On non-secant defectivity of Segre-Veronese varieties

TL;DR

This paper proves that Segre-Veronese varieties are asymptotically not defective for certain bounds on h, expanding understanding of their geometric properties in algebraic geometry.

Contribution

It establishes new asymptotic non-defectivity results for Segre-Veronese varieties, a significant class in algebraic geometry.

Findings

Segre-Veronese varieties are not h-defective for h up to a bound depending on n_1 and d.

Provides asymptotic bounds on defectivity for these varieties.

Enhances understanding of the geometric complexity of Segre-Veronese embeddings.

Abstract

Let $SV^{\pmb n}_{\pmb d}$ be the Segre-Veronese given as the image of the embedding induced by the line bundle $\mathcal{O}_{\mathbb{P}^{n_1}\times\dots\times\mathbb{P}^{n_r}}(d_1,\dots, d_r)$. We prove that asymptotically $SV^{\pmb n}_{\pmb d}$ is not $h$-defective for $h\leq n_1^{\lfloor \log_2(d-1)\rfloor}$.