Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Projective surfaces over a finite field

Authors: Roger Wiegand and William Krauter
Journal: Proc. Amer. Math. Soc. 83 (1981), 233-237
MSC: Primary 14J99
MathSciNet review: 624904
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ k$ be the algebraic closure of a finite field, and let $ X$ be an irreducible projective surface over $ k$. Let $ C$ be a curve on $ X$, and let $ \Omega $ be a finite set of closed points of $ X$ meeting each irreducible component of $ X$. We prove that there is an irreducible curve on $ X$ whose set-theoretic intersection with $ C$ is $ \Omega $. Using this theorem we characterize $ {\mathbf{P}}_k^2$ as a topological space, and we show that for any two irreducible plane curves $ C$, $ C'$ there is a homeomorphism from $ {\mathbf{P}}_k^2$ onto itself taking $ C$ onto $ C'$.

References [Enhancements On Off] (What's this?)

  • [A] M. Artin, Some numerical criteria for contractability of curves on algebraic surfaces, Amer. J. Math. 84 (1962), 485-496. MR 0146182 (26:3704)
  • [H] R. Hartshorne, Algebraic geometry, Graduate Texts in Math., vol. 52, Springer-Verlag, New York, 1977. MR 0463157 (57:3116)
  • [M] D. Mumford, The topology of normal singularities of an algebraic surface and a criterion for simplicity, Inst. Hautes Études Sci. Publ. Math. 9 (1961), 5-22. MR 0153682 (27:3643)
  • [S] R. G. Swan, A cancellation theorem for protective modules in the metastable range, Invent. Math. 27 (1974), 23-43. MR 0376681 (51:12856)
  • [W] R. Wiegand, Homeomorphisms of affine surfaces over a finite field, J. London Math. Soc. 18 (1978), 28-32. MR 0491732 (58:10933)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 14J99

Retrieve articles in all journals with MSC: 14J99

Additional Information

Article copyright: © Copyright 1981 American Mathematical Society

American Mathematical Society