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Math soc ns 12 1985 247249 mr776478 86f14028 25

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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 3 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 [2] that the group Γ is a hyperbolic complex crystallographic reflection group 4 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 4 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 o of the sublattice (SX )G . This idea was used by S. Kond¯ to construct an action of 6 9 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 4 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 = 6 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 o 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 = 8 hyperplane gives a complex reflection group in HC...
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