A well-known theorem by Alexander-Hirschowitz states that all the higher secant varieties of Vn,d (the d-uple embedding of P^n) have the expected dimension, with few known exceptions. We study here the same problem for Tn,d, the tangential variety to Vn,d, and prove a conjecture, which is the analogous of Alexander-Hirschowitz theorem, for n ≤ 9. Moreover, we prove that it holds for any n, d if it holds for d = 3. Then we generalize to the case of Ok,n,d, the k-osculating variety to Vn,d, proving, for n = 2, a conjecture that relates the defectivity of σ_s (Ok,n,d) to the Hilbert function of certain sets of fat points in P^n
Secant varieties to osculating varieties of Veronese embeddings of P^n
CATALISANO, MARIA VIRGINIA;
2009-01-01
Abstract
A well-known theorem by Alexander-Hirschowitz states that all the higher secant varieties of Vn,d (the d-uple embedding of P^n) have the expected dimension, with few known exceptions. We study here the same problem for Tn,d, the tangential variety to Vn,d, and prove a conjecture, which is the analogous of Alexander-Hirschowitz theorem, for n ≤ 9. Moreover, we prove that it holds for any n, d if it holds for d = 3. Then we generalize to the case of Ok,n,d, the k-osculating variety to Vn,d, proving, for n = 2, a conjecture that relates the defectivity of σ_s (Ok,n,d) to the Hilbert function of certain sets of fat points in P^n
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