Divisors on Rational Normal Scrolls

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dc.creatorKustin, Andrew R.
dc.creatorPolini, Claudia
dc.creatorUlrich, Bernd
dc.date2008-11-07
dc.date.accessioned2026-07-07T10:16:48Z
dc.date.available2026-07-07T10:16:48Z
dc.descriptionLet $A$ 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 $\ell$ catalecticant blocks. The class group of $A$ is cyclic, and is infinite provided $\ell$ 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 $n^{\text{th}}$ ordinary power, the $n^{th}$ symmetric power, and the $n^{th}$ symbolic power all coincide and therefore all three $n^{th}$ 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 Groebner 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.
dc.description32 pages
dc.identifierhttps://arxiv.org/abs/0811.1069
dc.identifierhttp://arxiv.org/abs/0811.1069
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/173636
dc.subjectCommutative Algebra
dc.subject13C20; 13P10
dc.titleDivisors on Rational Normal Scrolls
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