Skip to main content

On a Generalization of Mumford’s Result and Related Question

  • Chapter
Book cover Advances in Algebra and Geometry
  • 76 Accesses

Abstract

A well-known result of D. Mumford says that a normal analytic surface dominated by a smooth surface has at most quotient singularities. We pose an analogous question in dimension > 2. An affirmative answer to this question has some interesting consequences. We also discuss some related general results which support a positive answer to this question.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 52.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. A. Bartolo, I. Luengo and A. Melle-Hernandez, On a conjecture of W. Veys, Math. Ann. 317, no. 2 (2000), 325–327.

    Article  MathSciNet  MATH  Google Scholar 

  2. E. Brieskorn, Beispiele zur Differentialtopologie von Singularitaten, Invent. Math. 2 (1966), 1–14.

    Article  MathSciNet  MATH  Google Scholar 

  3. E. Brieskorn, Rationale Singularitaten Komplexer Flachen, Invent. Math. 4 (1968), 336–358.

    Article  MathSciNet  MATH  Google Scholar 

  4. H. Flenner, Rationale quasi-homogene singularitaten, Arch. Math. (Basel), 36 (1981), 35–44.

    Article  MathSciNet  MATH  Google Scholar 

  5. B. Giesecker, Simpliziale Zerlegung abzahlbarer analytische Raume, Math. Zeit. 83 (1964), 177–213.

    Article  MATH  Google Scholar 

  6. R.V. Gurjar, Topology of affine varieties dominated by an affine space, Invent. Math., 59 (1980), 221–225.

    Article  MathSciNet  MATH  Google Scholar 

  7. R.V. Gurjar, On Ramification of Self-maps of P2, Preprint.

    Google Scholar 

  8. R.V. Gurjar, A.J. Parameswaran, Open surfaces with non- positive Euler characteristic, Compositio Math. 99 (1995), 213–229.

    MathSciNet  MATH  Google Scholar 

  9. R.V. Gurjar, C.R. Pradeep, D.-Q. Zhang, On Gorenstein surfaces dominated by P2, To appear in Nagoya J. Math.

    Google Scholar 

  10. R.V. Gurjar and A.R. Shastri, The fundamental group at infinity of an affine surface, Comment. Math. Helvetici, 59 (1984), 459–484.

    Article  MathSciNet  MATH  Google Scholar 

  11. R.V. Gurjar and A.R. Shastri, A topological characterization of ℂ2/G, Journal of Math. Kyoto Univ., 25 (1984), 767–733.

    Article  MATH  Google Scholar 

  12. H. Hamm, Lokale topologische Eigenschaften komplexer Raume, Math. Annalen, 191 (1971), 235–252.

    Article  MathSciNet  MATH  Google Scholar 

  13. F. Hidaka and K. Watanabe, Normal Gorenstein Surfaces with Ample Anti-canonical Divisor, Tokyo J. Math., 4 (1981), 319–330.

    Article  MathSciNet  MATH  Google Scholar 

  14. V.Y. Lin, M.G. Zaidenberg, An irreducible simply-connected algebraic curve in2 is equivalent to a quasi-homogeneous curve, Soviet Math. Dokl. 28 (1983), 200–203.

    Google Scholar 

  15. J. Milnor, Singular Points Of Complex Hypersurfaces, Annals of Math. Studies, Princeton Univ. Press.

    Google Scholar 

  16. D. Mumford, The topology of normal singularities of an algebraic surface, Publ. Math. I.H.E.S. 9 (1961), 5–22.

    Article  MathSciNet  MATH  Google Scholar 

  17. M. Nori, Zariski’s conjecture and related problems, Ann. Scient. Ecole Norm. Sup. 4e Serie, t. 16 (1983), 305–344.

    MathSciNet  MATH  Google Scholar 

  18. V.L. Popov, Representations with a free module of covariants, Functional Anal. Appl. 10 (1976), 242–245.

    Article  MathSciNet  MATH  Google Scholar 

  19. D. Rotillon and K. Watanabe, Invariant subrings of ℂ[X, Y, Z] which are complete intersections, Manuscripta Math., 39 (1982), 339–357.

    Article  MathSciNet  MATH  Google Scholar 

  20. D.-Q. Zhang, Logarithmic del Pezzo surfaces of rank 1 with contractible boundaries, Osaka J. Math. 25 (1988), 461–497.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

C. Musili

Rights and permissions

Reprints and permissions

Copyright information

© 2003 Hindustan Book Agency

About this chapter

Cite this chapter

Gurjar, R.V. (2003). On a Generalization of Mumford’s Result and Related Question. In: Musili, C. (eds) Advances in Algebra and Geometry. Hindustan Book Agency, Gurgaon. https://doi.org/10.1007/978-93-86279-12-5_13

Download citation

Publish with us

Policies and ethics