Frobenius orbit maps and rank-four group schemes
We construct finite free group schemes of rank four and exponent eight as stabilizers in a smooth two-dimensional affine group. We first compute the complete local deformation ring of the invariant space of constants and squares in characteristic two, using six Grassmannian coordinates. For every Artinian specialization, evaluation at the identity has a finite free rank-four stabilizer. Its fourth-power morphism is $(x,y)\mapsto(2bxy,2axy)$, and its eighth-power morphism is trivial. The fourth power is nonzero on the third-order neighbourhood of the universal deformation ring and on a…
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