2024/04/29 by Julien Heyd, Heyd, Julien, Joël Merker +1
Mathematics · #Algebraic Geometry and Number Theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.2404.18565
openalex publication_date 2024/04/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We determine all affinely homogeneous hypersurfaces S3 in R4 whose Hessian is (invariantly) of constant rank 2, including the simply transitive ones. We find 34 inequivalent terminal branches yielding each to a nonempty moduli space of homogeneous models of hypersurfaces S3 in R4, sometimes parametrized by a certain complicated algebraic variety, especially for the 15 (over 34) families of models which are simply transitive. We employ the power series method of equivalence, which captures invariants at the origin, creates branches, and infinitesimalizes calculations. In Lie's original classification spirit, we describe the found homogeneous models by listing explicit Lie algebras of infinitesimal transformations, sometimes parametrized by absolute invariants satisfying certain algebraic equations.