Higher secant varieties of $\mathbb{P}^n \times \mathbb{P}^m$ embedded in bi-degree $(1,d)$

TL;DR

This paper investigates the dimensions of higher secant varieties of a specific Segre-Veronese embedding of product projective spaces, establishing conditions for non-defectiveness and expected dimension.

Contribution

It provides a comprehensive analysis of secant variety dimensions for the embedding $X^{(n,m)}_{(1,d)}$, identifying when they are non-defective and have expected dimension.

Findings

No defective secant varieties except possibly for certain $s$ values.

Secant varieties have expected dimension when $(m+d inom{d}{d})$ is divisible by $(m+n+1)$.

Results extend understanding of secant varieties in bi-degree embeddings.

Abstract

Let $X^{(n,m)}_{(1,d)}$ denote the Segre-Veronese embedding of $\mathbb{P}^n \times \mathbb{P}^m$ via the sections of the sheaf $\mathcal{O}(1,d)$. We study the dimensions of higher secant varieties of $X^{(n,m)}_{(1,d)}$ and we prove that there is no defective $s^{th}$ secant variety, except possibly for $n$ values of $s$. Moreover when ${m+d \choose d}$ is multiple of $(m+n+1)$, the $s^{th}$ secant variety of $X^{(n,m)}_{(1,d)}$ has the expected dimension for every $s$.