Higher secant varieties of $\mathbb P^n \times\mathbb P^ 1$ embedded in bi-degree $(a,b)$

TL;DR

This paper computes the dimensions of higher secant varieties of the Segre-Veronese embedding of b^n imes \u00bb^1 in bi-degree (a,b), relating it to Grassmann defectivity and classifying certain defective Veronese varieties.

Contribution

It provides a comprehensive dimension calculation for secant varieties of b^n imes bb^1 in any bi-degree, and classifies Grassmann defective Veronese varieties.

Findings

Dimension formulas for all higher secant varieties computed.

Relation established between secant varieties and Grassmann defectivity.

Complete classification of Grassmann (1,s-1)-defective Veronese varieties.

Abstract

In this paper we compute the dimension of all the higher secant varieties to the Segre-Veronese embedding of $\mathbb{P}^n\times \mathbb{P}^1$ via the section of the sheaf $\mathcal{O}(a,b)$ for any $n,a,b\in \mathbb{Z}^+$. We relate this result to the Grassmann Defectivity of Veronese varieties and we classify all the Grassmann $(1,s-1)$-defective Veronese varieties.