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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(E6) resp. a certain arithmetic group in PU(1,4). Along the way we present a certain central extension by 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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Cite

BibTeXRIS
BibTeX
@misc{allcock2026moduli,
  title = {{The moduli space of cubic surfaces and its orbifold fundamental group}},
  author = {Daniel Allcock and Tathagata Basak and Eduard Looijenga},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.16080},
  eprint = {2610.16080},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.16080},
}
RIS
TY  - UNPB
TI  - The moduli space of cubic surfaces and its orbifold fundamental group
AU  - Daniel Allcock
AU  - Tathagata Basak
AU  - Eduard Looijenga
PY  - 2026
DA  - 2026/10/12
PB  - arXiv
DO  - 10.48550/arxiv.2610.16080
UR  - https://arxiv.org/abs/2610.16080
AB  - 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 PU(1,4). Along the way we present a certain central extension by 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.
ER  -