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.