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

Let Vn,d⊆PN, for N:=(n+dn)−1, be the order-d Veronese embedding of Pn, Xn,d:=T(Vn,d)⊆PN the tangent developable of Vn,d, and Ss−1(Xn,d)⊆PN the s-secant variety of Xn,d, i.e. the closure in PN of the union of all (s−1)-linear spaces spanned by s points of Xn,d. Ss−1(Xn,d) has expected dimension min{N,(2n+1)s−1}. Catalisano, Geramita, and Gimigliano conjectured that Ss−1(Xn,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.