Unformatted text preview: interpreted as
saying that the moduli space of nonsingular cubic surfaces admits a compactiﬁcation
isomorphic to the weighted projective space P(1, 2, 3, 4, 5). Let F (T0 , T1 , T2 , T3 ) = 0
be an equation of a nonsingular cubic surface S . Adding the cube of a new variable
T4 , we obtain an equation
F (T0 , T1 , T2 , T3 ) + T4 = 0 of a nonsingular cubic hypersurface X in P4 (C). There is a construction of an
abelian variety of dimension 10 attached to X (the intermediate jacobian ) Jac(X ).
The variety X admits an obvious automorphism of order 3 deﬁned by multiplying
the last coordinate by a third root of unity. This makes Jac(X ) a principally
polarized abelian variety of dimension 10 with complex multiplication of certain
type.14 The moduli space of such varieties is known to be isomorphic to a quotient
of a 4-dimensional complex ball by a certain discrete subgroup Γ. It is proven
in  that the group Γ is a hyperbolic complex crystallographic reﬂection group
and the quotient HC /Γ is isomorphic to the moduli space of cubic surfaces with at
most ordinary double points as singularities. By adding one point one obtains a
compactiﬁcation of the moduli space isomorphic to the weighted projective space
P(1, 2, 3, 4, 5).
The geometric interpretation of reﬂection hyperplanes is also very nice; they form
one orbit representing singular surfaces. The group Γ contains a normal subgroup Γ
with quotient isomorphic to the Weyl group W (E6 ). The quotient subgroup HC /Γ
is the moduli space of marked nodal cubic surfaces. For a nonsingular surface a
marking is a ﬁxing of order on the set of 27 lines on the surface.
We mentioned before that some complex ball quotients appear as the moduli
space of K3 surfaces which admit an action of a cyclic group G with ﬁxed structure
of the sublattice (SX )G . This idea was used by S. Kond¯ to construct an action of
a crystallographic reﬂection group Γ in a complex ball HC (resp. HC ) with orbit
space containing the moduli space of nonsingular plane quartic curves of genus 3
(resp. moduli space of canonical curves of genus 4).15 In the ﬁrst case he assigns
to a plane quartic F (T0 , T1 , T2 ) = 0 the quartic K3-surface
F (T0 , T1 , T2 ) + T3 = 0
14 This beautiful idea of assigning to a cubic surface a certain abelian variety was independently
suggested by B. van Geemen and B. Hunt.
15 It is isomorphic to the moduli space of Del Pezzo surfaces of degree 2. 54 IGOR V. DOLGACHEV G
with automorphism of order 4 and SX ∼ U (2) ⊥ A1 (−1)6 . This leads to a new
example of a crystallographic reﬂection group in HC . It is known that a canonical
curve C of genus 4 is isomorphic to a complete intersection of a quadric and cubic
in P3 (C). To each such curve Kond¯ assigns the K3 surface isomorphic to the
triple cover of the quadric branched along the curve C . It has an action of a
cyclic group of order 3 with (SX )G ∼ U ⊥ A2 (−2). The stabilizer of a reﬂection
hyperplane gives a complex reﬂection group in HC...
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