Classification of Affinely Homogeneous Hessian Rank 2 Hypersurfaces S^3 in R^4

TL;DR

This paper classifies all affinely homogeneous hypersurfaces in R^4 with Hessian rank 2, identifying 34 inequivalent models and their moduli spaces using the power series method of equivalence and Lie algebra descriptions.

Contribution

It provides a complete classification of affinely homogeneous Hessian rank 2 hypersurfaces in R^4, including explicit Lie algebra models and moduli space descriptions.

Findings

Identified 34 inequivalent homogeneous models.

Described models using explicit Lie algebras and invariants.

Found moduli spaces parametrized by algebraic varieties.

Abstract

We determine all affinely homogeneous hypersurfaces S^3 in R^4 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 S^3 in R^4, 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.