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

Bernardi, Alessandra ; Carlini, Enrico ; Catalisano, Maria Virginia (2010) Higher secant varieties of $\mathbb{P}^n \times \mathbb{P}^m$ embedded in bi-degree $(1,d)$. [Preprint]
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## 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 $s$.

Abstract
Document type
Preprint
Creators
CreatorsAffiliationORCID
Bernardi, Alessandra
Carlini, Enrico
Catalisano, Maria Virginia
Subjects
DOI
Deposit date
16 Apr 2010 12:35