2 3 Se p 20 05 A FAMILY OF MARKED CUBIC SURFACES AND THE ROOT SYSTEM

We define and study a family of cubic surfaces in the projectivized tangent bundle over a four dimensional projective space associated to the root system D 5. The 27 lines are rational over the base and we determine the classifying map to the moduli space of marked cubic surfaces. This map has degree two and we use it to get short proofs for some results on the Chow group of the moduli space of marked cubic surfaces. A marked cubic surface is a smooth cubic surface S with an ordered set of six skew lines on S. The symmetry group of the configuration of the 27 lines on a smooth cubic surface is the Weyl group of the root system E 6. This group acts on the moduli space of marked cubic surfaces M. In various constructions of M certain subgroups of W (E 6) play a prominent role. Starting with the description of a marked cubic surface as the blow-up of the projective plane in six ordered points the action of the symmetric group S 6 ⊂ W (E 6) on M is very evident. In this approach, M is constructed as a double cover of a projective space branched along a quartic threefold, cf. [DGK], 2.11. The universal family of marked cubic surfaces is given there by the Cremona hexaedral form of a cubic surface (cf. [Co1], section 4, [D], section 3), this family has an obvious S 6-action. In his construction of M and its desingularisation C, Naruki [N] used the subgroup W (F 4), which is associated to a tritangent plane of a cubic surface. In his approach, M is a compacti-fication of the maximal torus of a complex Lie group of type D 4. The universal family is given explicitly, but it is rather complicated. In this paper we consider the subgroup W (D 5) of W (E 6). It corresponds to the stabiliser of a line on a cubic surface. The standard five dimensional represention of W (D 5) gives an action of this Weyl group on projective four space. We define a family of marked cubic surfaces X in the projectivized tangent bundle of this projective space. The family is defined by a very simple determinantal equation and W (D 5) acts on it. The 27 lines and the 45 tritangent planes are also given by simple expressions. Naruki's construction of M and …