Properties of Models of Algebraic Curves

Article
  • 17 Downloads

Abstract

Let \(\mathcal {O}_{K}\) be a Dedekind domain with fraction field K. In this paper, we present some results regarding models over \(\mathcal {O}_{K}\) of a smooth projective curve over K. We will focus on semi-stable models of marked curves, and give a description of semi-stable models (marked and unmarked) of the projective line.

Keywords

Algebraic curve regular model semi-stable model marked curve 

Mathematics Subject Classification

14H25 11G20 14G20 

References

  1. 1.
    Cornell, G., Silverman, J.: Arithmetic Geometry. Springer, Berlin (1986)CrossRefMATHGoogle Scholar
  2. 2.
    Deligne, P., Mumford, D.: The irreducibility of the space of curves of given genus. Publ. Math. IHES 36, 75–109 (1969)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Fresnel, J., van der Put, M.: Rigid Analytic Geometry and Its Applications, Progress in Mathematics, vol. 218. Birkhauser, Basel (2004)CrossRefMATHGoogle Scholar
  4. 4.
    Lehr, C.: Reduction of p-cyclic covers of the projective line. Manuscr. Math. 106, 151–175 (2001)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Liu, Q.: Algebraic Geometry and Arithmetic Curves, Oxford Graduate Texts in Mathematics, vol. 6. Oxford University Press, Oxford (2006)Google Scholar
  6. 6.
    Liu, Q.: Stable reduction of finite covers of curves. Compos. Math. 142, 101–118 (2006)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Mumford, D.: An analytic construction of degenerating curves over complete rings. Compos. Math. 24, 129–174 (1972)MathSciNetMATHGoogle Scholar
  8. 8.
    Raynaud, M.: Specialization des revetements en caracteristique \(p>0\). Ann. Sci. l’E.N.S 32, 87–126 (1999)MATHGoogle Scholar
  9. 9.
    Saito, T.: Vanishing cycles and geometry of curves over a discrete valuation ring. Am. J. Math. 109(6), 1043–1085 (1987)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Schmechta, T.: Mumford-tate curves. In: Bost, J.B., Loeser, F., Raynaud, M. (eds.) Courbes semi-stables et groupe fondamental en geometrie algebrique: Luminy, Decembre 1998, pp. 111–119. Birkhauser, Basel (2000)Google Scholar
  11. 11.
    Silverman, J.: Advanced Topics in the Arithmetic of Elliptic Curves. Springer, New-York (1986)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Research Faculty of Mathematics“Alexandru Ioan Cuza” UniversityBd. Carol I, 11 IasiRomania

Personalised recommendations