On Automorphism Groups of (1,2)-Surfaces

Abstract

Let S be a minimal surface of general type with K_S²=1 and p_g(S)=2, called a (1,2)-surface. We give a geometric recovery and refinement of D. Wen's bound |Aut(S)|≤ 200 using the canonical genus-two fibration and the double-cover model over the Hirzebruch surface Σ₂. Computing Aut(Σ₂) from its Cox ring separates the action on the base ℙ¹ from the vertical kernel. If K̅₀ denotes the vertical image on Σ₂, then K̅₀ is cyclic of order $5$, $4$, $2$, or $1$, and we prove the respective sharp bounds |Aut(S)|≤ 200, 192, 48, 80. The proof combines singularity indices of genus-two fibrations with invariant divisors arising from the branch equation. We also construct admissible branch divisors attaining all four bounds and show that their full automorphism groups are, respectively, (C₁₀× C₁₀)rtimes C₂, (BO₄₈× C₈)/⟨(−I,J⁴)⟩, (BT₂₄× C₄)/⟨(−I,J²)⟩, and C₄₀× C₂. Here BO₄₈ and BT₂₄ denote the binary octahedral and binary tetrahedral groups, respectively, and J denotes a generator of the corresponding cyclic factor.

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BibTeX RIS

BibTeX

@misc{zhao2026automorphism,
  title = {{On Automorphism Groups of (1,2)-Surfaces}},
  author = {Hang Zhao},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2606.25554},
  eprint = {2606.25554},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.25554},
}

RIS

TY  - UNPB
TI  - On Automorphism Groups of (1,2)-Surfaces
AU  - Hang Zhao
PY  - 2026
DA  - 2026/10/22
PB  - arXiv
DO  - 10.48550/arxiv.2606.25554
UR  - https://arxiv.org/abs/2606.25554
AB  - Let S be a minimal surface of general type with K_S^2=1 and p_g(S)=2, called a (1,2)-surface. We give a geometric recovery and refinement of D. Wen's bound |Aut(S)|≤ 200 using the canonical genus-two fibration and the double-cover model over the Hirzebruch surface Σ_2. Computing Aut(Σ_2) from its Cox ring separates the action on the base ℙ^1 from the vertical kernel. If K̅_0 denotes the vertical image on Σ_2, then K̅_0 is cyclic of order $5$, $4$, $2$, or $1$, and we prove the respective sharp bounds |Aut(S)|≤ 200, 192, 48, 80. The proof combines singularity indices of genus-two fibrations with invariant divisors arising from the branch equation. We also construct admissible branch divisors attaining all four bounds and show that their full automorphism groups are, respectively, (C_10× C_10)rtimes C_2, (BO_48× C_8)/⟨(−I,J^4)⟩, (BT_24× C_4)/⟨(−I,J^2)⟩, and C_40× C_2. Here BO_48 and BT_24 denote the binary octahedral and binary tetrahedral groups, respectively, and J denotes a generator of the corresponding cyclic factor.
ER  -

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