A well known theorem by Alexander-Hirschowitz states that all the higher secant varieties of $V_{n,d}$ (the $d$-uple embedding of $mathbb{P}^n$) 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 $nleq 9$. 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 $mathbb{P}^n$.

Secant varieties to Osculating Varieties of Veronese embeddings of P n / Bernardi, Alessandra; M. V., Catalisano; A., Gimigliano; M. I. d., À.. - In: JOURNAL OF ALGEBRA. - ISSN 0021-8693. - 321:(2009), pp. 982-1004. [10.1016/j.jalgebra.2008.10.020]

Secant varieties to Osculating Varieties of Veronese embeddings of P n

Bernardi, Alessandra;
2009

Abstract

A well known theorem by Alexander-Hirschowitz states that all the higher secant varieties of $V_{n,d}$ (the $d$-uple embedding of $mathbb{P}^n$) 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 $nleq 9$. 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 $mathbb{P}^n$.
Bernardi, Alessandra; M. V., Catalisano; A., Gimigliano; M. I. d., À.
Secant varieties to Osculating Varieties of Veronese embeddings of P n / Bernardi, Alessandra; M. V., Catalisano; A., Gimigliano; M. I. d., À.. - In: JOURNAL OF ALGEBRA. - ISSN 0021-8693. - 321:(2009), pp. 982-1004. [10.1016/j.jalgebra.2008.10.020]
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11572/134897
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