Moduli Space of Cubic Surfaces as Ball Quotient via Hypergeometric Functions

TL;DR

This paper links the moduli space of cubic surfaces to a specific hypergeometric ball quotient, expanding understanding of their geometric and monodromy group relationships.

Contribution

It demonstrates that the moduli space of cubic surfaces is a finite cover of a Deligne-Mostow hypergeometric ball quotient, answering a key question about their monodromy groups.

Findings

The moduli space of cubic surfaces is a finite cover of a specific hypergeometric ball quotient.

The hypergeometric functions for six points on P^2 are derived from those for seven points on P^1.

The monodromy group G_C is shown to be commensurable with Deligne-Mostow groups.

Abstract

We describe hypergeometric functions of Deligne-Mostow type for open subsets of the configuration space of six points on P^2, induced from those for seven points on P^1. The seven point ball quotient example DM(2^5,1^2) does not appear on Mostow's original list, but does appear on Thurston's corrected version. We show that DM(2^5,1^2) is a finite cover of the moduli space of cubic surfaces M_C endowed with the ball quotient structure G_C\B^4 of Allcock, Carlson, and Toledo. This answers a question of Allcock about the commensurability of G_C with the monodromy groups of Deligne-Mostow hypergeometric functions.