For any finite k-group scheme G acting rationally on a k-variety X, if the action is generically free then the dimension of Lie (G) is upper bounded by the dimension of the variety.
This inequality turns out to be also a sufficient condition for the existence of such actions, when k is a perfect field of positive characteristic and G is infinitesimal commutative trigonalizable. These group schemes are non-reduced and arise only in positive characteristic. After presenting the main objects involved and overviewing the motivation for this problem, we willl explain the result in the case of actions of the p-torsion of a supersingular elliptic curve.
