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 $\scrD$ be a versal deformation of $D$ over a smooth $k$-scheme $\scrA$ which is equidimensional of dimension $cd$. We show that there exists a reduced, locally closed subscheme $\grs_D(m)$ of $\scrA$ that has the following property: a point $y\in\scrA(k)$ belongs to $\grs_D(m)(k)$ if and only if $y^*(\scrD)[p^m]$ is isomorphic to $D[p^m]$. We prove that $\grs_D(m)$ is { \it regular and equidimensional} of { \it dimension} $cd-\dim(\pmb{ \text{ Aut}}(D[p^m]))$. We give a proof of { \it Traverso's formula} which for $m>>0$ computes the codimension of $\grs_D(m)$ in $\scrA$ (i.e., $\dim(\pmb{ \text{ Aut}}(D[p^m]))$) in terms of the Newton polygon of $D$. We also provide a criterion of when $\grs_D(m)$ satisfies the { \it purity property} (i.e., it is an affine $\scrA$-scheme). Similar results are proved for { \it 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 $(0,p)$.
Categories: math.NT math.AG