On the secant varieties to the tangent developable of a Veronese variety

Let V-n,V-d subset of P-N, for N := ((n+d)(n)) - 1, be the order-d Veronese embedding of P-n,P- X-n,X-d := T(V-n,V-d) subset of P-N the tangent developable of V-n,V-d, and Ss-1(X-n,X-d) subset of P-N the s-secant variety of X-n,X-d, i.e. the closure in P-N of the union of all (s - 1)-linear spaces spanned by s points Of X-n,X-d. Ss-1(X-n,X-d) has expected dimension min {N, (2n + 1)s - 1}. Catalisano, Geramita, and Gimigliano conjectured that Ss-l (X-n,X-d) has always the expected dimension, except when d = 2, n >= 2s or d = 3 and n = 2, 3, 4. In this paper we prove their conjecture when n = 2 and n = 3. (c) 2005 Elsevier Inc. All rights reserved.