On Automorphism Groups of (1,2)-Surfaces

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 $|\mathrm{Aut}(S)|\le 200$ using the canonical genus-two fibration and the double-cover model over the Hirzebruch surface ${\Sigma }_2$. Computing $\mathrm{Aut}({\Sigma }_2)$ 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 ${\Sigma }_2$, then ${\overline{K}}_0$ is cyclic of order $5$, $4$, $2$, or $1$, and we prove the respective sharp bounds $|\mathrm{Aut}(S)|\le 200,192,48,80$. 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, $(C_{10}\times C_{10})⋊C_2$, $(B{O}_{48}\times C_8)/⟨(-I,J^4)⟩$, $(B{T}_{24}\times C_4)/⟨(-I,J^2)⟩$, and $C_{40}\times C_2$. Here $B{O}_{48}$ and $B{T}_{24}$ denote the binary octahedral and binary tetrahedral groups, respectively, and $J$ denotes a generator of the corresponding cyclic factor.

arXiv comment19 pages. Corrected Theorem 4.4(iii) and the automorphism-group identifications in Proposition 5.1; added an example attaining the sharp bound 48