On algebraic surfaces associated to line arrangements

TL;DR

This paper studies algebraic surfaces derived from line arrangements in the complex projective plane, computing their Chern numbers, analyzing their geometric properties, and establishing conditions for their classification as surfaces of general type.

Contribution

It provides explicit calculations of Chern numbers for these surfaces and proves they are never ball quotients, also characterizing when they are of general type.

Findings

Chern numbers expressed via combinatorics of arrangements

Minimal resolution never a quotient of a ball

Surfaces are of general type with only nodes or triple points

Abstract

For a line arrangement in the complex projective plane $\mathbb{P}^2$, we investigate the compactification $\overline{F}$ of the affine Milnor fiber in $\mathbb{P}^3$ and its minimal resolution $\widetilde{F}$. We compute the Chern numbers in terms of the combinatorics of the line arrangement, then we show that the minimal resolution is never a quotient of a ball; in addition, we also prove that $\widetilde{F}$ is of general type when the arrangement has only nodes or triple points as singularities.