Modules of constant Jordan type, pullbacks of bundles and generic kernel filtrations

Let $kE$ denote the group algebra of an elementary abelian $p$-group of rank $r$ over an algebraically closed field of characteristic $p$. We investigate the functors $\mathcal{F}_i$ from $kE$-modules of constant Jordan type to vector bundles on $\mathbb{P}^{r-1}(k)$, constructed by Benson and Pevtsova. For a $kE$-module $M$ of constant Jordan type, we show that restricting the sheaf $\mathcal{F}_i(M)$ to a dimension $s-1$ linear subvariety of $\mathbb{P}^{r-1}(k)$ is equivalent to restricting $M$ along a corresponding rank $s$ shifted subgroup of $kE$ and then applying $\mathcal{F}_i$. In the case $r=2$, we examine the generic kernel filtration of $M$ in order to show that $\mathcal{F}_i(M)$ may be computed on certain subquotients of $M$ whose Loewy lengths are bounded in terms of $i$. More precise information is obtained by applying similar techniques to the $n$th power generic kernel filtration of $M$. The latter approach also allows us to generalise our results to higher ranks $r$.