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.
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Last modified by jac at 15:52 on 12/27/1997.