Secant non-defectivity via collisions of fat points

TL;DR

This paper introduces a novel degeneration technique to study secant defectivity of projective varieties, successfully proving a conjecture about the non-defectivity of certain Segre-Veronese embeddings.

Contribution

It develops a new approach allowing base points to collapse, leading to a proof of a conjecture on non-defectivity of Segre-Veronese embeddings for specific degrees.

Findings

Proves non-defectivity of Segre-Veronese embeddings for c,d ≥ 3

Introduces a technique for collapsing base points in degenerations

Provides a general result applicable to secant defectivity studies

Abstract

Secant defectivity of projective varieties is classically approached via dimensions of linear systems with multiple base points in general position. The latter can be studied via degenerations. We exploit a technique that allows some of the base points to collapse together. We deduce a general result which we apply to prove a conjecture by Abo and Brambilla: for $c \geq 3$ and $d \geq 3$, the Segre-Veronese embedding of $\mathbb{P}^m\times\mathbb{P}^n$ in bidegree $(c,d)$ is non-defective.