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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Modèles entiers des courbes hyperelliptiques
sur un corps de valuation discrète

Author: Qing Liu
Journal: Trans. Amer. Math. Soc. 348 (1996), 4577-4610
MSC (1991): Primary 11G20, 14H25; Secondary 14G20
MathSciNet review: 1363944
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Abstract: Let $C$ be a hyperelliptic curve of genus $g\ge 1$ over a discrete valuation field $K$. In this article we study the models of $C$ over the ring of integers $ \mathcal {O}_{K}$ of $K$. To each Weierstrass model (that is a projective model arising from a hyperelliptic equation of $C$ with integral coefficients), one can associate a (valuation of) discriminant. Then we give a criterion for a Weierstrass model to have minimal discriminant. We show also that in the most cases, the minimal regular model of $C$ over $ \mathcal {O}_{K}$ dominates every minimal Weierstrass model. Some classical facts concerning Weierstrass models over $ \mathcal {O}_{K}$ of elliptic curves are generalized to hyperelliptic curves, and some others are proved in this new setting.

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  • [Art] M. Artin, Lipman's proof of resolution of singularities for surfaces, Arithmetic geometry (Cornell and Silverman, eds.), Springer-Verlag, 1986, pp. 267--287. MR 86:1980
  • [Chi] T. Chinburg, Minimal models for curves over Dedekind rings, Arithmetic geometry (Cornell and Silverman, eds.), Springer-Verlag, 1986, pp. 309--326. MR 86:1982
  • [Des] M. Deschamps, Réduction semi-stable, Séminaire sur les pinceaux de courbes de genre au moins deux, Astérisque, vol. 86, 1981, pp. 1--34.
  • [Har] R. Hartshorne, Algebraic Geometry, Graduate Texts in Math., 52, Springer-Verlag, 1977. MR 57:3116
  • [Kau] I. Kausz, Eine Abschätzung der Selbstschnittzahl des kanonischen Divisors auf arithmetischen Flächen mit hyperelliptischer generischer Faser, Dissertation, Köln (1995).
  • [Kod] K. Kodaira, On compact analytic surfaces, II, Ann. of Math. 77 (1963), 563--626. MR 89m:11059
  • [Lan] S. Lang, Introduction to Arakelov Theory, Springer-Verlag, 1988.
  • [Lic1] S. Lichtenbaum, Duality theorems for curves over $p$-adic fields, Invent. Math. 7 (1969), 120--136. MR 39:4158
  • [Lic2] S. Lichtenbaum, Curves over discrete valuation rings, Amer. J. Math. 90 (1968), 380--405. MR 37:6284
  • [Lip] J. Lipman, Rational singularities, Publ. Math. 36 (1969), 195-279. MR 43:1986
  • [Liu1] Q. Liu, Modèles minimaux des courbes de genre deux,, J. Reine Angew. Math. 453 (1994), 137--164. MR 95k:14024
  • [Liu2] Q. Liu, Conducteur et discriminant minimal de courbes de genre $2$, Compositio Math. 94 (1994), 51--79. MR 96b:14038
  • [Loc] P. Lockhart, On the discriminant of hyperelliptic curve, Trans. Amer. Math.Soc. 342 (1994), 729--752. MR 94f:11054
  • [Mat] H. Matsumura, Commutative Algebra, second edition, Benjamin/Cummings, New York, 1980. MR 82i:13003
  • [Mil1] J. S. Milne, Jacobian varieties, Arithmetic geometry (Cornell & Silverman,eds.), Springer-Verlag, 1986, pp. 167--212. MR 86:1976
  • [Mil2] J. S. Milne, Étale cohomology, Princeton Univ. Press, Princeton, New Jersey, 1980. MR 81j:14002
  • [N-U] Y. Namikawa, K. Ueno, The complete classification of fibers in pencils of curves of genus two, Manuscripta Math. 9 (1973), 143--186. MR 51:5595
  • [Ner] A. Néron, Modèles minimaux de variétés abélienne, Publ. Math. IHES 21 (1964). MR 31:3423
  • [Ogg] A. P. Ogg, On pencils of curves of genus two, Topology 5 (1966), 355--362. MR 34:3423
  • [Sai] T. Saito, Conductor, discriminant, and the Noether formula of arithmetic surfaces, Duke Math. Jour. 57 (1988), 151-173. MR 89f:14024
  • [Sil] J. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, Graduate Texts in Math., 151, Springer-Verlag, 1994. MR 96b:11074
  • [Tat] J. Tate, Algorithm for determining the type of a singular fiber in an elliptic pencil, Lect. Notes in Math., vol. 476, Springer Verlag, 1975, pp. 33--52. MR 52:13850
  • [Uen] K. Ueno, Discriminants of curves of genus $2$ and arithmetic surfaces, Algebraic geometry and commutative algebra, in honor of Masayaoshi Nagata, vol. II (1987), 749-770. MR 90a:14040
  • [Vie] E. Viehweg, Invarianten der degenerierten Fasern in lokalen Familien von Kurven, J. Reine Angew. Math. 293 (1977), 284--308. MR 58:16655

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Additional Information

Qing Liu
Affiliation: CNRS, Laboratoire de Mathématiques Pures, Université Bordeaux I, 351, Cours de la Libération, 33405 Talence Cedex, France

Keywords: Courbe hyperelliptique, modèle de Weierstrass, discriminant
Received by editor(s): August 22, 1995
Article copyright: © Copyright 1996 American Mathematical Society