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TitleOn higher syzygies of ruled surfaces III
Author(s) Youngook Choi, Euisung Park
TypeArticle in Journal
AbstractAbstract In this paper, we study the minimal free resolution of homogeneous coordinate rings of a ruled surface S over a curve of genus g with the numerical invariant e < 0 and a minimal section C 0 . Let L &#8712; Pic X be a line bundle in the numerical class of a C 0 + b f such that a &#8805; 1 and 2 b &#8722; a e = 4 g &#8722; 1 + k for some k &#8805; max ( 2 , &#8722; e ) . We prove that the Green–Lazarsfeld index index ( S , L ) of ( S , L ) , i.e. the maximum p such that L satisfies condition N 2 , p , satisfies the inequalities k/2 &#8722; g &#8804; index ( S , L ) &#8804; k/2 &#8722; (a e + 3)/2 + max ( 0 , &#8968; (2 g &#8722; 3 + a e &#8722; k)/4 &#8969; ) . Also if S has an effective divisor D &#8801; 2 C 0 + e f , then we obtain another upper bound of index ( S , L ) , i.e., index ( S , L ) &#8804; k + max ( 0 , &#8968; (2 g &#8722; 4 &#8722; k)/2 &#8969; ) . This gives a better bound in case b is small compared to a. Finally, for each e &#8712; &#8722; g , … , &#8722; 1 we construct a ruled surface S with the numerical invariant e and a minimal section C 0 which has an effective divisor D &#8801; 2 C 0 + e f .
ISSN0022-4049
URL http://www.sciencedirect.com/science/article/pii/S002240491500064X
LanguageEnglish
JournalJournal of Pure and Applied Algebra
Volume219
Number10
Pages4653 - 4666
Year2015
Edition0
Translation No
Refereed No
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