### 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 language | English |
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Pages (from-to) | 4653-4666 |

Number of pages | 14 |

Journal | Journal of Pure and Applied Algebra |

Volume | 219 |

Issue number | 10 |

DOIs | |

Publication status | Published - 2015 Oct 1 |

### ASJC Scopus subject areas

- Algebra and Number Theory

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## Cite this

*Journal of Pure and Applied Algebra*,

*219*(10), 4653-4666. https://doi.org/10.1016/j.jpaa.2015.02.037