The density property for Gizatullin surfaces of type $[[0,0,-r_2,-r_3]]$

TL;DR

This paper proves the algebraic and holomorphic density properties for certain Gizatullin surfaces characterized by specific polynomial equations, advancing understanding of their automorphism groups and complex structures.

Contribution

It establishes the algebraic density property for smooth Gizatullin surfaces of type [[0,0,-r_2,-r_3]] and extends the density property to cases with holomorphic functions.

Findings

Proved algebraic density property for smooth Gizatullin surfaces of the specified type.

Extended density property to surfaces defined by holomorphic functions P and Q.

Enhanced understanding of automorphism groups of these surfaces.

Abstract

Gizatullin surfaces of type $[[0,0,-r_2,-r_3]]$ can be described by the equations $yu = x P(x)$, $xv = u Q(u)$ and $yv = P(x) Q(u)$ in $\mathbb{C}^4_{x,y,u,v}$ where $P$ and $Q$ are non-constant polynomials. We establish the algebraic density property for smooth Gizatullin surfaces of this type. Moreover we also prove the density property for smooth surfaces given by these equations when $P$ and $Q$ are holomorphic functions.