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On automorphism groups of smooth hypersurfaces

Song Yang, Xun Yu, Zigang Zhu

TL;DR

The paper resolves the problem of which smooth hypersurfaces in ${\mathbb P}^{n+1}$ admit the largest possible automorphism group. It develops canonical bounds via primitive constituents and introduces Fermat-test ratios to compare group size against Fermat-type extremals, enabling a broad partition into primitive, imprimitive, and reducible cases. The authors prove that the Fermat hypersurface $X_d^n$ is the extremal object for most $(n,d)$, and they exhaustively enumerate explicit exceptional cases with complete automorphism data. The results yield optimal upper bounds for automorphism groups of general-type hypersurfaces and provide a precise, finite list of non-Fermat exceptions with concrete equations, advancing the classification of symmetry in high-dimensional algebraic varieties.

Abstract

We show that smooth hypersurfaces in complex projective spaces with automorphism groups of maximum size are isomorphic to Fermat hypersurfaces, with a few exceptions. For the exceptions, we give explicitly the defining equations and automorphism groups.

On automorphism groups of smooth hypersurfaces

TL;DR

The paper resolves the problem of which smooth hypersurfaces in admit the largest possible automorphism group. It develops canonical bounds via primitive constituents and introduces Fermat-test ratios to compare group size against Fermat-type extremals, enabling a broad partition into primitive, imprimitive, and reducible cases. The authors prove that the Fermat hypersurface is the extremal object for most , and they exhaustively enumerate explicit exceptional cases with complete automorphism data. The results yield optimal upper bounds for automorphism groups of general-type hypersurfaces and provide a precise, finite list of non-Fermat exceptions with concrete equations, advancing the classification of symmetry in high-dimensional algebraic varieties.

Abstract

We show that smooth hypersurfaces in complex projective spaces with automorphism groups of maximum size are isomorphic to Fermat hypersurfaces, with a few exceptions. For the exceptions, we give explicitly the defining equations and automorphism groups.
Paper Structure (14 sections, 28 theorems, 51 equations, 1 table)

This paper contains 14 sections, 28 theorems, 51 equations, 1 table.

Table of Contents

  1. Introduction
  2. Notation and conventions
  3. Abelian subgroups and canonical bounds
  4. Primitive constituents and associated exact sequences
  5. Abelian subgroups and canonical bounds
  6. Fermat-test ratios and proof of most cases
  7. Fermat-test ratios
  8. Proof of most cases
  9. Refined bounds via special monomials
  10. Primitive groups of automorphisms
  11. Proof of the main theorem
  12. Imprimitive ${\rm Aut}(F)$
  13. Reducible ${\rm Aut}(F)$
  14. A stronger form of the main theorem

Key Result

Theorem 1.1

Fix integers $n\geq1$, $d\geq3$ with $(n,d)\neq(1,3),(2,4)$. Let $X\subset{\mathbb P}^{n+1}$ be a smooth hypersurface of degree $d$ with maximum $|{\rm Aut}(X)|$. Then with the following exceptions:

Theorems & Definitions (61)

  • Theorem 1.1
  • Corollary 1.2
  • Definition 3.1
  • Definition 3.3
  • Definition 3.4
  • Theorem 3.5
  • proof
  • Definition 3.6
  • Lemma 3.7
  • Example 3.8
  • ...and 51 more