The Complex Hyperbolic Geometry of the Moduli Space of Cubic Surfaces

TL;DR

This paper establishes a complex hyperbolic geometric description of the moduli space of stable cubic surfaces, showing it as a quotient of the complex 4-ball by an explicit arithmetic group, revealing structural insights and symmetry points.

Contribution

It extends the known hyperbolic geometric structure from cubic curves to cubic surfaces, providing an explicit arithmetic group action and geometric interpretation.

Findings

Moduli space of stable cubic surfaces is biholomorphic to a quotient of the complex 4-ball.

Identification of points in hyperbolic space corresponding to symmetric cubic surfaces.

Structural insights into the geometry and symmetries of the moduli space.

Abstract

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 generated by complex reflections. This identification gives interesting structural information on the moduli space and allows one to locate the points in complex hyperbolic 4-space corresponding to cubic surfaces with symmetry, e.g., the Fermat cubic surface. Related results, not quite as extensive, were announced in alg-geom/9709016.