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Some Results on Surfaces of General Type

Published online by Cambridge University Press:  20 November 2018

B. P. Purnaprajna
B. P. Purnaprajna*
Affiliation:
Department of Mathematics, University of Kansas, Lawrence, Kansas, USA 66045-2142, e-mail: purna@math.ku.edu
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Abstract

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In this article we prove some new results on projective normality, normal presentation and higher syzygies for surfaces of general type, not necessarily smooth, embedded by adjoint linear series. Some of the corollaries of more general results include: results on property ${{N}_{p}}$ associated to ${{K}_{S}}\,\otimes \,{{B}^{\otimes n}}$ where $B$ is base-point free and ample divisor with $B\,\otimes \,{{K}^{*}}\,nef,$ results for pluricanonical linear systems and results giving effective bounds for adjoint linear series associated to ample bundles. Examples in the last section show that the results are optimal.

Keywords

13D0214C2014J29

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 57 , Issue 4 , 01 August 2005 , pp. 724 - 749
Copyright
Copyright © Canadian Mathematical Society 2005

References

[Bo] Bombieri, E., Canonical models of surfaces of general type. Inst. Hautes Études Sci. Publ. Math. 42(1973) 171–219.Google Scholar
[BS] Beltrametti, M. C. and Sommese, A. J., Sharp Matsusaka-type theorems on surfaces. Math. Nachr. 191(1998), 5–17.Google Scholar
[Bu] Butler, D., Normal generation of vector bundles over a curve. J. Differential Geometry 39(1994), 1–34.Google Scholar
[Ca] Catanese, F., Canonical rings and special surfaces of general type. Proceedings of the Summer Research Institute on Algebraic Geometry at Bowdoin, 1985, Part 1, American Mathematical Society, Providence, RI, 1987, pp. 175–194.Google Scholar
[Ci] Ciliberto, C., Sul grado dei generatori dell’anello canonico di una superficie di tipo generale. Rend. Sem. Mat. Univ. Politec. Torino 41(1983), 83–111.Google Scholar
[EL] Ein, L. and Lazarsfeld, R., Koszul cohomology and syzygies of projective varieties. Invent.Math. 111(1993), 51–67.Google Scholar
[D] del Busto, G. Fernandez, A Matsusaka-type theorem on surfaces. Geom, J. Algebraic. 5(1996), 513–520.Google Scholar
[GP1] Gallego, F. J. and Purnaprajna, B. P., Some results on rational surfaces and Fano varieties. J. Reine Angew.Math. 538(2001), 25–55.Google Scholar
[GP2] Gallego, F. J. and Purnaprajna, B. P., Projective normality and syzygies of algebraic surfaces. J. Reine Angew.Math. 506(1999), 145–180.Google Scholar
[GP3] Gallego, F. J. and Purnaprajna, B. P., Syzygies of Projective Surfaces: An Overview. J. RamanujanMath. Soc. 14(1999), 65–93.Google Scholar
[GP4] Gallego, F. J. and Purnaprajna, B. P., Normal presentation on elliptic ruled surfaces. J. Algebra 186(1996), 597–625.Google Scholar
[GP5] Gallego, F. J. and Purnaprajna, B. P., Higher syzygies of elliptic ruled surfaces. J. Algebra 186(1996), 626–659.Google Scholar
[GLM] Giraldo, L., Lopez, A. and Munoz, R., On the projective normality of Enriques surfaces. Math. Ann. 324(2002), 135–158.Google Scholar
[G1] Green, M., On the canonical ring of a variety of general type. Duke Math J. 49(1982), 1087–1113.Google Scholar
[G2] Green, M., Koszul cohomology and the geometry of projective varieties. J. Differential Geometry 19(1984), 125–171.Google Scholar
[GL] Green, M. and Lazarsfeld, R., Some results on the syzygies of finite sets and algebraic curves. Compositio Math. 67(1989), 301–314.Google Scholar
[Hb] Harbourne, B., Birational morphisms of rational surfaces. J. Algebra 190(1997), 145–162.Google Scholar
[Ho1] Homma, Y., Projective normality and the defining equations of ample invertible sheaves on elliptic ruled surfaces with e ≥ 0. Natural Science Report, Ochanomizu Univ. 31(1980), 61–73.Google Scholar
[Ho2] Homma, Y., Projective normality and the defining equations of an elliptic ruled surface with negative invariant. Natural Science Report, Ochanomizu Univ. 33(1982), 17–26.Google Scholar
[KM] Kawachi, T. and Masek, V., Reider-type theorems on normal surfaces. J. Algebraic Geom. 7(1998), 239–249.Google Scholar
[K] Kempf, G., Projective coordinate rings of Abelian varieties. Algebraic Analysis, Geometry and Number Theory, the Johns Hopkins Univ. Press., 1989, 225–235.Google Scholar
[Mu] Mumford, D., Varieties defined by quadratic equations. Corso CIME in Questions on Algebraic Varieties, Rome, (1970), 30–100.Google Scholar
[OP] Ottaviani, G. and Paoletti, R., Syzygies of Veronese embeddings. Compositio Math. 125(2001), 31–37.Google Scholar
[Pa] Pareschi, G., Syzygies of abelian varieties. J. Amer.Math. Soc. (3) 13(2000), 651–664 (electronic).Google Scholar
[R] Reider, I., Vector bundles of rank 2 and linear systems on an algebraic surface. Ann. of Math. (2) 127(1988), 309–316.Google Scholar
[V] Vermeire, P., Some results on secant varieties leading to a geometric flip construction. Compositio Math. (3) 125(2001), 263–282.Google Scholar