Chebyshev curves, free resolutions and rational curve arrangements

TL;DR

This paper constructs a free resolution for the Milnor algebra of Chebyshev plane curves, characterizes when the Milnor algebra of nodal curves exhibits similar behavior, and provides explicit factorizations for these cases.

Contribution

It introduces a new free resolution for the Milnor algebra of Chebyshev curves and characterizes the algebra's behavior for nodal curves with rational components.

Findings

Dimensions of graded components are constant for degrees ≥ 2d-3.

Milnor algebra behavior characterizes rational components in nodal curves.

Explicit factorizations for Chebyshev curves are provided.

Abstract

First we construct a free resolution for the Milnor (or Jacobian) algebra $M(f)$ of a complex projective Chebyshev plane curve $\CC_d:f=0$ of degree $d$. In particular, this resolution implies that the dimensions of the graded components $M(f)_k$ are constant for $k \geq 2d-3.$ Then we show that the Milnor algebra of a nodal plane curve $C$ has such a behaviour if and only if all the irreducible components of $C$ are rational. For the Chebyshev curves, all of these components are in addition smooth, hence they are lines or conics and explicit factorizations are given in this case.