Small modules with interesting rank varieties

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 $\mathfrak{S}_{kp}$.