A W(E_6)-equivariant projective embedding of the moduli space of cubic surfaces

TL;DR

This paper constructs an explicit, symmetric projective embedding of the moduli space of marked cubic surfaces that respects the Weyl group E6 symmetry, revealing its algebraic structure in a low-dimensional setting.

Contribution

It provides the first explicit E6-equivariant embedding of the moduli space, characterized by linear and cubic equations, with a notably low-dimensional image.

Findings

Embedding is defined by linear and cubic equations.

Image lies in a 9-dimensional subspace.

Embedding respects E6 symmetry.

Abstract

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.