Bracket width of the Lie algebra of vector fields on a smooth affine curve

TL;DR

This paper establishes bounds on the bracket width of the Lie algebra of vector fields on smooth affine curves, revealing that the width varies with the curve's geometric properties and is at most three.

Contribution

It provides the first bounds on the bracket width of vector field Lie algebras for various classes of affine curves, including exact values in special cases.

Findings

Bracket width of Vec(C) is at most three for smooth irreducible affine curves.

If C is a plane curve, the bracket width is at most two.

For rational curves, the bracket width is exactly one.

Abstract

We prove that the bracket width of the simple Lie algebra of vector fields $\rm{Vec}(C)$ of a smooth irreducible affine curve $C$ with a trivial tangent sheaf is at most three. In addition, if $C$ is a plane curve, the bracket width of $\rm{Vec}(C)$ is at most two and if moreover $C$ has a unique place at infinity, the bracket width of $\rm{Vec}(C)$ is exactly two. We also show that in case $C$ is rational, the width of $\rm{Vec}(C)$ equals one.