On higher syzygies of ruled surfaces III

Youngook Choi, Euisung Park

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Abstract

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<inf>0</inf>. Let L∈PicX be a line bundle in the numerical class of aC<inf>0</inf>+bf such that a≥1 and 2b-ae=4g-1+k for some k≥max(2, -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<inf>2,p</inf>, satisfies the inequalitiesk2-g≤index(S,L)≤k2-ae+32+max(0,⌈2g-3+ae-k4⌉). Also if S has an effective divisor D≡2C0+ef, then we obtain another upper bound of index(S, L), i.e., index(S,L)≤k+max(0,⌈2g-4-k2⌉). This gives a better bound in case b is small compared to a. Finally, for each e∈{-g, . ., -1} we construct a ruled surface S with the numerical invariant e and a minimal section C<inf>0</inf> which has an effective divisor D≡2C0+ef.

Original languageEnglish
Pages (from-to)4653-4666
Number of pages14
JournalJournal of Pure and Applied Algebra
Volume219
Issue number10
DOIs
Publication statusPublished - 2015 Oct 1

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Syzygies
Ruled Surface
Divisor
Homogeneous coordinates
Minimal Free Resolution
Invariant
Line Bundle
Genus
Upper bound
Ring
Curve

ASJC Scopus subject areas

  • Algebra and Number Theory

Cite this

On higher syzygies of ruled surfaces III. / Choi, Youngook; Park, Euisung.

In: Journal of Pure and Applied Algebra, Vol. 219, No. 10, 01.10.2015, p. 4653-4666.

Research output: Contribution to journalArticle

Choi, Youngook ; Park, Euisung. / On higher syzygies of ruled surfaces III. In: Journal of Pure and Applied Algebra. 2015 ; Vol. 219, No. 10. pp. 4653-4666.
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