Uniruledness of some low-dimensional ball quotients

TL;DR

This paper demonstrates that certain low-dimensional complex ball quotients, including moduli spaces of points, are uniruled by employing reflective modular forms and geometric methods, revealing their rich algebraic structure.

Contribution

It introduces a method using reflective modular forms to establish uniruledness of specific low-dimensional ball quotients, including new examples related to Hermitian lattices.

Findings

Certain 3, 4, and 5-dimensional ball quotients are uniruled.

Examples include moduli space of 8 points on projective line.

Some compactifications are rationally chain connected.

Abstract

We define reflective modular forms on complex balls and use a method of Gritsenko and Hulek to show that some ball quotients of dimensions 3, 4 and 5 are uniruled. We give examples of Hermitian lattices over the rings of integers of imaginary quadratic fields $\mathbb{Q}(\sqrt{-1})$ and $\mathbb{Q}(\sqrt{-2})$ for which the associated ball quotients are uniruled. Our examples include the moduli space of 8 points on $\mathbb{P}^1$. Moreover, we find that some of their Satake-Baily-Borel compactifications are rationally chain connected modulo certain cusps.