What question did this study set out to answer?

To determine the bracket width of the Lie algebra of vector fields on smooth affine curves with different properties.

March 29, 2026Open Access

Bracket Width of the Lie Algebra of Vector Fields on a Smooth Affine Curve

Key Points

To determine the bracket width of the Lie algebra of vector fields on smooth affine curves with different properties.
Proved theoretical limits for the bracket width for various types of smooth affine curves.
Analyzed the case of plane curves and their unique places at infinity.
Examined rational curves and connected properties to bracket width.
Bracket width of the Lie algebra Vec(C) is at most three for smooth irreducible affine curves.
For plane curves, bracket width is at most two, and exactly two for curves with a unique place at infinity.
Rational curves have a bracket width of one.

Abstract

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

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