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Zariski chambers on surfaces of high Picard number

Part of: Cycles and subschemes Computational aspects in algebraic geometry

Published online by Cambridge University Press:  01 August 2012

Thomas Bauer  and
David Schmitz
Thomas Bauer
Affiliation:
Fachbereich Mathematik und Informatik, Philipps-Universität Marburg, Hans-Meerwein-Straße D-35032 Marburg, Germany (email: [email protected])
David Schmitz
Affiliation:
Fachbereich Mathematik und Informatik, Philipps-Universität Marburg, Hans-Meerwein-Straße D-35032 Marburg, Germany (email: [email protected])

Abstract

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We present an improved algorithm for the computation of Zariski chambers on algebraic surfaces. The new algorithm significantly outperforms the currently available method and therefore allows us to treat surfaces of high Picard number, where huge numbers of chambers occur. As an application, we efficiently compute the number of chambers supported by the lines on the Segre–Schur quartic.

MSC classification

Secondary: 14C20: Divisors, linear systems, invertible sheaves 14Q10: Surfaces, hypersurfaces
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 15 , May 2012 , pp. 219 - 230
Copyright
Copyright © London Mathematical Society 2012

References

[1]Bareiss, E., ‘Sylvester’s identity and multistep integer-preserving Gaussian elimination’, Math. Comput. 22 (1968) 565–578.Google Scholar
[2]Barth, W., ‘Lectures on K3- and Enriques surfaces’, Algebraic geometry (Sitges, 1983), Lecture Notes in Mathematics 1124 (Springer, New York, 1985) 21–57.Google Scholar
[3]Bauer, T., Funke, M. and Neumann, S., ‘Counting Zariski chambers on Del Pezzo surfaces’, J. Algebra 324 (2010) 92–101.CrossRefGoogle Scholar
[4]Bauer, T., Küronya, A. and Szemberg, T., ‘Zariski chambers, volumes, and stable base loci’, J. reine angew. Math. 576 (2004) 209–233.Google Scholar
[5]Boissière, S. and Sarti, A., ‘Counting lines on surfaces’, Ann. Sc. Norm. Super. Pisa Cl. Sci. 6 (2007) 39–52.Google Scholar
[6]Boissière, S. and Sarti, A., ‘On the Néron–Severi group of surfaces with many lines’, Proc. Amer. Math. Soc. 136 (2008) 3861–3867.CrossRefGoogle Scholar
[7]Caporaso, L., Harris, J. and Mazur, B., ‘How many rational points can a curve have?’, The moduli space of curves (Texel Island, 1994), Progress in Mathematics 129 (Birkhäuser, Boston, 1995) 13–31.CrossRefGoogle Scholar
[8]Kaltofen, E. and Villard, G., ‘On the complexity of computing determinants’, Comput. Complexity 13 (2004) 91–130.CrossRefGoogle Scholar
[9]Kovács, S. J., ‘The cone of curves of a K3 surface’, Math. Ann. 300 (1994) 681–691.CrossRefGoogle Scholar
[10]Schur, F., ‘Ueber eine besondre Classe von Flächen vierter Ordnung’, Math. Ann. 20 (1882) 254–296.CrossRefGoogle Scholar
[11]Segre, B., ‘The maximum number of lines lying on a quartic surface’, Oxf. Quart. J. 14 (1943) 86–96.CrossRefGoogle Scholar
[12]Segre, B., ‘On arithmetical properties of quartic surfaces’, Proc. Lond. Math. Soc. 49 (1944) 353–395.Google Scholar
[13]Shioda, T. and Inose, H., ‘On singular K3 surfaces’, Complex analysis and algebraic geometry: a collection of papers dedicated to K. Kodaira (Cambridge University Press, Cambridge, 1977) 119–136.CrossRefGoogle Scholar