Skip to main content
SpringerLink
Log in

On the singularities of weakly normal varieties

  • Published:
manuscripta mathematica Aims and scope Submit manuscript

Abstract

Among the weakly normal varieties (in the sense of Andreotti and Bombieri, [1]) are of particular interest those varieties such that the normalization morphism is unramified outside a subvariety of codimension not less than 2. We describe the singularities of these varieties (called here WN1) by means of analytic equations, tangent cones, analytic branches and we show that any irreducible projective variety is birationally equivalent to a WN1 hypersur face and that a Gorenstein variety is weakly normal if and only if it is WN1.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Bibliography

  1. ANDREOTTJ A., BOMBIERI, E.: Sugli omeomorfismi delle varietà algebriche. Ann. Sc. Norm. Sup. Pisa23, 430–450 (1969)

    Google Scholar 

  2. CUM NO, C.: Sulle sottovarietà singolari delle varietà algebriche. Boll. Un. Mat. Ital.15-B, 906–914 (1978)

    Google Scholar 

  3. CUMINO, C.: Sulla seminormalità delle superficie algebriche. Rend. Sem. Mat. Univ. Padova58, 91–100 (1977)

    Google Scholar 

  4. DAVIS, E.: On the geometric interpretation of seminormality. Proc. Am. Math. Soc.68, 1–5 (1978)

    Google Scholar 

  5. GIANNI, P.: Morfismi interi delle varietà algebriche. Boll. Un. Mat. Ital.15-A, 187–196 (1978)

    Google Scholar 

  6. GRECO, S.: On the theory of branches. Int. Symp. of Algebraic Geometry, Kyoto (1977), 477–493

  7. GRECO, S., TRAVERSO, C.: On seminormal schemes. Compositio Math.40, 325–365 (1980)

    Google Scholar 

  8. GROTHENDIECK, A., DIEUDONNE', J.: Eléments de Géometrie Algébrique. ch. IV, Inst. Haut. Etud. Sci., Pubbl. Math. 20, 24, 28, 32 (1964–1967)

    Google Scholar 

  9. HIRONAKA, H.: Resolution of singularities of an algebraic variety over a field of characteristic O. Ann. of Math.79, 109–326 (1964)

    Google Scholar 

  10. LIPMAN, J.: Relative Lipschitz saturation. Am. J. of Math.97, 791–813 (1975)

    Google Scholar 

  11. MANARESI, M.: Some properties of weakly normal varieties. Nagoya Math. J. 77, 61–74 (1980)

    Google Scholar 

  12. NAGATA, M.: Local rings. Interscience (1962)

  13. RAYNAUD, M.: Anneaux locaux henséliens. Lecture Notes Math.169, Berlin-Heidelberg-New York, Springer (1970)

    Google Scholar 

  14. ROBERTS, J.: Generic projections of algebraic varieties. Am. J. of Math.93, 191–214 (1971)

    Google Scholar 

  15. TRAVERSO, C.: Seminormality and Picard group. Ann. Sc. Norm. Sup. Pisa24, 585–595 (1970)

    Google Scholar 

  16. ZARISKI, O., SAMUEL, P.: Commutative Algebra. vol. II, New York-London, Van Nostrand (1960)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Istituto Matematico del Politecnico c. so Duca degli Abruzzi, 24-10129, Torino

    Caterina Cumino

  2. Istituto di Geometria dell'Università P. zza di Porta S. Donato, 5-40100, Bologna

    Mirella Manaresi

Authors
  1. Caterina Cumino
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mirella Manaresi
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

This research was done when the authors were members of G.N.S.A.G.A. of the C.N.R.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cumino, C., Manaresi, M. On the singularities of weakly normal varieties. Manuscripta Math 33, 283–313 (1981). https://doi.org/10.1007/BF01798229

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01798229

Keywords

Search

Navigation