The fundamental group of surfaces parametrizing cuboids

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.