Purity of the stratification by Newton polygons

de Jong, A.; Oort, F.

doi:10.1090/S0894-0347-99-00322-7

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

DOI: https://doi.org/10.1090/S0894-0347-99-00322-7

Published electronically: September 22, 1999

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Abstract:

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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