Geodesic
Purity of the stratification by Newton polygons
de Jong, A.1 ; Oort, F.2
1 Massachusetts Institute of Technology, Department of Mathematics, Building 2, Room 270, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139-4307
2 Universiteit Utrecht, Mathematisch Instituut, Budapestlaan 6, NL-3508 TA Utrecht, The Netherlands
Journal of the American Mathematical Society, Tome 13 (2000) no. 1, pp. 209-241

Voir la notice de l'article provenant de la source American Mathematical Society

Let $S$ be a variety in characteristic $p>0$. Suppose we are given a nondegenerate $F$-crystal over $S$, for example the $i$th relative crystalline cohomology sheaf of a family of smooth projective varieties over $S$. At each point $s$ of $S$ we have the Newton polygon associated to the action of $F$ on the fibre of the crystal at $s$. According to a theorem of Grothendieck the Newton polygon jumps up under specialization. The main theorem of this paper is that the jumps occur in codimension $1$ on $S$ (the Purity Theorem). As an application we prove some results on deformations of iso-simple $p$-divisible groups.
DOI : 10.1090/S0894-0347-99-00322-7

de Jong, A. 1 ; Oort, F. 2

1 Massachusetts Institute of Technology, Department of Mathematics, Building 2, Room 270, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139-4307
2 Universiteit Utrecht, Mathematisch Instituut, Budapestlaan 6, NL-3508 TA Utrecht, The Netherlands 
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de Jong, A.; Oort, F. Purity of the stratification by Newton polygons. Journal of the American Mathematical Society, Tome 13 (2000) no. 1, pp. 209-241. doi: 10.1090/S0894-0347-99-00322-7

[1] Artin, M. Algebraization of formal moduli. II. Existence of modifications Ann. of Math. (2) 1970 88 135

[2] Artin, M. Algebraic approximation of structures over complete local rings Inst. Hautes Études Sci. Publ. Math. 1969 23 58

[3] Berthelot, Pierre Cohomologie cristalline des schémas de caractéristique 𝑝>0 1974 604

[4] Berthelot, Pierre, Messing, William Théorie de Dieudonné cristalline. III. Théorèmes d’équivalence et de pleine fidélité 1990 173 247

[5] De Jong, A. J. Crystalline Dieudonné module theory via formal and rigid geometry Inst. Hautes Études Sci. Publ. Math. 1995

[6] De Jong, A. J. Smoothness, semi-stability and alterations Inst. Hautes Études Sci. Publ. Math. 1996 51 93

[7] Deligne, P., Mumford, D. The irreducibility of the space of curves of given genus Inst. Hautes Études Sci. Publ. Math. 1969 75 109

[8] Ferrand, Daniel, Raynaud, Michel Fibres formelles d’un anneau local noethérien Ann. Sci. École Norm. Sup. (4) 1970 295 311

[9] Gross, Benedict H. Ramification in 𝑝-adic Lie extensions 1979 81 102

[10] Grothendieck, Alexandre Groupes de Barsotti-Tate et cristaux de Dieudonné 1974 155

[11] Harbater, David Fundamental groups and embedding problems in characteristic 𝑝 1995 353 369

[12] Séminaire sur les pinceaux arithmétiques: la conjecture de Mordell 1985

[13] Katz, Nicholas M. 𝑝-adic properties of modular schemes and modular forms 1973 69 190

[14] Katz, Nicholas M. Slope filtration of 𝐹-crystals 1979 113 163

[15] Li, Ke Zheng Classification of supersingular abelian varieties Math. Ann. 1989 333 351

[16] Li, Ke-Zheng, Oort, Frans Moduli of supersingular abelian varieties 1998

[17] Lipman, Joseph Desingularization of two-dimensional schemes Ann. of Math. (2) 1978 151 207

[18] Matsumura, Hideyuki Commutative algebra 1980

[19] Messing, William The crystals associated to Barsotti-Tate groups: with applications to abelian schemes 1972

[20] Miki, Hiroo On 𝑍_{𝑝}-extensions of complete 𝑝-adic power series fields and function fields J. Fac. Sci. Univ. Tokyo Sect. IA Math. 1974 377 393

[21] Moret-Bailly, Laurent Pinceaux de variétés abéliennes Astérisque 1985 266

[22] Moret-Bailly, Laurent Un problème de descente Bull. Soc. Math. France 1996 559 585

[23] Mumford, David The topology of normal singularities of an algebraic surface and a criterion for simplicity Inst. Hautes Études Sci. Publ. Math. 1961 5 22

[24] Norman, Peter An algorithm for computing local moduli of abelian varieties Ann. of Math. (2) 1975 499 509

[25] Norman, Peter, Oort, Frans Moduli of abelian varieties Ann. of Math. (2) 1980 413 439

[26] Oort, Frans Moduli of abelian varieties and Newton polygons C. R. Acad. Sci. Paris Sér. I Math. 1991 385 389

[27] Pierce, Richard S. Associative algebras 1982

[28] Saavedra Rivano, Neantro Catégories Tannakiennes 1972

[29] Serre, Jean-Pierre Corps locaux 1962 243

[30] Serre, Jean-Pierre Groupes algébriques et corps de classes 1975 204

[31] Tate, J. T. 𝑝-divisible groups 1967 158 183

[32] Revêtements étales et groupe fondamental 1971

[33] Grothendieck, Alexander Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (𝑆𝐺𝐴 2) 1968

[34] Groupes de monodromie en géométrie algébrique. I 1972

[35] Grothendieck, A. Éléments de géométrie algébrique. I. Le langage des schémas Inst. Hautes Études Sci. Publ. Math. 1960 228

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