Affine Homogeneous Surfaces with Hessian rank 2 and Algebras of Differential Invariants

TL;DR

This paper classifies affine homogeneous surfaces with Hessian rank 2 by analyzing their differential invariants and algebraic conditions, organizing models into distinct inequivalent classes.

Contribution

It introduces algebraic conditions for homogeneity and systematically classifies models based on differential invariants, extending prior work on affine homogeneous surfaces.

Findings

Necessary algebraic conditions for homogeneity are established.

Homogeneous models are organized into inequivalent branches.

The approach extends classification methods using Olver's recurrence formulas.

Abstract

Consider a graphed holomorphic surface $u=F(x,y)$ in $\mathbb{C}^3_{x,y,u}$ under the action of the affine transformation group $A(3)$. In 1999, Eastwood and Ezhov obtained a list of homogeneous models by determining possible tangential vector fields. Inspired by Olver's recurrence formulas, we study the algebra of $A(3)$ differential invariants of surfaces. We obtain necessary conditions for homogeneity of algebraic nature. Solving these conditions, we organise homogeneous models in inequivalent branches.