Secant varieties to osculating varieties of Veronese embeddings of $\mathbb{P}^n$

Details Secant varieties to osculating varieties of Veronese embeddings of $\mathbb{P}^n$ Secant varieties to osculating varieties of Veronese embeddings of $\mathbb{P}^n$

2008-07-15

arxiv.org/abs/0807.2455v2

Journal of Algebra, 321, (2009), pp. 982-1004

Mathematics

Algebraic Geometry

23 pages

Scientific paper

10.1016/j.algebra.2008.10.020

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 $n\leq 9$. 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 $\mathbb{P}^n$.

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