Cubic surfaces and Borcherds products

Abstract.

We apply Borcherds' methods for constructing automorphic forms to embed the moduli space $\mathcal{M}$ of marked complex cubic surfaces into $\mathbb{C}P^9$. Specifically, we construct 270 automorphic forms on the complex 4-ball $\mathcal{B}_4$, automorphic with respect to a particular discrete group $\Gamma$. We use the identification from [ACT2] of $\mathcal{M}$ with the Baily-Borel compactification of $\mathcal{B}_4 / \Gamma$. Our forms span a 10-dimensional space, and we exhibit the image of $\mathcal{M}$ in $\mathbb{C}P^9$ as the intersection of 270 cubic hypersurfaces. Finally, we interpret the pairwise ratios of our forms as the original invariants of cubic surfaces, the cross-ratios introduced by Cayley. It turns out that this model of $\mathcal{M}$ was found by Coble [C] in an entirely different way; see [vG].