Elementary Theory of Del Pezzo Surfaces

  • Josef Schicho
Conference paper


Del Pezzo surfaces are certain algebraic surfaces in projective n-space of degree n. They contain an interesting configuration of lines and have a rational parametrization. We give an overview of the classification with an emphasis on algorithmic constructions (e.g. of the parametrization), on explicit computations, and on real algebraic geometry.


Elliptic Curve Minimal Degree Double Point Pezzo Surface Hilbert Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bruce, J.W., and Wall, C. T. C. On the classification of cubic surfaces. J. London Math. Soc. (2)19 (1979), 245–256.Google Scholar
  2. 2.
    Cayley, A. A memoir on cubic surfaces. Phil. Trans. Roy. Soc. London 159 (1869), 231–326.Google Scholar
  3. 3.
    Comesatti, A. Fondamenti per la geometria sopra le superficie rationali del punto di vista reale. Math. Ann. 73 (1912), 1–72.MathSciNetCrossRefGoogle Scholar
  4. 4.
    Conforto, F. Le superficie razionali. Zanichelli, 1939.Google Scholar
  5. 5.
    del Pezzo, P. On the surfaces of order n embedded in n-dimensional space. Rend. mat. Palermo 1 (1887), 241–271.zbMATHCrossRefGoogle Scholar
  6. 6.
    Fujita, T. On the structure of polarized manifolds with total deficiency one. J. Math. Soc. Japan 33 (1981), 415–434.zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Griffiths, P., and Harris, J. Principles of algebraic geometry. John Wiley, 1978.Google Scholar
  8. 8.
    Harris, J. Algebraic geometry, a First Course. Springer, 1992.Google Scholar
  9. 9.
    Hartshorne, R. Algebraic Geometry. Springer-Verlag, 1977.Google Scholar
  10. 10.
    Jung, G. Ricerche sui sistemi lineari di curve algebriche di genere qualunque. Annali di Mat. 2 (1888), 277–312.zbMATHGoogle Scholar
  11. 11.
    Krasauskas, R. Toric surface patches. In Advances in geometrical algorithms and representations (2002), vol. 17, pp. 89–113.zbMATHMathSciNetGoogle Scholar
  12. 12.
    Manin, Y. Cubic Forms. North-Holland, 1974.Google Scholar
  13. 13.
    Peternell, M. Rational parametrizations for envelopes of quadric families. PhD thesis, Techn. Univ. Vienna, 1997.Google Scholar
  14. 14.
    Reid, M. Graded rings and varieties in weighted projective space. Tech. rep., Math. Institute, University of Warwick, 2002. downloadable via Scholar
  15. 15.
    Russo, F. The antibirational involutions of the plane and the classification of real del Pezzo surfaces. In Algebraic Geometry, M. B. et al., Ed. de Gruyter, 2002, pp. 289–312.Google Scholar
  16. 16.
    Schicho, J. Inversion of rational maps with Gröbner bases. In Gröbner bases and applications (1998), B. Buchberger and F. Winkler, Eds., Cambridge Univ. Press, pp. 495–503.Google Scholar
  17. 17.
    Schicho, J. Rational parameterization of real algebraic surfaces. In Proc. ISSAC’98 (1998), ACM Press, pp. 302–308.Google Scholar
  18. 18.
    Schicho, J. Rational parametrization of surfaces. J. Symb. Comp. 26,1 (1998), 1–30.zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Sermenev, A. M. On some unirational surfaces. Mat. Zametki 5 (1969), 155–159.MathSciNetGoogle Scholar
  20. 20.
    Silhol, R. Real Algebraic Surfaces. Springer, 1980.Google Scholar
  21. 21.
    Wall, C. T. C. Real forms of smooth del Pezzo surfaces. J. Reine Angew. Math. 375/376 (1987), 47–66.zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Josef Schicho
    • 1
  1. 1.Radon Insitute for Computational and Applied MathematicsAustrian Academy of SciencesGermany

Personalised recommendations