On the geography of log-surfaces

TL;DR

This survey explores the geometric properties of log-surfaces, focusing on the distribution of their invariants and introducing new results on pairs involving K3 surfaces and rational curve arrangements.

Contribution

It provides a comprehensive overview of the geography problem for complex log-surfaces and presents original findings on log-surfaces with K3 surfaces and specific combinatorial conditions.

Findings

Conditions for log-Chern slope to equal 3

Explicit examples of log-surfaces with rational curve arrangements

New results on pairs involving K3 surfaces

Abstract

This survey focuses on the geometric problem of log-surfaces, which are pairs consisting of a smooth projective surface and a reduced non-empty boundary divisor. In the first part, we focus on the geography problem for complex log-surfaces associated with pairs of the form $(\mathbb{P}^{2}, C)$, where $C$ is an arrangement of smooth plane curves admitting ordinary singularities. Specifically, we focus on the case in which $C$ is an arrangement consisting of smooth rational curves as its irreducible components. In the second part, containing original new results, we study log-surfaces constructed as pairs consisting of a complex projective $K3$ surface and a rational curve arrangement. In particular, we provide some combinatorial conditions for such pairs to have the log-Chern slope equal to $3$. Our survey is illustrated with many explicit examples of log-surfaces.