[Paper Review] On Abelian Automorphism Groups of Hypersurfaces

Zhiwei Zheng|arXiv (Cornell University)|Apr 20, 2020

Algebraic Geometry and Number Theory7 references4 citations

TL;DR

This paper classifies all finite abelian groups acting faithfully and linearly on smooth hypersurfaces in complex projective space, under specific lifting and coprimality conditions. By introducing 'simple polynomials' of special monomial types and analyzing their automorphism groups via character theory and diagonal actions, the authors determine all possible orders of linear automorphisms for smooth hypersurfaces, proving that for cubic fourfolds, such orders must divide one of 21, 30, 32, 33, 36, or 48 — each realized uniquely up to isomorphism.

ABSTRACT

Given integers $d\ge 3$ and $N\ge 3$. Let $G$ be a finite abelian group acting faithfully and linearly on a smooth hypersurface of degree $d$ in the complex projective space $\mathbb{P}^{N-1}$. Suppose $G\subset PGL(N, \mathbb{C})$ can be lifted to a subgroup of $GL(N,\mathbb{C})$. Suppose moreover that there exists an element $g$ in $G$ such that $G/\langle g angle$ has order coprime to $d-1$. Then all possible $G$ are determined (Theorem 4.3). As an application, we derive (Theorem 4.8) all possible orders of linear automorphisms of smooth hypersurfaces for any given $(d,N)$. In particular, we show (Proposition 5.1) that the order of an automorphism of a smooth cubic fourfold is a factor of 21, 30, 32, 33, 36 or 48, and each of those 6 numbers is achieved by a unique (up to isomorphism) cubic fourfold.

Motivation & Objective

• To classify all finite abelian groups acting faithfully and linearly on smooth hypersurfaces of degree $d \geq 3$ in $\mathbb{P}^{N-1}$, under lifting and coprimality conditions.
• To determine all possible orders of linear automorphisms for smooth $(N-2)$-folds of degree $d$.
• To apply the classification to cubic fourfolds ($d=3$, $N=6$), identifying all possible automorphism orders and proving their uniqueness up to isomorphism.

Proposed method

• Represent abelian group actions via $N$ characters $\lambda_i: G \to \mathbb{C}^\times$, reducing the problem to monomial invariance under group action.
• Introduce 'simple polynomials' composed of monomials of type $K$ and $T$, which support smooth, $G$-invariant hypersurfaces.
• Use the condition that $G/\langle g \rangle$ has order coprime to $d-1$ to constrain the group structure and ensure liftable actions.
• Analyze diagonal automorphisms via eigenvalue assignments $\zeta_n^{\sigma_i}$, linking group orders to monomial invariance conditions.
• Apply the classification to $(d,N) = (3,6)$, showing that automorphism orders must divide 21, 30, 32, 33, 36, or 48.
• Prove uniqueness of cubic fourfolds realizing each of these orders by showing that any such automorphism forces the monomial support to match that of a unique simple polynomial.

Experimental results

Research questions

• RQ1 Which finite abelian groups can act faithfully, linearly, and liftable on a smooth hypersurface of degree $d \geq 3$ in $\mathbb{P}^{N-1}$, under the condition that $G/\langle g \rangle$ has order coprime to $d-1$?
• RQ2 What are all possible orders of linear automorphisms of smooth hypersurfaces for arbitrary $(d,N)$?
• RQ3 For cubic fourfolds ($d=3$, $N=6$), which integers can occur as orders of linear automorphisms, and are these orders realized uniquely?

Key findings

• All possible orders of linear automorphisms of smooth hypersurfaces are determined via Theorem 4.8, which classifies such orders for any $(d,N)$.
• For cubic fourfolds, the order of any linear automorphism must divide one of 21, 30, 32, 33, 36, or 48.
• Each of the six values 21, 30, 32, 33, 36, 48 is realized as the order of an automorphism of a unique (up to isomorphism) smooth cubic fourfold.
• The automorphism group of each such cubic fourfold is cyclic of order exactly one of these six values, with the group structure determined by the monomial type of the defining polynomial.
• The classification relies on constructing 'simple polynomials' whose monomial supports are preserved under specific diagonal actions, and proving that any such automorphism must preserve the same monomial structure.
• Uniqueness of the cubic fourfold for each order is established by showing that any smooth cubic polynomial admitting an automorphism of order $n$ in the set must have the same monomial support as the corresponding simple polynomial, up to coordinate permutation.