Real rank boundaries and loci of forms

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.

arXiv comment17 pp