Chebyshev curves, free resolutions and rational curve arrangements

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

On the syzygies and Alexander polynomials of nodal hypersurfaces

“…Proof. Note that Proposition 3.1 in 10 implies the claim for n = 2. Suppose first that r ≤ d − 3 and that the claim holds for n − 1 ≥ 2.…”

“…Consider the evaluation map where \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {C}[x_1,\ldots ,x_n]_{\le r}$\end{document} denotes the vector space of polynomials of degree at most r , \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathcal F}({\mathcal N}(n,d))$\end{document} denotes the vector space of \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {C}$\end{document}‐valued functions on the set \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathcal N}(n,d)$\end{document}, and a polynomial h is mapped to the function sending \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$a \in {\mathcal N}(n,d)$\end{document} to \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$h(a)\in \mathbb {C}$\end{document}. Then we have the following partial generalization of Proposition 3.1 in 10.…”

Saturation of Jacobian ideals: Some applications to nearly free curves, line arrangements and rational cuspidal plane curves

“…is given in [25,Section (6.21)]. Similarly, for the Chebyshev curves considered in [11], we get via a direct computation using Singular or CoCoA softwares, the following minimal resolution for S/I f , in the case d = 15:…”

On Hodge Theory of Singular Plane Curves

“…The Hodge theory of the complement of projective hypersurfaces have received a lot of attention, see for instance Griffiths [10] in the smooth case, Dimca-Saito [5] and Sernesi [12] in the singular case. In this paper we consider the case of plane curves and continue the study initiated by Dimca-Sticlaru [7] in the nodal case and the author [1] in the case of plane curves with ordinary singularities of multiplicity up to 3. In the second section we compute the Hodge-Deligne polynomial of a plane curve C, the irreducible case in Proposition 2.1 and the reducible case in Proposition 2.2. Using this we determine the Hodge-Deligne polynomial of U = P 2 \ C and then we deduce in Theorem 2.7 the dimensions of the graded quotients of H 2 (U ) with respect to the Hodge filtration.…”