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On the combinatorial classification of toric log del Pezzo surfaces

Part of: Polytopes and polyhedra Computational aspects in algebraic geometry Special varieties

Published online by Cambridge University Press:  01 January 2010

Alexander M. Kasprzyk ,
Maximilian Kreuzer  and
Benjamin Nill
Alexander M. Kasprzyk
Affiliation:
School of Mathematics and Statistics, University of Sydney, Sydney, NSW 2006, Australia (email: a.m.kasprzyk@usyd.edu.au)
Maximilian Kreuzer
Affiliation:
Institute for Theoretical Physics, Vienna University of Technology, Wiedner Hauptstrasse 8-10 1040 Vienna, Austria (email: maximilian.kreuzer@tuwien.ac.at)
Benjamin Nill
Affiliation:
Institut für Mathematik Arbeitsgruppe Gitterpolytope, Freie Universität Berlin, Arnimallee 3 14195 Berlin, Germany (email: nill@math.fu-berlin.de)

Abstract

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Toric log del Pezzo surfaces correspond to convex lattice polygons containing the origin in their interior and having only primitive vertices. Upper bounds on the volume and on the number of boundary lattice points of these polygons are derived in terms of the index ℓ. Techniques for classifying these polygons are also described: a direct classification for index two is given, and a classification for all ℓ≤16 is obtained.

MSC classification

Secondary: 52B20: Lattice polytopes (including relations with commutative algebra and algebraic geometry) 14M25: Toric varieties, Newton polyhedra 14Q10: Surfaces, hypersurfaces
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 13 , January 2010 , pp. 33 - 46
Copyright
Copyright © London Mathematical Society 2010

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