Unrestrictions and concise secant varieties
This work advances the algebraic geometry of tensors by providing new structural insights and characterizations that may impact tensor rank theory and its applications, though the results are theoretical and incremental in nature.
The paper introduces concise secant varieties, which are projective and birational to abstract secant varieties, with each point corresponding to a concise tensor of appropriate border rank. It provides a characterization of border rank ≤ r tensors as unrestrictions of minimal border rank r tensors, and discusses implications for tensor theory including cactus rank, border apolarity, and connections to defectivity and identifiability.
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.