On the cross ratio variety from the viewpoint of the root system of type $3A_{2}$

Abstract

We observe Naruki's cross ratio variety from the viewpoint of the root system of type $3A_{2}$. We construct some kind of models as moduli space of marked cubic surfaces on which the action of a Weyl subgroup of type $3A_{2}$ and its normalizer can be easily observed. We describe the structure of our model and its relationship to the cross ratio variety.