Complete algebraic vector fields on affine surfaces

TL;DR

This paper classifies all normal affine algebraic surfaces that are quasi-homogeneous under the subgroup generated by flows of complete algebraic vector fields, using dual graphs of their boundary divisors.

Contribution

It provides a complete classification of such surfaces based on the dual graphs of their boundary divisors, linking algebraic vector fields to surface geometry.

Findings

Classification of quasi-homogeneous affine surfaces under the subgroup aAutH(X)

Connection between algebraic vector fields and boundary dual graphs

Characterization of surfaces via SNC-completions

Abstract

Let $\AAutH (X)$ be the subgroup of the group $\AutH (X)$ of holomorphic automorphisms of a normal affine algebraic surface $X$ generated by elements of flows associated with complete algebraic vector fields. Our main result is a classification of all normal affine algebraic surfaces $X$ quasi-homogeneous under $\AAutH (X)$ in terms of the dual graphs of the boundaries $\bX \setminus X$ of their SNC-completions $\bX$.