The moduli space of cubic surfaces and its orbifold fundamental group

Abstract

We find a new presentation of the orbifold fundamental group of the moduli space of cubic surfaces, based on the Artin group of the Petersen graph. This group has two well-known quotients, which appear in the simplest possible way: killing the squares resp. sixth powers of our generators yields the Coxeter group $W(E_6)$ resp. a certain arithmetic group in $\operatorname{PU}(1,4)$. Along the way we present a certain central extension by $\mathbb{Z}$, of the genus zero mapping class group with any number of punctures. The $6$-puncture case is essential in this paper, and the general case may have independent interest.