Real rank boundaries and loci of forms

In this article, we study forbidden loci and typical ranks of forms with respect to the embeddings of P 1 × 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 P n × P 1 . Finally, in connection with real rank boundaries, we give a new interpretation of the 2 × n × n hyperdeterminant.