Rationality of the moduli spaces of Eisenstein K3 surfaces

TL;DR

This paper proves that twenty-two out of twenty-four moduli spaces of Eisenstein K3 surfaces, characterized by non-symplectic order 3 symmetry, are rational, advancing understanding of their geometric structure.

Contribution

It establishes the rationality of most moduli spaces of Eisenstein K3 surfaces, a significant step in classifying their geometric properties.

Findings

Twenty-two moduli spaces are proven to be rational.

The moduli spaces are associated with Eisenstein lattices and complex ball quotients.

The study enhances understanding of the geometric structure of K3 surfaces with non-symplectic symmetry.

Abstract

K3 surfaces with non-symplectic symmetry of order 3 are classified by open sets of twenty-four complex ball quotients associated to Eisenstein lattices. We show that twenty-two of those moduli spaces are rational.