Combinatorics and topology of line arrangements in the complex projective plane

Artal-Bartolo, Enrique

doi:10.1090/s0002-9939-1994-1189536-3

Cited by 10 publications

(11 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…, and l i is the number of lines in L passing through V i , for i = 1, 2, 3. Using this formula, we have an alternative proof of the result of Artal [AB94a]. Indeed, if we consider the (3, 2)configurations given in Figure 10 (called the Pappus and the non-Pappus configurations), we obtain that:…”

Section: Study Of Characteristic Varietiesmentioning

confidence: 94%

“…Remark 2.3. Classically, a weaker notion of combinatorics for line arrangements is also considered (see, for example [AB94a]), defined by the number of lines and the number of singular points of each multiplicity (globally and on each line).…”

Section: Combinatorics and Topology Of Arrangementsmentioning

confidence: 99%

“…In our examples of Zariski pairs, the simplicity of computation of the chamber weight and the possibility of constructing other examples using the present method are a significant progress for the fundamental questions relating combinatorial data and topology of line arrangements. As an illustration of these possibilities, at the end of the paper we give a sketch of an alternative proof of Artal's result [AB94a] stating that the characteristic varieties are 1 An appendix containing detailed figures of our first Zariski pair with 13 lines as well as those of its degeneration with two quintuple points can be found in the authors' websites. not determined by the weak combinatorics.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Configurations of points and topology of real line arrangements

Guerville-Ballé

Viu-Sos

2018

Math. Ann.

View full text Add to dashboard Cite

A central question in the study of line arrangements in the complex projective plane CP 2 is the following: when does the combinatorial data of the arrangement determine its topological properties? In the present work, we introduce a topological invariant of complexified real line arrangements, called the chamber weight. This invariant is based on the weight counting over the points of the associated dual configuration, located in particular chambers of the real projective plane RP 2 .Using this dual setting, we construct several examples of complexified real line arrangements with the same combinatorial data and different embeddings in CP 2 (i.e. Zariski pairs) which are distinguished by this invariant. In particular, we obtain new Zariski pairs of 13, 15 and 17 lines defined over Q and containing only double and triple points. For each one of our examples, we derive some degenerations containing points of multiplicity 2, 3 and 5, which are also Zariski pairs.We compute explicitly the moduli space of the combinatorics of one of these examples, and prove that it has exactly two connected components. We also obtain three geometric characterizations of these components: the existence of two smooth conics, one tangent to six lines and the other containing six triple points, as well as the collinearity of three specific triple points.

show abstract

Section: Study Of Characteristic Varietiesmentioning

confidence: 94%

Section: Combinatorics and Topology Of Arrangementsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Configurations of points and topology of real line arrangements

Guerville-Ballé

Viu-Sos

2018

Math. Ann.

View full text Add to dashboard Cite

show abstract

“…Indeed, in the real case, the braided wiring diagram can be identified with the usual drawing of the arrangement in lR. 2 .…”

Section: Presentations Of 11"1mentioning

confidence: 99%

On the homotopy theory of arrangements, II

Falk¹,

Randell²

Advanced Studies in Pure Mathematics

View full text Add to dashboard Cite

show abstract

“…Proposition 1.6. Let π : X → X be the (p, q)-blow-up at a point of type (d; a, b) as in (1). Consider two Q-divisors C and D on X(d; a, b).…”

Section: Intersection Theory On V-manifoldsmentioning

confidence: 99%

Embedded Q-resolutions for Yomdin-Lê surface singularities

Martín-Morales

2014

Isr. J. Math.

View full text Add to dashboard Cite

In a previous work we have introduced and studied the notion of embedded Q-resolution, which essentially consists in allowing the final ambient space to contain abelian quotient singularities. Here we explicitly compute an embedded Q-resolution of a Yomdin-Lê surface singularity (V, 0) in terms of a (global) embedded Q-resolution of their tangent cone by means of just weighted blow-ups at points. The generalized A'Campo's formula in this setting is applied so as to compute the characteristic polynomial. As a consequence, an exceptional divisor in the resolution of (V, 0), apart from the first one which might be special, contributes to its complex monodromy if and only if so does the corresponding divisor in the tangent cone. Thus the resolution obtained is optimal in the sense that the weights can be chosen so that every exceptional divisor in the Q-resolution of (V, 0), except perhaps the first one, contributes to its monodromy.

show abstract

Combinatorics and topology of line arrangements in the complex projective plane

Abstract: Abstract. We use some results about Betti numbers of coverings of complements of plane projective curves to discuss the problem of how combinatorics determine the topology of line arrangement, finding a counterexample to a conjecture of Orlik.

Cited by 10 publications

References 5 publications

Configurations of points and topology of real line arrangements

Configurations of points and topology of real line arrangements

On the homotopy theory of arrangements, II

Embedded Q-resolutions for Yomdin-Lê surface singularities

Contact Info

Product

Resources

About