Additive actions on projective hypersurfaces

TL;DR

This paper classifies additive group actions on projective hypersurfaces by linking them to invariant multilinear symmetric forms on local algebras, providing explicit classifications for certain quadrics.

Contribution

It establishes a correspondence between additive actions on hypersurfaces and invariant multilinear forms, extending previous descriptions to explicit classifications.

Findings

Classifies additive actions on non-degenerate quadrics.

Classifies additive actions on quadrics of corank one.

Provides explicit descriptions of these actions.

Abstract

By an additive action on a hypersurface H in the projective space P^{n+1} we mean an effective action of a commutative unipotent group on P^{n+1} which leaves H invariant and acts on H with an open orbit. Brendan Hassett and Yuri Tschinkel have shown that actions of commutative unipotent groups on projective spaces can be described in terms of local algebras with some additional data. We prove that additive actions on projective hypersurfaces correspond to invariant multilinear symmetric forms on local algebras. It allows us to obtain explicit classification results for non-degenerate quadrics and quadrics of corank one.