Divisors on rational normal scrolls

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• 作者：Andrew R. Kustin ; Claudia Polini ; Bernd Ulrich
• 关键词：Divisor class group ; Filtration ; Grö ; bner basis ; Rational normal scroll ; Regularity ; Resolution ; Symbolic Rees ring
• 刊名：Journal of Algebra
• 出版年：2009
• 出版时间：1 September 2009
• 年：2009
• 卷：322
• 期：5
• 页码：1748-1773
• 全文大小：315 K

文摘

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.