Commutative actions on smooth projective quadrics

Abstract

By a commutative action on a smooth quadric Qn in we mean an effective action of a commutative connected algebraic group on Qn with an open orbit. We show that for all commutative actions on Qn are additive actions described by Sharoiko in 2009. So there is a unique commutative action on Qn up to equivalence. For n = 2 there are three commutative actions on Q 2 up to equivalence, for n = 1 there are two commutative actions on Q 1 up to equivalence.