### Abstract

We study higher syzygies of a ruled surface X over a curve of genus g with the numerical invariant e. Let L ∈ PicX be a line bundle in the numerical class of aC_{0} + bf. We prove that for 0 ≤ e ≤ g - 3, L satisfies property N_{p}if a ≥ p+2 and b - ae ≥ 3g - 1 - e+p, and for e ≥ g - 2, L satisfies property N_{p} if a ≥ p+2 and b - ae ≥ 2g + 1+ p. By using these facts, we obtain Mukai-type results. For ample line bundles A_{i}, we show that K_{X}+A_{1}+⋯+A _{q} satisfies property N_{p} when 0 ≤ e < g-3/2 and q ≥ g -2e+1+p or when e ≥ g-3/2 and q ≥ p + 4. Therefore we prove Mukai's conjecture for ruled surface with e ≥ g-3/2. We also prove that when X is an elliptic ruled surface with e ≥ 0, L satisfies property N_{p} if and only if a ≥ 1 and b - ae ≥ 3 + p.

Original language | English |
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Pages (from-to) | 3733-3749 |

Number of pages | 17 |

Journal | Transactions of the American Mathematical Society |

Volume | 358 |

Issue number | 8 |

DOIs | |

Publication status | Published - 2006 Aug 1 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematics(all)