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arXiv · 2610.16080

The moduli space of cubic surfaces and its orbifold fundamental group

Daniel Allcock·Tathagata Basak·Eduard Looijenga
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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.

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Daniel Allcock, Tathagata Basak, Eduard Looijenga. 2026-10-12. The moduli space of cubic surfaces and its orbifold fundamental group. https://arxiv.org/abs/2610.16080

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