On the defectivity of segre-veronese varieties via collapsing points

The study of dimensions of secant varieties is a very classical subject which regained a lot of interest in the last part of the last century due to its relation with the study of tensor decompositions. The celebrated Alexander-Hirschowitz Theorem of 1995 completed the classification of Veronese varieties whose secant varieties have dimension less than the expected. Since then, a great literature has been dedicated to similar classifications for Segre and Segre-Veronese varieties. In 2013, Abo and Brambilla conjectured that Segre-Veronese embeddings of P×P in bidegree (c, d) are never defective if both c and d are larger or equal than three. They also proved the inductive step of a possible proof, namely they showed that if they are non-defective in the cases (3, 3), (3, 4) and (4, 4), then they are non-defective for higher bidegrees. In this paper we solve the case (3, 3). Following a classical approach, we turn our attention to the equivalent problem of computing the dimensions of linear systems of divisors of bi-degree (3, 3) in P × P with general 2-fat base points. The novelty is to use a degeneration technique that allows some of the base points to collapse together. The latter technique proved its power in a recent work of the first author and Mella in the context of identifiability for Waring decompositions of general polynomials. The present work has not yet been submitted to the arXiv nor to a journal since we are still trying to use these methods to approach the cases (3, 4) and (4, 4). At the moment of the submission to participate at MEGA2021, this should be considered as a work-in-progress, but we might submit it to a journal during the next few months before the conference.