Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Level $ m$ stratifications of versal deformations of $ p$-divisible groups

Author: Adrian Vasiu
Journal: J. Algebraic Geom. 17 (2008), 599-641
Published electronically: February 6, 2008
MathSciNet review: 2424922
Full-text PDF

Abstract | References | Additional Information

Abstract: Let $ k$ be an algebraically closed field of characteristic $ p>0$. Let $ c,d,m$ be positive integers. Let $ D$ be a $ p$-divisible group of codimension $ c$ and dimension $ d$ over $ k$. Let $ \mathscr{D}$ be a versal deformation of $ D$ over a smooth $ k$-scheme $ \mathscr{A}$ which is equidimensional of dimension $ cd$. 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 $ D[p^m]$. 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 $ D$. 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 $ p$-varieties of Hodge type that generalize the special fibres of good integral models of Shimura varieties of Hodge type in unramified mixed characteristic $ (0,p)$.

References [Enhancements On Off] (What's this?)

Additional Information

Adrian Vasiu
Affiliation: Department of Mathematical Sciences, Binghamton University, Binghamton, New York 13902-6000

Received by editor(s): June 14, 2006
Received by editor(s) in revised form: May 15, 2007
Published electronically: February 6, 2008
Dedicated: To Carlo Traverso, for his 62nd birthday

Journal of Algebraic Geometry
The Journal of Algebraic Geometry
is sponsored by the Department of Mathematical Sciences
of Tsinghua University
and is distributed by the American Mathematical Society
for University Press, Inc.
Online ISSN 1534-7486; Print ISSN 1056-3911
© 2016 University Press, Inc.
AMS Website