Period Map of Triple Coverings of P${}^2$ and Mixed Hodge Structures

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 is a two-dimensional variety, the minimal models of the coverings form a one-parameter family of K3 surfaces. We extract extra one-dimensional information from the mixed Hodge structure on the second relative homology group. We define the period map from the moduli space of marked configurations to the domain $\mathbf{B}\times {\mathbf{C}}^2$, where $\mathbf{B}$ is the right half-plane, and give a defining equation of its image by a theta function. We write down the inverse of the period map using theta functions.