Divisors on Rational Normal Scrolls

Let A be the homogeneous coordinate ring of a rational normal scroll. The ring A is equal to the quotient of a polynomial ring S by the ideal generated by the two by two minors of a scroll matrix ψ with two rows and l catalecticant blocks. The class group of A is cyclic, and is infinite provided l is at least two. One generator of the class group is [J], where J is the ideal of A generated by the entries of the first column of ψ. The positive powers of J are well-understood, in the sense that the nth ordinary power, the nth symmetric power, and the nth symbolic power all coincide and therefore all three nth powers are resolved by a generalized Eagon-Northcott complex. The inverse of [J] in the class group of A is [K], where K is the ideal generated by the entries of the first row of ψ. We study the positive powers of [K]. We obtain a minimal generating set and a Gröbner basis for the preimage in S of the symbolic power K(n). We describe a filtration of K(n) in which all of the factors are Cohen-Macaulay S-modules resolved by generalized Eagon-Northcott complexes. We use this filtration to describe the modules in a finely graded resolution of K(n) by free S-modules. We calculate the regularity of the graded S-module K(n) and we show that the symbolic Rees ring of K is Noetherian. Introduction. Fix a field k and positive integers l and σ1 ≥ σ2 ≥ · · · ≥ σl ≥ 1. The rational normal scroll Scroll(σ1, . . . , σl) is the image of the map Σ : (A \ {0})× (A \ {0}) → P , where N = l− 1 + l ∑ i=1 σi and Σ(x, y; t1, . . . , tl) = (x t1, x yt1, . . . , y t1, x t2, x yt2, . . . , y ltl). 1Supported in part by the National Security Agency. 2Supported in part by the National Science Foundation and the National Security Agency. 3Supported in part by the National Science Foundation. 2000 Mathematics Subject Classification. Primary: 13C20, Secondary: 13P10.