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CUBIC SURFACES AND BORCHERDS PRODUCTS DANIEL ALLCOCK AND EBERHARD FREITAG
 

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 .
Speci cally, we construct 270 automorphic forms on the complex 4-ball B 4 ,
automorphic with respect to a particular discrete group . We use the identi-
cation from [ACT2] of M with the Baily-Borel compacti cation of B 4 =. Our
forms span a 10-dimensional 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 cross-ratios
introduced by Cayley. It turns out that this model of M was found by Coble
[C] in an entirely di erent way; see [vG].
1. Introduction
The moduli space M of marked cubic surfaces can be identi ed with the Baily-
Borel compacti cation of B 4 =, where B 4 denotes the complex 4-ball 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 9-dimensional projective space P 9 (C ),
whose image is the intersection of 270 explicitly known cubic 8-folds. This map is
compatible with the actions of the Weyl group W (E 6 ) on M and P 9 . The former

  

Source: Allcock, Daniel - Department of Mathematics, University of Texas at Austin

 

Collections: Mathematics