Secant varieties to osculating varieties of Veronese embeddings of P^n

Alessandra Bernardi 1 Maria Virgina Catalisano 2 Alessandro Gimigliano 3 Monica Idà 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 : A well known theorem by Alexander-Hirschowitz states that all the higher secant varieties of $V_{n,d}$ (the $d$-uple embedding of $\PP n$) have the expected dimension, with few known exceptions. We study here the same problem for $T_{n,d}$, the tangential variety to $V_{n,d}$, and prove a conjecture, which is the analogous of Alexander-Hirschowitz theorem, for $n\leq 9$. Moreover. we prove that it holds for any $n,d$ if it holds for $d=3$. Then we generalize to the case of $O_{k,n,d}$, the $k$-osculating variety to $V_{n,d}$, proving, for $n=2$, a conjecture that relates the defectivity of $\sigma_s(O_{k,n,d})$ to the Hilbert function of certain sets of fat points in $\PP n$.
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Alessandra Bernardi, Maria Virgina Catalisano, Alessandro Gimigliano, Monica Idà. Secant varieties to osculating varieties of Veronese embeddings of P^n. Journal of Algebra, Elsevier, 2009, 321 (3), pp.982-1004. ⟨10.1016/j.jalgebra.2008.10.020⟩. ⟨hal-00645970⟩

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