Smooth minimal surfaces of general type with $p_g=0, K^2=7$ and involutions

Abstract

Lee and the second named author studied involutions on smooth minimal surfaces $S$ of general type with $p_g(S)=0$ and $K_S^2=7$. They gave the possibilities of the birational models $W$ of the quotients and the branch divisors $B_0$ induced by involutions $\sigma$ on the surfaces $S$. In this paper we improve and refine the results of Lee and the second named author. We exclude the case of the Kodaira dimension $\kappa(W)=1$ when the number $k$ of isolated fixed points of an involution $\sigma$ on $S$ is nine. The possibilities of branch divisors $B_0$ are reduced for the case $k=9$, and are newly given for the case $k=11$. Moreover, we show that if the branch divisor $B_0$ has three irreducible components, then $S$ is an Inoue surface.