Year: 2026
Type: preprint
Abstract: 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 $|\mathrm{Aut}(S)|\leq 200$ using the canonical genus-two fibration and the double-cover model over the Hirzebruch surface $Σ_2$. Computing $\mathrm{Aut}(Σ_2)$ from its Cox ring separates the action on the base $\mathbb{P}^1$ from the vertical kernel. If $\overline K_0$ denotes the vertical image on $Σ_2$, then $\overline K_0$ is cyclic of order $5$, $4$, $2$, or $1$, and we prove the respective sharp bounds $|\mathrm{Aut}(S)|\leq 200, 192, 48, 80$. 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, $(C_{10}\times C_{10})\rtimes C_2$, $(BO_{48}\times C_8)/\langle(-I,J^4)\rangle$, $(BT_{24}\times C_4)/\langle(-I,J^2)\rangle$, and $C_{40}\times C_2$. 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.
Source: arXiv (Cornell University)
Author: Zhao, Hang
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: Shandong University
OpenAlex ID: W7165907812