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

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.

• The paper constructs an infinite series of finite-dimensional Gorenstein local algebras failing the affine homogeneity property using explicit automorphism and derivation proofs.
• It employs a rigorous Gröbner basis approach to detail the structure, invariant subspaces, and combinatorial properties of the algebras.
• The findings reveal significant implications for additive group actions on projective hypersurfaces and resolve open questions in algebraic geometry.

Infinite Series of Gorenstein Local Algebras Lacking Affine Homogeneity

Introduction and Background

This paper constructs and analyzes an explicit family of commutative, finite-dimensional, local Gorenstein $K$-algebras $A_n$ (for $n \geq 2$) over an arbitrary field $K$—subject to mild characteristic constraints—that fail the affine homogeneity property (AH). This property is central in the study of the geometry and symmetry of associated nil-hypersurfaces and has substantial implications in the theory of additive group actions on projective hypersurfaces, via the generalized Hassett–Tschinkel correspondence.

The automorphism group of a local Gorenstein algebra acts naturally on the set of complementary hyperplanes to the socle in the maximal ideal. Property (AH) corresponds to the transitivity of this action: if for every pair of such hyperplanes $U_1, U_2 \subset m$ there exists an algebra automorphism $\phi$ such that $\phi(U_1) = U_2$, then $A$ is said to possess the affine homogeneity property (see [Fels–Kaup] and [Isaev]). This geometric criterion is further linked, in characteristic zero, to the existence and uniqueness (up to isomorphism) of additive group actions on nondegenerate projective hypersurfaces associated to $A$ by the Hassett–Tschinkel correspondence.

Up to this point, examples of Gorenstein local algebras failing property (AH) were isolated and often verified by computer algebra, lacking a general, constructive framework. This work remedies that by presenting a whole infinite series $A_n$ for which (AH) fails, and by providing completely elementary and explicit proofs.

Construction and Structure of the Algebras $A_n$0

For each integer $A_n$1, the algebra

$A_n$2

is considered. This quotient of the polynomial ring by a three-generated ideal is shown to be finite-dimensional, local, and Gorenstein, with socle generated by $A_n$3. The presentation is crafted to allow systematic analysis of monomials, their relations, and their images under automorphisms and derivations.

A Grőbner basis argument precisely characterizes the $A_n$4-basis and multiplicative structure of $A_n$5. The algebra is two-generated, with the maximal ideal $A_n$6 generated by $A_n$7 and residue field $A_n$8. All relations in $A_n$9 among basis monomials are made explicit and visualized, facilitating computations with automorphisms and derivations.

Failure of Affine Homogeneity: Main Results

Two distinct, independent proofs are given, one using automorphism groups and the other using derivations (i.e., the Lie algebra of the automorphism group). Both are elementary and entirely constructive.

Automorphism Group Action

The first approach computes the explicit form of automorphisms of $n \geq 2$0. For $n \geq 2$1 and $n \geq 2$2 either zero or not dividing $n \geq 2$3 or $n \geq 2$4, it is proven that

• The one-dimensional subspace $n \geq 2$5 is invariant under all algebra automorphisms.
• Specifically, every automorphism acts on $n \geq 2$6 via multiplication by a root of unity determined by $n \geq 2$7 and the characteristic.

This strict invariance leads to the existence of complementary hyperplanes $n \geq 2$8, $n \geq 2$9 in $K$0 which cannot be mapped onto each other by any automorphism of $K$1 (i.e., $K$2, $K$3). Thus $K$4 fails property (AH).

Derivation Approach

The derivation-based proof is more succinct and establishes that for all field characteristics as above, every derivation of $K$5 annihilates $K$6. In characteristic zero, this implies that $K$7 is fixed by the identity component of the automorphism group, so $K$8 cannot act transitively on the set of complementary hyperplanes; (AH) necessarily fails.

Implications for Additive Group Actions on Hypersurfaces

The failure of (AH) has substantial consequences for the theory of induced additive actions on projective hypersurfaces. The generalized Hassett–Tschinkel correspondence associates to each H-pair $K$9—where $U_1, U_2 \subset m$0 is a complementary hyperplane in the maximal ideal of a local Gorenstein algebra $U_1, U_2 \subset m$1—a projective hypersurface $U_1, U_2 \subset m$2 possessing an effective open orbit of an additive group action.

This paper constructs, for each $U_1, U_2 \subset m$3, explicit pairs of complementary hyperplanes giving rise to non-isomorphic nondegenerate projective hypersurfaces $U_1, U_2 \subset m$4, $U_1, U_2 \subset m$5 with distinct induced additive actions, but arising from the same algebra $U_1, U_2 \subset m$6. This resolves an open problem highlighted in [Arzhantsev–Zaitseva 2022] regarding whether such non-equivalent additive actions for the same Gorenstein algebra can occur. In the concrete case $U_1, U_2 \subset m$7, the equations for $U_1, U_2 \subset m$8 and $U_1, U_2 \subset m$9 in $\phi$0 are given explicitly, with generators for their defining ideals computed via symbolic algebra.

Broader Theoretical and Practical Implications

The construction provides a parametrized family of finite-dimensional, explicit counterexamples to affine homogeneity in the Gorenstein context, closing several gaps in the literature and showing that the phenomenon is robust, not sporadic. From a group-theoretic perspective, it shows that invariants beyond the socle may fundamentally obstruct automorphic flexibility, and that roots of unity constraints in automorphism groups can have geometric ramifications.

From an algebraic geometry viewpoint, the result highlights that the landscape of induced equivariant additive actions on projective hypersurfaces is richer and more nuanced than previously appreciated. The correspondence between the combinatorics of the basis and the geometry of the associated hypersurface/action pairings becomes explicit and computable.

Future directions may include:

• Classification of further families of finite-dimensional Gorenstein algebras failing (AH).
• Exploring connections to deformation theory and algebraic group orbits in representation spaces.
• Applications to the structure and classification of degenerations of projective spaces and their group actions.

Conclusion

This work presents an infinite series of explicit, elementary, and combinatorially tractable Gorenstein local algebras $\phi$1 that fail the affine homogeneity property. Both automorphism- and derivation-based arguments are provided. The geometric outcome is the existence, for any $\phi$2, of infinitely many non-isomorphic (as algebraic varieties) projective hypersurfaces associated to distinct H-pairs for the same algebra, each carrying distinct non-equivalent induced additive group actions. This fundamentally advances the understanding of the interplay between algebra automorphism groups and the geometry of equivariant compactifications.

Reference: "An infinite series of Gorenstein local algebras failing the affine homogeneity property" (2604.04793)