Secant varieties to osculating varieties of Veronese embeddings of P n

A well-known theorem by Alexander–Hirschowitz states that all the higher secant varieties of V n , d (the d-uple embedding of 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 ⩽ 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 σ s ( O k , n , d ) to the Hilbert function of certain sets of fat points in P n .