Projective normality of ruled surfaces
Author:
Euisung Park
Journal:
Proc. Amer. Math. Soc. 136 (2008), 839847
MSC (2000):
Primary 14J26
Published electronically:
November 30, 2007
MathSciNet review:
2361855
Fulltext PDF Free Access
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Abstract: In this article we study normal generation of irrational ruled surfaces. When is a smooth curve of genus , Green and Lazarsfeld proved that a very ample line bundle Pic with deg Cliff is normally generated where Cliff denotes the Clifford index of the curve (Green and Lazarsfeld, 1986). We generalize this to line bundles on a ruled surface over .
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 David C. Butler, Normal generation of vector bundles over a curve, J. Differ. Geom. 39 (1994), 134. MR 1258911 (94k:14024)
 [B2]
 David C. Butler, Global sections and tensor products of line bundles over a curve, Math. Z. 231 (1999), 397407. MR 1704986 (2000e:14006)
 [C]
 G. Castelnuovo, Sui multipli di une serie lineare di gruppi di punti appartenente ad une curva algebraic, Rend. Circ. Mat. Palermo (2) 7 (1893), 89110.
 [FP1]
 Luis Fuentes Garcıa and Manuel Pedreira, The projective theory of ruled surfaces, Note di Mat. 24 (2005), no. 1, 2563. MR 2199622 (2006k:14065)
 [FP2]
 Luis Fuentes Garcıa and Manuel Pedreira, Canonical geometrically ruled surfaces, Math. Nachr. 278, no. 3, 240257. MR 2110530 (2005k:14076)
 [GP1]
 F. J. Gallego and B. P. Purnaprajna, Normal Presentation on Elliptic Ruled Surfaces, J. Algebra 186 (1996), 597625. MR 1423277 (98c:14030)
 [GP2]
 F. J. Gallego and B. P. Purnaprajna, Some results on rational surfaces and Fano varieties, J. reine angew. Math. 538 (2001), 2555. MR 1855753 (2002f:14024)
 [GL]
 M. Green and R. Lazarsfeld, On the projective normality of complete linear series on an algebraic curve, Inv. Math. 83 (1986), 7390. MR 813583 (87g:14022)
 [Ha]
 Robin Hartshorne, Algebraic Geometry, no. 52, SpringerVelag, New York (1977). MR 0463157 (57:3116)
 [Ho1]
 Y. Homma, Projective normality and the defining equations of ample invertible sheaves on elliptic ruled surfaces with , Natural Sci. Rep. Ochanomizu Univ. 31 (1980), 6173. MR 610593 (82e:14044)
 [Ho2]
 Y. Homma, Projective normality and the defining equations of an elliptic ruled surface with negative invariant, Natural Sci. Rep. Ochanomizu Univ. 33 (1982), 1726. MR 703959 (85a:14027)
 [L]
 R. Lazarsfeld, Positivity in Algebraic Geometry I. Classical setting: line bundles and linear series, A Series of Modern Surveys in Mathematics, no. 48, SpringerVerlag, Berlin, 2004. MR 2095471 (2005k:14001a)
 [M1]
 David Mumford, Lectures on curves on an algebraic surface, Ann. of Math. Studies 59 (1966). MR 0209285 (35:187)
 [P1]
 Euisung Park, On higher syzygies of ruled surfaces, Trans. Amer. Math. Soc. 358 (2006), 37333749. MR 2218997 (2007e:13025)
 [P2]
 Euisung Park, On higher syzygies of ruled surfaces II, Journal of Algebra 294 (2005) 590608. MR 2183366 (2007a:14043)
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Additional Information
Euisung Park
Affiliation:
Department of Mathematics, Korea University, Seoul 136701, Republic of Korea
Email:
euisungpark@korea.ac.kr
DOI:
http://dx.doi.org/10.1090/S0002993907091216
PII:
S 00029939(07)091216
Received by editor(s):
July 15, 2005
Received by editor(s) in revised form:
February 19, 2007
Published electronically:
November 30, 2007
Communicated by:
Michael Stillman
Article copyright:
© Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
