### 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 |
---|---|

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 |

### Fingerprint

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Journal of Pure and Applied Algebra*,

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

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

Research output: Contribution to journal › Article

*Journal of Pure and Applied Algebra*, vol. 219, no. 10, pp. 4653-4666. https://doi.org/10.1016/j.jpaa.2015.02.037

}

TY - JOUR

T1 - On higher syzygies of ruled surfaces III

AU - Choi, Youngook

AU - Park, Euisung

PY - 2015/10/1

Y1 - 2015/10/1

N2 - 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 C0. Let L∈PicX be a line bundle in the numerical class of aC0+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 N2,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 C0 which has an effective divisor D≡2C0+ef.

AB - 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 C0. Let L∈PicX be a line bundle in the numerical class of aC0+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 N2,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 C0 which has an effective divisor D≡2C0+ef.

UR - http://www.scopus.com/inward/record.url?scp=84929518650&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84929518650&partnerID=8YFLogxK

U2 - 10.1016/j.jpaa.2015.02.037

DO - 10.1016/j.jpaa.2015.02.037

M3 - Article

AN - SCOPUS:84929518650

VL - 219

SP - 4653

EP - 4666

JO - Journal of Pure and Applied Algebra

JF - Journal of Pure and Applied Algebra

SN - 0022-4049

IS - 10

ER -