Abstract for the Complex Hyperbolic Geometry of the Moduli Space of Cubic Surfaces.
Recall that the moduli space of smooth (that is, stable) cubic curves is isomorphic to the quotient of the upper half plane by the group of fractional linear transformations with integer coefficients. We establish a similar result for stable cubic surfaces: the moduli space is biholomorphic to a quotient of the compex 4-ball by an explict arithmetic group geneated by complex reflections. This identification gives interesting structural information on the moduli space and, for example, allows one to locate the points in complex hyperbolic 4-space corresponding to cubic surfaces with symmetry, e.g., the Fermat cubic surface.
Related resuls, not quite as extensive, and not discussed in the
present level of detail were announced in Complex Hyperbolic Structure for Moduli
of Cubic Surfaces listed on the preceding page.