Notes on automorphisms of surfaces of general type with \(p_g = 0\) and \(K^2 = 7\) (Q2828020)

Let \(S\) be a minimal complex surface of general type with \(p_g(S)=0\) and \(K^2_S=7\) and let \(\sigma\) be an involution of \(S\). The author proves that \(\sigma\) is in the center of the automorphism group of \(S\); as a corollary, he shows that the involutions of \(S\) form a subgroup of \(\text{Aut}(S)\) of order at most 4.NEWLINENEWLINEThis bound on the number of involutions of \(S\) is sharp, since there exists a family of examples, due to \textit{M. Inoue} [Tokyo J. Math. 17, No. 2, 295--319 (1994; Zbl 0836.14020)], that have an action of \(\mathbb Z_2^2\). The authors determines the automorphism group of this family of examples, showing that it is isomorphic either to \(\mathbb Z_2^2\) or to \(\mathbb Z_2\times\mathbb Z_4\), and that the former case occurs for a general surface in the family.