An infinite series of Gorenstein local algebras failing the affine homogeneity property

TL;DR

The paper constructs an infinite series of Gorenstein local algebras that do not satisfy the affine homogeneity property, with implications for additive actions on projective hypersurfaces.

Contribution

It introduces a new infinite series of Gorenstein local algebras that fail affine homogeneity, providing elementary proofs and discussing related geometric consequences.

Findings

Algebras $A_n$ have a one-dimensional subspace invariant under automorphisms, different from the socle.

These algebras fail the affine homogeneity property.

Implications for additive actions on projective hypersurfaces and the generalized Hassett-Tschinkel correspondence.

Abstract

We provide an infinite series of commutative finite-dimensional Gorenstein local algebras $A_n$ for $n \ge 2$. We give an elementary proof that the maximal ideal of every algebra $A_n$ possesses a one-dimensional subspace that is different from the socle and invariant under the automorphism group of $A_n$. The latter implies that the algebras $A_n$ fail the affine homogeneity property. We also discuss some consequences concerning additive actions on projective hypersurfaces, related to the generalized Hassett-Tschinkel correspondence for these algebras.