Real rank boundaries and loci of forms

TL;DR

This paper investigates the structure of real rank boundaries and loci of forms on certain algebraic varieties, introducing new loci, analyzing their properties, and connecting to hyperdeterminants, with implications for understanding tensor ranks.

Contribution

It introduces the Ranestad-Schreyer locus for non-reduced apolar schemes, analyzes forbidden loci and real rank boundaries, and extends results to broader embeddings, providing new insights into tensor rank geometry.

Findings

Ranestad-Schreyer locus is contained in the forbidden locus.

Identified a hypersurface dividing minimal and higher typical ranks.

Connected real rank boundaries to the hyperdeterminant of a specific tensor format.

Abstract

In this article we study forbidden loci and typical ranks of forms with respect to the embeddings of $\mathbb P^1\times \mathbb P^1$ given by the line bundles $(2,2d)$. We introduce the Ranestad-Schreyer locus corresponding to supports of non-reduced apolar schemes. We show that, in those cases, this is contained in the forbidden locus. Furthermore, for these embeddings, we give a component of the real rank boundary, the hypersurface dividing the minimal typical rank from higher ones. These results generalize to a class of embeddings of $\mathbb P^n\times \mathbb P^1$. Finally, in connection with real rank boundaries, we give a new interpretation of the $2\times n \times n$ hyperdeterminant.