On syzygies of divisors on rational normal scrolls

Abstract

In this paper, we study the minimal free resolution of a nondegenerate projective variety X⊂Pr when X is contained in a variety Y of minimal degree as a divisor. Such a variety is of interest because of its extremal behavior with respect to various properties. The graded Betti diagram of X has been completely known only when X is arithmetically Cohen‐Macaulay. Our main result in the present paper provides a detailed description of the graded Betti diagram of X for the case where X is not arithmetically Cohen‐Macaulay.

On curves lying on a rational normal surface scroll

“…(2), we get reg(E(r, a 1 , q)) = q−1 a 1 + 1. Also reg(X) = a + 1 + b−1 a 1 by [P2,Theorem 4.3]. Therefore reg(X) = reg(E(r, a 1 , q)).…”

“…Therefore the integer δ(X) can be regarded as a measure of how far X is from the arithmetically Cohen-Macaulay property since a curve linearly equivalent to X + ℓC 0 is ACM if and only if ℓ = δ. Indeed, see [P2,Theorem 4.3] for the cases where a ≥ 1 or a = 0 and b > a 2 + 1. Also, if a = 0 and b = a 2 + 1 then δ(X) = 1 and X + C 0 is a rational normal curve which is apparently arithmetically Cohen-Macaulay.…”

“…Thus the divisor class group of S is freely generated by the hyperplane section H and a ruling line F of S. When X is linearly equivalent to aH + bF , it is nondegenerate in P r if and only if either a = 0 and b > a 2 or a = 1 and b ≥ 1 or a ≥ 2 and b ≥ −aa 2 (1.1) (cf. [P2,Lemma 2.2]). Concerned with the minimal free resolution of X, it is an interesting and important property that β(X) is invariant inside the divisor class of X.…”

“…That is, if X ′ is a curve in S and X ′ ≡ X, then β(X) = β(X ′ ) (cf. [P2,Proposition 4.1]). Finally, note that the graded Betti numbers of X are completely known when a ≥ 1 and b ≤ 1 (cf.…”

On the Minimal Free Resolution of Curves of Maximal Regularity

“…Note that the graded Betti numbers of C of type I are uniquely determined by r and d (cf. Proposition 4.1 in [8]). On the other hand, curves of type I and curves of type II must have different Betti tables by Green's K p,1 Theorem (cf.…”