On the Problem of Irreducibility of the Algebraic System of Irreducible Plane Curves of a Given Order and Having a Given Number of Nodes

Zariski, Oscar

doi:10.1007/978-1-4757-9286-7_19

Oscar Zariski³

Part of the book series: Progress in Mathematics ((PM,volume 36))

2091 Accesses

Abstract

Let k be an algebraically closed group field of characteristic zero. The following result is well known (see Severi [3], Anhang F):

If σ_n d is a maximal irreducible algebraic system, defined over k, of plane algebraic (not necessarily irreducible) curves of a given order n, and if the general curve C* of σ_n d/k has d nodes (and no other singularities), then the dimension of σ_n d EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCaiabg2 % da9maalaaabaGaaiikaiaad6gacqGHsislcaaIXaGaaiykamaabmaa % baGaamOBaiabgkHiTiaaikdaaiaawIcacaGLPaaaaeaacaaIYaaaaa % aa!40D4!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$p = \frac{{(n - 1)\left( {n - 2} \right)}}{2}$$ is equal to 3n + p − 1, where is the “effective” genus of C*.

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

Access this chapter

Log in via an institution

Chapter: USD 29.95; Price excludes VAT (USA)

eBook: USD 39.99; Price excludes VAT (USA)

Softcover Book: USD 54.99; Price excludes VAT (USA)

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

References

P. Deligne, “Le groupe fondamental du complément d’une courbe plane n’ayant que des points doubles ordinaires est abélien,” Séminaire Bourbaki (November, 1979 ).
Google Scholar
W. Fulton, “On the fundamental group of the complement of a node curve,” Ann. of Math., 111 (1980) pp. 407–409.
Article MathSciNet MATH Google Scholar
F. Severi, Vorlesungen fiber algebraische Geometrie, B. G. Teubner, Leipzig (1921).
Google Scholar
O. Zariski, “On the problem of existence of algebraic functions of two variables possessing a given branch curve,” Amer. J. Math., 51 (1929) pp. 305–328.
Article MathSciNet MATH Google Scholar
O. Zariski, Algebraic Surfaces, second edition (1971), Springer-Verlag, Berlin, Heidelberg, and New York.
Google Scholar
O. Zariski, “Dimension-theoretic characterization of maximal irreducible algebraic systems of plane nodal curves of a given order n and with a given number d of nodes,” Amer. J. Math., 1O (1982) pp. 209–226.
Article MathSciNet Google Scholar
and v. 2 (1960), D. Van Nostrand Company, Princeton, also Springer-Verlag, Berlin, Heidelberg, and New York.
Google Scholar

Download references

Author information

Authors and Affiliations

Department of Mathematics, Harvard University, Cambridge, Massachusetts, 02138, USA
Professor Oscar Zariski

Authors

Professor Oscar Zariski
View author publications
You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

Mathematics Department, Massachusetts Institute of Technology, 02139, Cambridge, MA, USA
Michael Artin
Mathematics Department, Harvard University, 02138, Cambridge, MA, USA
John Tate

Rights and permissions

Reprints and permissions

Copyright information

About this chapter

Cite this chapter

Zariski, O. (1983). On the Problem of Irreducibility of the Algebraic System of Irreducible Plane Curves of a Given Order and Having a Given Number of Nodes. In: Artin, M., Tate, J. (eds) Arithmetic and Geometry. Progress in Mathematics, vol 36. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4757-9286-7_19

Download citation

DOI: https://doi.org/10.1007/978-1-4757-9286-7_19
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-0-8176-3133-8
Online ISBN: 978-1-4757-9286-7
eBook Packages: Springer Book Archive

Publish with us

Policies and ethics