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Journal of the American Mathematical Society

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Purity of the stratification by Newton polygons
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by A. J. de Jong and F. Oort
J. Amer. Math. Soc. 13 (2000), 209-241
Published electronically: September 22, 1999


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.
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Bibliographic Information
  • A. J. de Jong
  • Affiliation: Massachusetts Institute of Technology, Department of Mathematics, Building 2, Room 270, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139-4307
  • Email:
  • F. Oort
  • Affiliation: Universiteit Utrecht, Mathematisch Instituut, Budapestlaan 6, NL-3508 TA Utrecht, The Netherlands
  • Email:
  • Received by editor(s): October 28, 1998
  • Received by editor(s) in revised form: July 27, 1999
  • Published electronically: September 22, 1999
  • Additional Notes: The research of Dr. A.J. de Jong has been made possible by a fellowship of the Royal Netherlands Academy of Arts and Sciences.
  • © Copyright 1999 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 13 (2000), 209-241
  • MSC (2000): Primary 14L05, 14B05
  • DOI:
  • MathSciNet review: 1703336