Journal of Algebraic Geometry

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A simple proof for the existence of Zariski decompositions on surfaces

Author(s): Thomas Bauer
Journal: J. Algebraic Geom.
Posted: March 4, 2008
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Abstract | References | Additional information

Abstract: In this note we give a quick and simple proof of the existence (and uniqueness) of Zariski decompositions on surfaces. While Zariski's original proof employs a rather sophisticated procedure to construct the negative part of the decomposition, the present approach is based on the idea that the positive part can be constructed from a maximality condition. It may also be useful that this approach yields a practical algorithm for the computation of the positive part.

References:

1.: Badescu, L.: Algebraic Surfaces. Springer-Verlag, 2001. MR 1805816 (2001k:14068)
2.: Fujita, T.: On Zariski problem. Proc. Japan Acad. 55, Ser. A, 106-110 (1979). MR 531454 (80j:14029)
3.: Nakayama, N.: Zariski decomposition and abundance. Memoir, Math. Soc. Japan, 2004. MR 2104208 (2005h:14015)
4.: Lazarsfeld, R.: Positivity in Algebraic Geometry I. Springer-Verlag, 2004. MR 2095471 (2005k:14001a)
5.: Zariski, O.: The theorem of Riemann-Roch for high multiples of an effective divisor on an algebraic surface. Ann. Math. 76, 560-615 (1962). MR 0141668 (25:5065)

Additional Information:

Thomas Bauer
Affiliation: Fachbereich Mathematik und Informatik, Philipps-Universität Marburg, Hans-Meerwein-Stra{ß}e, D-35032~Marburg, Germany
Email: tbauer@mathematik.uni-marburg.de

PII: S 1056-3911(08)00509-2
Received by editor(s): August 9, 2007
Received by editor(s) in revised form: November 7, 2007
Posted: March 4, 2008
Additional Notes: The author was partially supported by DFG grant BA1559/4-3

The Journal of Algebraic Geometry
is distributed by the American Mathematical Society
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Online ISSN 1534-7486; Print ISSN 1056-3911
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