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Groups of components of Néron models of Jacobians. (English) Zbl 0737.14008
Let $$A$$ be a complete discrete valuation ring with quotient field $$K$$, and algebraically closed residue field $$k$$. Let $$X$$ be a complete non- singular curve over $$K$$, $${\mathcal X}$$ be the minimal model of $$X$$ over $$A$$, and $$J$$ be the Jacobian variety of $$X$$. Moreover let $${\mathcal J}$$ be the Néron model of $$J$$. In this paper, the author considers the order $$\varphi$$ of the étale group scheme consisting of the components of the special fibre of $${\mathcal J}$$, and he gives an estimation of $$\varphi$$ using the difference between the Euler-Poincaré characteristics of the special fibre $${\mathcal X}_s$$ and the generic fibre $${\mathcal X}_\eta$$ of $${\mathcal X}$$, under the condition that the gcd of the multiplicities of the irreducible components of $${\mathcal X}_s$$ is equal to one. In the proof, he uses a result of M. Raynaud [Publ. Math., Inst. Hautes Étud. Sci. 38, 27–76 (1970; Zbl 0207.51602)]. Moreover, he gives an improvement for a result of H. W. Lenstra jun. and F. Oort [J. Pure Appl. Algebra 36, 281–298 (1985; Zbl 0557.14022)].

##### MSC:
 14H40 Jacobians, Prym varieties 14H25 Arithmetic ground fields for curves 14E30 Minimal model program (Mori theory, extremal rays)
##### Keywords:
minimal model; Jacobian variety; Néron model
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##### References:
 [1] M. Artin and G. Winters , Degenerate fibers and stable reduction of curves . Topology 10 (1971), 373-383. · Zbl 0196.24403 [2] G. Cornell and J. Silverman, Eds. Arithmetic Geometry . Spinger-Verlag, 1986. · Zbl 0596.00007 [3] M. Deschamps , Réduction semi-stable . In Séminaire sur les pinceaux de courbes de genre au moins deux (1981), L. Szpiro, Ed., pp. 1-34. · Zbl 0505.14008 [4] I. Dolgačev , On the purity of the degeneration loci of families of curves . Invent. Math. 8 (1969), 34-54. · Zbl 0176.18503 [5] A. Grothendieck , Exposés I and IX in the Séminaire de géométrie algébrique SGA 7 I . Lect. Notes Math. 288, Springer-Verlag, 1970. [6] S. Lang , Introduction to Arakelov Theory . Springer-Verlag, 1988. · Zbl 0667.14001 [7] H. Lenstra , and F. Oort , Abelian varieties having purely additive reduction . J. Pure and Applied Alg. 36 (1985), 281-298. · Zbl 0557.14022 [8] J. Lipman , Introduction to resolution of singularities . In Algebraic Geometry, Arcata 1974 (1975), AMS Proc. Symp. Pure Math. 29. · Zbl 0306.14007 [9] D. Lorenzini , Arithmetical graphs . To appear in Math. Annalen. · Zbl 0662.14008 [10] D. Lorenzini , Dual graphs of degenerating curves . To appear in Math. Annalen. · Zbl 0668.14019 [11] W Mccallum , The component group of a Néron model . Preprint. [12] T. Oda and C. Seshadri , Compactification of the generalized jacobian variety . Trans. of AMS 253 (1979), 1-90. · Zbl 0418.14019 [13] A. Ogg , Elliptic curves and wild ramification . Amer. J. Math. 89 (1967), 1-21. · Zbl 0147.39803 [14] F. Oort , A construction of generalized jacobians varieties by group extensions . Math. Annalen 147 (1962), 277-286. · Zbl 0101.38502 [15] F. Oort , Good and stable reduction of abelian varieties . Manuscr. Math. 11 (1974), 171-197. · Zbl 0266.14016 [16] M. Raynaud , Spécialisation du foncteur de Picard . Publ. Inst. Hautes Etudes Sci., Paris 38 (1970), 27-76. · Zbl 0207.51602 [17] T. Saito , Vanishing cycles and the geometry of curves over a discrete valuation ring . Amer. J. Math. 109 (1987), 1043-1085. · Zbl 0673.14014 [18] J.-P. Serre , Propriétés galoisiennes des points d’ordre fini des courbes elliptiques . Invent. Math. 15 (1972), 259-331. · Zbl 0235.14012 [19] J.-P. Serre and J. Tate , Good reduction of abelian varieties . Ann. of Math. 88 (1968), 492-517. · Zbl 0172.46101 [20] J. Silverman , The Néron fiber of abelian varieties with potential good reduction . Math. Annalen 264 (1983), 1-3. · Zbl 0497.14016 [21] J. Silverman , The Arithmetic of Elliptic Curves . Spinger-Verlag, 1986. · Zbl 0585.14026 [22] J. Tate , Algorithm for determining the type of a singular fiber in an elliptic pencil . In Modular functions in one variable IV (1975), Springer Lect. Notes in Math. vol. 476. · Zbl 1214.14020 [23] E. Viehweg , Invarianten der degenerierten Fasem in lokalen Familien von Kurven . J. Reine Math. 293 (1977), 284-308. · Zbl 0349.14017 [24] G. Winters , On the existence of certain families of curves . Amer. J. Math. 96 (1974), 215-228. · Zbl 0334.14004
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