Period map of triple coverings of $\mathbf P^2$ and mixed Hodge structures

TL;DR

This paper investigates the period map for triple coverings of the projective plane branched along specific six-line configurations, revealing a one-parameter family of K3 surfaces through mixed Hodge structures.

Contribution

It introduces a novel approach to analyze the period map using mixed Hodge structures, uncovering additional one-dimensional information beyond the moduli space.

Findings

Identifies a one-parameter family of K3 surfaces from triple coverings.

Extracts extra information from mixed Hodge structures on relative homology.

Connects the moduli of line configurations with the geometry of K3 surfaces.

Abstract

We study a period map for triple coverings of $\mathbf P^2$branching along special configurations of $6$ lines. Though the moduli space of special configurations isa two dimensional variety,the minimal models of the coverings form a oneparameter family of K3 surfaces.We extract extra one dimensionalinformation from the mixed Hodge structure on the second relative homology group.