On integral Zariski decompositions of pseudoeffective divisors on algebraic surfaces

B. Harbourne; P. Pokora; H. Tutaj-Gasińska

doi:10.3934/era.2015.22.103

Article Contents

2015, Volume 22: 103-108. Doi: 10.3934/era.2015.22.103

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On integral Zariski decompositions of pseudoeffective divisors on algebraic surfaces

1.
Department of Mathematics, University of Nebraska, Lincoln, NE 68588-0130
2.
Institute of Mathematics, Pedagogical University of Cracow, Podchorążych 2, PL-30-084, Kraków
3.
Departament of Mathematics and Computer Sciences, Jagiellonian University, Łojasiewicza 6, PL-30-348 Kraków

Received: August 2015

Revised: November 2015

Published: January 2015

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Abstract

In this note we consider the problem of integrality of Zariski decompositions for pseudoeffective integral divisors on algebraic surfaces. We show that while sometimes integrality of Zariski decompositions forces all negative curves to be $(-1)$-curves, there are examples where this is not true.

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Mathematics Subject Classification: 14C20, 14M25.

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References

[1]	T. Bauer, P. Pokora and D. Schmitz, On the boundedness of the denominators in the Zariski decomposition on surfaces, Journal für die reine und angewandte Mathematik, (2015).doi: 10.1515/crelle-2015-0058.
[2]	T. Bauer, S. Di Rocco, B. Harbourne, J. Huizenga, A. Lundman, P. Pokora and T. Szemberg, Bounded negativity and arrangements of lines, Int. Math. Res. Notices IMRN, 2015 (2015), 9456-9471.
[3]	T. De Fernex, Negative curves on very general blow-ups of $\mathbbP^2$, in Projective Varieties with Unexpected Properties, Walter de Gruyter GmbH & Co. KG, Berlin, 2005, 199-207.
[4]	M. Dumnicki, B. Harbourne, U. Nagel, A. Seceleanu, T. Szemberg and H. Tutaj-Gasińska, Resurgences for ideals of special point configurations in $P^N$ coming from hyperplane arrangements, J. Algebra, 443 (2015), 383-394.doi: 10.1016/j.jalgebra.2015.07.022.
[5]	T. Fujita, On Zariski problem, Proc. Japan Acad. Ser. A Math. Sci., 55 (1979), 106-110.doi: 10.3792/pjaa.55.106.
[6]	B. Harbourne, The geometry of rational surfaces and Hilbert functions of points in the plane, in Proceedings of the 1984 Vancouver Conference in Algebraic Geometry, CMS Conf. Proc., 6, Amer. Math. Soc., Providence, RI, 1986, 95-111.
[7]	A. L. Knutsen, Smooth curves on projective $K3$ surfaces, Math. Scand., 90 (2002), 215-231.
[8]	O. 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.doi: 10.2307/1970376.