$\mathbb{Q}$-Homology Plane pairs with Logarithmic Kodaira dimension 1

TL;DR

This paper studies pairs of algebraic surfaces and curves with specific homology properties, focusing on their singularities and logarithmic Kodaira dimension 1, and provides new results and examples in this classification problem.

Contribution

It establishes new results on the structure of singular $Q$-homology plane pairs with Kodaira dimension 1, including properties of rational curves and singularities, and presents an explicit example.

Findings

Results on smooth rational curves on $S$

Characterization of singularities of $S$

An explicit example of such pairs

Abstract

A pair $(S,C)$ is called a singular $\mathbb{Q}$-homology plane pair if $S$ is a singular projective surface with only quotient singularities having the same rational homology as $\mathbb{p}^2$ and $C \subset S$ has the same rational homology as $\mathbb{p}^1$. We will prove results concerning smooth rational curves on $S$ and the singularities of $S$ such that $\overline \kappa(S^0)=1$ and $\overline \kappa(S-C) \neq -\infty$. We end with an example of such pairs.