Modèles entiers des courbes hyperelliptiques sur un corps de valuation discrète

Liu, Qing

doi:10.1090/S0002-9947-96-01684-4

Modèles entiers des courbes hyperelliptiques sur un corps de valuation discrète
HTML articles powered by AMS MathViewer

by Qing Liu PDF

Trans. Amer. Math. Soc. 348 (1996), 4577-4610 Request permission

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.

References

R. H. J. Germay, Généralisation de l’équation de Hesse, Ann. Soc. Sci. Bruxelles Sér. I 59 (1939), 139–144 (French). MR 86
R. H. J. Germay, Généralisation de l’équation de Hesse, Ann. Soc. Sci. Bruxelles Sér. I 59 (1939), 139–144 (French). MR 86
M. Deschamps, Réduction semi-stable, Séminaire sur les pinceaux de courbes de genre au moins deux, Astérisque, vol. 86, 1981, pp. 1–34.
Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157, DOI 10.1007/978-1-4757-3849-0
I. Kausz, Eine Abschätzung der Selbstschnittzahl des kanonischen Divisors auf arithmetischen Flächen mit hyperelliptischer generischer Faser, Dissertation, Köln (1995).
Serge Lang, Introduction to Arakelov theory, Springer-Verlag, New York, 1988. MR 969124, DOI 10.1007/978-1-4612-1031-3
S. Lang, Introduction to Arakelov Theory, Springer-Verlag, 1988.
Stephen Lichtenbaum, Duality theorems for curves over $p$-adic fields, Invent. Math. 7 (1969), 120–136. MR 242831, DOI 10.1007/BF01389795
Stephen Lichtenbaum, Curves over discrete valuation rings, Amer. J. Math. 90 (1968), 380–405. MR 230724, DOI 10.2307/2373535
Joseph Lipman, Rational singularities, with applications to algebraic surfaces and unique factorization, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 195–279. MR 276239, DOI 10.1007/BF02684604
Qing Liu, Modèles minimaux des courbes de genre deux, J. Reine Angew. Math. 453 (1994), 137–164 (French). MR 1285783, DOI 10.1515/crll.1994.453.137
Qing Liu, Conducteur et discriminant minimal de courbes de genre $2$, Compositio Math. 94 (1994), no. 1, 51–79 (French). MR 1302311
P. Lockhart, On the discriminant of a hyperelliptic curve, Trans. Amer. Math. Soc. 342 (1994), no. 2, 729–752. MR 1195511, DOI 10.1090/S0002-9947-1994-1195511-X
Hideyuki Matsumura, Commutative algebra, 2nd ed., Mathematics Lecture Note Series, vol. 56, Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980. MR 575344
R. H. J. Germay, Généralisation de l’équation de Hesse, Ann. Soc. Sci. Bruxelles Sér. I 59 (1939), 139–144 (French). MR 86
James S. Milne, Étale cohomology, Princeton Mathematical Series, No. 33, Princeton University Press, Princeton, N.J., 1980. MR 559531
Yukihiko Namikawa and Kenji Ueno, The complete classification of fibres in pencils of curves of genus two, Manuscripta Math. 9 (1973), 143–186. MR 369362, DOI 10.1007/BF01297652
André Néron, Modèles minimaux des variétés abéliennes sur les corps locaux et globaux, Inst. Hautes Études Sci. Publ. Math. 21 (1964), 128 (French). MR 179172, DOI 10.1007/bf02684271
Maria Tallini Scafati, Sui $\{k,\,n\}$-archi di un piano grafico finito, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 40 (1966), 373–378 (Italian, with English summary). MR 203573
Takeshi Saito, Conductor, discriminant, and the Noether formula of arithmetic surfaces, Duke Math. J. 57 (1988), no. 1, 151–173. MR 952229, DOI 10.1215/S0012-7094-88-05706-7
Joseph H. Silverman, Advanced topics in the arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 151, Springer-Verlag, New York, 1994. MR 1312368, DOI 10.1007/978-1-4612-0851-8
J. Tate, Algorithm for determining the type of a singular fiber in an elliptic pencil, Modular functions of one variable, IV (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Lecture Notes in Math., Vol. 476, Springer, Berlin, 1975, pp. 33–52. MR 0393039
Kenji Ueno, Discriminants of curves of genus $2$ and arithmetic surfaces, Algebraic geometry and commutative algebra, Vol. II, Kinokuniya, Tokyo, 1988, pp. 749–770. MR 977781
Esther Ferrari, On general topological spaces, Publ. Inst. Mat. Univ. Nac. Litoral 6 (1946), 183–189 (Spanish). MR 16655