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

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 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Journal of Algebra*,

*296*(1), 267-284. https://doi.org/10.1016/j.jalgebra.2005.08.015

**Higher syzygies of ruled varieties over a curve.** / Park, Euisung.

Research output: Contribution to journal › Article

*Journal of Algebra*, vol. 296, no. 1, pp. 267-284. https://doi.org/10.1016/j.jalgebra.2005.08.015

}

TY - JOUR

T1 - Higher syzygies of ruled varieties over a curve

AU - Park, Euisung

PY - 2006/2/1

Y1 - 2006/2/1

N2 - 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 Npof 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 Np. We prove that line bundles on X of the form aH + π* B satisfy property Np 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.

AB - 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 Npof 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 Np. We prove that line bundles on X of the form aH + π* B satisfy property Np 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.

UR - http://www.scopus.com/inward/record.url?scp=30344480477&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=30344480477&partnerID=8YFLogxK

U2 - 10.1016/j.jalgebra.2005.08.015

DO - 10.1016/j.jalgebra.2005.08.015

M3 - Article

AN - SCOPUS:30344480477

VL - 296

SP - 267

EP - 284

JO - Journal of Algebra

JF - Journal of Algebra

SN - 0021-8693

IS - 1

ER -