Notes on Automorphisms of Surfaces of General Type With And

Abstract

Let$S$be a smooth minimal complex surface of general type with$p_{g}=0$and$K^{2}=7$. We prove that any involution on$S$is in the center of the automorphism group of$S$. As an application, we show that the automorphism group of an Inoue surface with$K^{2}=7$is isomorphic to$\mathbb{Z}_{2}^{2}$or$\mathbb{Z}_{2}\times \mathbb{Z}_{4}$. We construct a$2$-dimensional family of Inoue surfaces with automorphism groups isomorphic to$\mathbb{Z}_{2}\times \mathbb{Z}_{4}$.

A characterization of Inoue surfaces with $$p_g=0$$ p g = 0 and $$K^2=7$$ K 2 = 7

“…Then the subgroup σ, τ of the automorphism group of S is isomorphic to Z 2 × Z 2 and S is an Inoue surface.Proof. Since deg ϕ = 2, the birational involutions σ is contained in the center of the automorphism group Aut(S) of S (see[7, Theorem 1.2] for a general statement). Therefore σ, τ = {1, σ, τ, στ } and σ, τ ∼ = Z 2 × Z 2 .…”