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The complex hyperbolic geometry of the moduli space of cubic surfaces. (English) Zbl 1080.14532
Summary: 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 complex 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.

MSC:
14J10 Families, moduli, classification: algebraic theory
32Q45 Hyperbolic and Kobayashi hyperbolic manifolds
32G20 Period matrices, variation of Hodge structure; degenerations
11F23 Relations with algebraic geometry and topology
14J26 Rational and ruled surfaces
14K30 Picard schemes, higher Jacobians
32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)
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