January  2015, 22: 103-108. doi: 10.3934/era.2015.22.103

On integral Zariski decompositions of pseudoeffective divisors on algebraic surfaces

1. 

Department of Mathematics, University of Nebraska, Lincoln, NE 68588-0130, United States

2. 

Institute of Mathematics, Pedagogical University of Cracow, Podchorążych 2, PL-30-084, Kraków, Poland

3. 

Departament of Mathematics and Computer Sciences, Jagiellonian University, Łojasiewicza 6, PL-30-348 Kraków, Poland

Received  August 2015 Revised  November 2015 Published  December 2015

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.
Citation: B. Harbourne, P. Pokora, H. Tutaj-Gasińska. On integral Zariski decompositions of pseudoeffective divisors on algebraic surfaces. Electronic Research Announcements, 2015, 22: 103-108. doi: 10.3934/era.2015.22.103
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. Google Scholar

[2]

T. Bauer, S. Di Rocco, B. Harbourne, J. Huizenga, A. Lundman, P. Pokora and T. Szemberg, Bounded negativity and arrangements of lines,, \emph{Int. Math. Res. Notices IMRN}, 2015 (2015), 9456. Google Scholar

[3]

T. De Fernex, Negative curves on very general blow-ups of $\mathbbP^2$,, in \emph{Projective Varieties with Unexpected Properties}, (2005), 199. Google Scholar

[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,, \emph{J. Algebra}, 443 (2015), 383. doi: 10.1016/j.jalgebra.2015.07.022. Google Scholar

[5]

T. Fujita, On Zariski problem,, \emph{Proc. Japan Acad. Ser. A Math. Sci.}, 55 (1979), 106. doi: 10.3792/pjaa.55.106. Google Scholar

[6]

B. Harbourne, The geometry of rational surfaces and Hilbert functions of points in the plane,, in \emph{Proceedings of the 1984 Vancouver Conference in Algebraic Geometry}, (1984), 95. Google Scholar

[7]

A. L. Knutsen, Smooth curves on projective $K3$ surfaces,, \emph{Math. Scand.}, 90 (2002), 215. Google Scholar

[8]

O. Zariski, The theorem of Riemann-Roch for high multiples of an effective divisor on an algebraic surface,, \emph{Ann. of Math. (2)}, 76 (1962), 560. doi: 10.2307/1970376. Google Scholar

show all references

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. Google Scholar

[2]

T. Bauer, S. Di Rocco, B. Harbourne, J. Huizenga, A. Lundman, P. Pokora and T. Szemberg, Bounded negativity and arrangements of lines,, \emph{Int. Math. Res. Notices IMRN}, 2015 (2015), 9456. Google Scholar

[3]

T. De Fernex, Negative curves on very general blow-ups of $\mathbbP^2$,, in \emph{Projective Varieties with Unexpected Properties}, (2005), 199. Google Scholar

[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,, \emph{J. Algebra}, 443 (2015), 383. doi: 10.1016/j.jalgebra.2015.07.022. Google Scholar

[5]

T. Fujita, On Zariski problem,, \emph{Proc. Japan Acad. Ser. A Math. Sci.}, 55 (1979), 106. doi: 10.3792/pjaa.55.106. Google Scholar

[6]

B. Harbourne, The geometry of rational surfaces and Hilbert functions of points in the plane,, in \emph{Proceedings of the 1984 Vancouver Conference in Algebraic Geometry}, (1984), 95. Google Scholar

[7]

A. L. Knutsen, Smooth curves on projective $K3$ surfaces,, \emph{Math. Scand.}, 90 (2002), 215. Google Scholar

[8]

O. Zariski, The theorem of Riemann-Roch for high multiples of an effective divisor on an algebraic surface,, \emph{Ann. of Math. (2)}, 76 (1962), 560. doi: 10.2307/1970376. Google Scholar

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