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Alessandro Oneto: Secant non-defectivity of Segre–Veronese varieties via collapsing points

Time: Mon 2021-10-11 15.00 - 16.00

Location: Zoom, meeting ID: 698 1231 6151

Lecturer: Alessandro Oneto (Trento)


It is an open problem to classify which embeddings of the Segre product of projective spaces \(\mathbb{P}^m \times \mathbb{P}^n\) via the linear systems of divisors of bi-degree (c,d) are defective, i.e., have a secant variety of dimension strictly smaller than the expected. Such a result would be a generalisation of the celebrated Alexander–Hirschowitz classification for Veronese varieties to the case of Segre–Veronese varieties with two factors.

In this bi-graded setting, defective cases are known for low bi-degrees and, in 2013, Abo and Brambilla conjectured that there are no-defective cases for degrees (c,d) whenever both c and d are at least three. In their work, they gave an inductive argument which reduced the problem to the only cases (3,3), (3,4) and (4,4). In a recent work with Francesco Galuppi (Trieste), we solved these three base cases, completing the proof of Abo and Brambilla. Following a classical approach, we study dimensions of linear systems of divisors with singular base points in general position in \(\mathbb{P}^m \times \mathbb{P}^n\). The novelty is to use in this setting a degeneration technique that allows some of the base points to collapse together. This result for Segre–Veronese varieties is actually a by-product of a more general certificate for non-defectivity of secant varieties.