Hilbert function and resolution of the powers of the ideal of the rational normal curve

Let P be the ideal of R = K[x(0,...,)x(n)] generated by the 2-minors of the Hankel matrix [GRAPHICS] It is well known that P is the defining ideal of the rational normal curve of P-n, that is, the Veronese embedding of P-1 in P-n. The minimal free resolution of RIP is the "generic" one, i.e. the Eagon-Northcott resolution. The resolution of the powers of generic maximal miners has been described by Akin et al. (Adv. Math. 39 (1981) 1-30) and it is linear. It is easy to see that the powers of P do not have the "generic" resolution if n greater than or equal to 5. The goal of this note is to show that R/P-h has a linear resolution for all h. We determine also the Hilbert function (and hence the Betti numbers) of R/P-h for all h. We compute the Hilbert function of R/P-(h) if either h less than or equal to 3 or n less than or equal to 4. Here P-(h) denotes the hth symbolic power of P which in this case coincides with the saturation of P-h. This yields a formula for the Hilbert function of the module pf Kahler differentials Omega(A/K) of A = R/P. Just to avoid trivial cases we will always assume that n greater than or equal to 3. (C) 2000 Elsevier Science B.V. All rights reserved.