Representations of finite group schemes and morphisms of projective varieties

Given a finite group scheme $\text{cG}$ over an algebraically closed field $k$ of characteristic $\text{Char}(k)=p>0$, we introduce new invariants for a $\text{cG}$-module $M$ by associating certain morphisms $\deg^j_M : U_M \lra \Gr_d(M) \
(1!\le!j!\le! p!-!1)$to$M$thattakevaluesinGrassmanniansof$M$. These maps are studied for two classes of finite algebraic groups, infinitesimal group schemes and elementary abelian group schemes. The maps associated to the so-called modules of constant $j$-rank have a well-defined degree ranging between $0$ and $j{\text{rk}}^j(M)$, where ${\text{rk}}^j(M)$ is the generic $j$-rank of $M$. The extreme values are attained when the module $M$ has the equal images property or the equal kernels property. We establish a formula linking the $j$-degrees of $M$ and its dual $M^{\ast }$. For a self-dual module $M$ of constant Jordan type this provides information concerning the indecomposable constituents of the pull-back ${\alpha }^{\ast }(M)$ of $M$ along a $p$-point $\alpha :k[X]\mathrm{/}(X^p)\text{lra}k\text{cG}$.