Secant varieties to osculating varieties of Veronese embeddings of Pn

A well-known theorem by Alexander–Hirschowitz states that all the higher secant varieties of Vn,d (the d-uple embedding of Pn) 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 σs(Ok,n,d) to the Hilbert function of certain sets of fat points in Pn.