Secant varieties of Segre-Veronese varieties P^m x P^n embedded by O(1,2)

TL;DR

This paper investigates the dimensions of secant varieties of Segre-Veronese varieties embedded by O(1,2), providing bounds for when these secant varieties have expected dimensions and conjecturing a complete list of defective cases.

Contribution

It introduces bounds for secant variety dimensions of Segre-Veronese varieties with O(1,2) embedding and proposes a conjecture on all defective cases.

Findings

Established bounds for secant variety dimensions

Identified conditions for expected dimension

Conjectured a complete list of defective cases

Abstract

Let $X_{m,n}$ be the Segre-Veronese variety $\mathbb{P}^m \times \mathbb{P}^n$ embedded by the morphism given by $\mathcal{O}(1,2)$. In this paper, we provide two functions $\underline{s}(m,n)\le \bar{s}(m,n)$ such that the $s^{\mathrm{th}}$ secant variety of $X_{m,n}$ has the expected dimension if $s \leq \underline{s}(m,n)$ or $ \bar{s}(m,n) \leq s$. We also present a conjecturally complete list of defective secant varieties of such Segre-Veronese varieties.