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, commonly referred to as a (1,2)-surface. The automorphism groups of such surfaces have been classified by David Wen using algebraic methods via the canonical ring, establishing the bound |Aut(S)| <= 200. In this paper, we provide a geometric recovery of this bound from the double cover of the Hirzebruch surface Sigma_2. We compute the automorphisms of Sigma_2 from its Cox ring and analyze the induced action on the base P^1 together with its vertical kernel. Applying this to automorphisms preserving the branch divisor R = Delta_0 + R_0, R_0 in |5 Delta_0 + 10 Gamma|, gives a geometric framework for the vertical and horizontal parts of the automorphism group and recovers Wen&#39;s bound |Aut(S)| <= 200.