Cubic surfaces and Borcherds products

By Daniel Allcock, Eberhard Freitag. 2000

50 citations recorded by the source corpus.

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].

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b1a8c344ebe1803555983101a668ceca9dba0d27

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

1. Igor Dolgachev
2. Shigeyuki Kondo
3. Bert van Geemen
4. D. Huybrechts
5. Serge Cantat
6. J. Blanc

In Mathematical Physics

1. Igor Dolgachev
2. Shigeyuki Kondo
3. Bert van Geemen
4. Serge Cantat
5. J. Blanc

Citations in the corpus

• Classical Algebraic Geometry: A Modern View (2012, background): “The embedding of the moduli spaces in P(9) by means of automorphic forms on the 4-dimensional complex ball is discussed in [7] and [226].”
• A complex ball uniformization of the moduli space of cubic surfaces via periods of K 3 surfaces (2003, methodology): “It is proven in [ AF ], Proposition 2.1 that th ere are”