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

TL;DR

This paper refines the classification of smooth minimal surfaces of general type with specific invariants, focusing on involutions and their fixed points, leading to new restrictions and characterizations of such surfaces.

Contribution

It improves previous results by excluding certain cases and providing new classifications for surfaces with specific involution fixed points and branch divisors.

Findings

Excluded the case of Kodaira dimension 1 for certain involutions.

Reduced possibilities for branch divisors when fixed points are nine.

Identified surfaces with three irreducible branch components as Inoue surfaces.

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.