Infinitesimal commutative unipotent group schemes with one-dimensional Lie algebra

Abstract We prove that over an algebraically closed field of characteristic $$p&gt;0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> there are exactly, up to isomorphism, n infinitesimal commutative unipotent k -group schemes of order $$p^n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>p</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:math> with one-dimensional Lie algebra, and we explicitly describe them. We consequently obtain an explicit description of all infinitesimal subgroup schemes of any supersingular elliptic curve over an algebraically closed field, recovering all their $$p^n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>p</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:math> -torsions as well. Finally, we use these results to answer a question of Brion on rational actions of infinitesimal commutative unipotent group schemes on curves.