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

Let P be the ideal of R=K[x0,…,xn] generated by the 2-minors of the Hankel matrix It is well known that P is the defining ideal of the rational normal curve of Pn, that is, the Veronese embedding of P1 in Pn. The minimal free resolution of R/P is the “generic” one, i.e. the Eagon–Northcott resolution. The resolution of the powers of generic maximal minors 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≥5. The goal of this note is to show that R/Ph has a linear resolution for all h. We determine also the Hilbert function (and hence the Betti numbers) of R/Ph for all h. We compute the Hilbert function of R/P(h) if either h≤3 or n≤4. Here P(h) denotes the hth symbolic power of P which in this case coincides with the saturation of Ph. This yields a formula for the Hilbert function of the module of Kähler differentials ΩA/K of A=R/P. Just to avoid trivial cases we will always assume that n≥3.