Moduli Space of Cubic Surfaces as Ball Quotient via Hypergeometric Functions

2004/04/04 by Brent Doran, Brent R. Doran, Doran, Brent R. · 1 citation
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Mathematics and Applications #Polynomial and algebraic computation #math.AG

paper · pdf · doi:10.48550/arxiv.math/0404062

arxiv created 2004/04/04 · openalex publication_date 2004/04/04 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We describe hypergeometric functions of Deligne-Mostow type for open subsets of the configuration space of six points on P², induced from those for seven points on P¹. The seven point ball quotient example DM(2⁵,1²) does not appear on Mostow's original list, but does appear on Thurston's corrected version. We show that DM(2⁵,1²) is a finite cover of the moduli space of cubic surfaces M_C endowed with the ball quotient structure G_C\B⁴ 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.

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