  This paper focuses on the rank varieties for modules over a group algebra
$\mathbb{F}E$ where $E$ is an elementary abelian $p$-group and $p$ is the
characteristic of an algebraically closed field $\mathbb{F}$. In the first
part, we give a sufficient condition for a Green vertex of an indecomposable
module containing an elementary abelian $p$-group $E$ in terms of the rank
variety of the module restricted to $E$. In the second part, given a
homogeneous algebraic variety $V$ , we explore the problem on finding a small
module with rank variety $V$ . In particular, we examine the simple module
$D^{(kp-p+1,1^{p-1})}$ for the symmetric group $\mathfrak{S}_{kp}$.
