Commutative actions on smooth projective quadrics

Abstract

By a commutative action on a smooth quadric Q_n in ${\mathbb{P}}^{n+1},$ 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 Q₂ up to equivalence, for n = 1 there are two commutative actions on Q₁ up to equivalence.

Keywords:

• Algebraic group action
• commutative algebra
• commutative group
• projective quadric

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 14L30
• 14J50
• Secondary: 13A50
• 14J70

Additional information

Funding

Viktoriia Borovik and Sergey Gaifullin were supported by RSF grant 19-11-00172.