Topological invariants of the complement to arrangements of rational plane curves

Cogolludo-Agustín, José Ignacio

doi:10.1090/memo/0756

Cited by 19 publications

(21 citation statements)

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“…It turns out that these curves are unions of lines and conics in general position, see Corollaries 2•2 and 2•3. Hence their study may be of interest in the theory of rational curve arrangements, see for instance [2], [6], [18]. There are interesting relations with knot theory as well, see [16].…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Chebyshev curves, free resolutions and rational curve arrangements

Dimca¹,

Sticlaru²

2012

Math. Proc. Camb. Phil. Soc.

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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

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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Chebyshev curves, free resolutions and rational curve arrangements

Dimca¹,

Sticlaru²

2012

Math. Proc. Camb. Phil. Soc.

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“…The exceptional divisor E = π Using the results in [25] we can generalize Definition 2.7 in [13] for a non-normal crossing Q-divisor in X.…”

Section: Definition 24 ([19]mentioning

confidence: 99%

Numerical adjunction formulas for weighted projective planes and lattice point counting

Cogolludo-Agustín¹,

Martín-Morales²,

Ortigas-Galindo³

2016

Kyoto J. Math.

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“…Examples 4.8 and 4.10 illustrate this approach by looking at some arrangements of lines and conics in the plane. Though these examples may be treated using the results by 3506 ALEXANDRU DIMCA AND LAURENTIU MAXIM Cogolludo in [4], we feel that our approach is more general and hence more likely to extend to other situations.…”

Section: Introductionmentioning

confidence: 99%

“…Alexander invariants in the form of Alexander modules, characteristic varieties and Alexander polynomials have been recently intensively studied, in particular in relation to the twisted cohomology of hypersurface arrangement complements, see for instance [1], [4], [5], [6], [18], [20], [21], [22], [25], [31], [32], [35], [41], [46].…”

Section: Introductionmentioning

confidence: 99%

Multivariable Alexander invariants of hypersurface complements

Dimca

Maxim

2007

Trans. Amer. Math. Soc.

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Abstract. We start with a discussion on Alexander invariants, and then prove some general results concerning the divisibility of the Alexander polynomials and the supports of the Alexander modules, via Artin's vanishing theorem for perverse sheaves. We conclude with explicit computations of twisted cohomology following an idea already exploited in the hyperplane arrangement case, which combines the degeneration of the Hodge to de Rham spectral sequence with the purity of some cohomology groups.

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Topological invariants of the complement to arrangements of rational plane curves

Cited by 19 publications

References 0 publications

Chebyshev curves, free resolutions and rational curve arrangements

Chebyshev curves, free resolutions and rational curve arrangements

Numerical adjunction formulas for weighted projective planes and lattice point counting

Multivariable Alexander invariants of hypersurface complements

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