## Abstract

Let X be a ruled variety over a smooth projective curve C with the projection morphism π : X → C. In this paper we study higher syzygies of very ample line bundles on X. Each embedding of X is fiberwise a Veronese embedding. And our first result is to clarify the relation between property N_{p}of very ample line bundles on X and that of the Veronese embedding. More precisely, letting H be the tautological line bundle of X, assume that the a-uple Veronese embedding of a fiber satisfies property N_{p}. We prove that line bundles on X of the form aH + π* B satisfy property N_{p} if deg (B) is sufficiently large (Theorem 1.1). Also we get some partial answer for the converse (Corollary 3.7). From this observation, we improve Butler's result in [D.C. Butler, Normal generation of vector bundles over a curve, J. Differential Geom. 39 (1994) 1-34] for ruled scrolls, ruled surfaces and Veronese surface fibrations.

Original language | English |
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Pages (from-to) | 267-284 |

Number of pages | 18 |

Journal | Journal of Algebra |

Volume | 296 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2006 Feb 1 |

### Bibliographical note

Funding Information:✩ Supported by Korea Research Foundation Grant (KRF-2002-070-C00003). E-mail address: [email protected].

## ASJC Scopus subject areas

- Algebra and Number Theory