Unrestrictions and concise secant varieties

2026-04-27

AI Summary

This study investigates the border rank and cactus rank of tensors along with their underlying algebraic-geometric structures. To this end, the authors introduce the “concise secant variety”—a partial desingularization of the Segre-embedded secant variety modulo equivalence—whose points are in bijection with concise tensors of a prescribed border rank. By employing unrestricted limits, they characterize tensors of higher border rank as limits of those of minimal border rank, extending this framework to Veronese and Segre–Veronese settings. The work further develops concise analogues of border apolarity theory and Varieties of Sums of Powers (VSP). These contributions provide a geometric characterization of both border and cactus ranks, advance the analysis of identifiability and defectivity for Segre varieties, and offer new insights into related conjectures such as 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.

Problem

• border rank
• cactus rank
• secant varieties
• concise tensors
• Segre embeddings

Innovation

• concise secant varieties
• border rank
• unrestrictions
• cactus rank
• border apolarity