Nagoya Mathematical Journal

On Gorenstein surfaces dominated by {${\bf P}\sp 2$}

R. V. Gurjar, C. R. Pradeep, and D.-Q. Zhang

Full-text: Open access


In this paper we prove that a normal Gorenstein surface dominated by $\mathbf{P}2$ is isomorphic to a quotient $\mathbf{P}^2/G$, where $G$ is a finite group of automorphisms of $\mathbf{P}^2$ (except possibly for one surface $V_8'$). We can completely classify all such quotients. Some natural conjectures when the surface is not Gorenstein are also stated.

Article information

Nagoya Math. J., Volume 168 (2002), 41-63.

First available in Project Euclid: 27 April 2005

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14J26: Rational and ruled surfaces
Secondary: 14E20: Coverings [See also 14H30]


Gurjar, R. V.; Pradeep, C. R.; Zhang, D.-Q. On Gorenstein surfaces dominated by {${\bf P}\sp 2$}. Nagoya Math. J. 168 (2002), 41--63.

Export citation


  • Alekseev, V. A. and Kolpakov-Miroshnichenko, I. Ya., On quotient surfaces of $\proj^2$ by a finite group , Russian Math. Surveys, 43-5 (1988), 207–208.
  • Bredon, G. E., Introduction to Compact Transformation Groups, Pure and Applied Mathematics, Vol. 46, Academic Press (1972).
  • Demazure, M., Surfaces de del Pezzo, Lecture Notes in Math. Vol. 777, Springer, Berlin-Heidelberg-New York (1980).
  • Hidaka, F. and Watanabe, K., Normal Gorenstein surfaces with ample anti-canonical divisor , Tokyo J. Math., 4 (1981), 319–330.
  • Gurjar, R. V. and Shastri, A. R., The fundamental group at infinity of affine surfaces , Comment. Math. Helvitici, 59 (1984), 459–484.
  • Mohan Kumar, N., Rational double points on a rational surface , Invent. Math., 65 (1981/82), 251–268.
  • Lazarsfeld, R., Some applications of the theory of positive vector bundles, Complete intersections, Acireale (1983, Lecture Notes in Mathematics, Vol. 1092 ), 29–61, Springer Verlag.
  • Miyanishi, M., Normal Affine Subalgebras of a Polynomial Ring , Algebraic and Topological Theories – to the memory of T. Miyata, 37-51, Kinokuniya, Tokyo, (1985).
  • Miyanishi, M. and Zhang, D. -Q., Gorenstein log del Pezzo surfaces of rank one , J. Algebra, 118 (1988), 63–84.
  • Mumford, D., The topology of normal singularities of an algebraic surface and a criterion for simplicity , Publ. Math. IHES, 9 (1961), 5–22.
  • Seshadri, C. S., Quotient spaces modulo reductive algebraic groups , Ann. Math., 97 (1972), 511–556.
  • Samuel, P., On Unique Factorization Domains, Tata Institute of Fundamental Research, Lectures on Mathematics and Physics, No. 30 .