TY - JOUR

T1 - Higher syzygies of ruled varieties over a curve

AU - Park, Euisung

N1 - Funding Information:
✩ Supported by Korea Research Foundation Grant (KRF-2002-070-C00003). E-mail address: puserdos@kias.re.kr.

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.

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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 -