## 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_{0}. Let L∈PicX be a line bundle in the numerical class of aC_{0}+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_{2,p}, 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_{0} which has an effective divisor D≡2C0+ef.

Original language | English |
---|---|

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 |

### Bibliographical note

Publisher Copyright:© 2015 Elsevier B.V.

## ASJC Scopus subject areas

- Algebra and Number Theory