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 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.
| 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
Higher syzygies of ruled varieties over a curve. / Park, Euisung.
In: Journal of Algebra, Vol. 296, No. 1, 01.02.2006, p. 267-284.Research output: Contribution to journal › Article
}
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 -