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

Abstract

We prove that the bracket width of the simple Lie algebra of vector fields $\mathrm{Vec}(\mathrm{C})$ 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 $\mathrm{Vec}(\mathrm{C})$ is at most two and if moreover $C$ has a unique place at infinity, the bracket width of $\mathrm{Vec}(\mathrm{C})$ is exactly two. We also show that in case $C$ is rational, the width of $\mathrm{Vec}(\mathrm{C})$ equals one.