Graded analogues of one-parameter subgroups and applications to the cohomology of $G{L}_{m\mathrm{\mid }n(r)}$

We introduce a family ${\mathbb{M}}_{r;f,\eta }$ of infinitesimal supergroup schemes, which we call multiparameter supergroups, that generalize the infinitesimal Frobenius kernels ${\mathbb{G}}_{a(r)}$ of the additive group scheme ${\mathbb{G}}_a$. Then, following the approach of Suslin, Friedlander, and Bendel, we use functor cohomology to define characteristic extension classes for the general linear supergroup $G{L}_{m\mathrm{\mid }n}$, and we calculate how these classes restrict along homomorphisms $\rho: $\mathbb{M}${r;f,\eta} \rightarrow GL{m|n}.$ Finally, we apply our calculations to describe (up to a finite surjective morphism) the spectrum of the cohomology ring of the $r$-th Frobenius kernel $G{L}_{m\mathrm{\mid }n(r)}$ of the general linear supergroup $G{L}_{m\mathrm{\mid }n}$.