\author{Dave Benson}
\author{Julia Pevtsova$^{*}$}
\title[Modules of constant Jordan type and vector bundles]
{A realization theorem for modules of constant Jordan type and vector bundles}
\thanks{ $^{*}$ partially supported by the NSF}
\begin{document}
\begin{abstract}
Let $E$ be an elementary abelian $p$-group of rank $r$ and let
$k$ be a field of characteristic $p$.
We introduce functors $\cF_i$ from finitely generated 
$kE$-modules of constant Jordan type to vector bundles over projective space
$\bP^{r-1}$. 
The fibers of the functors $\cF_i$ encode complete information about
the Jordan type of the module.

We prove that given any vector bundle $\cF$
of rank $s$ on $\bP^{r-1}$,
there is a $kE$-module $M$ of stable constant Jordan type
$[1]^s$ such that $\cF_1(M)\cong \cF$
if $p=2$, and such that $\cF_1(M) \cong F^*(\cF)$ if $p$ is odd. Here,
$F\colon\bP^{r-1}\to\bP^{r-1}$ is the Frobenius map. 
We prove that the theorem cannot be improved if $p$ is odd, because
if $M$ is any module of stable constant Jordan type $[1]^s$ 
then the Chern numbers $c_1,\dots,c_{p-2}$
of $\cF_1(M)$ are divisible by $p$.
\end{abstract}