Commutative actions on smooth projective quadrics

Abstract

By a commutative action on a smooth quadric $Q_n$ in $P^{n+1}$ we mean an effective action of a commutative connected algebraic group on $Q_n$ with an open orbit. We show that for $n\ge 3$ all commutative actions on $Q_n$ are additive actions described by Sharoiko in 2009. So there is a unique commutative action on $Q_n$ 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.