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 catalecticant blocks. The class group of A is cyclic, and is infinite provided 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 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.

Rational normal scrolls and the defining equations of Rees algebras

“…According to Theorem 1.11, we need to identify generators for the ideal L in S with y n LA = gK (n) . The following minimal generating set for K (n) is calculated in [20,Prop. 1.20].…”

“…A related invariant, the postulation number of F (I), is computed in Corollary 6.9. Most of the results are collected in Theorem 4.4; these results are proved, in a more general setting, in [20]; see Theorem 4.5. The main result of this section is Theorem 4.6 where we calculate the reduction number, r(I), of I when ρ = 2.…”

“…The map Π induces an epimorphism A −→ R(I) , whose kernel is a height one prime ideal AA of the normal domain A, and hence gives rise to an element of the divisor class group group of A. Now in [20] we study divisors on rational normal scrolls of arbitrary dimension -most notably, for any given divisor class we describe an explicit monomial generating set of an unmixed ideal representing it, and we investigate these ideals in detail by providing free resolutions. Exhibiting an isomorphism between AA and the much simpler, monomial representative of its divisor class we obtain closed formulas for the defining equations of R(I) (Theorem 3.6), which turn out to be tremendously complicated despite the seemingly strong assumptions on I!…”

Equations of Rees algebras of ideals in two variables

“…They showed that the Rees algebra is a quotient of a ring A, which is the coordinate ring of a rational normal scroll, by a height one prime ideal KA. The equations of the scroll being easy to compute, this reduces the problem to computing the generators of the divisor KA on A, which was carried out in [21]. The equations are even given explicitly, assuming ϕ is put into a canonical form.…”

Rees algebras of truncations of complete intersections

“…Our goal is then reduced to explicitly describing ideals that represent the elements in the divisor class group of R(M ). The approach is very much inspired by [21] and [20]. For r = 2 and d ≥ d 1 + d 2 − 1 or r ≥ 3 and d ≥ d 1 + d 2 , the ideal I has a linear presentation and Q turns out to be a linear ideal in the T ′ s. Using this we prove that the defining ideal of R(I) has a quadratic Gröbner basis and hence R(I) is a Koszul algebra as well.…”