 
Summary: CUBIC SURFACES AND BORCHERDS PRODUCTS
DANIEL ALLCOCK AND EBERHARD FREITAG
Abstract. We apply Borcherds' methods for constructing automorphic forms
to embed the moduli space M of marked complex cubic surfaces into CP 9 .
Specically, we construct 270 automorphic forms on the complex 4ball B 4 ,
automorphic with respect to a particular discrete group . We use the identi
cation from [ACT2] of M with the BailyBorel compactication of B 4 =. Our
forms span a 10dimensional space, and we exhibit the image of M in CP 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 crossratios
introduced by Cayley. It turns out that this model of M was found by Coble
[C] in an entirely dierent way; see [vG].
1. Introduction
The moduli space M of marked cubic surfaces can be identied with the Baily
Borel compactication of B 4 =, where B 4 denotes the complex 4ball and is a
certain arithmetic re
ection group. (See [ACT2] and also [ACT1].) In this paper
we use the methods of R. Borcherds to construct automorphic forms on B 4 . We
will obtain an embedding of M into the 9dimensional projective space P 9 (C ),
whose image is the intersection of 270 explicitly known cubic 8folds. This map is
compatible with the actions of the Weyl group W (E 6 ) on M and P 9 . The former
