Unrestrictions and concise secant varieties

TL;DR

This paper introduces concise secant varieties, offering a modular approach to desingularize secant varieties of Segre embeddings, with implications for tensor rank characterization and algebraic geometry.

Contribution

It defines concise secant varieties as partial desingularizations, linking them to tensor border rank and providing new characterizations and tools in tensor theory.

Findings

Characterization of border rank ≤ r tensors as unrestrictions of minimal border rank r tensors.

Concise versions of border apolarity and Varieties of Sums of Powers.

Connections to defectivity, identifiability, and the Salmon conjecture.

Abstract

We introduce the concise secant varieties, which are, informally speaking, modular partial desingularisations of secant varieties to Segre embeddings. More precisely, they are projective and birational to the abstract secant varieties, yet each of their points corresponds to a concise tensor of appropriate border rank (that is, to a minimal border rank tensor). We discuss implications throughout the theory of tensors, including a characterisation of border rank $\leq r$ tensors as unrestrictions of minimal border rank $r$ tensors (also in the Veronese and Segre-Veronese cases), a characterisation of tensors with cactus rank $\leq r$, concise versions of border apolarity including the fixed point theorem, concise Varieties of Sums of Powers, counting points on the second secant variety, connections to defectivity and identifiability in the Segre case, to the Salmon conjecture etc.