A W (E6)-equivariant projective embedding of the moduli space of cubic surfaces

An explicit projective embedding of the moduli space of marked cubic surfaces is given. This embedding is equivariant under the Weyl group of type E6. The image is defined by a system of linear and cubic equations. To express the embedding in a most symmetric way, the target would be 79-dimensional, however the image lies in a 9-dimensional linear subspace. 1 The moduli space of cubic surfaces 1.1 The space M and the action of the group G We first fix some notation and recall a few known facts on the moduli space of marked cubic surfaces. The moduli space of marked cubic surfaces, which we denote by M , is studied for example in [5] and [8]. Since any nonsingular cubic surface can be obtained by blowing up the projective plane P at six points, it can be represented by a 3× 6-matrix of which columns give homogeneous coordinates of the six points. In order to get a smooth cubic surface from six points, we assume that no three points are collinear and the six points are not on a conic. On the set of 3 × 6 matrices, we have a cannical action of GL3 on the left and the group C × acts naturally on homogeneous coordinates. By killing such ambiguity of coordinates, we get the following expression