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 < 10. 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.
A.Bernardi , M.V.Catalisano , A.Gimigliano, M.Idà. (2009). Secant varieties to Osculating Varieties of Veronese embeddings of P n. JOURNAL OF ALGEBRA, 321, 982-1004 [10.1016/j.jalgebra.2008.10.020].
Secant varieties to Osculating Varieties of Veronese embeddings of P n
BERNARDI, ALESSANDRA;GIMIGLIANO, ALESSANDRO;IDA', MONICA
2009
Abstract
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 < 10. 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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
