On defectivity of joins, reducible secants and Fröberg's conjecture

We introduce the notion of r-defectivity for a vector bundle on a quasi-projective variety. Using this tool, we prove several previously unknown cases of Fröberg’s conjecture and also of the postulation problem for fat point schemes. Our techniques also allow us to study r-nondefectivity for joins and secants of reducible varieties. As a consequence, we derive results for mixture distributions and partition ranks.