Algebra & Number Theory

Wild models of curves

Dino Lorenzini

Full-text: Open access

Abstract

Let K be a complete discrete valuation field with ring of integers OK and algebraically closed residue field k of characteristic p>0. Let XK be a smooth proper geometrically connected curve of genus g>0 with X(K) if g=1. Assume that XK does not have good reduction and that it obtains good reduction over a Galois extension LK of degree p. Let YOL be the smooth model of XLL. Let H:= Gal(LK).

In this article, we provide information on the regular model of XK obtained by desingularizing the wild quotient singularities of the quotient YH. The most precise information on the resolution of these quotient singularities is obtained when the special fiber Ykk is ordinary. As a corollary, we are able to produce for each odd prime p an infinite class of wild quotient singularities having pairwise distinct resolution graphs. The information on the regular model of XK also allows us to gather insight into the p-part of the component group of the Néron model of the Jacobian of X.

Article information

Source
Algebra Number Theory, Volume 8, Number 2 (2014), 331-367.

Dates
Received: 3 January 2013
Revised: 6 June 2013
Accepted: 16 July 2013
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.ant/1513730153

Digital Object Identifier
doi:10.2140/ant.2014.8.331

Mathematical Reviews number (MathSciNet)
MR3212859

Zentralblatt MATH identifier
1332.14029

Subjects
Primary: 14G20: Local ground fields
Secondary: 14G17: Positive characteristic ground fields 14K15: Arithmetic ground fields [See also 11Dxx, 11Fxx, 11G10, 14Gxx] 14J17: Singularities [See also 14B05, 14E15]

Keywords
model of a curve ordinary curve cyclic quotient singularity wild ramification arithmetical tree resolution graph component group Néron model

Citation

Lorenzini, Dino. Wild models of curves. Algebra Number Theory 8 (2014), no. 2, 331--367. doi:10.2140/ant.2014.8.331. https://projecteuclid.org/euclid.ant/1513730153


Export citation

References

  • A. Altman and S. Kleiman, Introduction to Grothendieck duality theory, Lecture Notes in Mathematics 146, Springer, Berlin, 1970.
  • M. Artin, “Wildly ramified $Z/2$ actions in dimension two”, Proc. Amer. Math. Soc. 52 (1975), 60–64.
  • S. Bosch and D. Lorenzini, “Grothendieck's pairing on component groups of Jacobians”, Invent. Math. 148:2 (2002), 353–396.
  • S. Bosch, W. Lütkebohmert, and M. Raynaud, Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 21, Springer, Berlin, 1990.
  • B. Edixhoven, Q. Liu, and D. Lorenzini, “The $p$-part of the group of components of a Néron model”, J. Algebraic Geom. 5:4 (1996), 801–813.
  • O. Gabber, Q. Liu, and D. Lorenzini, “The index of an algebraic variety”, Invent. Math. 192:3 (2013), 567–626.
  • L. H. Halle, “Stable reduction of curves and tame ramification”, Math. Z. 265:3 (2010), 529–550.
  • H. Hasse, “Theorie der relativ-zyklischen algebraischen Funktionenkörper, insbesondere bei endlichem Konstantenkörper”, J. Reine Angew. Math. 172 (1934), 37–54.
  • T. Katsura, “On Kummer surfaces in characteristic $2$”, pp. 525–542 in Proceedings of the International Symposium on Algebraic Geometry (Kyoto, 1977), edited by M. Nagata, Kinokuniya Book Store, Tokyo, 1978.
  • J. Lipman, “Rational singularities, with applications to algebraic surfaces and unique factorization”, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 195–279.
  • Q. Liu and D. Lorenzini, “Special fibers of Néron models and wild ramification”, J. Reine Angew. Math. 532 (2001), 179–222.
  • Q. Liu, D. Lorenzini, and M. Raynaud, “Néron models, Lie algebras, and reduction of curves of genus one”, Invent. Math. 157:3 (2004), 455–518.
  • D. Lorenzini, “Arithmetical graphs”, Math. Ann. 285:3 (1989), 481–501. http://msp.org/idx/mr/91b:14026MR 91b:14026
  • D. Lorenzini, “Dual graphs of degenerating curves”, Math. Ann. 287:1 (1990), 135–150.
  • D. Lorenzini, “Groups of components of Néron models of Jacobians”, Compositio Math. 73:2 (1990), 145–160.
  • D. Lorenzini, “On the group of components of a Néron model”, J. Reine Angew. Math. 445 (1993), 109–160.
  • D. Lorenzini, “Reduction of points in the group of components of the Néron model of a Jacobian”, J. Reine Angew. Math. 527 (2000), 117–150.
  • D. Lorenzini, “Models of curves and wild ramification”, Pure Appl. Math. Q. 6:1 (2010), 41–82.
  • D. Lorenzini, “Wild quotient singularities of surfaces”, Math. Z. 275:1–2 (2013), 211–232.
  • D. Lorenzini, “Wild quotients of products of curves”, preprint, 2013, http://www.math.uga.edu/~lorenz/Paper3.pdf.
  • D. Lorenzini and T. J. Tucker, “Thue equations and the method of Chabauty–Coleman”, Invent. Math. 148:1 (2002), 47–77.
  • D. Penniston, “Unipotent groups and curves of genus two”, Math. Ann. 317:1 (2000), 57–78.
  • B. R. Peskin, “Quotient-singularities and wild $p$-cyclic actions”, J. Algebra 81:1 (1983), 72–99.
  • M. Raynaud, “Spécialisation du foncteur de Picard”, Inst. Hautes Études Sci. Publ. Math. 38 (1970), 27–76.
  • T. Sekiguchi, F. Oort, and N. Suwa, “On the deformation of Artin–Schreier to Kummer”, Ann. Sci. École Norm. Sup. $(4)$ 22:3 (1989), 345–375.
  • J.-P. Serre, Groupes algébriques et corps de classes, Publications de l'institut de mathématique de l'université de Nancago, VII. Hermann, Paris, 1959.
  • B. Singh, “On the group of automorphisms of function field of genus at least two”, J. Pure Appl. Algebra 4 (1974), 205–229.
  • D. Subrao, “The $p$-rank of Artin–Schreier curves”, Manuscripta Math. 16:2 (1975), 169–193.
  • E. Viehweg, “Invarianten der degenerierten Fasern in lokalen Familien von Kurven”, J. Reine Angew. Math. 293/294 (1977), 284–308.
  • G. B. Winters, “On the existence of certain families of curves”, Amer. J. Math. 96 (1974), 215–228.