The complex hyperbolic geometry of the moduli space of cubic surfaces

By Daniel Allcock, J. Carlson, D. Toledo. 2000

166 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/740f10e46e3098255d8bc470f725511de08a4d9a

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

The Moduli Space of Cubic Threefolds As a Ball Quotient (2006, methodology): “In [4], we showed that there is an analogous result for cubic surfaces; the analogy is clearest when we view the upper half plane as complex hyperbolic 1-space, that is, as the unit disk.”
New non-arithmetic complex hyperbolic lattices (2014, background): “The few other examples of moduli spaces that are known to admit a complex hyperbolic uniformization (see [ACT02], [Kon00] for instance) produce arithmetic lattices.”
The period map for cubic threefolds (2006, unknown): “The second author is supported by the Netherlands Organisation for Scientific Research (NWO).”
Classical Algebraic Geometry: A Modern View (2012, methodology): “We also refer to a recent construction of the GIT-moduli space of cubic surfaces as a quotient of a complex 4-dimensional ball by a reflection group [6],[186].”
Reflection groups in algebraic geometry (2006, background): “Toledo of a complex ball uniformization of the moduli space of cubic surfaces [2].”
Compactifications defined by arrangements I: The Ball Quotient case (2001, background): “Allcock, Carlson, and Toledo gave in [ 3 ] and [4] a 4-ball quotient description of this moduli space by assigning to a smooth cubic surface S P3 first the µ3-cover K! P3 totally ram-”
Moduli of K3 Surfaces and Complex Ball Quotients (2007, background): “The topic of the talks was an introduction to recent work of various people on the complex ball uniformization of the moduli spaces of del Pezzo surfaces and algebraic curves of lower genus ([ ACT ], [Vo], [K1]–[K4], [HL]).”
Special cycles on unitary Shimura varieties II: global theory (2009, methodology): “,1) ∗, which is ﬁnite over SpecZ\S, and contracts lines in the ﬁbersof M(t;1,1)∗,S over primes p∈S. 15. Images of period maps Our aim in this section is to connect some of the ”hidden” period maps of [1, 2, 54, 55, 56, 57] to our moduli spaces and identify the complements of their images with our sp”
o-minimal GAGA and a conjecture of Griffiths (2018, background): “ddition d1 = ··· = dc then MT is an aﬃne scheme, [5, Theorem 1.1.i)], while if c>1 and d2 = ··· = dc, MT is quasi-projective by [5, Corollary 1.2]. Finally, for T= (3;2), MT is quasi-projective by [1]. Corollary 7.5. For all T6= (2; n), the coarse moduli space MT is quasi-projective. Proof. This ”
COMPLEX MULTIPLICATION, GRIFFITHS-YUKAWA COUPLINGS, AND RIGIDITY FOR FAMILIES OF HYPERSURFACES (2003, background): “By [1] the moduli stack M3,3 has this property.”
Cubic surfaces and Borcherds products (2000, methodology): “automorphic forms on the complex 4-ball B4, automorphic with respect to a particular discrete group i. We use the identiflcation from [ ACT2 ] of M with the Baily{Borel compactiflcation of B4=i.”
The moduli space of cubic fourfolds (2007, background): “the moduli spaces of cubic surfaces and cubic threefolds – see [3], [4] and [18]).”
A complex ball uniformization of the moduli space of cubic surfaces via periods of K 3 surfaces (2003, methodology): “Proof. If we consider T as a free Hermitian module � over Z[ζ3] (see 6.9), then [ ACT , AF] define an isotropic vector, a short vector and a long vector as a vector with Hermitian square equal to 0, −1, −2, respectively.”
The moduli space of cubic threefolds (2003, methodology): “The current work, together with refinements of the techniques of [1], should provide much more detailed information, such as exactly which discrete group, which periods arise from smooth threefolds, and how degeneration to singular threefolds is reflected in the ball quotient.”