The fundamental group of surfaces parametrizing cuboids

2023/10/19 by Enriquez, Benjamin, Jarossay, David, David Jarossay +4
Mathematics · #14G05 11G99 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Geometric and Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2310.12710

openalex publication_date 2023/10/19 · openalex created_date 2023/10/21 · openalex updated_date 2026/07/28

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 Υ 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 S₁ and S₂ of the surface parametrizing face cuboids, show that each has fundamental group isomorphic to \mathbbF₃\ltimes ℤ², 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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