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.