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arXiv · 2310.12710

The fundamental group of surfaces parametrizing cuboids

Benjamin Enriquez·David Jarossay·Francesco Maria Saettone·Yotam Svoray
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Abstract

We prove that an irreducible projective complete intersection of dimension at least two with isolated singularities has trivial fundamental group. As an application, the surface $\Upsilon$ parametrizing cuboids and its minimal resolution of singularities are simply connected. By an independent argument we also show that the surface $V$ parametrizing face cuboids and its resolution are simply connected as well. We then introduce two smooth open subvarieties $S_{1}$ and $S_{2}$ of the surface parametrizing face cuboids, show that each has fundamental group isomorphic to $\mathbb{F}_{3}\ltimes \mathbb{Z}^{2}$, and prove that their Malcev completions reduce to the free pro-unipotent group on three generators. In an appendix we treat the corresponding real loci, whose fundamental groups, in contrast, are far from trivial.

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Benjamin Enriquez, David Jarossay, Francesco Maria Saettone, Yotam Svoray. 2023-10-19. The fundamental group of surfaces parametrizing cuboids. https://arxiv.org/abs/2310.12710

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