### Abstract

In this article we continue the study of property N_{p} of irrational ruled surfaces begun in [E. Park, On higher syzygies of ruled surfaces, math.AG/0401100, Trans. Amer. Math. Soc., in press]. Let X be a ruled surface over a curve of genus g ≥ 1 with a minimal section C_{0} and the numerical invariant e. When X is an elliptic ruled surface with e = -1, it is shown in [F.J. Gallego, B.P. Purnaprajna, Higher syzygies of elliptic ruled surfaces, J. Algebra 186 (1996) 626-659] that there is a smooth elliptic curve E ⊂ X such that E ≡ 2C_{0} - f. And we prove that if L ∈ Pic X is in the numerical class of aC_{0} + bf and satisfies property N_{p}, then (C, L _{C0}) and (E, L _{E}) satisfy property N_{p} and hence a + b ≥ 3 + p and a + 2b ≥ 3 + p. This gives a proof of the relevant part of Gallego-Purnaprajna' conjecture in [F.J. Gallego, B.P. Purnaprajna, Higher syzygies of elliptic ruled surfaces, J. Algebra 186 (1996) 626-659]. When g ≥ 2 and e ≥ 0 we prove some effective results about property N_{p}. Let L ∈ Pic X be a line bundle in the numerical class of aC_{0} + bf. Our main result is about the relation between higher syzygies of (X, L) and those of (C, L_{C}) where L_{C} is the restriction of L to C_{0}. In particular, we show the followings: (1) If e ≥ g - 2 and b - ae ≥ 3g - 2, then L satisfies property N_{p} if and only if b - ae ≥ 2g + 1 + p. (2) When C is a hyperelliptic curve of genus g ≥ 2, L is normally generated if and only if b - ae ≥ 2g + 1 and normally presented if and only if b - ae ≥ 2g + 2. Also if e ≥ g - 2, then L satisfies property N_{p} if and only if a ≥ 1 and b - ae ≥ 2g + 1 + p.

Original language | English |
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Pages (from-to) | 590-608 |

Number of pages | 19 |

Journal | Journal of Algebra |

Volume | 294 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2005 Dec 15 |

### ASJC Scopus subject areas

- Algebra and Number Theory

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## Cite this

*Journal of Algebra*,

*294*(2), 590-608. https://doi.org/10.1016/j.jalgebra.2005.05.022