Surfaces with $\chi=5, K^{2}=9$ and a canonical involution

TL;DR

This paper classifies minimal surfaces of general type with specific invariants, focusing on those with a canonical involution, and identifies six families with detailed moduli space dimensions.

Contribution

It provides a classification of such surfaces with $ ext{chi}=5$, $K^2=9$, and canonical involution, including the dimensions of their moduli space components.

Findings

Six families of surfaces classified with explicit moduli dimensions

One family forms an irreducible component with a genus 2 fibration

Dimensions of moduli space components range from 27 to 33

Abstract

In this paper, we classify the minimal surfaces of general type with $\chi=5$, $K^{2}=9$ whose canonical map is composed with an involution. We obtain 6 families, whose dimensions in the moduli space are 28, 27, 33, 32, 31 and 32 respectively. Among them, the family of surfaces having a genus 2 fibration forms an irreducible component of $\mathfrak{M}_{\chi=5, K^{2}=9}$.