Chebyshev curves, free resolutions and rational curve arrangements

Details Chebyshev curves, free resolutions and rational curve arrangements Chebyshev curves, free resolutions and rational curve arrangements

2011-08-03

arxiv.org/abs/1108.0798v3

Mathematics

Algebraic Geometry

14 pages, version 3: two misprints are corrected on p.11 and a reference to further work by the authors in this direction is a

Scientific paper

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.

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