Advertisement

Journal of Algebraic Combinatorics

, Volume 34, Issue 2, pp 213–236 | Cite as

Lattice polygons and families of curves on rational surfaces

  • Niels Lubbes
  • Josef Schicho
Article

Abstract

First we solve the problem of finding minimal degree families on toric surfaces by reducing it to lattice geometry. Then we describe how to find minimal degree families on, more generally, rational complex projective surfaces.

Keywords

Algebraic geometry Toric geometry Lattice polygons Families of curves Surfaces 

References

  1. 1.
    Cox, D.: What is a toric variety. In: Topics in Algebraic Geometry and Geometric Modeling. Contemp. Math., vol. 334, pp. 203–223. Am. Math. Soc., Providence (2003) Google Scholar
  2. 2.
    Draisma, J., McAllister, T.B., Nill B.: Lattice width directions and Minkowski’s 3d-theorem. Technical Report, arXiv:0901.1375v1 [math.CO] (2009)
  3. 3.
    Günter, Ewald: Combinatorial Convexity and Algebraic Geometry. Graduate Texts in Mathematics, vol. 168. Springer, New York (1996). ISBN 0-387-94755-8 zbMATHGoogle Scholar
  4. 4.
    Fulton, W.: Introduction to Toric Varieties. Annals of Mathematics Studies, vol. 131. Princeton University Press, Princeton (1993) zbMATHGoogle Scholar
  5. 5.
    Haase, C., Schicho, J.: Lattice polygons and the number 2i+7. Math. Mon. (2009) Google Scholar
  6. 6.
    Halphen, G.H.: On plane curves of degree six through nine double points (in french). Bull. Soc. Math. Fr. 10, 162–172 (1882) MathSciNetGoogle Scholar
  7. 7.
    Hartshorne, R.: Algebraic Geometry. Graduate Texts in Mathematics, vol. 52. Springer, New York (1977). zbMATHGoogle Scholar
  8. 8.
    Manin, Ju.I.: Rational surfaces over perfect fields. Publ. Math. Inst. Hautes Etudes Sci. 30, 55–113 (1966) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Matsuki, K.: Introduction to the Mori Program. Universitext. Springer, New York (2002) zbMATHGoogle Scholar
  10. 10.
    Schicho, J.: Rational parametrization of surfaces. J. Symb. Comput. 26(1), 1–30 (1998) MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Schicho, J.: The multiple conical surfaces. Beitrage Algebra Geom. 42, 71–87 (2001) MathSciNetzbMATHGoogle Scholar
  12. 12.
    Schicho, J.: Simplification of surface parameterizations—a lattice polygon approach. J. Symb. Comput. 36, 535–554 (2003) MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Johann Radon InstitutÖsterreichische Akademie der WissenschaftenLinzAustria

Personalised recommendations