A simple proof for the existence of Zariski decompositions on surfaces
Author:
Thomas Bauer
Journal:
J. Algebraic Geom. 18 (2009), 789-793
DOI:
https://doi.org/10.1090/S1056-3911-08-00509-2
Published electronically:
March 4, 2008
MathSciNet review:
2524598
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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
- Lucian Bădescu, Algebraic surfaces, Universitext, Springer-Verlag, New York, 2001. Translated from the 1981 Romanian original by Vladimir Maşek and revised by the author. MR 1805816
- Takao Fujita, On Zariski problem, Proc. Japan Acad. Ser. A Math. Sci. 55 (1979), no. 3, 106–110. MR 531454
- Noboru Nakayama, Zariski-decomposition and abundance, MSJ Memoirs, vol. 14, Mathematical Society of Japan, Tokyo, 2004. MR 2104208
- Robert Lazarsfeld, Positivity in algebraic geometry. I, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 48, Springer-Verlag, Berlin, 2004. Classical setting: line bundles and linear series. MR 2095471
- Oscar Zariski, The theorem of Riemann-Roch for high multiples of an effective divisor on an algebraic surface, Ann. of Math. (2) 76 (1962), 560–615. MR 141668, DOI https://doi.org/10.2307/1970376
References
- Badescu, L.: Algebraic Surfaces. Springer-Verlag, 2001. MR 1805816 (2001k:14068)
- Fujita, T.: On Zariski problem. Proc. Japan Acad. 55, Ser. A, 106-110 (1979). MR 531454 (80j:14029)
- Nakayama, N.: Zariski decomposition and abundance. Memoir, Math. Soc. Japan, 2004. MR 2104208 (2005h:14015)
- Lazarsfeld, R.: Positivity in Algebraic Geometry I. Springer-Verlag, 2004. MR 2095471 (2005k:14001a)
- 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-Straße, D-35032 Marburg, Germany
Email:
tbauer@mathematik.uni-marburg.de
Received by editor(s):
August 9, 2007
Received by editor(s) in revised form:
November 7, 2007
Published electronically:
March 4, 2008
Additional Notes:
The author was partially supported by DFG grant BA 1559/4-3
