ for $k > g$, where $g$ is the generic rank.
• To bound the dimensions of high-rank loci and establish containments between them and secant varieties.
• To determine the maximal possible rank for specific classes of varieties, including curves and homogeneous spaces.
• To provide a lower bound on the dimension of the $\mathrm{GL}_n$-orbit of a concise homogeneous form, even when the hypersurface is singular or reducible.

Proposed method

• Define the rank of a point $p$ with respect to $X$ as the minimal number of points on $X$ whose linear span contains $p$.
• Introduce the loci $W_k$ as the closure of the set of points of rank exactly $k$, extending the classical notion of secant varieties beyond the generic rank.
• Establish a key nesting result: for $k > g$, the join of $W_k$ and $X$ is contained in $W_{k-1}$, implying $W_k \subseteq W_{k-1}$.
• Use this nesting to derive upper bounds on the dimensions of $W_k$ for $k > g$.
• Apply representation-theoretic and geometric techniques to analyze orbits of homogeneous forms under $\mathrm{GL}_n$ action.
• Use projection and adjunction arguments on quadric surfaces to study rank loci for curves of bidegree $(a,b)$.

Experimental results

Research questions

• RQ1 What is the structure of the locus $W_k$ for $k > g$, the generic rank, and how do these loci relate to secant varieties?
• RQ2 Can the dimension of $W_k$ for $k > g$ be bounded, and what are the implications for the geometry of high-rank points?
• RQ3 What is the maximal possible rank of a tensor or symmetric tensor with respect to a given variety $X$, particularly when $X$ is a curve or a homogeneous space?
• RQ4 How large is the locus of points achieving the maximal rank, and can a lower bound on its dimension be established?
• RQ5 What is the dimension of the $\mathrm{GL}_n$-orbit of a concise homogeneous form, even when the associated hypersurface is singular or non-reduced?

Key findings

• The high-rank loci $W_k$ for $k > g$ are nested: $W_k \subseteq W_{k-1}$, and more strongly, the join of $W_k$ and $X$ is contained in $W_{k-1}$.
• For a general curve of bidegree $(a,b)$ on a smooth quadric surface with $a \geq 4$, $b \geq 1$, the locus $W_3$ is empty, so the maximal rank is 2.
• For a general elliptic quartic curve in $\mathbb{P}^3$ (bidegree $(2,2)$), $W_3$ is a curve of degree 8, disjoint from the original curve, with exactly 4 points of rank 2.
• The locus of symmetric tensors of maximal rank has dimension at least $\dim \mathrm{GL}_n / \mathrm{Stab}(F)$, and this is bounded below even when $F$ is not smooth.
• The dimension of the locus of $2 \times 2n \times 2n$ tensors of maximal rank is computed explicitly.
• For $2 \times 4 \times 4$ tensors, the paper fully characterizes all points of rank greater than the generic rank, showing they are contained in a specific algebraic set.