Higher secant varieties of P^n × P^m embedded in bi-degree (1,d)

Alessandra Bernardi 1 Enrico Carlini 2 Maria Virgina Catalisano 3
1 GALAAD - Geometry, algebra, algorithms
CRISAM - Inria Sophia Antipolis - Méditerranée , UNS - Université Nice Sophia Antipolis, CNRS - Centre National de la Recherche Scientifique : UMR6621
Abstract : Let $X^{(n,m)}_{(1,d)}$ denote the Segre\/-Veronese embedding of $\PP n \times \PP m$ 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 a multiple of $(m+n+1)$, the $s^{th}$ secant variety of $X^{(n,m)}_{(1,d)}$ has the expected dimension for every $s$.
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Alessandra Bernardi, Enrico Carlini, Maria Virgina Catalisano. Higher secant varieties of P^n × P^m embedded in bi-degree (1,d). Journal of Pure and Applied Algebra, Elsevier, 2011, 215 (12), pp.2853-2858. ⟨10.1016/j.jpaa.2011.04.005⟩. ⟨hal-00645976⟩

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