The Complex Hyperbolic Geometry of the Moduli Space of Cubic Surfaces

Details The Complex Hyperbolic Geometry of the Moduli Space of Cubic Surfaces The Complex Hyperbolic Geometry of the Moduli Space of Cubic Surfaces

2000-07-09

arxiv.org/abs/math/0007048v1

Mathematics

Algebraic Geometry

Also available at http://www.math.utah.edu/~carlson/research/eprints

Scientific paper

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.

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