Divisors on rational normal scrolls

Abstract

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; if ℓ is at least two, then the nth ordinary power, the nth symmetric power, and the nth symbolic power 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.