Abstract for Cubic Surfaces Note

We show that the moduli space M of marked cubic surfaces is biholomorphic to the quotient by a discrete group generated by complex reflections of the complex four-ball minus the reflection hyperplanes of the group. Thus M carries a complex hyperbolic structure: an (incomplete) metric of constant holomorphic sectional curvature.

As a consequence we show that the fundamental group G of M contains a normal subgroup K that is not finitely generated. Indeed, G is an extension of the complex reflection group mentioned above by K. We also show that G is not a lattice in any semisimple Lie group.