Automorphisms of affine Veronese surfaces

TL;DR

This paper investigates the automorphisms and derivations of affine Veronese surfaces, showing they are induced by those of the polynomial algebra in two variables and revealing their group structure.

Contribution

It proves that all automorphisms and derivations of certain subalgebras are induced by those of the polynomial algebra, and describes their automorphism group structure.

Findings

Automorphisms are induced by automorphisms of K[x,y]

Derivations are induced by derivations of K[x,y]

Automorphism group has an amalgamated free product structure

Abstract

We prove that every derivation and every locally nilpotent derivation of the subalgebra $K[x^n, x^{n-1}y,\ldots,xy^{n-1}, y^n]$, where $n\geq 2$, of the polynomial algebra $K[x,y]$ in two variables over a field $K$ of characteristic zero is induced by a derivation and a locally nilpotent derivation of $K[x,y]$, respectively. Moreover, we prove that every automorphism of $K[x^n, x^{n-1}y,\ldots,xy^{n-1}, y^n]$ over an algebraically closed field $K$ of characteristic zero is induced by an automorphism of $K[x,y]$. We also show that the group of automorphisms of $K[x^n, x^{n-1}y,\ldots,xy^{n-1}, y^n]$ admits an amalgamated free product structure.