Details:
Title  Divisors on rational normal scrolls  Author(s)  Andrew R. Kustin, Claudia Polini, Bernd Ulrich  Type  Article in Journal  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 wellunderstood; 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 Smodules resolved by generalized Eagon–Northcott complexes. We use this filtration to describe the modules in a finely graded resolution of K ( n ) by free Smodules. We calculate the regularity of the graded Smodule K ( n ) and we show that the symbolic Rees ring of K is Noetherian.  Keywords  Divisor class group, Filtrationm, Gröbner basis, Rational normal scroll, Regularity, Resolution, Symbolic Rees ring  ISSN  00218693 
URL 
http://www.sciencedirect.com/science/article/pii/S002186930900283X 
Language  English  Journal  Journal of Algebra  Volume  322  Number  5  Pages  1748  1773  Year  2009  Edition  0  Translation 
No  Refereed 
No 
