Journal of Algebraic Geometry

Author(s): Adrian Vasiu
Journal: J. Algebraic Geom. 17 (2008), 599-641.
Posted: February 6, 2008
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Abstract: Let

be an algebraically closed field of characteristic

. Let

be positive integers. Let

be a

-divisible group of codimension

and dimension

over

. Let $\mathscr{D}$ be a versal deformation of

over a smooth

-scheme $\mathscr{A}$ which is equidimensional of dimension

. We show that there exists a reduced, locally closed subscheme $\mathfrak{s}_D(m)$ of $\mathscr{A}$ that has the following property: a point $y\in\mathscr{A}(k)$ belongs to $\mathfrak{s}_D(m)(k)$ if and only if $y^*(\mathscr{D})[p^m]$ is isomorphic to

. We prove that $\mathfrak{s}_D(m)$ is regular and equidimensional of dimension $cd-\dim(\pmb{\mathrm{Aut}}(D[p^m]))$ . We give a proof of Traverso's formula which for $m\gg0$ computes the codimension of $\mathfrak{s}_D(m)$ in $\mathscr{A}$ (i.e., $\dim(\pmb{\mathrm{Aut}}(D[p^m]))$ ) in terms of the Newton polygon of

. We also provide a criterion of when $\mathfrak{s}_D(m)$ satisfies the purity property (i.e., it is an affine $\mathscr{A}$ -scheme). Similar results are proved for quasi Shimura -varieties of Hodge type that generalize the special fibres of good integral models of Shimura varieties of Hodge type in unramified mixed characteristic

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