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2014 2 Benoît Guerville-Ballé Topological invariants of line arrangements Departamento Director/es Instituto Universitario de Investigación en Matemáticas y sus Aplicaciones Artal Bartolo, Enrique Manuel Florens, Vincent Vallès, Jean Director/es Tesis Doctoral Autor Repositorio de la Universidad de Zaragoza – Zaguan http://zaguan.unizar.es UNIVERSIDAD DE ZARAGOZA Departamento Director/es Benoît Guerville-Ballé TOPOLOGICAL INVARIANTS OF LINE ARRANGEMENTS Director/es Instituto Universitario de Investigación en Matemáticas y sus Aplicaciones Artal Bartolo, Enrique Manuel Florens, Vincent Vallès, Jean Tesis Doctoral Autor 2013 Repositorio de la Universidad de Zaragoza – Zaguan http://zaguan.unizar.es UNIVERSIDAD DE ZARAGOZA Departamento Director/es Director/es Tesis Doctoral Autor Repositorio de la Universidad de Zaragoza – Zaguan http://zaguan.unizar.es UNIVERSIDAD DE ZARAGOZA Benoît GUERVILLE-Ballé TOPOLOGy Topological invariants of line arrangements advisors enrique Artal bartolo Vincent FLORENS Jean Vallès Université de Pau et des pays de l’adour universidad de Zaragoza Thesis submitted for the degree of Doctor of Philosophy Thesis prepared within the Laboratory of Applied Mathematics of Pau in the doctoral school of exact sciences and their applications -ED 211- Jury Advisors: •Enrique Manuel Artal Bartolo Catedratico, Universidy of Zaragoza, Spain •Vincent Florens Maître de Conférence, University of Pau (U.P.P.A.), France •Jean Vallès Maître de Conférence (HDR), University of Pau (U.P.P.A.), France Reviewers: •Alexander Degtyarev Associate Professor, Bilkent University, Ankara, Turkey •Alexandru Dimca Professeur, University of Nice - Sophia Antipolis, France •Masahiko Yoshinaga Associate Professor, Hokkaido University, Japan Examinators: •Sebastian Baader Tenure-track Professor, University of Bern, Switzerland •Arnaud Bodin Maître de Conférence (HDR), University of Lille I, France •Pierrette Cassou-Noguès Professeur Emérite, University of Bordeaux I, France •José-Ignacio Cogolludo Agustín Titular, Universidy of Zaragoza, Spain •Louis Paris (President) Professeur, University of Bourgogne, France "Je ne cherche pas à connaître les réponses, je cherche à comprendre les questions." Confucius 6 Prime combinatorics, computation & examples 93 6.1 Primecombinatorics .............................. 94 6.1.1 Depth & characters . . . . . . . . . . . . . . . . . . . . . . . . . . 94 6.1.2 Prime combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . 95 6.1.3 Combinatorics & realizations . . . . . . . . . . . . . . . . . . . . . 96 6.2 Computation&output............................. 97 6.2.1 Sageprogram.............................. 97 6.2.2 Output ................................. 98 6.3 Examples .................................... 99 6.3.1 Firsttests................................ 99 6.3.2 Prime combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . 99 6.4 NC-Zariskipairs ................................105 6.4.1 Oriented NC-Zariski pair . . . . . . . . . . . . . . . . . . . . . . . 105 6.4.2 NC-Zariskipair.............................107 Conclusion and future works 111 English ........................................111 French.........................................113 Spanish ........................................115 A Code 117 A.1 Wiringdiagram.................................117 A.2 Inner-cyclic combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 A.3 Space of realization ΣC............................128 Bibliography 131 Abstract 139 English ........................................139 French.........................................140 Spanish ........................................141 xiii xiv List of Figures 1 Ceva’s arrangement (1678) . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Incidence graph of the Ceva’s arrangement . . . . . . . . . . . . . . . . . . 1 3 Example of wiring diagram . . . . . . . . . . . . . . . . . . . . . . . . . . 3 4 Arrangement de Ceva (1678) . . . . . . . . . . . . . . . . . . . . . . . . . 9 5 Graphe d’incidence de l’arrangement de Ceva . . . . . . . . . . . . . . . . 9 6 Exemple de diagramme de câblage . . . . . . . . . . . . . . . . . . . . . . 11 7 Configuración de Ceva (1678) . . . . . . . . . . . . . . . . . . . . . . . . . 17 8 Grafo de incidencia de la configuración de Ceva . . . . . . . . . . . . . . . 17 9 Ejemplo de diagrama de cableado . . . . . . . . . . . . . . . . . . . . . . . 19 1.1 Falk’sexample ................................. 27 1.2 Pappusarrangement.............................. 29 1.3 Example of an incidence graph . . . . . . . . . . . . . . . . . . . . . . . . 31 1.4 Local blow-up of a singular point . . . . . . . . . . . . . . . . . . . . . . 34 2.1 Example of braided wiring diagram . . . . . . . . . . . . . . . . . . . . . . 39 2.2 Computation of Arvola’s words . . . . . . . . . . . . . . . . . . . . . . . . 40 2.3 Construction of γ0............................... 41 2.4 Example of problem with a non-generic projection . . . . . . . . . . . . . 44 2.5 Ordered MacLane combinatorics, viewed in F2 3............... 45 2.6 Braided wiring diagram of MacLane positive . . . . . . . . . . . . . . . . 46 2.7 Braided wiring diagram of MacLane negative . . . . . . . . . . . . . . . . 46 2.8 Braided wiring diagram of Fan real . . . . . . . . . . . . . . . . . . . . . 47 2.9 Braided wiring diagram of Fan Positive . . . . . . . . . . . . . . . . . . . 47 2.10 Braided wiring diagram of Fan negative . . . . . . . . . . . . . . . . . . . 47 3.1 Handlegluing ................................. 52 3.2 Construction of δ(ε).............................. 56 3.3 Construction of δ(ε)near a singular point . . . . . . . . . . . . . . . . . . 56 3.4 Indexation of a crossing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.5 Incidence graph of the MacLane arrangement Q+.............. 61 3.6 Wiring diagram of the positive MacLane arrangement . . . . . . . . . . . 62 4.1 The extended Ceva’s arrangement . . . . . . . . . . . . . . . . . . . . . . 69 6.1 Real picture of the first three prime arrangements with nine lines . . . . 101 6.2 The graph ˆ ΓUξfor the fourth prime combinatorics with nine lines: C4. . 101 6.3 Braided wiring diagram of the positive realization A+of C4........102 6.4 The graph ˆ ΓUξfor the combinatorics C5...................103 6.5 A prime pair of arrangements with ten lines . . . . . . . . . . . . . . . . . 104 6.6 Braided wiring diagram of positive arrangement A+............105 6.7 Braided wiring diagram of positive arrangement B+............105 xvi Introduction The first drawing of any child is a line. What is more natural than lines? What is the difference between two drawings of lines? This is roughly speaking the subject of this thesis. The support is slightly different of a sheet of paper, since we are focusing on line arrangements in the complex projective plane. Figure 1: Ceva’s arrangement (1678) The first idea to describe a line arrangement is to define the set of lines, the set of singular points, and the incidence relations between them. This combinatorial information can be summarized in graphs, like Hasse’s graph [57] or the incidence graph. But, as shown in [71, 60, 7, 6], this description is not sufficient to determine an arrangement. L1L2L3L4L5L6 P1P2P3P4P5P6P7 Figure 2: Incidence graph of the Ceva’s arrangement In mathematics, the study of line arrangements is a domain with a history reaching INTRODUCTION back only forty-five years. Of course the prehistory goes back much further in time with the Greeks like Pappus of Alexandria [55], or in the XVIIth century with G. Ceva [16]. Its first materialization in modern mathematics is with E. Brieskorn [11, 12] where he shows that the braid group can be constructed as the fundamental group of the complement of a complex arrangement of hyperplanes. This relative young area takes thus its origins in the study of discriminants, combinatorics, configuration spaces, braids, etc. Owing to its multiple origins, the study of arrangements admits different facets: combinatorial, geometrical, arithmetical and topological. The interaction between all these facets is its most attractive feature. Furthermore, arrangements provide a fertile source of examples and problems. Techniques developed for their studies have been successfully exported to other settings. One main interesting facet of an arrangement is determined by its ground field. We take an interest in line arrangements defined over the complex field Cand their topologies. Our general aim is to investigate the relationship between their combinatorics and their topology. The motivations Definition. The topological type of an arrangement Ain CP2is the homeomorphism type of the pair (CP2,A). From the beginning of the study of arrangements, questions about their topologies were frequent. In 1969, V.I. Arnold gives a presentation of the cohomology of the complement as a graded algebra, see [2]. In 1973, P. Deligne [27] shows that the complement of any arrangement associated with a real reflection groups is aspherical. P. Orlik and L. Solomon [56] proved in 1980, that the cohomology ring of the complement only depends on the combinatorics of the arrangement. Question. What is the influence of the combinatorics on the topology of arrangements? Theoretically, a line arrangement is a specific case of algebraic plane curve. Fundamental results on curves are obtained by O. Zariski [71]. He shows the existence of a pair of curves admitting the same combinatorial data and different topological types. It is then natural to define a Zariski pair: Definition ([4]).AZariski pair of arrangement is a pair (A1,A2)such that A1and A2 admit the same combinatorics but different topological types. For technical reasons, we consider also blow-ups of arrangements at their multiple points. A nc-Zariski pair is a pair of arrangements with the same combinatorics and blow-ups with different topological types, –nc stands for normal crossing–. In [60], G. Rybnikov shows the existence of a Zariski pair in the case of line arrangements, and gives an explicit example. Studying the result of G. Rybnikov, the authors of [7, 6]: E. Artal, J. Carmona, J.I. Cogolludo and M.A. Marco, obtain a Zariski pair of complexified real arrangements. Examples of nc-Zariski pairs are obtained by A. Degtyarev [25] in the case of sextic curves. 2 INTRODUCTION The principal topological invariant of an arrangement Ais it complement EA: EA=CP2\A. From it, several invariants are constructed, the major of them is its fundamental group, see [46]. It is a strong invariant as shown in [60, 22, 21]. In [65], E.R. van Kampen, gives a method to compute a presentation of the fundamental group of an algebraic plane curve complement (known as the Zariski-van Kampen method). He uses the position of the curve relative to pencils of lines in CP2. This notion, the braid monodromy, receives its first modern approach in studies by D. Chéniot [17], O. Chisini [18] and B. Moishezon [51]. In [43], A. Libgober constructs a 2-dimensional CW-complex from the braid monodromy, admitting the same homotopy type as the complement of the curve. There are other contributions by M. Salvetti [63], D. Cohen and A. Suciu [21], V.S. Kulikov and M. Ta˘ıkher [41] and J. Carmona [15] on the relationship between topology and braid monodromy. The Zariski-van Kampen method is studied by R. Randell, in [58], in the case of complexified real arrangement. W. Arvola generalizes, in [9], the algorithm of R. Randell to the case of any complex line arrangement using the braided wiring diagram that encodes the braid monodromy of the arrangement. Figure 3: Example of wiring diagram Since the problem of group isomorphism is undecidable, related topological invariants are studied. One of the main invariant we study is the set of characteristic varieties, which are subsets of the character torus T(A), the group of characters of the fundamental group in C∗. They are first introduced by A. Libgober in the case of curves. They are cohomological invariants defined as the jumping loci of twisted cohomology of rank one local system on the variety, and they only depend on its fundamental group. Definition. The characteristic varieties of Aare: Vk(A) = nξ∈T(A)|dimCH1(EA;Lξ)≥ko, where Lξis the local system of coefficient defined by ξ. The depth of a character ξis dimCH1(EA;Lξ). The first general structure theorem for these loci is discovered by D. Arapura, who described them in terms of fibrations over curves, see [1]; the structure was completed by several authors, E. Artal, J.I.Cogolludo and D. Matei [8], N. Budur [14], A. Dimca [29] 3 INTRODUCTION and A. Libgober[47]. The characteristic varieties of a space can be seen as a generalization of the Alexander polynomial. This invariant has been extensively studied from different perspectives, in general case by S. Novikov [54], W. Dwyer and D. Fried [32] and A. Libgober [44]. In the case of arrangements, A. Libgober [45] gives an algorithm on how to find irreducible components of Vk(A), E. Hironaka [39] shows the equality of the characteristic varieties with a stratification of the character torus of π1(EA)coming from the Fox derivatives, and D. Cohen and A. Suciu [23] prove that all the irreducible components of Vk(A)which pass through the identity element are combinatorially determined. They are related with the Green-Lazarsfeld set [36, 35]. Question. Are the characteristic varieties of arrangements combinatorially determined? More recent works of A. Libgober and A. Dimca, D. Ibadula and D.A. Macinic show that for diagonal characters the projective depth is of combinatorial nature if the singular points of Ahave multiplicity less than 3 (see [48]), and in the case with multiplicity less than 5 if Acontains less than 14 lines (see [30]). The study of the boundary manifold is another approach of this question of the relationship between combinatorics and topology. It provides partial answers to this question. Indeed, this variety only depends on the combinatorics and can be constructed as a graph manifold over the incidence graph. E. Westlund describes, in his thesis [68], the graph manifold structure of the boundary manifold, and D. Cohen and A. Suciu in [24] obtain several results on its topology. The boundary manifold is intimately related to the complement of arrangements. In [40], E. Hironaka studies its inclusion in the complement in the case of complexified real arrangements, and shows that the complement can be constructed –up to homotopy– as a quotient of the boundary manifold. Our advances and results This thesis is at the intersection point of these two directions: the boundary manifold and the characteristic varieties. We have studied the inclusion map of the boundary manifold in the complement, see [34]. From it, attempts were made to find new topological invariants. This application was also used to study the non combinatorially part of the quasi-projective part of characteristic varieties. Indeed, E. Artal has developed an algorithm to compute the quasi-projective part of the depth of a character, using the image of some particular cycles of the boundary manifold in the complement, see [5]. We have then worked on the combinatoriality of the characteristic varieties using the combinatorial part of the complement given by the boundary manifold, and the relationship –without complete algorithm– with the characteristic varieties obtained by E. Artal. Even if we have not found the answer to this difficult question, this study allows to construct an explicit and complete method to compute the quasi-projective depth of a character, and a new topological invariant permitting to distinguish some nc-Zariski pairs of line arrangements. 4 INTRODUCTION In the first part of this work, we generalize the results obtained by E. Hironaka in [40] to the case of any complex line arrangement. To get around the problem due to the case of non complexified real arrangement we have studied the braided wiring diagram, and used it fully fledged as a topological object. So we have to develop a Sage program to compute it from the equation of the complex line arrangement. This diagram allows to give two explicit descriptions of the map: i∗:π1(BA)−→ π1(EA). From these descriptions, see Theorem 3.2.2 and Theorem 3.2.7, we obtain two new presentations of the fundamental group of the complement, see Corollary 3.2.5 and Corollary 3.2.8. The second corollary is, in fact, a generalization of Randell’s Theorem [58]. In the next step of our work, we study the map induced on the first homology groups. We obtain simple descriptions of this application, see Theorem 4.1.1 and Theorem 4.1.3. Then inspired by ideas of J.I. Cogolludo, we give a canonical description of the homology of the boundary manifold as the product of the 1-homology with the 2-cohomology of the complement: Theorem (4.3.2).Let EAbe the complement of Aand let BAbe its boundary manifold, then we have a split exact sequence: 0→H2(EA;C)−→ H1(BA;C)i∗ −−→ H1(EA;C)→0. Finally, we obtain an isomorphism between the 2-cohomology of the complement with the 1-homology of the incidence graph of the arrangement. This provides an intrinsic decomposition of H1(BA;C)as H1(EA;C)⊕H1(ΓA;C). In the second part, we apply this work to the study of characters on the complement. We start from the results of E. Artal [5] on the computation of the depth of a character. This depth can be decomposed into a projective term and a quasi-projective term, vanishing for characters that ramify along all the lines. An algorithm to compute the projective part is given by A. Libgober [48]. E. Artal focuses on the quasi-projective part and gives a method to compute it from the image by the character of certain cycles of the complement. We use our results on the inclusion map of the boundary manifold to determine these cycles explicitly. Combined with the work of E. Artal we obtain an algorithm to compute the quasi-projective depth of any character. Theorem 5.2.23 gives a combinatorial condition on characters to admit a quasiprojective depth potentially not determined by the combinatorics. With this property, we define the inner-cyclic characters. From their study, we observe a strong condition on the combinatorics of an arrangement to have only characters with null quasi-projective depth, see Proposition 6.1.1. Related to this, in order to reduce the number of computations, we introduce the notion of prime combinatorics. If a combinatorics is not prime, then the characteristics varieties of its realizations are completely determined by realization of a prime combinatorics with fewer lines. 5 INTRODUCTION In parallel, we observe that the composition of the map induced by the inclusion with specific characters provide topological invariants of the blow-up of arrangements, see Theorem 5.3.14. We show in Theorem 6.4.6 and Corollary 6.4.7 that the invariant captures more than combinatorial information. Thereby, we detect two new examples of nc-Zariski pairs. The first, is an oriented and ordered pair, see Theorem 6.3.2. The second, comes from the following combinatorics: C= [ [1,2],[1,3,5,7,12],[1,4,6,8],[1,9],[1,10,11],[2,3,6,9], [2,4,5,10],[2,7,11],[2,8],[2,12],[3,4],[3,8,11],[3,10],[4,7], [4,9,11],[4,12],[5,6],[5,8,9],[5,11],[6,7,10],[6,11],[6,12], [7,8],[7,9],[8,10],[8,12],[9,10],[9,12],[10,12],[11,12] ]. It admits four realizations denoted by A±and B±. Theorem (6.4.10 and 6.4.13).The pairs (A±,B±)are nc-Zariski pairs, and the 4tuplet (A+,B+,A−,B−)is an oriented nc-Zariski 4-tuplet. This provides the third example, known in the literature, of nc-Zariski pair. The two other being also Zariski pairs are obtained in [60, 6]. Thesis plan This manuscript is composed of six chapters, following our work: The first Chapter is essentially composed of basic tools. We define the notion of line arrangements in the complex projective plane. Then we explain the notions of combinatorics and their realizations. In the second section, we recall the desingularization of a space using the blow-up method, and apply it on line arrangements. In Chapter 2, we study the braided wiring diagram, introduced by W. Arvola in [9] to compute the fundamental group of an arrangement. The first section contains the construction of this diagram. Then we show: how to use it to compute the fundamental group, how it is related to the braid monodromy, and how to find the braided wiring diagram of the conjugate arrangement. Then we explain the construction of a Sage program to compute the diagram. In the last section, we compute wiring diagrams of MacLane’s arrangements and Fan’s arrangements. The generalization of the result of E. Hironaka in done in Chapter 3. First, we describe the boundary manifold as a graph manifold over the incidence graph of the arrangement, and we deduce a presentation of its fundamental groups. In the second section, we give an explicit description of the map induced by the inclusion of the boundary manifold in the complement on the fundamental groups. We obtain two new presentations of the fundamental group and a new proof of the Randell theorem. To conclude this Chapter, we give the detailed example of the positive MacLane arrangement. 6 INTRODUCTION (FRENCH) Nous avons ainsi travaillé sur la combinatorialité des variétés caractéristiques en utilisant la partie combinatoire du complémentaire donnée par la variété bord, et sur l’algorithme –non complet– d’E. Artal reliant la combinatoire et les variétés caractéristiques. Même si nous n’avons pas trouvé la réponse à cette question difficile qu’est le lien entre topologie et combinatoire, ces travaux ont permis de construire une méthode complète et explicite pour calculer la partie quasi-projective de la profondeur d’un caractère, ainsi que de découvrir un nouvel invariant permettant de distinguer des nc-paires de Zariski. Dans la première partie de ce travail, nous avons généralisé le résultat obtenu par E. Hironaka dans [40] au cas des arrangements complexes. Pour contourner les problèmes dus au cas des arrangements non complexifiés, nous avons étudié le diagramme de câblage, et l’avons considéré comme un objet topologique à part entière. Afin de l’utiliser de façon systématique, nous avons développé un programme sous Sage permettant de le construire à partir des équations des droites de l’arrangement. Ce diagramme nous a permis de donner deux descriptions explicites de l’application : i∗:BA−→ EA. De ces deux descriptions, cf Théorème 3.2.2 et Théorème 3.2.7, nous avons obtenu deux nouvelles présentations du groupe fondamental du complémentaire, cf Corollaire 3.2.5 et Corollaire 3.2.8. Le second est une généralisation du Théorème de R. Randell [58]. Dans une deuxième étape, nous avons étudié l’application induite par l’inclusion sur les premiers groupes d’homologies. Nous avons donc obtenu deux descriptions simples de cette application, cf Théorème 4.1.1 et Théorème 4.1.3. Enfin, en s’inspirant des travaux de J.I. Cogolludo, nous avons trouvé une décomposition canonique du premier groupe d’homologie de la variété bord comme le produit de la 1-homologie et de la 2-cohomologie du complémentaire : Theorem (4.3.2).Soient EAle complémentaire de Aet BAsa variété bord, on a alors la suite exacte scindée : 0→H2(EA;C)−→ H1(BA;C)i∗ −−→ H1(EA;C)→0. Pour finir, nous avons obtenu un isomorphisme entre la 2-cohomologie du complémentaire et la 1-homologie du graphe d’incidence de l’arrangement. Ce qui fournit une décomposition intrinsèque de H1(BA;C)en H1(EA;C)⊕H1(ΓA;C). Dans la seconde partie de ce travail, nous avons appliqué les résultats précédents à l’étude des caractères sur le complémentaire. Notre point de départ fût les résultats d’E. Artal [5] sur le calcul de la profondeur d’un caractère. Cette profondeur se décompose en un terme projectif et un terme quasi-projectif. Un algorithme pour calculer la partie projective a été donné par A. Libgober [48]. E. Artal s’est concentré sur la partie quasi-projective, et a donné une méthode pour la calculer à partir de l’image de certains cycles du complémentaire. Nous avons donc utilisé le travail précédemment fait pour 13 INTRODUCTION (FRENCH) calculer explicitement ces cycles. En combinant ces deux travaux, nous avons obtenu un algorithme explicite pour calculer la profondeur quasi-projective d’un caractère. Théorème 5.2.23 donne une condition combinatoire pour qu’un caractère admette une profondeur quasi-projective potentiellement non combinatoire. De cette propriété, nous avons défini la notion de caractère inner-cyclic. De leur étude, nous avons observé une condition forte sur la combinatoire pour n’avoir que des caractères de profondeur quasi-projective nulle, cf Proposition 6.1.1. Dans le but de réduire le nombre de calculs, nous avons introduit la notion de combinatoire première. Si une combinatoire n’est pas première, alors les variétés caractéristiques de ses réalisations sont complètement déterminées par une réalisation d’une combinatoire première avec moins de droites. En parallèle, nous avons observé que la composition de l’application induite par l’inclusion avec un caractère spécifique fournissait des invariants topologiques du blowup de l’arrangement, cf Théorème 5.3.14. Nous avons donc prouvé dans Théorème 6.4.6 et Corollaire 6.4.7, que cet invariant capture plus que la combinatoire. En effet, nous avons détecté deux nouveaux exemples de nc-paires de Zariksi. Le premier est une paire orientée et ordonnée, cf Théorème 6.3.2. Le second provient de la combinatoire suivante : C= [ [1,2],[1,3,5,7,12],[1,4,6,8],[1,9],[1,10,11],[2,3,6,9], [2,4,5,10],[2,7,11],[2,8],[2,12],[3,4],[3,8,11],[3,10],[4,7], [4,9,11],[4,12],[5,6],[5,8,9],[5,11],[6,7,10],[6,11],[6,12], [7,8],[7,9],[8,10],[8,12],[9,10],[9,12],[10,12],[11,12] ]. Elle admet quatre réalisations notées A±et B±. Theorem (6.4.10 et 6.4.13).Les paires (A±,B±)sont des nc-paires de Zariski, et le 4-tuplet (A+,B+,A−,B−)est un nc-4-tuplet de Zariski orienté. Cela nous fournit le troisième exemple, connu dans la littérature, de nc-paire de Zariski. Les deux autres, étant aussi des paires de Zariski, sont obtenus dans [60, 6]. Plan de la thèse Ce manuscrit est composé de six chapitres suivant le fil de notre travail : Le premier chapitre est essentiellement composé des outils de base. Nous y définissons la notion d’arrangement de droites dans le plan projectif complexe. Ensuite, nous expliquons la notion de combinatoire et de réalisation. Dans la seconde section, nous rappelons la désingularisation d’un espace en utilisant la méthode du blow-up, et nous l’appliquons aux arrangements de droites. Dans le Chapitre 2, nous étudions les diagrammes de câblage. La première section contient la construction théorique de ce diagramme. Ensuite, nous montrons comment l’utiliser pour calculer le groupe fondamental du complémentaire, comment en extraire la monodromie de tresse de l’arrangement, et comment construire le diagramme de câblage 14 INTRODUCTION (FRENCH) de l’arrangement conjugué. Dans la deuxième section, nous expliquons la construction d’un programme sous Sage afin de calculer ce diagramme. Dans la dernière section, nous calculons les diagrammes de câblage des arrangements de MacLane et de Fan. La généralisation du résultat d’E. Hironaka est faite dans le Chapitre 3. Tout d’abord, nous décrivons la variété bord comme une variété graphée au-dessus du graphe d’incidence. De cette description, nous en déduisons une présentation de son groupe fondamental. Dans la deuxième section, nous donnons une description explicite de l’application induite par l’inclusion sur les groupes fondamentaux de la variété bord et du complémentaire. On en déduit deux nouvelles présentations du groupe fondamental du complémentaire, ainsi qu’une nouvelle preuve du Théorème de Randell. Pour conclure ce Chapitre, nous donnons le détail des calculs de cette application dans le cas de l’arrangement de MacLane positif. Le Chapitre 4 est composé de la version homologique de l’application inclusion décrite dans le Chapitre précédent. Deux exemples sont détaillés dans la seconde section : Ceva-7 et MacLane (positif et négatif). La troisième section est composée d’une décomposition canonique du premier groupe d’homologie du complémentaire, ainsi que de l’étude du lien entre la 2-cohomologie du complémentaire et la 1-homologie du graphe d’incidence. Les résultats d’E. Artal forment la partie principale du Chapitre 5. Dans la première section, nous rappelons la définition et des propriétés des variétés caractèristiques. La deuxième section est une description détaillée des résultats d’E. Artal. Dans la dernière section, nous définissons la notion de combinatoire inner-cyclic et de caractère innercyclic. Ensuite nous prouvons que l’image de cycle particulier de la variété bord par un caractère particulier est un invariant topologique du blow-up de l’arrangement. Le dernier Chapitre est le point d’intersection des travaux précédents. La première section contient la définition d’une combinatoire première, et les raisons de leur étude. Dans la second section nous présentons deux programmes. L’un sous Sage pour détecter les combinatoires premières réalisables, et le second sous Maple qui donne les équations d’un arrangement à partir de sa combinatoire. Avec ces deux programmes, nous avons testé un grand nombre d’exemples, et nous présentons dans la troisième section les plus intéressants d’entre eux. La quatrième section est, quant à elle, composée de l’étude d’un exemple à onze droites, permettant la construction d’une nc-paire de Zariski. 15 INTRODUCTION (FRENCH) 16 Introducción (Spanish) El primer dibujo que hace cualquier niño es una línea. ¿Qué es más natural que una recta?, ¿cuál es la diferencia entre dos dibujos de rectas?. Este es grosso modo el tema de estudio de la presente tesis. El soporte sobre el que vamos a trabajar es por supuesto ligeramente diferente a una hoja de papel, ya que trabajaremos sobre el plano proyectivo complejo. Figura 7: Configuración de Ceva (1678) La primera idea para describir una configuración de rectas es definir el conjunto de rectas, el conjunto de puntos singulares y las relaciones de incidencia entre ambos conjuntos. Esta información combinatoria puede ser representada por grafos, como el grafo de Hasse [57] o el grafo de incidencia. Pero, como se puede ver en [71, 60, 7, 6], esta descripción no es suficiente para determinar una configuración. L1L2L3L4L5L6 P1P2P3P4P5P6P7 Figura 8: Grafo de incidencia de la configuración de Ceva INTRODUCCIÓN (SPANISH) En matemáticas, el estudio de configuraciones de rectas es un dominio con sólo cuarenta y cinco años de historia. Por supuesto, la prehistoria de esta rama va mucho más lejos en el tiempo llegando al mundo griego, con Pappus de Alejandría [55], o en el siglo XVII con G. Ceva [16]. La primera materialización en las matemáticas modernas se debe a E. Brieskorn [11, 12], donde muestra que el grupo de trenzas puede ser construido como el grupo fundamental del complemento de una configuración de hiperplanos complejos. Este área relativamente joven de las matemáticas comienza en sus orígenes con el estudio de discriminantes, combinatoria, espacios de configuración, trenzas, etc. Debido a sus múltiples orígenes, el estudio de configuraciones de hiperplanos admite diferentes aproximaciones: combinatorio, geométrico, aritmético y topológico. La interacción entre tales áreas constituye el rasgo más atractivo de este estudio. Más aun, las configuraciones de hiperplanos proporcionan una fuente fértil de ejemplos y problemas interesantes. Las técnicas desarrolladas para este estudio han sido satisfactoriamente exportadas a otras áreas. Las características de una configuración dependen del cuerpo sobre el que esta se define. Tenemos especial interés en el estudio de configuraciones de rectas definidas sobre el cuerpo complejo Cy sus topologías. Nuestro propósito general es el de observar la relación entre sus combinatorias y sus topologías. Motivaciones Definition. El tipo topológico de una configuración de rectas Aen CP2es el tipo de homeomorfismo del par (CP2,A). Las cuestiones en torno a la topología de las configuraciones son frecuentes desde el inicio de su estudio. En 1969, V.I. Arnold da una presentación de la cohomología del complementario como un álgebra graduada [2]. En 1973, P. Deligne [27] muestra que el complementario de una configuración de rectas asociada a un grupo de reflexión real es un espacio de Eilenberg-McLane. P. Orlik y L. Solomon [56] probaron en 1980 que el anillo de cohomología del complementario depende únicamente de la combinatoria de la configuración. Question. ¿Cuál es la influencia de la combinatoria en la topología de las configuraciones? Teóricamente, una configuración de rectas es un caso particular de curvas algebraicas, cuyos principales resultados fueron obtenidos por O. Zariski [71]. Zariski mostró la existencia de pares de curvas admitiendo la misma combinatoria pero diferentes tipos topológicos, que son definidos como pares de Zariski: Definition ([4]).Un par de Zariski de una configuración de rectas es un par (A1,A2) tal que A1yA2admiten la misma estructura combinatoria pero poseen diferentes tipos topológicos. Por razones técnicas, vamos a considerar también blow-ups oexplosiones de una configuración de rectas en sus puntos múltiples. Un nc-par de Zariski es un par de 18 INTRODUCCIÓN (SPANISH) configuraciones de rectas con la misma estructura combinatoria pero cuyos blow-ups poseen diferentes tipos topológicos, – donde nc proviene de los cruces normales (normal crossing)–. En [60], G. Rybnikov muestra la existencia de un par de Zariski en el caso de configuraciones de rectas dando un ejemplo explícito. Estudiando el resultado de G. Rybnikov, los autores de [7, 6]: E. Artal, J. Carmona, J.I. Cogolludo y M.A. Marco, obtienen un par de Zariski de configuraciones reales complexificadas. Ejemplos de nc-pares de Zariski fueron obtenidos por A. Degtyarev [25] en el caso de curvas séxticas. El principal invariante topológico de una configuración Aes su complementario EA: EA=CP2\A. Numerosos invariantes están construidos a partir del complementario, siendo el más importante el grupo fundamental de la configuración [46]. Como se puede ver en [60, 22, 21] el grupo fundamental es un invariante fuerte de la configuración. En [65], E.R. van Kampen, da un método para calcular la presentación del grupo fundamental del complementario de una curva algebraica plana (conocido como el método de Zariski-van Kampen). En este método, utiliza la posición relativa de haces de rectas en CP2. Esta noción, llamada monodromía de trenzas, tiene su primer estudio moderno de la mano de D. Chéniot [17], O. Chisini [18] y B. Moishezon [51]. En [43], A. Libgober construye un CW-complejo 2-dimensional a partir de la monodromía de trenzas admitiendo el mismo tipo de homotopía que el complementario de una cierta curva. Existen numerosas contribuciones dadas por M. Salvetti [63], D. Cohen y A. Suciu [21], V.S. Kulikov y M. Ta˘ıkher [41] y J. Carmona [15] sobre la relación entre la topología y la monodromía de trenzas. El método de Zariski-van Kampen para el caso de configuraciones de rectas reales complexificadas es estudiado por R. Randell en [58]. W. Arvola generaliza, en [9], el algoritmo de R. Randell para el caso de configuraciones de rectas complejas utilizando el diagrama de cableado de trenzas que codifica la monodromía de trenzas de la configuración. Figura 9: Ejemplo de diagrama de cableado Ya que el problema de saber si dos grupos son isomorfos a partir de dos presentaciones es indecidible, se opta por estudiar los invariantes topológicos asociados. Uno de los principales invariantes que estudiamos es el conjunto de variedades características, que son subconjuntos del toro de caracteres T(A), el grupo de caracteres en C∗del grupo 19 INTRODUCCIÓN (SPANISH) fundamental de la configuración. Éstos fueron introducidos por primera vez por A. Libgober en el caso de curvas. Son invariantes cohomológicos definidos como los lugares de saltos de la cohomología torcida de sistemas locales de rango uno en la variedad, quienes sólo dependen del grupo fundamental. Definition. Las variedades características de Ason: Vk(A) = nξ∈T(A)|dimCH1(EA;Lξ)≥ko, donde Lξes el sistema local de coeficientes definido por ξ. La profundidad de un carácter ξes dimCH1(EA;Lξ). El primer teorema general de estructura para estos lugares de salto fue descubierto por D. Arapura, quien los describió en términos de fibraciones sobre curvas [1]; esta estructura fue completamente descrita por varios autores, E. Artal, J.I.Cogolludo y D. Matei [8], N. Budur [14], A. Dimca [29] y A. Libgober[47]. Las variedades características de un espacio pueden verse como una generalización del polinomio de Alexander. Este invariante ha sido ampliamente estudiado desde diversos puntos de vista y en el caso general por S. Novik [54], W. Dwyer y D. Fried [32] y A. Libgober [44]. En el caso de configuraciones, A. Libgober [45] da un algoritmo para encontrar las componentes irreducible de Vk(A), E. Hironaka [39] demuestra la igualdad entre variedades características con una estratificación del toro de caracteres de π1(EA)a partir de las derivadas de Fox, y D. Cohen y A. Suciu [23] prueba que todas las componentes irreducibles de Vk(A) conteniendo el elemento trivial están combinatoriamente determinadas. Las variedades características están relacionadas con el conjunto de Green–Lazarsfeld [36, 35]. Question. ¿Las variedades características de las configuraciones de rectas están determinadas combinatoriamente? Trabajos más recientes de A. Libgober y A. Dimca, D. Ibadula y D.A. Macinic muestran que para el carácter diagonal la profundidad proyectiva es de naturaleza combinatoria si el lugar singular de Atiene multiplicidad inferior a tres (ver [48]), y si tiene multiplicidad inferior a 5 y Acontiene menos de 14 rectas (ver [30]). El estudio de la variedad borde es otro enfoque de esta cuestión sobre la relación entre la combinatoria y la topología, pero nos da una respuesta parcial a esta cuestión. En efecto, esta variedad depende únicamente de la combinatoria y puede ser construida como una variedad grafo sobre el grafo de incidencia. E. Westlund describió, en su tesis [68], la estructura de variedad grafo de la variedad borde, así como D. Cohen y A. Suciu in [24] obtuvieron diversos resultados sobre su topología. La variedad borde está íntimamente relacionada con el complementario de una configuración. En [40], E. Hironaka estudia su inclusión en el complementario de configuraciones reales complexificados, también demuestra que el complementario puede ser construido –salvo homotopía– como cociente de la variedad borde. 20 INTRODUCCIÓN (SPANISH) Nuestros avances y resultados Esta tesis estudia problemas que se encuentran en la intersección de dos objetos de estudio: la variedad borde y las variedades características. Hemos estudiado la aplicación inclusión de la variedad borde en el complementario, ver [34]. El objetivo de este estudio era el de encontrar y definir nuevos invariantes topológicos. Esta aplicación nos ha ayudado también a estudiar la parte no combinatoria de la parte cuasi-proyectiva de las variedades características. De hecho, E. Artal ha desarrollado un algoritmo para calcular la parte cuasi-proyectiva de la profundidad de un carácter utilizando la imagen de ciclos particulares de la variedad borde en el complementario, ver [5]. Hemos trabajado también sobre la combinatorialidad de las variedades características utilizando la parte combinatoria del complementario dado por la variedad borde, y sobre el algoritmo –no completo– de E. Artal relacionando la combinatoria y las variedades características. Aunque no hayamos encontrado una respuesta a esta difícil cuestión que es la relación entre la topología y la combinatoria, estos estudios nos han permitido construir un método completo y explícito para calcular la parte cuasi-projectiva de la profundidad de un carácter, así como descubrir un nuevo invariante permitiendo distinguir algunos nc-pares de Zariski de configuraciones de rectas. En la primera parte de este trabajo, hemos generalizado el resultado obtenido por E. Hironaka en [40] para el caso de configuraciones de rectas complejas. Para resolver los problemas provenientes de configuraciones no complexificadas, hemos estudiado el diagrama de cableado considerándolo enteramente como un objeto topológico. Con el fin de utilizarlo de forma sistemática, hemos desarrollado un programa en Sage para construirlo a partir de las ecuaciones de las rectas de la configuración. Este diagrama nos ha permitido dar dos expresiones explícitas de la aplicación: i∗:π1(BA)−→ π1(EA). A partir de estas descripciones, ver Teorema 3.2.2 y Teorema 3.2.7, obtenemos dos nuevas presentaciones del grupo fundamental del complementario, ver Corolario 3.2.5 y Corolario 3.2.8. El segundo corolario es, de hecho, una generalización del Teorema de Randell [58]. En una segunda parte de este trabajo, hemos estudiado la aplicación inducida por la inclusión sobre los primeros grupos de homología, obteniendo descripciones simples de esta aplicación, ver Teorema 4.1.1 y Teorema 4.1.3. Luego, inspirados en los trabajos de J.I. Cogolludo, hemos encontrado una descomposición canónica del primer grupo de homología de la variedad borde como el producto de la 1-homología y de la 2-cohomología del complementario: Theorem (4.3.2).Sea EAel complementario de AyBAsu variedad borde, tenemos entonces una sucesión exacta escindida: 0→H2(EA;C)−→ H1(BA;C)i∗ −−→ H1(EA;C)→0. 21 INTRODUCCIÓN (SPANISH) Finalmente, hemos obtenido un isomorfismo entre la 2-cohomología del complementario con la 1-homología del grafo de incidencia de la configuración. Esto nos da una descomposición intrínseca de H1(BA;C)as H1(EA;C)⊕H1(ΓA;C). En una segunda parte de este trabajo, hemos aplicado los resultados precedentes en el estudio de los caracteres sobre el complementario. Nuestro punto de partida viene de resultados de E. Artal [5] en el cálculo de la profundidad de un carácter. Esta profundidad puede descomponerse en una parte proyectiva y una parte cuasi-proyectiva; esta última es nula en los caracteres que ramifican a lo largo de todas las rectas. Para calcular la parte proyectiva, A. Libgober da un algoritmo en [48]. E. Artal se centra en la parte cuasi-proyectiva y da un método para calcularla a partir de la imagen de ciertos ciclos del complementario. Usando estos resultados sobre la aplicación inclusión de la variedad borde, podemos determinar estos ciclos explícitamente. Combinando estos dos estudios, hemos obtenido un algoritmo explícito para calcular la profundidad cuasi-proyectiva de un carácter. El Teorema 5.2.23 nos da una condición combinatoria para que un carácter admita una profundidad cuasi-proyectiva potencialmente no combinatoria. A partir esta propiedad, hemos definido la noción de carácter inner-cyclic. A partir de su estudio, hemos observado una condición fuerte sobre la combinatoria para que una configuración pueda tener caracteres en los que la profundidad cuasi-proyectiva es no nula, ver Proposición 6.1.1. En relación a tal condición y con la intención de reducir el número de cálculos, hemos introducido la noción de combinatoria prima. Si una combinatoria no es prima, entonces las variedades características de sus realizaciones están definidas por las de una subconfiguración con un número menor de rectas. Paralelamente a este estudio, hemos observado que la composición de la aplicación inducida por la inclusión sobre el primer grupo de homología con un carácter proporciona un invariante topológico de configuraciones de rectas desingularizadas, ver Teorema 5.3.14. También mostramos en el Teorema 6.4.6 y el Corolario 6.4.7 que este invariante contiene más información que la combinatoria. De hecho, hemos podido detectar dos nuevos ejemplos de nc-pares de Zariski. El primero es un par orientado y ordenado, ver Teorema 6.3.2. El segundo proviene de la siguiente combinatoria: C= [ [1,2],[1,3,5,7,12],[1,4,6,8],[1,9],[1,10,11],[2,3,6,9], [2,4,5,10],[2,7,11],[2,8],[2,12],[3,4],[3,8,11],[3,10],[4,7], [4,9,11],[4,12],[5,6],[5,8,9],[5,11],[6,7,10],[6,11],[6,12], [7,8],[7,9],[8,10],[8,12],[9,10],[9,12],[10,12],[11,12] ]. Esta combinatoria admite cuatro realizaciones denotadas por A±yB±. Theorem (6.4.10 y 6.4.13).Los pares (A±,B±)son nc-pares de Zariski, y la 4-tupla (A+,B+,A−,B−)es un nc-4-tupla de Zariski orientado. Esto nos da el tercer ejemplo conocido en la literatura de nc-par de Zariski. Los otros dos –que son también pares de Zariski– fueron obtenidos en [60, 6]. 22 1.1. DEFINITIONS L0=∞ L1 L2 L3 L7 L6 L5L4 Figure 1.2: Pappus arrangement Remark 1.1.18. These are the only two examples currently known. Definition 1.1.19 ([28]).AZariski pair is a pair of arrangements A1and A2with the same combinatorics, but with different topological type. Definition 1.1.20. Arealization of a line combinatorics C= (L,P,∈), is a line arrangement Awith combinatorics C. An ordered realization of an ordered line combinatorics is defined accordingly. Notation. The space of all realizations of a line combinatorics Cis denoted by Σ(C). If Cis ordered, we denote by Σord(C)the space of all ordered realizations of C. There is a natural action of PGL3(C)on Σ(C)and Σord(C). This justifies the following definition. Definition 1.1.21. The moduli space of a combinatorics Cis the quotient: M(C) = Σ(C)/PGL3(C). The ordered moduli space Mord(C)of an ordered combinatorics Cis defined accordingly. For more details about moduli space, see [52]. Remark 1.1.22. Let us consider an ordered line combinatorics C,Aut(C)acts on Mord(C). Proposition 1.1.23. All realizable combinatorics admit at least a realization in a number field. Proof. Let Li=[x:y:z]∈CP2|aix+biy+ciz= 0for i= 1,2,3be lines of CP2. They are concurrent in CP2if and only if:  a1a2a3 b1b2b3 c1c2c3  = 0. 29 CHAPTER 1. PRELIMINARIES Then the equations satisfied by the coefficients of the lines of Aare polynomials of degree 3. Thus there is at least one solution in a number filed. Remark 1.1.24. There are combinatorics that do not admit realization. Consider the combinatorics Cof the Pappus arrangement shown in Example 1.1.16. A simple computation shows that Pappus’ arrangement is –up to deformation– the only one realization of C. Add a line L8to C, and assume that it passes through P2,7and P3,5, and that it generically intersects the other lines. We then obtain: [ [0,1],[0,2,3],[0,4,5],[0,6,7],[0,8],[1,2,4],[1,3,6],[1,5,7], [1,8],[2,5],[2,6],[2,7,8],[3,4],[3,5,8],[3,7],[4,6],[4,7],[4,8],[5,6],[7,8] ]. Pappus’ Theorem implies that the three points marked with a •in Figure 1.2 (i.e. P2,7, P3,5and P4,6) are aligned. But the line L8added to Cpasses through by P2,7and P3,5, but not by P4,6. Then this combinatorics is not realizable. 1.1.3 Incidence graph The incidence graph is a tool to encode the combinatorial information of an arrangement, see [57] for details. It is equivalent to the combinatorics. For P∈ Q, let us recall that AP={L∈ A | P∈L}. Definition 1.1.25. The incidence graph ΓAof Ais a non-oriented bipartite graph where the set of vertices V(A)decomposes as VP(A)qVL(A), with VP(A) = {vP|P∈ Q},and VL(A) = {vL|L∈ A}. The vertices of VP(A)are called point-vertices and those of VL(A)are called line-vertices. The edges of ΓAjoin vLto vPif and only if L∈ AP. They are denoted by e(L, P). Since two lines always intersect, then it is not necessary to conserve this information in the incidence graph if it is not in a singular point of multiplicity greater than 3. Then we can define the reduced incidence graph: Definition 1.1.26. The reduced incidence graph Γ0 Aof Ais a non-oriented graph where the set of vertices V0(A)decomposes as V0 P(A)qV0 L(A), with V0 P(A) = {vP|P∈ Q, mP≥3} ⊂ VP, V 0 L(A) = {vL|L∈ A} =VL. The edges of Γ0 Aare of two types: - It joins vLto vPif and only if L∈ APand mP≥3. - It joins vLito vLj, with i6=j, if and only if the point P=Li∩Ljhas multiplicity 2. 30 1.2. BLOW-UP L0 L1 L2 L3 P0,1,2 P0,3 P1,3 P2,3 L0 L1 L2 L3 Figure 1.3: Example of an incidence graph Example 1.1.27. The incidence graph of the arrangement composed of three concurrent lines L0, L1, L2and a generic line L3is pictured in Figure 1.3. An isomorphism of incidence graphs sends elements of VP(A)(resp. VL(A)) to elements of VP(A)(resp. VL(A)), and preserve the incidence relations. The automorphisms of an incidence graph induce automorphisms of the corresponding line combinatorics. Proposition 1.1.28. Let Aand A0be two topologically equivalent line arrangements, then Aand A0have the same combinatorics, and isomorphic incidence graphs. Proof. By the definition of topologically equivalent arrangement, there exists a homeomorphism Φsending Aon A0. This implies that they have the same combinatorics. Therefore, they have isomorphic incidence graphs. 1.2 Blow-up 1.2.1 Description The blow-up is a geometric transformation of the projective plane in a projective surface which helps to simplify the singularities of the curves in the surface. Initially, we define the blow-up of C2at the point (0,0). Since, the blow-up is a local modification of C2, using local charts on a complex surface, we then define the blow-up of a surface Min a point P∈M. Consider the quasi-projective space C2×CP1, where (x1, x2)are the complex affine coordinates of C2, and [z1:z2]the homogeneous coordinates of CP1. Definition 1.2.1. The blow-up of C2at the point (0,0) is the subset Xof C2×CP1 defined by: X={((x1, x2),[z1:z2]) |xizj=xjzi, i, j ∈ {1,2}}. 31 CHAPTER 1. PRELIMINARIES Remark 1.2.2. Since the projective line is the set of lines in C2passing through the origin, the blow-up of C2at the point (0,0), can be defined as the set: X=n(P, L)|P∈C2, P ∈L, (0,0) ∈Lo. Remark 1.2.3. To blow-up C2in a point Qdifferent of P= (0,0), we make a linear change of coordinates such that the new coordinates of Qare (0,0). We have a natural projection σ:X→C2×CP1defined by the restriction of the usual projection C2×CP1→C2. It can be pictured by the following commutative diagram: X $$$$ I I I I I I I I I I //C2×CP1  C2 Property 1.2.4. The blow-up map σ:X→C2×CP1, is a birational map, which is an isomorphism outside the origin. Definition 1.2.5. If Uis an open set of C2containing P= (0,0). The blow-up ˆ Uof U at the point Pis defined by: ˆ U=σ−1(U), where σ:X→C2is the blow-up of C2at the point P. Let Mbe a complex surface and Pa point of M. Consider a local chart c:V→U⊂C2, where Vis an open neighborhood of Pand Uan open set of C2containing (0,0). Assume that the local chart cis centered at P(i.e. c(P) = (0,0)). Then, the blow-up ˆ Mof M at the point Pis: ˆ M= (M\{P})Gˆ U, where ˆ Uis the blow-up of Uover (0,0) and the points of V\{P}are identified with the points of ˆ U\σ−1((0,0)). Remark 1.2.6. To blow-up a surface at a point P, we replace its neighborhood Vby the blow-up at the point (0,0) of the local chart. Consider the blow-up ˆ Mat the point P. Let Q∈M\ {P}and let Ube any neighborhood of Qwhich did not contain P, then σ−1(U)≃U. In fact, we have M\{P} ≃ σ−1(M\{P})), and σ−1(P)≃CP1. Remark 1.2.7. The points of σ−1(P)are in one-to-one correspondence with the set of all the complex directions passing through P. Since the blow-up of Mat the point Pmake only a local modification around P, it is possible to blow-up any other points of X. 32 1.2. BLOW-UP 1.2.2 Case of arrangements Let us recall the notion of divisor (also called Weil’s divisor). If Xis an algebraic variety, a divisor on Xis a finite formal sum with integer coefficients of closed and irreducible sub-varieties of codimension 1. It is an effective divisor if all the coefficients are positive. It is a normal crossing divisor if the irreducible sub-varieties are smooth, intersect pairwise in a transversal way, and all points of Xare at most in two irreducible sub-varieties. Proposition 1.2.8. If we blow-up one time each singular point of multiplicity greater than 3 of an arrangement A, then the preimage ˆ Aof Ais an effective divisor with normal crossings. Proof. The case of a singular point of multiplicity 3 is detailed in Example 1.2.11. With this equation it is clear that ˆ Ais a divisor with normal crossings. The case of points with greater multiplicity are similar. Definition 1.2.9. Let Abe a line arrangement. The blow-up of Ais the pair (X, ˆ A), where Xis the blow-up of CP2over all the singular points of Awith multiplicity greater than 3, and ˆ Ais preimage of Aby the blow-up. Definition 1.2.10. Let σ:X→CP2be the blow-up of CP2over the singular points of multiplicity greater than 3. The strict transformation of L∈ A is the inverse image of the smooth part of Lby σ. The exceptional component associated with a singular point P∈ Q of multiplicity greater than 3, is σ−1(P). Example 1.2.11. The blow-up of a singular point of multiplicity 3 is pictured in Figure 1.4 and the computation is detailed below. Consider a pencil of three lines A={L0, L1, L2}defined by the polynomial: PA(x1, x2)=(x1+x2)x2(x1−x2). The blow-up Xof C2is given by the equation x1z2=x2z1. It is homeomorphic to C2 except that the point (0,0) has been replaced by CP1. To obtain the total inverse image of Ain X, we consider the equations in C2×CP1: (x1+x2)x2(x1−x2)=0and x1z2=x2z1. Since the coordinates of CP1are homogenous, either z1or z2is non-zero. Suppose z16= 0, then we may assume that z1= 1 and we have the equations (x1+x2)x2(x1−x2)=0and x1z2=x2. Substituting, we obtain: x3 1z2(1 + z2)(1 −z2)=0. Hence the total inverse image ˆ Aof Ais composed of the four irreducible components: σ−1(P):x1= 0,x2= 0,z2arbitrary, corresponds to the inverse image of P. ˆ L0:z2=−1,x2arbitrary, x1=−x2, corresponds to the transformation of L0. ˆ L1:z2= 0,x2= 0,x2arbitrary, corresponds to the transformation of L1. ˆ L2:z2= 1,x2arbitrary, x1=x2, corresponds to the transformation of L2. 33 CHAPTER 1. PRELIMINARIES L0 L1 L2 Pσ Blow-up ˆ L0 ˆ L1 ˆ L2 σ−1(P) Figure 1.4: Local blow-up of a singular point Definition 1.2.12. Anc-Zariski pair is a pair of arrangements A1and A2with the same combinatorics and such that there is no homeomorphism between the couples (X1,ˆ A1) and (X2,ˆ A2), preserving strict transform of lines and exceptional components. Remark 1.2.13. The initials nc mean normal crossing. In both definitions of Zariski pair (usual and nc) we can consider oriented (resp. ordered) pair, in this case, we are only interested in homeomorphism preserving the orientation (resp. order). Proposition 1.2.14. If a pair A1and A2of arrangement is a Zariski pair, then it is an nc-Zariski pair. Proof. We proceed by contradiction. Assume that A1and A2is not a nc-Zariski pair, then there is a homeomorphism φfrom (X1,ˆ A1)to (X2,ˆ A2). Since the map σi:Xi→ CP2preserve strict transformations and exceptional components and are continuous, then φinduce a homeomorphism from (CP2,A1)to (CP2,A2). Question. Let A1,A2be a nc-Zariski pair, is it a Zariski pair ? 1.2.3 Dual Graph Definition 1.2.15. Let ˆ Abe the blow-up of an arrangement A. The dual graph of ˆ A is defined as follows: •The set of vertices is in a one-to-one correspondence with the irreductible components of ˆ A. •An edge joins two vertices if and only if the corresponding components intersect. The dual graph is denoted by ˆ ΓA. Proposition 1.2.16. The graphs Γ0 Aand ˆ ΓAare isomorphic. 34 1.2. BLOW-UP Proof. An irreducible component of ˆ Acan be of two types: •It corresponds to a line Liof A, and it is denoted ˆ Li. •It corresponds to a singular point Pj∈ Q of A, and it is denoted by Ej. The isomorphism sends each ˆ Lito vLiand each Ejto vPj. Thus the edges of ˆ ΓAare sent on the edges of Γ0 A. Remark 1.2.17. If we consider the construction of ˆ Ain which we blow-up all the singular points, then the dual graph is isomorphic to the incidence graph ΓA. 35 CHAPTER 1. PRELIMINARIES 36 Chapter 2 Wiring diagram One of the difficult points in the study of objects in the complex plane is the comprehension of a 4-dimensional space. To overcome this difficulty, we consider smaller dimensional objects reflecting their properties. The braid monodromy is one of the more powerful of such objects. It has been first initiated by O. Zariski in [71] and E.R. van Kampen in [65], where he gives a method to compute the fundamental group of the complement of an algebraic curve (known as the Zariski-van Kampen method) using the positions of the curves along pencils of lines. Refinements of van Kampen’s algorithm were given by O. Chisini [18]. In the early 80’s, B. Moishezon [51] introduced the modern notion of braid monodromy, which he used to recover van Kampen’s presentation. In [43], A. Libgober introduces a 2-dimensional CW-complex constructed from the braid monodromy admitting the same homotopy type than the complement. In [9], W. Arvola generalizes the method of R. Randell in [58] to compute a Zariskivan Kampen–like presentation of the fundamental group of a complex line arrangement. For that, he introduces the notion of braided wiring diagram, which is a singular braid derived from the braid monodromy of the arrangement. First, we give the definition and explain how to construct this diagram. Then we describe Arvola’s algorithm to compute the presentation of the fundamental group of EA, and we give some other properties of the braided wiring diagram. In Section 2.2, we describe the construction of a Sage program to compute the braided wiring diagram from the equations of the lines of Aaff. Finally, in Section 2.3, we picture the braided wiring diagram of two particular examples: MacLane’s arrangements, and Fan’s arrangements –transmit by K.M. Fan to E. Artal–. CHAPTER 2. WIRING DIAGRAM 2.1 Braided Wiring Diagram 2.1.1 Braided wiring diagram Up to a change of coordinates, let us assume that L0={z= 0}, and consider the chart n[x:y:z]∈CP2|z6= 0o≃C2. The affine part Aaff ⊂C2of Acan be viewed as the set of the affine part of the lines L1,··· , Ln. Let π:C2→C, be a generic linear projection, in the sense that: - For all i∈ {1,··· , n}, the restriction of π|Liis a homeomorphism (i.e. there is no vertical line). - Each multiple point lies in a different fiber of π. We suppose also that the points xi=π(Pi)have distinct real parts, then we can order the points of π(P)by increasing real parts, so that <(x1)<<(x2)··· <<(xk). A path γ: [0,1] →Cemanating from x0with <(x0)<<(x1), passing through x1,··· , xk in order, with no self-intersection, and horizontal (i.e. with constant imaginary part) in a neighborhood of each xiis said to be admissible. Definition 2.1.1. The wiring diagram associated to an admissible path γis defined by: WA={(x, y)∈ A | ∃t∈[0,1], π(x, y) = γ(t)}=π−1(γ([0,1])) ∩A. The trace ωi=WA∩Liis the wire associated to the line Li. Note that if Ais a real complexified arrangement, then we can take γ= [x0−η, xk+ η]⊂R; and WA≃ A∩R2. Remark 2.1.2. We can define also γon Rinstead of [0,1]. In this case, WAis the generalization of the trace Aaff ∩R2for a complexified real arrangement. Remark 2.1.3. i) The wiring diagram depends on the path γ, and on the projection π. ii) The set of singular points Pis contained in WA. We re-index the lines L1,··· , Lnsuch that: Ii> Ij⇐⇒ i < j, where Ii=Im(Li∩π−1(x0)). On the representation of the braided wiring diagram described below, this re-indexation implies that the lines are ordered, at the left of the diagram, from top to bottom. Since the xcoordinates of the points of WAare parametrized by γ, the wiring diagram can be seen as a one dimensional object inside R3≃[0,1] ×C. It is then called braided wiring diagram. Consider its image by a generic projection γ([0,1])×C→R2. If we take 38 2.3. EXAMPLES •Add a function to extract the braid monodromy of the braided wiring diagram. This function needs that the previous one already exists. •Draw the wiring diagram (when it is possible, see Remark 2.2.3). •Obtain the L A TEXcode for the picture using the package Tikz. 2.3 Examples 2.3.1 MacLane arrangements The MacLane arrangements are two conjugated arrangements coming from MacLane’s matroid [50]. It is the arrangement with a minimal number of lines such that the combinatorics admits a realization over Cbut not over R(see, [69]). These arrangements are constructed as follows. Let us consider the 2-dimensional vector space on the field F3of three elements. Such a plane contains 9 points and 12 lines, 4 of them pass through the origin. Consider L=F2 3\{(0,0)}and P, the set of lines in F2 3(i.e. the set of lines in F2 3not passing through the origin). This provides a line combinatorics (L,P,b), where for all `∈ L, P ∈ P, we have Pb`⇔(`∈P, in F2 3). Figure 2.5 represents the ordered MacLane’s combinatorics viewed in F2 3. As ordered combinatorics, it admits two ordered realizations, denoted by Q+and Q−. They are defined by the following equations: L0:z= 0 L1:z−x= 0 L2:x= 0 L3:y= 0 L4:z+ω2x+ωy = 0 L5:y−x= 0 L6:z−x−ω2y= 0 L7:z+ωy = 0 where ω= exp(2iπ 3)for Q+, and ω= exp(−2iπ 3)for Q−. • • • • • • • • 5 0 7 2 4 1 3 6 Figure 2.5: Ordered MacLane combinatorics, viewed in F2 3 The braided wiring diagram of Q+, is pictured in Figure 2.6. It is computed using the Sage program described in Subsection 2.2.1, with the lines L1and L2taken vertically (i.e. they are fibers of the projection). The diagram is then pictured with a small perturbation of the projection, such that L1and L2are generics relative to the new projection. Using Subsection 2.1.4, we have computed the braided wiring diagram of Q−, pictured in Figure 2.7. As they are minimal (in terms of number of lines such that the combinatorics admits a realization over Cbut not over R), these arrangements are good examples of complex 45 CHAPTER 2. WIRING DIAGRAM L1 L2 L3 L4 L5 L6 L7 Figure 2.6: Braided wiring diagram of MacLane positive L1 L2 L3 L4 L5 L6 L7 Figure 2.7: Braided wiring diagram of MacLane negative arrangements. Then Q+is used as example in the next chapter, to illustrate the method developed. 2.3.2 Fan’s arrangements K.M. Fan transmitted to E. Artal a combinatorics admitting three different realizations (one real and two complex). The question about them is: Question. Is the real arrangement topologically different from the complex realizations ? Their equations are: L0:z= 0 L1:x= 0 L2:x−z= 0 L3:y= 0 L4:y−z= 0 L5:x−y= 0 L6:x−y−z= 0 L7:x−αz = 0 L8:αy + (2 −α)x−αz = 0 L9: (3α−4)y+ (2 −α)x−αz = 0 L10 : (3α−4)y+ (2 −α)x−2α(α−1)z= 0 where αis a solution of the equation x3−7 2x2+ 6x−4=0. The realization associated to the real solution is denoted by F0, and F+(resp. F−) is the realization associated to the complex solution with a positive (resp. negative) imaginary part. The braided wiring diagram of F0, is its trace in R2. It is pictured in Figure 2.8. The braided wiring diagram of the positive Fan’s arrangement F+is pictured in Figure 2.9. It is computed with the Sage program with a non generic projection. And the negative case is obtained using Subsection 2.1.4, and it is pictured in Figure 2.10. 46 2.3. EXAMPLES L8 L4 L3 L5 L6 L10 L9 L1L2L7 Figure 2.8: Braided wiring diagram of Fan real L4 L3 L8 L10 L9 L5 L6 L1L2L7 Figure 2.9: Braided wiring diagram of Fan Positive L4 L3 L8 L10 L9 L5 L6 L1L2L7 Figure 2.10: Braided wiring diagram of Fan negative As F+and F−are conjugated arrangement, then their fundamental groups are isomorphic. These wiring diagrams allow to compute some topological invariants. But they give no hint about topological equivalence. 47 CHAPTER 2. WIRING DIAGRAM 48 Chapter 3 Boundary manifold The boundary manifold BAis a compact graph manifold, in the sense of F. Waldhausen [66]. It is determined by the combinatorics and modeled on the incidence graph of the arrangement. In [40], E. Hironaka describes the embedding of BAin EAin the case of the complexified real arrangement. In this Chapter, we generalize this result to the case of complex line arrangements using the braided wiring diagram. This work gives rise to a submitted paper [34]. In Section 3.1, we describe the procedure to construct the boundary manifold from the incidence graph. Then we use this description to compute its fundamental group. In Section 3.2, we generalize the results of E. Hironaka [40] and give the explicit algorithm to compute the map induced by the inclusion on the fundamental groups. From this map, we deduce a new minimal presentation of π1(EA)such that the CWcomplex built on it has the same homotopical type that EA. After being given a simpler version of the previous result, we recover the presentation of the fundamental group of EAobtained by R. Randell in [58]. To conclude this Chapter, in Section 3.3, we explicitly compute the fundamental group of the positive MacLane’s arrangement, hence also of the negative one. 3.1 Boundary manifold 3.1.1 Definition & construction Let Ube a compact regular neighborhood of Aas constructed in Subsection 1.1.1. We define the boundary manifold BAas ∂U. Remark that –up to orientation– BAis also the boundary of ExtA. This manifold is combinatorially determined and can be computed from the incidence graph ΓAas follows: For every singular point P∈ Q of A, consider a 4-ball BPof radius η, centered in CHAPTER 3. BOUNDARY MANIFOLD P. Let SP=∂(BP)\T, where Tis an open regular neighborhood of the link LP= (∂BP∩A). The boundary of SPis a union of disjoint tori Tindexed by the lines Li passing through P, and TL= (L∩∂BP)×S1. Definition 3.1.1. Let P∈ Q and L∈ A be such that P∈L. The meridian mLand the longitude lLof the torus TLare the pair of oriented simple closed curves in TL=∂T which are determined up to isotopy by the homology and linking relations: mL∼0, lL∼(L∩T)in H1(T); `(mL, L ∩T)=1, `(lL, L ∩T)=0, where `(·,·)denotes the linking number in ∂BP≃S3. Consider the surface: F=A\ a P∈QA∩ ◦ BP, it is obtained by removing of Aopen discs from the BP’s. One sees that Fis a union n ` i=0 Fiwhere each Ficorresponds to the line Liof A. Let Ni=Fi×S1whose boundary is a union of disjoint tori Tindexed by the points P∈ P ∩Li. Let Dbe a generic line (i.e. for all P∈ Q,P /∈D), and consider Das the line at infinity. We decompose Niin the solid torus Ni∩T(D) = T∞ i, where T(D)is regular neighborhood of D, and the affine part Naff idefined as the closure of Ni\T∞ i. Viewed as the affine part, Naff iadmits a natural trivialization in the affine space, and we choose a section sof Naff i. Definition 3.1.2. Let Li∈ A and P∈ P be such that P∈Li. The longitude lPof the torus TPis the intersection of the section swith TP. The meridian mPof TPis the class in H1(TP)of {∗}×S1. Remark 3.1.3. To reconstruct Nifrom Naff iand T∞ i, we glue the boundary component of Naff idifferent of the TP’s with ∂T∞ i. The gluing is done identifying the intersection of the section in this component with the sum of a longitude of ∂T∞ i(i.e. a curve in ∂T∞ i homologically equivalent to Li∩T∞ iin T∞ i) and a meridian; and the meridian with a fiber of the S1-fibration of T∞ i. For each edge e(Li, P)of ΓA, glue SPwith Nialong TLiand TPidentifying meridian with meridian, and longitude with longitude. The manifold then obtained is the boundary manifold of A. From this construction of BA, we deduce its structure of graph manifold. Proposition 3.1.4 ([68, 40]).Let Abe a complex line arrangement. The boundary manifold BAis a graph manifold over the incidence graph ΓA. Remark 3.1.5. The construction previously done is different of the classic one using plumbing graph as defined by W.D. Neumann in [53]. With elementary computations, one may show that two constructions coincide, though gluings are described differently. Indeed, to obtain this construction from the one present here, we need to blow-up the singular points, but this transformation does not change the SP’s. 50 3.1. BOUNDARY MANIFOLD Corollary 3.1.6. The boundary manifold of a complex line arrangement depends only on the combinatorics of the arrangement. Proof. The plumbing used to compute BAas a graph manifold over ΓAis combinatorial. Then the equivalence of ΓAand the combinatorics of Ainduce the result. A similar construction is done by [68] and [24] but using the blow-up ˆ Aof A, and its dual graph ˆ ΓA. Since the incidence graph of Aand the dual graph of ˆ Aare equivalent, using a result of Waldhausen [66], they obtain homeomorphic manifold. However these two constructions give different presentations of the fundamental group of the boundary manifold. 3.1.2 Combinatorial groups In [65], E.R. van Kampen explains how compute the fundamental group of the gluing of topological spaces (known as Seifert-Van Kampen method). Definition 3.1.7. Let Gand Hbe two groups admitting the following presentations G=hg1,··· , gm|r1,···rniand H=hh1,··· , hk|s1,··· , sli. Suppose that A⊂Gand B⊂Hare subgroups with an isomorphism Φ : A→B. Then the free product of Gand Hwith an amalgamation of subgroups Aand Bby Φis the group with presentation: hg1,··· , gn, h1,··· , hk|r1,··· , rn, s1,··· , sl, a = Φ(a),∀a∈Ai, abbreviated as hG∗H|A=B, Φi. Definition 3.1.8. Let Aand Bbe subgroups of G, with an isomorphism Φ : A→B. The Higman-Neumann-Neumann (HNN) extension of G, with stable letter tand associated subgroups Aand Bis the group: hG, t |t−1·a·t= Φ(a),∀a∈Ai. That is, the presentation contains all the generators and relators of G, with an additional generator t, and a set of new relations of the form t−1·a·t= Φ(a). a) Product with amalgamation and fundamental group of a union Suppose that Xand Yare path-connected spaces having open path-connected subspaces Uand V, respectively. Suppose Φ : U→Vis a homeomorphism. Let u∈Ube a base point for the fundamental groups of Uand X, and Φ(u)∈Vbe a base point for the fundamental groups of Vand Y. The homeomorphism Φinduces an isomorphism Φ∗:π1(U, u)→π1(V, Φ(u)). Let Zbe the space obtained by identifying Uand Vvia Φ. Proposition 3.1.9 (Seifert-Van Kampen).With the previous notations, we have: π1(Z) = hπ1(X)∗π1(Y)|π1(U) = π1(V),Φ∗i. If U=V=X∩Y, then this gives π1(X∪Y). 51 CHAPTER 3. BOUNDARY MANIFOLD b) HNN extension and fundamental group of a space with a handle Now suppose Uand Vare disjoint homeomorphic path-connected subspaces in a pathconnected space X. Let Ibe the unit interval [0,1]. We attach a handle U×Ito X by identifying U×{0}with Uand U×{1}with V, using the homeomorphism Φ(see Figure 3.1). We call Zthis new space. Id U X U×I Φ V Figure 3.1: Handle gluing Proposition 3.1.10 (Seifert-Van Kampen).With the previous notations, the fundamental group of Zis the HNN extension of π1(X)with stable letter tand associated subgroups π1(U)and π1(V). In other terms: π1(Z) = hπ1(X), t |t−1·π1(U)·t=π1(V),Φ∗i. Remark 3.1.11. The stable letter tobtained is a cycle arising from the gluing of the handle. 3.1.3 Fundamental group of BA The fundamental group of BAis the group associated to the incidence graph, see [68, 24]. Two types of generators naturally appear: the meridians of the lines and the cycles related to the graph. Definition 3.1.12. Let Lbe a line in CP2, and b∈CP2\ A. A homotopy class α∈π1(CP2\A, b)is a meridian of Lif αhas a representative δconstructed as follows: •there is a smooth complex analytic disc ∆⊂CP2transverse to Lat a smooth point of Aand such that ∆∩L={b0} ⊂ L, and pick out a point b00 ∈∂∆. •there is a path ain CP2\A from bto b00 ∈∂∆; •δ=a−1·β·a, where βis the closed path based in b0given by ∂∆(in the positive direction). 52 3.1. BOUNDARY MANIFOLD Choose arbitrarily a line L0of the arrangement. Note that a meridian of L0is the product of the inverse of some meridians of the lines L1,··· , Ln, in EA. Let Pbe the set of singular points of the affine arrangement Aaff =A\L0. We fix a braided wiring diagram WAand assume that Ais ordered as described in Subsection 2.1.1. Definition 3.1.13. Acycle of the incidence graph ΓAis an element of π1(ΓA, vL0). Remark 3.1.14. The group π1(ΓA, vL0)is a free group on b1(ΓA)generators. We construct a generating system Eof cycles of ΓAas follows. Let Tbe the maximal tree of ΓAcontaining the following edges: •e(L, P)for all P∈L0, and L∈ A; •e(Lν(P), P)for all P∈ P and ν(P) = min{j|Li∈ AP}. Remark 3.1.15. Up to the choice of a projection and an admissible path for the wiring diagram WA, and then of an order on A, this maximal tree is uniquely determined. An edge in ΓA\ T is of the form e(Lj, P), with P∈ P and Lj∈ A \ L0. By definition of a maximal tree, there exists a unique path λP,j in Tjoining vPand vLj. The unique cycle of ΓAcontaining the three line-vertices vL0,vLν(P)and vLj, and no other line-vertex, is denoted by: ξν(P),j =λP,j ∪e(Lj, P). Let Ebe the set of cycles of ΓAof the form ξs,t. To each ξs,t in Ewill correspond an element (a stable letter) of π1(BA, X0)(where X0∈ N0), that we denote es,t. Notation. As in Chapter 2, we denote [a1,··· , am]the equality of all the cyclic permutations a1···am=a2···ama1=··· =ama1···am−1. For i= 0,··· , n, let αibe a meridian of Licontained in the boundary of a regular neighborhood of L0, and for ξs,t ∈E, let es,t be a non trivial cycle contained in the boundary of a regular neighborhood of L0∪Ls∪Lt(it comes from the gluing over the edge e(Lt, P)where Pis Ls∩Lt). By Definition 3.1.12, αiis ai·βi·a−1 i. We assume that es,t can be decomposed in at·−1τ·as, where τis a path in the boundary of a regular neighborhood of Ls∪Lt. We also supposed that for (s, t)6= (s0, t0),es,t ∩es0,t0=X0. Proposition 3.1.16. Let αiand es,t be as previously defined. For any singular point P=Pi1,···,imwith multiplicity mand i1=ν(P), let RP= [αcim im,··· , αci2 i2, αi1],where cij=ei1,ijfor all j= 2,··· , m. The fundamental group of the boundary manifold BAadmits the following presentation: π1(BA, X0) = hα0, α1,··· , αn,es1,t1,··· ,esl,tl|[ P∈P RPi. 53 CHAPTER 3. BOUNDARY MANIFOLD It is worth noticing that the es,t are not uniquely defined (see details in the proof). Proof. Consider P∈ Q. Assume that P=Pi1,···,im. Let yP,i1, . . . , yP,imbe the ’local’ meridians of the line Liin ∂BP. We have the following presentation of π1(SP): π1(SP) = hyP,i1,··· , yP,im|[yP,im,··· , yP,i1]i. Remark that, according with Definition 3.1.1, yP,ijis a meridian of Tijand a longitude is the product of the other yP,ik. Consider k∈ {0,··· , n}. Let Q∩Lk={Pk1,··· , Pkl}. Let gk,kibe the image of a meridian in Fkaround Pki, viewed in Fk×{1} ⊂ Nk, and αk∈π1(Nk)a meridian of Lkcontained in a regular neighborhood of L0. We have the following presentation of π1(Nk): π1(Nk) = hgk,k1,··· , gk,kl, αk| ∀i∈ {1,··· , l}, α−1 k·gk,ki·αk=gk,kii. Remark that according with Definition 3.1.2, gk,kiis a longitude and αkis a meridian of TPi. In a first step, we only glue the Nk’s and the SP’s over the edges of T. To do this, we use Proposition 3.1.9, and we consider a contractible set Θhomeomorphic to Tand joining the base points of the Nk’s and the SP’s. The fundamental group of BAis compute relatively to Θ. In a second step, we glue over the edges of ΓA− T (or equivalently the elements of E). Then we use Proposition 3.1.10, and we denote es,t the stable letters coming from the glue due to the edge e(Lt, P), with P=Ls∩Lt. Note that if Pi∈Lj, then the meridian αjis identified with yPi,j and gj,i with the product of generators of π1(SPi)different of yPi,j. Then after a first simplification, we obtain the following presentation of the fundamental group of the boundary manifold: π1(BA) = hα0, α1,··· , αn,es1,t1,··· ,esl,tl|[ P∈QRPi. Remark that the presentation of π1(N0)implies that α0commutes with the g0,i and by identification with αi, for i∈ {1,··· , n}. By construction of T, the relators RP, for P∈L0, are of the form: [α0m, α0m−1,··· , α02, α0]. Then they are all trivial. 3.2 Inclusion map Let WAbe the wiring diagram associated to the choice of a generic projection πand an admissible path γ. We start by choosing a generating system E={ξs,t}of cycles of the incidence graph ΓA. These cycles can be directly seen in WA, since it contains all the singular points and the vertices of ΓAcan be identified with their corresponding wires between two singular points. Then, the first step is to "push" each cycle ξs,t from 54 3.3. EXAMPLE: MACLANE Proposition 3.2.10. The presentation of the fundamental group of EAis minimal, and the 2-complex built on it has also the same homotopy type with the exterior of A. As consequence of this corollary, we found the theorem of Randell in the case of real complexified arrangement. Theorem 3.2.11 (Randell, [58]).Let Abe a real complexified line arrangement. For i= 1,··· , n, let αibe the meridians of the lines Li. The fundamental group of EA admits the following presentation: π1(EA) = hα1,··· , αn|[ Pi∈P [αim,··· , αi2, αi1],with P=Li1∩···∩Limi, with Li1,··· , Limare taken from top to bottom at the left of Piin WA. 3.3 Example: MacLane In this section, we illustrate Theorem 3.2.2 with MacLane’s arrangement Q+defined in Subsection 2.3.1. The incidence graph Γof Q+is given in Figure 3.5. L0 P0,1,2 P0,3,4 P0,5,6 P0,7L7 L6 L5 L4 L3 L2 L1 P1,5,7 P1,3 P1,4,6 P2,3,5 P2,4,7 P2,6 P3,6,7 P4,5 Figure 3.5: Incidence graph of the MacLane arrangement Q+ It is worth mentioning that Q+is one of the only two ordered realizations of this combinatorial data by an arrangement in CP2. 3.3.1 Generating set of cycles of ΓQ+ Consider the maximal tree Tin ΓQ+indicated with thick lines in Figure 3.5. Let Ebe the generating system of cycles induced by T(it is in one-to-one correspondence with the dotted lines in Figure 3.5): E={ξ2,3, ξ2,5, ξ2,4, ξ2,7, ξ2,6, ξ4,5, ξ3,6, ξ3,7, ξ1,5, ξ1,7, ξ1,3, ξ1,4, ξ1,6}. 61 CHAPTER 3. BOUNDARY MANIFOLD 3.3.2 Group of the boundary manifold By Section 3.2.1, the images εof the cycles ξby the application σprovide a family of cycles in π1(BQ+). Proposition 3.1.16 applies to this explicit family, and π1(BQ+) admits a presentation with generators: {α0, α1, α2, α3, α4, α5, α6, α7}∪{ε2,3, ε2,5, ε2,4, ε2,7, ε2,6, ε4,5, ε3,6, ε3,7, ε1,5, ε1,7, ε1,3, ε1,4, ε1,6}, and relations: [αε1,7 7, αε1,5 5, α1],[αε1,3 3, α1],[αε1,6 6, αε1,4 4, α1],[αε2,5 5, αε2,3 3, α2], [αε2,7 7, αε2,4 4, α2],[αε2,6 6, α2],[αε3,7 7, αε3,6 6, α3],[αε4,5 5, α4]. 3.3.3 Geometric cycles and unknotting map 1 2 3 4 5 6 7 ς1ς2ς3 Figure 3.6: Wiring diagram of the positive MacLane arrangement Let WQ+be the braided wiring diagram of Q+given in Figure 3.6. Note that WQ+ differs from the wiring diagram considered in [24] by an axial symmetry and a local move on the wires corresponding to L3, L5, L7. The diagram WQ+is used to compute the unknotting map δ, and the images of the cycles εin terms of geometric cycles, see Proposition 3.2.1. The thick lines in Figure 3.6 represent the cycle ξ4,5, divided into two arcs of L4and L5. - The first arc L4meets the triple point vP2,4,7. This gives δl 4,5=ε−1 2,4α−1 2ε2,4. - The second arc L5meets vP2,3,5, and δr 4,5=ε−1 2,5(ε2,3α3ε−1 2,3)α2ε2,5. This implies that δ(ε4,5) = ε−1 2,4α−1 2ε2,4. ε4,5.hε−1 2,5ε2,3α3ε−1 2,3α2ε2,5i. 62 3.3. EXAMPLE: MACLANE Similarly, one computes: δ(ε2,3) = ε2,3, δ(ε2,5) = ε2,5, δ(ε2,4) = ε2,4, δ(ε2,7) = ε2,7, δ(ε2,6) = ε2,6, δ(ε4,5) = ε−1 2,4α−1 2ε2,4. ε4,5.hε−1 2,5ε2,3α3ε−1 2,3α2ε2,5i δ(ε3,6) = ε−1 2,3α−1 2ε2,3. ε3.6.ε−1 2,6α2ε2,6 δ(ε3,7) = ε−1 2,3α−1 2ε2,3. ε3,7.hε−1 2,7ε2,4α4ε−1 2,4α2ε2,7i δ(ε1,5) = ε1,5.hε−1 4,5α4ε4,5ε−1 2,5(ε2,3α3ε2,3)−1α2ε2,5i δ(ε1,7) = ε1,7.hε−1 3,7ε3,6α6ε−1 3,6α3ε3,7ε−1 2,7ε2,4α4ε−1 2,4α2ε2,7i δ(ε1,3) = ε1,3.ε−1 2,3α2ε2,3 δ(ε1,4) = ε1,4.ε−1 2,4α2ε2,4 δ(ε1,6) = ε1,6.hε−1 3,6α3ε3,6ε−1 2,6α2ε2,6i 3.3.4 Retractions of geometric cycles We now compute the family of µs,t, required to obtain the inclusion map, see Subsection 3.2.2. The arcs of the wiring diagram WQ+are labeled by the algorithm of Arvola, see Subsection 2.1.2. The case of µ4,5is drawn in thick in Figure 3.6. The over arcs ς1,ς2and ς3are dotted in Figure 3.6. Arvola’s labellings of these arcs are respectively : aς1=α4,aς2=α7and aς3=α−1 7α4α7. Furthermore, sgn(ς1) = −1,sgn(ς2) = 1 and sgn(ς3) = 1. We obtain: µ4,5=α−1 7α4α7.α7.α−1 4, which gives µ4,5=α−1 7α4α7. α7. α−1 4. Similarly: µ2,3= 1, µ2,5=−α4, µ2,4= 1, µ2,7= 1, µ2,6=α7, µ4,5=α−1 7α4α7. α7. α−1 4, 63 CHAPTER 3. BOUNDARY MANIFOLD µ3,6=hα−1 4α5α4α−1 7α−1 7α4α2 7α−1 4α−1 5α4α−2 7α−1 4α7(α7)i.hα−1 7α−1 7α−1 4α7(α7)i, µ3,7=hα−1 4α5α4α−1 7α−1 7α4α2 7α−1 4α−1 5α4α−2 7α−1 4α7(α7)i, µ1,5=α−1 7α−1 7α4α7(α7)α−1 4, µ1,7= 1, µ1,3=α−1 7α−1 4α2 7α−1 6α−2 7α4α7α−1 7α−1 7α4α2 7α−1 4α5α4α−2 7α−1 4α7(α7)α−1 4α−1 5α4, µ1,4= 1, µ1,6=α−1 7α4α7α−1 7α−1 7α−1 4α7(α7). 3.3.5 Images in the group of the complement Following Theorem 3.2.2, we can compute i∗:π1(BQ+)π1(EQ+). The computations above describe the relations induced by the images of the cycles εin π1(EQ+). By the previous computations, ε2,3, ε2,4, ε2,7are equal to 1(i.e. they are contractible in EQ+). They are the relations r2,3, r2,4and r2,7. Without additional computation, we obtain: r2,5:ε2,5=α−1 4, r2,6:ε2,6=α7. The case of r4,5: r4,5: (ε−1 2,4α−1 2ε2,4). ε4,5.[ε−1 2,5(ε2,3α3ε−1 2,3)α2ε2,5] = α−1 7α4α7. α7. α−1 4. Then using r2,4,r2,5and r2,3, we obtain that: r4,5:ε4,5= (α2).α−1 7α4α7. α7. α−1 4.α4α−1 2α−1 3α−1 4. The others relations can be computed by the same way. And from the proof of Corollary 3.2.5, we obtain: Property 3.3.1. The fundamental group of EQ+admits the following presentation: π1(E(A+)) = hα1, α2, α3, α4, α5, α6, α7, ε2,3, ε2,5, ε2,4, ε2,7, ε2,6, ε4,5, ε3,6, ε3,7, ε1,5, ε1,7, ε1,3, ε1,4, ε1,6| r2,3, r2,5, r2,4, r2,7, r2,6, r4,5, r3,6, r3,7, r1,5, r1,7, r1,3, r1,4, r1,6, [αε1,7 7, αε1,5 5, α1],[αε1,3 3, α1],[αε1,6 6, αε1,4 4, α1],[αε2,5 5, αε2,3 3, α2], [αε2,7 7, αε2,4 4, α2],[αε2,6 6, α2],[αε3,7 7, αε3,6 6, α3],[αε4,5 5, α4], α0···αni. 64 Chapter 4 Inclusion in homology The homology groups of a topological space are important invariants. It seems natural to compute the application induced by the inclusion of BAin EAon the first homology group. This computation is done in Section 4.1, and will reveal to be very useful in the results of E. Artal [5] to compute a part of the depth of a character (see Chapter 5). On the other hand, it allows to construct a topological invariant of the blow-up of an arrangement. In Section 4.2, as examples, we explicitly compute the map H1(BA;Z)→H1(EA;Z) in the case of extended Ceva’s arrangement and MacLanes’s arrangements. In Section 4.3, we present a canonical decomposition of the first homology group of the boundary manifold into the 1-homology and the 2-cohomology of the complement. This result goes in the sense of the decomposition of the generators of π1(EA)done in Chapter 3. Using Poincaré’s residue, we obtain an explicit description of an isomorphism between H2(EA;C)and H1(ΓA;C). 4.1 Inclusion map 4.1.1 Homotopy vs homology Consider the homology groups of the topological space, and more precisely, the first homology group. It is well known that it can be defined as the abelianised of the fundamental group. Then it contains some information present in the fundamental group, and considering the homology with integer coefficients, we have an abelian group; and with complex coefficients, we obtain vector spaces. In both cases, they admit better properties than finitely presented groups. To conserve a lot of properties coming from the fundamental group, we study the presentation of the first group of homology inherited from the presentation of the fun- CHAPTER 4. INCLUSION IN HOMOLOGY damental group. Indeed these presentations have geometrical interpretation. 4.1.2 The map To describe the inclusion map in homology, we compute the framed cycles and the geometric cycles in H1(EA). In this section, we consider the homology with integer coefficients. Notation. Let x∈π1(EA), the homological class of xis denoted by e x∈H1(EA;Z). Let e δ: H1(BA;Z)→H1(BA;Z)be the application induced by the unknotting maps (see Subsection 3.2.1) on the homology groups. We have: e δ:(H1(BA;Z)−→ H1(BA;Z) e εs,t 7−→ e δl s,t +e εs,t +e δr s,t. Notation. To simplifiy the notation of the homological unknotting map we denote: e δs,t =e δl s,t +e δr s,t. As in Chapter 3, by considering framed or geometric cycles (i.e. εs,t or Es,t), we obtain two possible descriptions of the application induced by the inclusion on the first homology groups. Let εs,t ∈π1(BA)be framed cycles of BAas described in Section 3.2. Since, the εs,t with the αi’s generate π1(BA), then the e εs,t’s with e αi’s generate H1(BA;Z). Theorem 4.1.1. The map induced by the inclusion of the boundary manifold in the complement of the arrangement Aon the first homology groups can be described by: H1(BA;Z)−→ H1(EA;Z) e αi7−→ e αi e εs,t 7−→ −e δs,t +e µs,t where e δs,t and e µs,t are described in Proposition 4.1.5 and Proposition 4.1.6, respectively. Proof. It is a direct consequence of Theorem 3.2.2. Corollary 4.1.2. If Ais a real complexified lines arrangement, then we have: H1(BA;Z)−→ H1(EA;Z) e αi7−→ e αi e εs,t 7−→ −e δs,t Let Es,t ∈π1(BA)be the geometric cycles as describe in Section 3.2, by definition we have e Es,t =e εs,t +e δ,t. By Subsection 3.2.5, the e Es,t’s with the e αi’s generate H1(BA). 66 4.1. INCLUSION MAP Theorem 4.1.3. The map induced by the inclusion of the boundary manifold in the complement of the arrangement Aon the first homology groups can be described by: H1(BA;Z)−→ H1(EA;Z) e αi7−→ e αi e Es,t 7−→ e µs,t where e µs,t is described in Proposition 4.1.6. Proof. It is a direct consequence of Theorem 3.2.7. Remark 4.1.4. Since we control the position of the εs,t (in contrast with the Es,t), Theorem 4.1.1 is the most useful of the both Theorems (4.1.1 and 4.1.3). It plays a fundamental role in the following Chapters. Let εs,t be a framed cycle of BA, coming from a maximal tree T0of the incidence graph ΓA, as described in Section 3.2. We consider the set of points on εs,t and we extract from εs,t two sets: the set of singular points on Lsand the set of singular points on Lt. Pt s={Pi∈εs,t |Pi∈Ls}and Ps t={Pi∈εs,t |Pi∈Lt}. Let P∈ P and L∈ A such that P∈L. We define the index of Lin P(denoted by Ind(L, P)): Ind(L, P) = m(P)−ord(L, P), where m(P)is the multiplicity of P, and ord(L, P )the position of the line Lat the left of the singular point Pon the picture of WA. We denote e αP,j the homological meridian around the jst line at the left of P. Proposition 4.1.5. The homological unknotting map e δcan be described as follows: e δ:(H1(BA;Z)−→ H1(BA;Z) e εs,t 7−→ e εs,t +e δs,t where e δs,t is:   kt X P∈Ps t Ind(Lt,P )−1 X j=1 e αP,j −  ks X P∈Pt s Ind(Ls,P )−1 X j=1 e αP,j . Proof. By Proposition 3.2.1, we know that the contribution of Plto δl s,t in homotopy is: ε−1 l1,lhα−1 l1εl1,l2α−1 l2ε−1 l1,l2···εl1,lh−1α−1 lh−1ε−1 l1,lh−1εl1,lh, where h=Ind(Ls, Pl), and ljis the jst index at the left of Pl. Since the homology is the abelianised of the homotopy, then the previous expression becomes: Ind(Ls,Pi)−1 X j=1 e αPi,j. 67 CHAPTER 4. INCLUSION IN HOMOLOGY And by the definition of the δl s,t, we obtain that: e δl s,t =− ks X P∈Pt s Ind(Ls,P )−1 X j=1 e αP,j. By the same way, we compute: e δr s,t = kt X P∈Ps t Ind(Lt,P )−1 X j=1 e αP,j. Let εs,t be a framed cycle of BA, as in Subsection 3.2.2, we denote sgn(ς)∈ {±1} the sign of the base {ς, εs,t}(in this order). Let αςbe the meridian associated with the line supporting the arc ς. Proposition 4.1.6. Let Ss,t be the set of the arcs over going εs,t in WA, then: e µs,t =X ς∈Ss,t sgn(ς)e ας. Proof. In the definition of µs,t, we consider the Arvola’s word associated to the arc ς. But, by construction, it is a conjugate of the meridian associated to the line supporting the arc ς. Then, in homology, the Arvola’s word becomes a meridian. We define µs,t =P ς∈Ss,t sgn(ς)e ας, where Ss,t is the set of the arcs under going εs,t (in WA). Let φthe isomorphism between H1(EA;Z)and H1(EA;Z)defined by: φ:(H1(EA;Z)→H1(EA;Z), e αi7→ e αi. Using the result of Subsection 2.1.4 on the complex conjugate of an arrangement, we have: Proposition 4.1.7. Let Abe a complex line arrangement, and Aits conjugate. The following diagram is commutative: H1(BA;Z)i∗// Id  H1(EA;Z) H1(BA;Z)j∗//H1(EA;Z) φ OO where j∗is defined by: j∗:     H1(BA;Z)−→ H1(EA;Z) f αi7−→ f αi e εs,t 7−→ −e δs,t +µs,t 68 4.2. EXAMPLES Proof. Since Aand Ahave the same combinatorics, then BA≃BA. The map i∗sends e αion e αi, the composition map φ◦j∗◦Id too. The map i∗sends e εs,t on −e δs,t +e µs,t. To obtain the braided wiring diagram of ¯ Awe inverse the virtual crossings of the braided wiring diagram of A. Then, the unknotting map is not changed when we pass from A to ¯ A. Since the arcs underpassing εs,t in WAare the arcs overpassing εs,t is W¯ A, then: φ◦j∗◦Id(e εs,t) = −e δs,t +e µs,t. 4.2 Examples By abuse of notation, the meridians and the cycles will be denoted in the same way whether they are in the fundamental group or in the first homology group. 4.2.1 Ceva-7 Consider four points in CP2in a generic position. The Ceva’s arrangement is the arrangement composed of the six lines of CP2passing through the four points. This arrangement admits four triple points and three double points. The extended Ceva’s arrangement, also called Ceva-7, is the usual Ceva’s arrangement with one additional line passing through two of the double points. Its real part is pictured in Figure 4.1. L1=∞ L2 L3 L4 L5 L6 L7 Figure 4.1: The extended Ceva’s arrangement As the Ceva’s arrangement, the Ceva-7 arrangement is a real complexified arrangement. Then the inclusion map in homology can be computed from Corollary 4.1.2. Considering L1as the line at the infinity, we obtain that ε3,4,ε3,5,ε3,6and ε3,7are 0 in H1(EA;Z); and that: ε4,7=−α6, ε2,5=−α3−α4, ε2,7=−α4−α3−α6, ε2,4=−α3, ε2,6=−α3. 69 CHAPTER 4. INCLUSION IN HOMOLOGY 4.2.2 MacLane The wiring diagrams of the MacLane’s arrangements Q+and Q−are pictured in Figure 2.6 and Figure 2.7. To compute the application induced in homology, we only consider the equations obtained in Section 3.3, which will be abelianised. Then for the positive MacLane arrangement Q+, the map induced by the inclusion is: ε2,3= 0, ε2,5=−α4, ε2,4= 0, ε2,7= 0, ε2,6=α7, ε4,5=−α3+α7, ε3,6=−α4, ε3,7=−α4, ε1,5=−α4−α3−α2, ε1,7=−α6−α4−α3−α2, ε1,3=−α2−α6, ε1,4=−α2, ε1,6=−α3−α2. To compute the inclusion map in the case of Q−, we use Proposition 4.1.7, and then: ε2,3= 0, ε2,5= 0, ε2,4=α5, ε2,7=−α6, ε2,6= 0, ε4,5= 0, ε3,6= 0, ε3,7=−α4, ε1,5=−α4−α3−α2, ε1,7=−α6−α4−α3−α2, ε1,3=−α2, ε1,4=−α2, ε1,6=−α3−α2−α6. 4.3 The 2-cohomology In this section, we work with homology and cohomology with complex coefficients. Since H1(EA;Z)and H1(BA;Z)are torsion free, by passing to complex coefficients, we conserve essentially all the information present in the homology with integer coefficients. The 2-cohomology of line arrangement is studied, among others, by J.I. Cogolludo [19] and M. Yoshinaga [70]. 4.3.1 Canonical decomposition of the complement Let us recall some notations of Chapter 1. Let Xbe the minimal blow-up of CP2 such that the blow-up ˆ Aof Ais a normal crossing divisor. And consider Ua regular neighborhood of ˆ A. By construction, it is homeomorphic to the regular neighborhood of Aconstructed in Subsection 1.1.1. Lemma 4.3.1. There is a natural injection of H1(EA;C)in H1(BA;C). Proof. Let Lbe a line generic to A. Since Uis compact, we may assume that Lis 70 Chapter 5 Cycles & characters The characteristic varieties are classical invariants of the fundamental group of a topological space. They first appeared in a more general context than line arrangements in [54, 32, 44], and have been recently studied in [1, 39, 45]. Some important questions are still opened about this subject. Question. Are the characteristic varieties of a line arrangement of a combinatorial nature ? A. Libgober [44] gives a method to compute almost all irreducible components of characteristic varieties; this method has a high computational complexity. E. Artal [5] found a method to compute the remaining components (which are only isolated torsion points). He obtains a description of the quasi-projective terms of the depth. We show in this work that his method, useful for the computation of characteristic varieties, is interesting in its own and provides topological invariants. In Section 5.1, after defining the twisted cohomology, we give definitions about characteristic varieties and some structural properties. In Section 5.2, we detail the method developed by E. Artal in [5] to compute the quasi-projective depth of a character, in the aim of using it in Chapter 6. This section is concluded by several results about the combinatoriality (or the non combinatoriality) of the quasi-projective depth. In Section 5.3, we introduce the notion of inner-cyclic characters, and illustrate their importance. Then we show that the image of particular cycles of the complement of A by an associated character is a topological invariant of the pair (X, ˆ A). CHAPTER 5. CYCLES & CHARACTERS 5.1 Characteristic varieties 5.1.1 Definitions As mentioned at the beginning of Chapter 4, the fundamental group of an arrangement is a very important invariant, but it is very difficult in general to get properties directly from a presentation of this group. Several weaker, but effective, invariants can be used. We focus our attention on the characteristic varieties. By the work of A. Libgober [43] the CW-complex K(A)construct on the presentation given in [9], [21] or [34] have the same fundamental group as EA(we may use any other CW-complex with the same fundamental group). Choosing the dual basis Bfor the cells of K(A)of the associated cochain complex: C∗(A):0−→ C0(A)A1 −→ C1(A)A2 −→ C2(A)−→ 0, the matrices (with integer coefficients) of A1and A2completely determine C∗(A). Remark 5.1.1. The rank of H1(EA;Z)is n. The abelianisation map πab :π1(EA)→H1(EA;Z)induces an infinite abelian cover ˇ E(A)→EA; the manifold ˇ E(A)is analytic but in general it will not be algebraic. Lifting the cells of K(A), we construct an infinite CW-complex ˇ K(A)with the same homotopy type than ˇ E(A). The group H1(EA;Z)acts on these cells and the cochain complex of ˇ K(A)acquires a structure of ΛC-module for which a basis is determined by an arbitrary choice of liftings for any cell in B, where ΛC=Chα±1 1,··· , α±1 ni. ˇ C∗(A) : 0 −→ ˇ C0(A)ˇ A1 −→ ˇ C1(A)ˇ A2 −→ ˇ C2(A)−→ 0. The ΛC-matrices ˇ A1and ˇ A2determine the cochain complex ˇ C∗(A). Remark 5.1.2. -dimCC∗(A) = rkΛCˇ C∗(A), with respect of the graduation. - The evaluation of ˇ A1(resp. ˇ A2), with xi= 1,∀i∈ {1,··· , n}, is A1(resp. A2). Definition 5.1.3. The character torus of Ais defined as: T(A)=H1(EA;C∗) = Hom(H1(EA;Z),C∗) = Hom(π1(EA),C∗). From a character ξ∈T(A)we define a twisted complex as follows. The character ξinduces an evaluation map evξ: ΛC→C, defining a structure of ΛC-module on C, denoted by Cξ. Consider the cochain complex: ˇ C∗(A)⊗ΛCCξ. 78 5.1. CHARACTERISTIC VARIETIES Definition 5.1.4. The twisted cohomology of EAis the cohomology of the previous cochain complex. Remark 5.1.5. The twisted cohomology is the cohomology of EAon the local system of coefficient Lξdefined by ξ. Definition 5.1.6. The characteristic varieties of Aare: Vk(A) = nξ∈T(A)|dimCH1(EA;Lξ)≥ko. Definition 5.1.7. The depth of a character ξ∈T(A)is: depth(ξ) = max {k∈N|ξ∈ Vk}= dimCH1(EA;Cξ). 5.1.2 Properties The very construction of the characteristic varieties implies that they are algebraic sub-varieties of T(A)defined over Q. Theorem 5.1.8 ([1, 8]).Let Abe an arrangement. The irreducible components of Vk(A) are sub-tori translated by torsion elements. This Theorem implies that we can reduce the study of characteristic varieties to the study of torsion character. Remark 5.1.9. This Theorem holds for quasi-projective varieties. For more references on Theorem 5.1.8, see [14, 29, 47]. Corollary 5.1.10. The varieties Vk(A)are determined by their torsion points. Proof. A subtorus is the closure of its torsion points. And it is also true for torsiontranslated tori. Moreover, the work of D. Cohen and A. Suciu [23] or A. Libgober and S. Yuzvinski [49] implies that all the components of the characteristic torus containing 1have a combinatorial description. Using Corollary 5.1.10, we focus our study only on the torsion characters. Since the meridians generate H1(EA), let tibe the image of the meridian associated with Liby ξ. Definition 5.1.11. The subset of T(A)composed of the torsion character is denoted by Ttors(A). Let ξ∈Ttors(A)be a torsion character of order N, and let ρ:Eξ A→EAbe the unramified N-fold cyclic cover associated to ρ. It is explicitly described in Subsection 5.2.1. We put ˇ ξa generator of the group of deck automorphisms of Eξ A, and ˇ ξ∗the induced map on the cohomology groups. We denote: H•,ξ = kerˇ ξ∗−exp(2iπ/N).IdH•(Eξ A;C). Proposition 5.1.12 ([62]).There is an isomorphism: H1,ξ ≃H1(EA;Lξ). 79 CHAPTER 5. CYCLES & CHARACTERS 5.2 Depth & quasi-projective depth Consider a character ξon the first homology group of the complement of A: ξ∈H1(EA;C∗) = Hom(H1(EA,Z);C∗). It is completely determined by the images ti∈C∗of the meridians associated to Li, where t0is t−1 n. . . t−1 1. Definition 5.2.1. A torsion character ξis said to be fully-ramified, if all the tiare different of 1. Let A0 ξ={Li∈ A | ti= 1}. If ξis fully-ramified then A0 ξ=∅. 5.2.1 Covers If ξis a torsion character of order Nthen the application ρ:π1(EA)→µN⊂C∗ induced by ξis onto. We construct a smooth model of the branched cyclic cover of X –the blow-up of CP2on the more than double points of A, see Section 1.2– induced by ξ. Let us recall that EA≃X\ˆ A. Definition 5.2.2. Let v= (v1,··· , vk)∈Nk, with the vicoprime, the weighted projective complex space Pk−1 vis defined by the natural structure of normal variety defined on Ck\{0}/∼, where ∼is the following equivalence relation: x∼y⇔[x1:··· :xk]=[tv1x1:··· :tvkxk]. The equivalence class will be denoted by [x1:. . . :xk]v(quasi-homogeneous coordinates). We have ξ(xi) = tiand let us denote kjthe element of {0,1,··· , N −1}such that tj= exp(2iπkj/N). Since Qxi= 1 then there exists ksuch that Pkj=kN. Let v= (1,1,1, k), we consider the hypersurface Pξ⊂P3 vdefined by a quasi-homogeneous polynomial: Pξ=  [x:y:z:T]|TN− n Y j=0 Fkj j(x, y, z)=0  . Remark 5.2.3. The space P3 vhas a unique singular point [0:0:0:1]and it is not in Pξ. Hence, Pξis contained in the smooth part of P3 v. Proposition 5.2.4. The space Pξis a ramified cyclic cover of CP2associated with ξ; it is unramified outside A. Remark 5.2.5. We consider P3 vinstead of CP3to obtain a ramified cover. 80 5.2. DEPTH & QUASI-PROJECTIVE DEPTH We have the stereographic projection of center [0:0:0:1]of P3 vin CP2(i.e. [x:y:z:T]→[x:y:z]), which restricts as a morphism of Pξ: P3 v π  Pξ ||zzzzzzzz ?_ oo CP2 We define Aξas the inverse image of Aby the restriction of the projection πto Pξ (i.e. Aξ=π−1(A)∩Pξ). By construction, Aξ≃ A is a reduced divisor of Pξ. Proposition 5.2.6. The space Pξ\Aξis a model for Eξ A. Consider the fibered product Xξ fb of Xand Pξover CP2: Xξ fb // πfb  Pξ π  Xσ//CP2 We define the inverse image Aξ fb of ˆ Aby πfb. We have Aξ fb ≃ˆ A, and Xξ fb \Aξ fb is smooth and still isomorphic to Eξ A. However, the projective variety Xξ fb has singularities only along Aξ fb. The points of ˆ Aare of two types: a) The point P∈ˆ Ahas {u= 0} ⊂ C2as local equation (i.e. Pis a smooth point of ˆ A). b) The point P∈ˆ Ahas {uv = 0} ⊂ C2as local equation (i.e. Pis a singular point of ˆ A). We denote by Qthe inverse image of Pin Xξ fb by the application πfb. Let us give the equations of Xξ fb,Aξ fb and Qin both cases a) and b). Case a) •Xξ fb :nTN=ulo⊂C3, with: l=kjif Pis on the component Djcoming from a line of A. l=Pkjif Pis on the exceptional component of ˆ Acoming from the point TLj. • Aξ fb :nTN= 0, u = 0o. •Q: (0,0). 81 CHAPTER 5. CYCLES & CHARACTERS Case b) •Xξ fb :nTN=ul1vl2o⊂C3, with similar notations as in case a). • Aξ fb :nTN= 0, uv = 0o. •Q: (0,0). We normalize Xξ fb to construct Xξ. For each singular point, its preimage is as many points as local branches. The normalization splits the dbranches and resolves each one by parametrizations. The normalization has a second step: adding holomorphic functions. In case a), the germs become smooth, in case b), we need to quotient by a cyclic group to obtain smooth germs. The space Xξthen obtained is a normal manifold with quotient singularities over the double points of ˆ A. Let us construct Xξ: Case a) We denote d= gcd(N, l),N0=N/d and l0=l/d. The equation of Xξ fb previously given becomes: d−1 Q i=0 (TN0−ζiul0), where ζis a d-primitive root of unity. The normalization is defined over the dpreimages of P(with d=Nif ξunramified over P). Its parametrization is defined by the following dlocal maps: (t, s)7→ (c1tN0, s, c2tl0), where ciare appropriate constants. Case b) We denote d= gcd(N, l1, l2),N0=N/d,l0 1=l1/d and l0 2=l2/d. The equation of Xξ fb previously given becomes: d−1 Q i=0 (TN1−ζiul0 1vl0 2), where ζis a d-primitive root of unity. The parametrization of the normalization is: (t, s)7→ (c1tN0, sN0, c2tl0 1sl0 2), where ciare appropriate constants. The map defines a parametrization as far as the source is seen as the quotient of a neighborhood of 0in C2by the action of a suitable cyclic group (of order the quotient of N0by the gcd’s with l0 i). This gives dnon-smooth ramified points over P(unramified and then smooth if d=N). It is easily seen that the map is unramified if and only if d=N; in that case, the source is smooth. Finally, in order to obtain Xξ, we resolve the singular points of Xξto obtain smooth points over the double points of ˆ A. The inverse image of a double point of ˆ Ais given by dfinite linear sequences of at most N00 =N0/(gcd(N0, l0 1)gcd(N0, l0 2)) copies of CP1(if N00 = 1 then the point is already smooth and no resolution is needed). Then we have: Eξ A ρ N:1  iN//Xξ ρ N:1  EAi//X 82 5.2. DEPTH & QUASI-PROJECTIVE DEPTH 5.2.2 Quasi-projective depth The map iNinduces an application in cohomology: i∗ N: H1(Xξ;C)−→ H1(Eξ A;C).(5.1) Notation. Let ζbe a primitive root of unity of order N, we denote with an exponent ˇ ξ∗the eigenspace associated with the eigenvalue ζ. The map i∗ Nrestricts to the eigenspace associate to ˇ ξ∗, and we define: i∗ N,ξ : H1(Xξ;C)ˇ ξ∗−→ H1(Eξ A;C)ˇ ξ∗= H1,ξ Theorem 5.2.7. ([42]) If ξis fully ramified (i.e. A0 ξ=∅) then, i∗ N,ξ is an isomorphism. When ξis not fully ramified, the default of isomorphism of i∗ N,ξ (i.e. the dimension of coker(i∗ N,ξ)) and the dimension of H1(Xξ;C)ˇ ξ∗determine the depth of ξ. Indeed, by Proposition 5.1.12, the map i∗ N,ξ can be considered with values in H1(EA,Lξ). Let KAbe the cokernel of i∗ N, and Kˇ ξ∗ Athe restriction to the eigenspace of ˇ ξ∗associated with the eigenvalue ζ, then we have: 0−→ H1(Xξ;C)ˇ ξ∗i∗ N,ξ −−−−→ H1(Eξ A;C)ˇ ξ∗−→ Kˇ ξ∗ A−→ 0, where the exactness at the left comes from the injectivity of iN. Definition 5.2.8. Let ξ∈Ttors(A), we define the quasi-projective depth of ξby: depth(ξ) = dim Kˇ ξ∗ A. Proposition 5.2.9. Let ξ∈Ttors(A), then we have: depth = depth + dim H1(Xξ;C)ˇ ξ∗. Definition 5.2.10. The dimension of H1(Xξ;C)ˇ ξ∗is the projective depth of ξ. Corollary 5.2.11. If ξis fully-ramified then: depth(ξ)=0. Proof. By Theorem 5.2.7, dim H1(Xξ;C)ˇ ξ∗= H1(Eξ A;C)ˇ ξ∗, then Proposition 5.2.9 implies that depth(ξ)=0. Consider ˆ Aξ=Xξ\Eξ A=ρ−1(ˆ A), and remark that ˆ Aξis a normal crossing divisor (n.c.d.). An irreducible component Dof ˆ Aξcan be of two types: (1)    ρ(D) = ndouble point of ˆ Ao, then Dcomes from a double point of A, ρ(D) = Li, then Dcomes from a line of A, 83 CHAPTER 5. CYCLES & CHARACTERS (2) ρ(D) = EP, then Dcomes from an exceptional component of ˆ A. Definition 5.2.12. A component L∈ˆ Aξis of single type if ρ(L) = {P}. If ρ(L)is a component H∈ˆ Athen Lis a H-component. A divisor D∈ˆ Aξinduces a 2-cycle in homology. By the previous description of the divisors of ˆ Aξ, they are sent to ρon a singular point of ˆ Aor to an irreducible component of ˆ A. Then we have the map: M D∈ˆ Aξ ChDiΦ −−→ H2(Xξ,C). Using Poincaré duality, it is equivalent to: M D∈ˆ Aξ ChDi −→ H2(Xξ,C). Proposition 5.2.13. The kernel of Φis KA. Proof. A classical result of Pure Hodge Theory (see for example [37]) implies that: H1(Xξ;C)≃H1(Xξ;OXξ)⊕H0(Xξ;Ω1 Xξ).(5.2) In the other hand, the Deligne’s Mixed Hodge Theory for quasi-projective varieties (see [26]) implies that: H1(Eξ A;C)≃H1(Xξ;OXξ)⊕H0(Xξ;Ω1 Xξlog( ˆ Aξ)).(5.3) As the first term of the both decompositions are the same, then: KA= cokerH0(Xξ;Ω1 Xξ)→H0(Xξ;Ω1 Xξlog( ˆ Aξ)).(5.4) In [61], K. Saito shows that: 0→Ω1 Xξ→Ω1 Xξlog( ˆ Aξ)→M D∈ˆ Aξ i∗OD→0.(5.5) Since the functor H∗(Xξ;•)is covariant, then we have: 0→H0(Xξ;Ω1 Xξ)→H0(Xξ;Ω1 Xξlog( ˆ Aξ)) →M D∈ˆ Aξ i∗OD→H1(Xξ;Ω1 Xξ). From equation 5.4, we can deduce the following exact sequence: 0→KA→M D∈ˆ Aξ i∗OD→H1(Xξ;Ω1 Xξ). As each D∈ˆ Aξis an irreducible divisor then H0(D;OD)≃C, and as H1(Xξ; Ω1 Xξ)⊂ H2(Xξ;C), then: KA≃ker M D∈ˆ Aξ ChDi → H2(Xξ;C) . 84 5.2. DEPTH & QUASI-PROJECTIVE DEPTH Since the analytic structure is not needed, we have also the following possible proof. Proof. (A. Degtyarev) Let Ube a regular tubular neighborhood of ˆ A. We have the following diagram: H1(X)i∗ //H1(EA)//H2(X, EA)//H2(X) H2(U)//H2(X) , where the two vertical isomorphism come from Poincarà c -Lefschetz duality. Then, we have: cokerH1(X)→H1(EA)≃ker(H2(U)→H2(X)). Passing to the eigenspace and since ˆ A ∼ U, we obtain the result. Remark 5.2.14. The previous Proposition can be restricted to the eigenspace of ˇ ξ: Kˇ ξ∗ A≃ker   M D∈ˆ Aξ ChDi  ˇ ξ∗ →(H2(Xξ;C))ˇ ξ∗  , Proposition 5.2.15. Let ξ∈TA, then: depth(ξ) = dim ker   M D∈ˆ Aξ ChDi  ˇ ξ∗ →(H2(Xξ;C))ˇ ξ∗  , Proof. It is a direct consequence of Proposition 5.2.8 and the previous remark. 5.2.3 Twisted intersection form We compute the quasi-projective depth of a character using Proposition 5.2.15. Definition 5.2.16. A component H∈ˆ Ais unramified if ξ(xH)=1. It is inner unramified if ξtakes value 1 for all the meridians of its neighbors in ˆ ΓA. The set of the inner unramified components of ˆ Ais denoted by Uξ⊂ˆ A. Lemma 5.2.17. Let Uξ=ChUξi. For each H∈ Uξ, fix a lift D0of Hin ˆ Aξ. And we denote Dj= (ˇ ξ∗)j(D0), with j∈ {1,··· , N −1}, the H-components. Then, the map: Ψ :            Uξ−→ L D∈ˆ Aξ ChDi!ˇ ξ∗ H7−→ 1 √N N−1 P j=0 exp2iπj NDj, is an isomorphism. 85 CHAPTER 5. CYCLES & CHARACTERS Proof. As His in Uξ, then the cover Xξ→Xis not ramified over H. Furthermore, at the singular points of Hthe inner unramification implies that the cover is not ramified at the singular point of H. Since His unramified, then outside the singular the cover is also unramified on H. Then ρ−1(H)admits Nconnected components: D0,··· , DN−1. The action of ˇ ξ∗is transitive on ρ−1(H). The definition of Ψimplies that Ψ(H)is in the eigenspace:  M D∈ˆ Aξ ChDi  ˇ ξ∗ . Conversely the elements of Uξare the only ones such that Xξis not ramified over them. Consider the intersection form on H2(Xξ;Z), it induces a non-degenerate hermitian form in H2(Xξ;C), and it induces an hermitian form on L D∈ˆ Aξ ChDi. Proposition 5.2.18. (i) For the previous hermitian form, the decomposition of H2(Xξ;C)in eigenspaces for ˇ ξ∗is orthogonal. (ii) For the previous hermitian form, the decomposition of L D∈ˆ Aξ ChDiin eigenspaces for ˇ ξ∗is orthogonal. Definition 5.2.19. Using the isomorphism of Lemma 5.2.17, we defined an intersection form on Uξnoted ·ξ. The matrix of this form for an order in the basis Uξis denoted by Aξ. Theorem 5.2.20. The twisted intersection form determines depth(ξ): depth(ξ) = corank Aξ. Proof. The Hodge Index Theorem implies that the signature of the intersection form on H1,1(Xξ;C)is (1, b2(Xξ)−1), i.e. it can be diagonalized with only one 1. Since the part of H2(Xξ;C)coming from divisors is contained in H1,1(Xξ;C), then it is also true in H2(Xξ;C). Let Lbe a generic line in CP2, then it self intersection in CP2is: (L·L)CP2= 1. The strict transformation ˇ Lof Lin Xdoes not pass through the centers of the blow-ups. Then the transform self-intersection (ˇ L·ˇ L)Xis also 1. Fix a preimage ˇ Lξof ˇ Lin Xξ. By the Poincaré duality, it defines an element in the 1-eigenspace of H2(Xξ;C), and (ˇ Lξ·ˇ Lξ)Xξ=h > 0. By Proposition 5.2.18, the decomposition in eigenspaces of H2(Xξ;C)is orthogonal for the intersection form. Then its restriction to H2(Xξ;C)ˇ ξ∗is negative definite, and so non-degenerate. 86 Chapter 6 Prime combinatorics, computation & examples Now that we know how to compute the quasi-projective depth of a character, and that we have constructed a topological invariant, the time has come to apply it. But on which arrangements shall we test our computations? The first ones are naturally the classical arrangements, like Ceva-7, MacLane, Rybnikov, Fan, etc... And then? To find more interesting arrangements, we consider prime combinatorics. The non prime combinatorics provide arrangement with characteristic varieties computable from an arrangement with fewer lines. This definition greatly reduces the combinatorics to test, and they provide a lot of interesting examples. In Section 6.1, we first discuss about the two methods presented in Chapter 5 and consider conditions on a character to have a positive depth, see Proposition 6.1.1. In the second part of this section, we introduce the notion of prime combinatorics in order to reduce the number of computations. The last subsection concerns other properties. In Section 6.2, we present a Sage program to select inner-cyclic or prime realizable combinatorics. A Maple program is also described, that computes realizations. In Section 6.3, we give a selection of examples coming from our computations: - three prime arrangements with nine lines, - an ordered, oriented nc-Zariski pair with nine lines, - a pair of complexified real prime arrangement conjugate in C[√5] with ten lines and admitting three inner-cyclic cycles, - a prime combinatorics admitting four realizations with eleven lines. In the last section, we present the detailed computation of the prime combinatorics with eleven lines given in the previous section. From this example we construct explicit CHAPTER 6. PRIME COMBINATORICS, COMPUTATION & EXAMPLES examples of nc-Zariski pairs, and a nc-Zariski 4-tuplet. 6.1 Prime combinatorics 6.1.1 Depth & characters The characteristics varieties are algebraic invariants. We do not know an algebraic method to compute this quasi-projective depth. But the algorithm developed in Subsection 5.2.2, is topological. The ultimate result is to discover a Zariski pair of line arrangements of a different nature than the previous [60, 7]. It means to detect them with a new invariant. To do this, we use the methods developed in Chapter 5. The goal of the first approach is to distinguish two arrangements with the same combinatorics by finding a character with distinct quasi-projective depths. To detect a pair with such a method would allow to solve the problem of the combinatoriality of the characteristics varieties. Unfortunately, the following proposition is often sufficient to show that this method does not give a positive answer. Proposition 6.1.1. Let Abe an arrangement, and ξbe a torsion character of A. Assume that ˆ Γξadmit only three vertices L1, L2, L3with a self intersection Li·Li≤ −2. If we denote γthe cycle of ˆ Γξthen: depth(ξ)>0⇔Li·Li=−2and ξ(γ)=1. Proof. In such a situation, the intersection matrix is: Aξ=   L1·L11 1 1L2·L2ξ(γ) 1ξ(γ)−1L3·L3   . Then, we have: det(Aξ) = ξ(γ) + ξ(γ)−1+L2·L2+L3+L3−L1·L1[(L2·L2)(L3·L3)−1]. A simple computation shows that det(Aξ) = 0 if and only if we have Li·Li=−2and ξ(γ)=1. Nevertheless, with the second approach, we differentiate two combinatorially equivalent arrangements by different images for a same nearby cycle. Such a pair gives an example of oriented ordered nc-Zariski pair, and shows that the maps induced by the inclusion on the fundamental groups (or the homology) are not of combinatorial nature. In Section 6.3, some positive cases are obtained with this method. 94 6.1. PRIME COMBINATORICS 6.1.2 Prime combinatorics Let us remark the following fact: consider an inner-cyclic combinatorics C0with realization A0. If we add a line Lto A0(and then obtain a new arrangement A), then Ais also inner-cyclic (keep the same character on the common lines of Aand A0, and send the meridian associated with Lon 1). Furthermore, the inner-cyclic cycles of A0 correspond to some inner-cyclic cycles of A. Definition 6.1.2. Let Cbe an inner-cyclic combinatorics, and ξan inner-cyclic character of a realization of C. The pair (C, ξ)is said prime if all the unramified lines of C are inner unramified (i.e. Uξ=A0 ξ). Remark 6.1.3. By an abuse of language, a combinatorics Cis said to be prime if it does admit a character ξsuch that (C, ξ)is prime. If (C, ξ)(with realization A) is not prime then there is a pair (C0, ξ0)prime (with realization A0) and a set of lines {L1,··· , Lk}such that A=A0SLi,Liis unramified for ξbut not inner unramified. Then the inner-cyclic cycles of Care in correspondence with the inner-cyclic cycles of C0, and their image by ξare completely determined by A0. Proposition 6.1.4. Let Cbe a combinatorics, and Aa realization of C. If Cis not prime, then the characteristics varieties of Aare completely determined by realizations of prime combinatorics with less lines than A. Proof. Let ξa character of A. Consider the construction doing before (i.e. A=A0SLi). Here, remark that A0depends of the character ξ. For each Lithree cases are possible: 1. Liis generic with the lines of the support of the inner-cyclic cycle. Then Aξ=Aξ0, so depth(ξ) = depth(ξ0). 2. Adding Licreated a singular point with multiplicity greater than 2 on a line of the support. Then, the support of the inner-cyclic cycle admits an additional line (i.e. the exceptional line coming from the new point). Then usual operations on the lines and the rows of Aξshow that: Aξ= Id 0 0Aξ0!. Then depth(ξ) = depth(ξ0). 3. The line passes through the intersection of two lines of the support, then once again we add a line to the support of the inner-cyclic cycle. By the same argument as in the previous case we have depth(ξ) = depth(ξ0). Since the ramified lines do not appear in the computation of the projective depth, then adding or deleting such lines does not change the projective depth. This proves the result. Remark 6.1.5. It is enough to study the prime character of a combinatorics in order to compute its characteristic varieties. 95 CHAPTER 6. PRIME COMBINATORICS, COMPUTATION & EXAMPLES 6.1.3 Combinatorics & realizations To simplify the computation, we are interested by particular combinatorics. The best case is the combinatorics such that AutCis trivial. Indeed, if it is not, there exists a non trivial σ∈AutC; and then we can find to different realizations A1and A2such that σ(A1) = A2. This implies that EA1=EA2. For the definition of ΣCand M(C), see Subsection 1.1.2. Definition 6.1.6. An automorphism of the incidence graph Γis an automorphism of the bipartite graph Γrespecting the two sets of vertices. The set of such automorphisms is denoted by AutΓ. Property 6.1.7. Let AutΓbe the automorphism group of the incidence graph Γof a combinatorics C, then AutCand AutΓare isomorphic. Proposition 6.1.8. Let A1and A2be two realizations of the same combinatorics C. If they are in the same connected component of ΣC, then they have the same topological type. Proof. It is a direct consequence of Property 1.1.9 and [59]. Proposition 6.1.9. Let A1and A2be two realizations of C. Assume that they are in distinct connected components of ΣC. If there exists an element σ∈AutCinducing a linear application of CP2sending A1on A2, then all the realizations of the two connected components have the same topological type. Proof. If σinduces a linear application φof CP2sending A1on A2, then φis a homeomorphism. By Definition 1.1.5, A1and A2have the same topological type. Remark 6.1.10. If there exists an element of AutCrealizable by a linear application of CP2then A1and A2are in the same connected component of M(C). Proposition 6.1.11. Let A1and A2be two realizations of C. Assume that they are in the same connected component of M(C). Then all the realizations of the connected component have the same topological type. Let us define the rigidity of a combinatorics: Definition 6.1.12. Let Abe an arrangement and Cits combinatorics. The rigidity of Ais the dimension of a connected component of ΣCcontaining A. An arrangement is rigid if its rigidity is zero. We focus on combinatorics with a small (by the cardinality) automorphism group of the combinatorics. Indeed if it is trivial then the notion of ordered topology is equivalent to the topology. If it is not, then we search for combinatorics where M(C)is not connected, so that there is no evident obstacle to have different topological types. 96 6.2. COMPUTATION & OUTPUT 6.2 Computation & output 6.2.1 Sage program a) Inner-cyclic & prime combinatorics program In Annexe A.2, we give a program to determine if a combinatorics is inner-cyclic. The entry data of this program is a combinatorics, and the algorithm can be describe as follows: •(function our_cycle_basis) In the data entries of this function, we choose a line at infinity. And we compute the basis of cycle Edefined in Sub-section 3.1.3. •(functions cycle_matrix_color,corner_intersection) We compute here the matrix of conditions associated to the combinatorics, and a cycle Eof E. •(function is_a_cycle) This short function distinguishes the true cycle and the pencil of lines. •(function inner-cyclic_cycle) Using the color matrix defined with the function cycle_matrix_color, we check if the cycle Eis an inner-cyclic cycle for some characters of order a prime number less than 50. •(function is_a_prime) We check if a character is inner-cyclic (only in the case of a single cycle). •(function inner-cyclic_combinatorics) For a combinatorics C, we test every cycle, for all infinity lines, to know if it is an inner-cyclic cycle. If not, then the function returns false. Else it returns the set of inner-cyclic cycles with the corresponding character and its order, as additional information. •(function check) This function was written with M.A. Marco-Buzunáriz: we use the Gröbner basis to know if a combinatorics is realizable, see [67] for details about the Gröbner basis. To optimize the computation, we suppose that the combinatorics admit four lines in generic position. •(function generic_lines) As previously explained, we need to find four lines intersecting in six different points. This function finds (if it exists) such a 4-tuplet of lines in C. •(function is_actual) This function uses the two previously described functions to know if the combinatorics Cadmits a realization. We use the Sage function fork to stop the computation if it is too long. •(functions industrial_test and display_combinatorics) These functions permit to check a lot of combinatorics, and display the result. The parameter spermits to initialize the counter at a precised value. It was useful in the case of combinatorics with nine and ten lines. 97 CHAPTER 6. PRIME COMBINATORICS, COMPUTATION & EXAMPLES Remark 6.2.1. Since the program can forget some combinatorics (due to the complexity of computation of the realizability), we need to study this combinatorics with another method than the Gröbner bases. To do that, we have written a Maple program (see Appendix A.3) to obtain directly a set –as smaller as possible– containing ΣCfrom a given combinatorics. b) Realization program (computation of ΣC) The program, given in Appendix A.3, returns the possible equations to realize a combinatorics C. It is developed on Maple to make formal computation on C. It is composed as follows: •(function equations) Returns the set of polynomial equations defining the set ΣC. •(function fixed_lines) Fixes the equations of five lines to rigidify the arrangement. •(function rigidification) Permits to add the rigidification previously defined to the set of polynomials obtained by the function equations. •(function lines_number) Returns the index of five lines in a rigid position. •(function reduc) Deletes the result with double lines on the rigid lines previously defined. •(function realization) Regroups the previous functions. c) Possible ameliorations In the first program, the function is_a_prime, only finds the case of inner-cyclic character with a single cycle. Thus it is possible to apply Proposition 6.1.1. But with it, we omit the case with several cycles. Then we need to develop a function to detect such combinatorics and apply it to the case of combinatorics with eleven lines. Remark 6.2.2. For the case of combinatorics with seven, eight, nine and ten lines, we have computed all the primes combinatorics because they were extracted from lists containing all the inner-cyclic combinatorics. A possible amelioration of the Maple program is to delete also the results with double lines. With such a improvement this program will return exactly the set ΣC 6.2.2 Output As our program was constructed to make a lot of examples, we have used the lists of all possible combinatorics with less than eleven lines given by M.A. Marco-Buzunàriz: http://riemann.unizar.es/combinatorias/ The results of the computations are present at: 98 6.3. EXAMPLES https://www.benoit-guervilleballe.com/publications/ By this computation on all the possible combinatorics, we have shown that the extended Ceva arrangement is the smallest arrangement (i.e. with a minimal number of lines) admitting a non trivial inner-cyclic character, and then its combinatorics is the first prime one. With all the programs present in this thesis, we can know if a combinatorics is innercyclic, and compute the quasi-projective depth of its inner-cyclic characters. Indeed, with the previous Sage program, we can determine if a combinatorics is inner-cyclic; with the Maple program, we obtain the equations of the realizations; and the Sage program on the wiring diagram permits to compute the image by the inner-cyclic character of the inner-cyclic cycles. Thus we have computed the quasi-projective depth if the inner-cyclic character. 6.3 Examples 6.3.1 First tests Before making a complete analysis, we have studied some particular examples. First, we have tested MacLane’s arrangements. Indeed they are the smallest (in terms of number of lines) without real realizations. And we have also tested Fan’s arrangements. But they do not admit an inner-cyclic character. In a second time, we have tested the two arrangements obtained by G. Rybnikov in [60]. Indeed, it is a Zariski pair, therefore we would like to test them in the prospect of showing that our method is able to distinguish a Zariski pair. But once again they do not admit some inner-cyclic character. The first non completely negative test comes from the pair of arrangements obtained in [7]. Indeed they admit an inner-cyclic character, but the images by the character of the inner-cyclic cycle are exactly the same. Then we have used the lists obtained by the Sage program. We have tested some examples admitting a non-connected realization space. But in all the cases the quasiprojective depth of the inner-cyclic character was similar. After all these tests, we have tried to construct from the known Zariski pairs an innercyclic combinatorics (without success). It is here that the notion of prime combinatorics has appeared. 6.3.2 Prime combinatorics a) Ceva-7 The extended arrangement of Ceva (pictured in Figure 4.1), is the first known case of arrangement with an inner-cyclic character. As previously said in Subsection 6.2.2, it is the smallest (in terms of number of lines) admitting an inner-cyclic character ξ, and then also the first prime pair (C, ξ). Remark also that it is the first example of arrangement 99 CHAPTER 6. PRIME COMBINATORICS, COMPUTATION & EXAMPLES with essential component (see [23]). Furthermore, the computation of its inner-cyclic cycle γby ξ, shows that ξ(γ) = −1then: Aξ=  −1 1 1 1−1−1 1−1−1  , this implies that depth(ξ) = 2. This is in harmony with the result of D. Cohen and A. Suciu [23]. But the combinatorics of Ceva-7 admits a single realization, so it cannot give any information about the combinatoriality of i∗. Property 6.3.1. All the inner-cyclic combinatorics with eight lines come from Ceva-7 with an additional unramified line. And no one is prime. b) Nine lines There are exactly four prime combinatorics with nine lines. The first three are: C1= [[1,2],[1,3,4],[1,5,6],[1,7,8],[1,9],[2,3,5],[2,4,7],[2,6,8], [2,9],[3,6],[3,7],[3,8,9],[4,5],[4,6,9],[4,8],[5,7,9],[5,8],[6,7]] C2= [[1,2],[1,3,4],[1,5,6],[1,7,8],[1,9],[2,3,5],[2,4,7],[2,6,8], [2,9],[3,6,9],[3,7],[3,8],[4,5],[4,6],[4,8,9],[5,7,9],[5,8],[6,7]] C3= [[1,2,3],[1,4,5],[1,6,7],[1,8,9],[2,4,6],[2,5,8],[2,7,9], [3,4,9],[3,5,7],[3,6,8],[4,7],[4,8],[5,6],[5,9],[6,9],[7,8]] They all admit a single realization pictured in Figure 6.1. Furthermore, the fourth is: C4= [[1,2,3,7],[1,4,8],[1,5,9],[1,6],[2,4],[2,5,8],[2,6,9], [3,4,9],[3,5],[3,6,8],[4,5,6,7],[7,8],[7,9],[8,9]] and it admits two different realizations conjugated in C. The equations of the realizations are: L1:z= 0 L2:x= 0 L3:x−z= 0 L4:y= 0 L5:αx −y+z= 0 L6:αx +α2y+z= 0 L7:x+α2= 0 L8:y−z= 0 L9:x−α2y−z= 0 with α=−1±i√3 2. The character ξsuch that (C4, ξ)is prime is (ζ, ζ, ζ, ζ2, ζ2, ζ2,1,1,1) with ζa primitive 3-root of unity. Its inner-cyclic cycle is then supported by the lines L7, L8, L9. The graph ˆ ΓUξis pictured in Figure 6.2. 100 6.3. EXAMPLES L1=∞ L2 L3L4 L5 L6 L7 L8 L9 A1 L1=∞L2 L3L4 L5 L6 L7L8 L9 A2 L1=∞ L2L3 L4 L5 L6 L7 L8 L9 A3 Figure 6.1: Real picture of the first three prime arrangements with nine lines L7 L8L9 Figure 6.2: The graph ˆ ΓUξfor the fourth prime combinatorics with nine lines: C4 The automorphism group AutΓof the incidence graph is of order 12, and it is gen101 CHAPTER 6. PRIME COMBINATORICS, COMPUTATION & EXAMPLES L4 L8 L6 L9 L5 L2L3L7 Figure 6.3: Braided wiring diagram of the positive realization A+of C4 erated by: {(2,3)(4,5)(8,9),(1,2,3)(4,5,6),(1,5)(2,4)(3,6) }. The braided wiring diagram of A+is pictured in Figure 6.3. We obtain the image of the inner-cyclic cycle γ±in the two cases (using Proposition 4.1.7or Proposition 5.3.16 for γ−): ξ(γ+) = ξ(−α5−α6+α6) = ζ, ξ(γ−) = ξ(−α5−α6) = ζ2=ζ. It is the first case with different values by the character of a same cycle in two different realizations. But they are conjugates, then they do not permit to distinguish the two realizations. Theorem 6.3.2. There is no ordered and oriented homeomorphism from (CP2,A+)to (CP2,A−). In other terms (A+,A−)is an ordered and oriented nc-Zariski pair. Proof. By Theorem 5.3.14, ξ(γ)is an invariant of the oriented and ordered topological type of an arrangement. Then the previous computation shows the result. Remark 6.3.3. We may apply the construction of G. Rybnikov [60] to this example to this pair of arrangements as future work. c) Ten lines With ten lines, there is no prime combinatorics admitting only three lines in Uξ, with ξinner-cyclic. But, it is an example of two arrangements admitting four lines inner unramified and conjugate in Q[√5]. Their combinatorics is: C5= [[1,2,3],[1,4,7],[1,6,9],[2,4,8],[2,5,7],[2,9,10],[3,5,9],[3,6,8],[3,4,10],[4,5,6], [7,8,9],[6,7,10],[1,5],[1,8],[1,10],[2,6],[3,7],[4,9],[5,8],[5,10],[8,10]]. 102 6.4. NC-ZARISKI PAIRS Proof. Since the line L12 is sent on 1by ξthen it does not change the values of the inner-cyclic cycle. By Theorem 6.4.8, A+and B+form an ordered nc-Zariski pair. And Proposition 6.4.9 implies that there is no homeomorphism of X+ Ain X+ Bsending A+on B+and preserving the ordered. Proposition 6.4.11. By analogue construction with A−and B−to obtain A−and B−, the four pairs (A±,B±)are nc-Zariski pairs. Lemma 6.4.12. The pairs (A+,A−)and (B+,B−)are oriented nc-Zariski pairs. Proof. By Theorem 5.3.14, there is no oriented and ordered homeomorphism between the pairs (XA+,A+)and (XA−,A−). But by construction there is no automorphism of the combinatorics C. Then there is no oriented homeomorphism from (XA+,A+)to (XA−,A−). Theorem 6.4.13. The arrangements A+,A−,B+and B−form an oriented ncZariski 4-tuple. Proof. It is a consequence of Proposition 6.4.11 and Lemma 6.4.12. 109 CHAPTER 6. PRIME COMBINATORICS, COMPUTATION & EXAMPLES 110 Conclusion and future works And after... During all this thesis, we have studied how the topology and the combinatorics of complex line arrangements are related. Even if it seems out of reach to describe explicitly the relationship between this two facets of line arrangements, we have obtained some results in this direction. The description of the map induced by the inclusion of the boundary manifold in the complement of an arrangement on the Poincaré’s groups and on the first homology groups has been a first important step in the comprehension of this link. The study of the characteristic varieties, with the method due to E. Artal, was the second important step. Indeed it allows to develop a geometrical method to compute the quasi-projective part of the depth of any character. It also puts us on the right track to define the new invariant and permits us to discover nc-Zariski pairs. A possible future work following this thesis, will be to carry on with the study of this relationship, and to focus on the non quasi-projective part of the depth. Another way, is to extend a part of the results obtained here to the case of algebraic plane curves, some reflexion has already started. As mentioned in Chapter 2 and 6, some ameliorations are possible on the programs developed during this thesis. Optimizing the computation of the braided wiring diagram, developing a function to picture it, another to extract from it the braid monodromy, developing a better algorithm to detect all the prime combinatorics, completing the Maple program to compute the space realization of a combinatorics, this is as much work to do after this thesis. Directions of the future works The difference between a nc-Zariski pair and a usual pair of Zariski is very thin, then we may think that the arrangements A+and B+form a Zariski pair. CONCLUSION AND FUTURE WORKS Some of the results presented in the last chapter give a hope to find an example of a pair of arrangements distinguished by different characteristic varieties. The present work and all the tested arrangements tend towards the fact that if Ais a complexified real arrangement then the quasi-projective part of its characteristic varieties are of combinatorial nature. Indeed, in all the examples of pairs of complexified real arrangements the image of the inner-cyclic cycle is always the same for each arrangement of the pair. In contrast to the complex line arrangements where we have found conjugate images (see the example with nine lines), or completely different (see example with eleven lines). Furthermore, each time we have constructed a pair of complexified real arrangements, it always appeared that to be inner-cyclic, the image of the cycle by the character in both cases needs to be the same. For the case of complex line arrangements, no rule emerges to know if the characteristic varieties are of combinatorial nature. The examples previously presented give prospect that they are not combinatorial. But Proposition 6.1.1, gives very binding conditions on the combinatorics of such an example. 112 Conclusion et travaux futurs Et après... Durant cette thèse, nous avons étudié comment la topologie et la combinatoire d’un arrangement complexe sont reliées. Même s’il semble hors de portée de décrire explicitement la relation entre ces deux facettes, nous avons obtenu des résultats dans cette direction. La description de l’application induite par inclusion de la variété bord d’un arrangement dans son complémentaire sur les groupes de Poincaré et sur les premiers groupes d’homologies fut une étape importante dans la compréhension de ce lien. L’étude des variétés caractéristiques, avec la méthode due à E. Artal, a été la seconde étape importante. En effet, elle nous a permis de développer une méthode géométrique pour calculer la profondeur quasi-projective d’un caractère. Cela nous a aussi mis sur la bonne voie pour définir le nouvel invariant topologique et nous a donc permis de découvrir une nc-paire de Zariski. Etudier cette relation, et s’intéresser à la partie projective de la profondeur pourraient être des travaux possibles pour la suite de cette thèse. Une autre extension possible serait de traduire nos résultats dans le cas encore plus général des courbes algébriques planes. Des premières réflexions ont déjà été faites à ce sujet. Comme nous l’avons mentionné dans les Chapitres 2 et 6, des améliorations sont possibles sur les programmes développés durant cette thèse. Optimiser le calcul du diagramme de câblage, développer une fonction pour les dessiner directement, une autre pour extraire la monodromie de tresse, développer un meilleur algorithme pour détecter les combinatoires premières, compléter le programme sous Maple pour calculer l’espace des réalisations d’une combinatoire, voilà autant de travaux possibles à la suite de cette thèse. CONCLUSION ET TRAVAUX FUTURS Orientations des futurs travaux La différence entre une nc-paire de Zariski et une paire de Zariski classique est très faible, il semblerait donc naturel de penser que A+et B+forment une paire de Zariski. Certains des résultats présentés dans le dernier chapitre donnent un espoir de trouver un exemple de paire d’arrangements distingués par des variétés caractéristiques différentes. Les travaux développés dans cette thèse, ainsi que les nombreux exemples que nous avons testés tendent à dire que si Aest un arrangement réel complexifié, alors la partie quasi-projective de ses variétés caractéristiques est déterminée par la combinatoire. En effet, dans tous les exemples d’arrangements réels complexifiés, l’image des cycles inner-cyclic est toujours la même pour chaque arrangement de la paire. Ceci contraste avec le cas d’arrangement complexe (non réel complexifié) où l’on trouve des images complexes conjuguées, cf exemple à neuf droites, ou même complètement différentes, cf exemple à onze droites. De plus, à chaque tentative de construction d’un exemple de paire d’arrangements réels complexifiés, pour obtenir une combinatoire inner-cyclic, l’image des méridians par le caractère devenait toujours la même. Pour le cas des arrangements complexes en général, aucune règle ne semble se dessiner sur la nature combinatoire des variétés caractéristiques (ni même sur leurs parties quasi-projectives). Les exemples donnés dans le dernier chapitre encouragent à croire en l’existence d’une paire distinguée par un caractère dont les profondeurs seraient différentes. En revanche, la Proposition 6.1.1 impose des conditions fortes sur la combinatoire d’un tel exemple. 114 Conclusiones y trabajo futuro Y después ... Durante la realización de esta tesis, hemos estudiado la forma en la que están relacionadas la topología y la combinatoria en las configuraciones de rectas complejas. Aunque parece completamente inalcanzable llegar a describir explícitamente la relación entre estas dos facetas de las configuraciones de rectas, hemos obtenido algunos resultados en esta dirección. La descripción de la aplicación inducida por la inclusión de la variedad borde en el complementario de una configuración sobre los grupos de Poincaré y los primeros grupos de homología ha sido un primer paso importante en la comprensión de esta relación. El estudio de variedades características con el método dado por E. Artal fue el segundo paso fundamental. De hecho, esto nos permitió desarrollar un método geométrico para calcular la parte cuasi-proyectiva de la profundidad de un carácter cualquiera y nos puso en el buen camino para definir el nuevo invariante así como para descubrir nuevos nc–pares de Zariski. Un posible trabajo futuro a esta tesis pasa por la profundización del estudio de esta relación así como en el estudio de la parte no cuasi-proyectiva de la profundidad de un carácter. Otro posible camino es el de extender los resultados aquí obtenidos al caso de las curvas algebraicas planas, caso sobre el cual ya hemos comenzado a reflexionar. Como se ha mencionado en los capítulos 2 y 6, existes múltiples mejoras a realizar en los programas de cálculo desarrollados durante esta tesis: optimizar el cálculo de los diagramas de cableado, programar una función para dibujarlo o para extraerlo directamente de la monodromía de trenzas, desarrollar un algoritmo más eficaz para detectar combinatorias primas o completar el programa en Maple para el cálculo del espacio de realizaciones de las combinatorias. CONCLUSIONES Y TRABAJO FUTURO Orientaciones para un trabajo futuro La diferencia entre un nc–par de Zariski y un par de Zariski usual es muy fina por lo que podemos pensar que las configuraciones A+yB+encontradas pueden formar muy posiblemente un par de Zariski. Algunos de los resultados presentados en el último capítulo de la tesis nos dan la esperanza de encontrar un ejemplo de un par de configuraciones distinguidas por diferentes variedades características. El presente trabajo y todas las configuraciones estudiadas nos inclinan a pensar el hecho de que si Aes una configuración real complexificada entonces la parte cuasi-proyectiva de su variedad característica tiene naturaleza combinatoria. De hecho, en todos los ejemplos de pares de configuraciones reales complexificadas la imagen del ciclo inner-cyclic es siempre el mismo para toda configuración del par, en contraste con las configuraciones de rectas complejas, donde hemos encontrado o bien imágenes conjugadas (véase el ejemplo de nueve rectas) o bien imágenes completamente diferentes (véase el ejemplo de nueve rectas). Es mas, cada vez que hemos construido un par de configuraciones reales complexificadas, siempre ha sucedido que para que la imagen de un ciclo por el carácter fuera inner-cyclic, en ambos casos debía de ser la misma. Para el caso de configuraciones de rectas complejas, no aparece ninguna regla que nos permita saber si las variedades características son de naturaleza combinatoria. Los ejemplos previamente presentados nos hacen pensar que no son combinatorias. Sin embargo, la Proposición 6.1.1 nos da una condiciones muy específicas sobre las combinatorias que deben poseer tales ejemplos. 116 Appendix A Code A.1 Wiring diagram This program is coded in Sage v.5.8. reset() # d e f i n i t i o n of the arrangement A from the equations . var ( ’x , y ’ ) var ( ’a ’ ) # Copy a ve ctor in another def copy (G) : g=[] for iin range (0 , len (G) ) : g . append (G[ i ] ) return g # s i n g u l a r i t i e s computation def sing (E) : S=[] for iin range (0 , len (E) −1): for jin range ( i +1, len (E) ) : s=solve ( [E[ i ] ,E[ j ] ] , x , y) i f s < >[]: i f not ( s [ 0 ] in S ) : S . append ( [ x==(s [ 0 ] [ 0 ] . rhs ()) , y==(s [ 0 ] [ 1 ] . rhs ( ) ) ] ) return S # projection on the axes APPENDIX A. CODE def Xprojection(S): X=[] for iin range (0 , len (S ) ) : X. append (S [ i ] [ 0 ] . rhs ()) return X # p r o j ec t i o n on r e a l and imaginary part def Rprojection(S): R=[] for sin S : R. append( real ( s )) return R def Iprojection(S): I =[] for sin S : I . append (imag( s )) return I # ordering of the projection along x def sort (R,L ) : S1=[] S2=[] r=copy (R) l=copy (L) for iin range (0 , len (R) ) : min=0 for jin range (0 , len ( r ) ) : i f r [ j ]<r [ min ] : min=j S1 . append ( r [ min ] ) S2 . append ( l [ min ] ) del r [ min ] del l [ min ] return S1 , S2 def order (L ) : e=0.000001 R=Rprojection(L) I=Iprojection (L) ( I ,R)=sort ( I ,R) #ordering the y (R, I)=sort (R, I ) #ordering the x m=[] n=[] for iin range (0 , len (R) −1): i f abs (R[ i ]−R[ i +1])>e or abs ( I [ i ]−I [ i +1])>e : m. append (R[ i ] ) n . append ( I [ i ] ) m. append(R[ len (R) −1]) 118 A.2. INNER-CYCLIC COMBINATORICS def inner−cyclic_combinatorics (C, prime=f a l s e ) : R=[] n=max( f latt en (C)) for infini in range (1 ,n−1): BE=our_cycle_basis ( in fi n i , n) for Ein BE: e=copy (E) e . append ( i n f i n i ) i f ( in fi ni <E[ 0 ] ) and ( inf in i <E[ 1 ] ) and ( is_a_cycle (C, e )) : IE=inner−cyclic_cycle (C,E, i n f i ni , n , prime ) i f not ( IE==f a l s e ) : e =[[ in f ini ,E[ 0 ] ,E [ 1 ] ] ] e . append(IE) R. append ( e ) i f len (R)>0: return R else : return f a l s e # Check fo r an arrangement with 4 " generic " l i n e s def check (comb , l1 , l2 , l3 , l4 ) : n=max( f latt en (comb)) R=PolynomialRing (QQ,3∗n , ’x ’ ) M=matrix (n ,3 ,R. gens ( )) r el a ti o ns =[R. gen (3∗l1 −3)−1,R. gen (3∗l1 −2),R. gen (3∗l1 −1), R. gen (3∗l2 −3) ,R. gen (3∗l2 −2)−1,R. gen (3∗l2 −1),R. gen (3∗l3 −3), R. gen (3∗l3 −2) ,R. gen (3∗l3 −1)−1,R. gen (3∗l4 −3)−1, R. gen (3∗l4 −2)−1,R. gen (3∗l4 −1)−1] M=M. apply_map(lambda a : a . reduce ( r elat ion s )) for iin comb : i f len ( i )>2: N=M. matrix_from_rows ([ j−1for jin i ] ) rel ati ons+=N. minors (3) I=R. ideal ( re lat ion s ) J=I . radical () for iin Set ( range (n ) ) . subsets (2) : N=M. matrix_from_rows ( l i s t ( i )) Min=N. minors (2) i f Min [ 0 ] in Jand Min [ 1 ] in Jand Min [ 2 ] in J : return False cond=R(1) for iin Set ( range (n ) ) . subsets (3) : comprobar=True for kin comb : i f i . i ss ubs et ( Set ( [ s−1for sin k ] ) ) : comprobar=False i f comprobar: N=M. matrix_from_rows ( l i s t ( i )) 125 APPENDIX A. CODE cond=(cond∗N. determinant ( ) ) . reduce (J) return not cond in J # Research of 4 " generic " ’ l i n e s def generic_lines(C): G=[] c =[] n=max( f latt en (C)) for iin range ( len (C) ) : c . append ( Set (C[ i ] ) ) G. append (C[ 0 ] [ 0 ] ) G. append (C[ 0 ] [ 1 ] ) G=Set (G) test=true i=1 while test : i f (G. in terse ction (c [ i ] ) ) . cardinality ()>0 : te st=f a l s e A=(c [ i ] . dif fere nc e (G) ) [ 0 ] G=G. union ( Set ( [A] ) ) else : i=i+1 i f i>len (C) : te st=f a l s e j=0 te st=f a l s e L=Set ( range (1 ,n+1)) L=L. dif fer enc e (G) while ( j <(L. cardinality ()) and not( test )) : k=true for iin range ( len (C) ) : i f (G. union ( Set ([L[ j ] ] ) ) . in terse cti on (c [ i ] ) ) . cardinality ()>2 : k=f a l s e i f k : test=true G=G. union ( Set ( [L[ j ] ] ) ) j=j+1 G=G. l i s t () i f len (G)<4: return f a l s e else : return G # Complete fu nctio n @fork ( timeout =500, verbose=True ) def is_actual(C): f i n i s h=f a l s e G=generic_lines (C) i f G==f a l s e : 126 A.2. INNER-CYCLIC COMBINATORICS R=’bad ’ else : R=check (C,G[ 0 ] ,G[ 1 ] ,G[ 2 ] ,G[ 3 ] ) f i n i s h=true return [R, f i n i s h ] def display_combinatorics (C, IC , i ) : print( str ( i)+"/␣␣␣␣␣␣ "+str (C)) for jin range ( len (IC ) ) : print( " ␣␣␣␣Cycle␣ : ␣"+str (IC [ j ] [ 0 ] ) ) for kin range ( len (IC [ j ] [ 1 ] ) ) : print("␣␣␣␣␣␣␣␣−␣Modulo : ␣"+str (IC [ j ] [ 1 ] [ k ] [ 0 ] ) + "␣␣␣␣Character : ␣"+str (IC [ j ] [ 1 ] [ k ] [ 1 ] [ 0 ] ) ) print( "␣␣ " ) def in du st ri al_t es t (L, s ) : R=[] c=s Bug=[] for iin range ( len (L ) ) : n=max( fl at te n (L[ i ] ) ) r=inner−cyclic_combinatorics(L[ i ] ,true) i f not ( r==f a l s e ) : realisable=false f i n i s h=f a l s e IA=is_actual (L[ i ] ) i f not (IA==’NO␣DATA␣ ( timed␣out ) ’ ) : realisable=IA[0] f i n i s h=IA [ 1 ] i f realisable and finish: R. append (L[ i ] ) display_combinatorics (L[ i ] , r , c ) c=c+1 i f not ( f i n i s h ) : Bug. append( i ) return [R, Bug] 127 APPENDIX A. CODE A.3 Space of realization ΣC This program is coded in Maple v.7. r es ta rt : with ( l i na l g ) : equations :=proc (C) > l oc al i , j , k ,m,EQ; > k:=0; >for ifrom 1 to nops (C) do >i f nops (C[ i ])>2 then >for jfrom 3 to nops (C[ i ] ) do > k:=k+1; > m:=matrix ( 3 ,3 ,[[ a [C[ i ] [ 1 ] ] , b [C[ i ] [ 1 ] ] , c [C[ i ] [ 1 ] ] ] , [ a [C[ i ] [ 2 ] ] , b [C[ i ] [ 2 ] ] , c [C[ i ] [ 2 ] ] ] , [ a [C[ i ] [ j ] ] , b [C[ i ] [ j ] ] , c [C[ i ] [ j ] ] ] ] ) ; > EQ[ k]:=( det (m)=0); > od ; > f i ; > od ; > RETURN( convert (EQ, set ) ) ; end : fixed_lines :=proc ( l1 , l2 , l3 , l4 , l5 ) > loc al i ,EQ; > EQ[1]:=( a [ l1 ]=0);EQ[2]:=(b [ l1 ]=0);EQ[3]:=( c [ l1 ]=1); > EQ[4]:=( a [ l2 ]=1);EQ[5]:=(b [ l2 ]=0);EQ[6]:=( c [ l2 ]=0); > EQ[7]:=( a [ l3 ]=1);EQ[8]:=(b [ l3 ]=0);EQ[9]:=( c [ l3 ]=−1); > EQ[10]:=( a [ l4 ]=0);EQ[11]:=(b [ l4 ]=1);EQ[12]:=( c [ l4 ]=0); > EQ[13]:=( a [ l5 ]=0);EQ[14]:=( b [ l5 ]=1);EQ[15]:=( c [ l5 ]=−1); > RETURN( convert (EQ, set ) ) ; end : r i g i d i f i c a t i o n :=proc (C) > loc al i , j ,R,L, J ,K1,K2, Bool : > R: = [ ] : > L : = [ ] ; >for ifrom 1 to nops (C) do >i f nops (C[ i ])>2 then L:=[ op(L) ,C[ i ] ] : end i f : > end do : > Bool:= true : >for ifrom 1 to nops (L)−1 do >for jfrom ( i +1) to nops (L) do >i f Bool then > J:=convert (L[ i ] , set ) i nte rse ct convert (L[ j ] , set ) ; >i f nops (J)>0 then > R:=[op(R) , J [ 1 ] ] ; > K1:= convert (L[ i ] , set ) minus J ; > R:=[ op(R) ,K1 [ 1 ] ,K1 [ 2 ] ] ; 128 A.3. SPACE OF REALIZATION ΣC > K2:= convert (L[ j ] , set ) minus J ; > R:=[ op(R) ,K2 [ 1 ] ,K2 [ 2 ] ] ; > Bool:= f a l s e ; > RETURN(R) : > end i f : > end i f : > end do : > end do : end : lines_number:=proc (C) > l oc al i , j , n : > n:=0: >for ifrom 1 to nops (C) do >for jfrom 1 to nops (C[ i ] ) do >i f C[ i ] [ j ]>n then n:=C[ i ] [ j ] : end i f : > end do : > end do : > RETURN( eval (n ) ) : end : reduc:=proc (S , l1 , l2 , l3 , l4 , l5 , n) > loc al i ,R, t , good , j ; > t :=1; >for ifrom 1 to nops (S) do > good:= true : >for jfrom 1 to n do >i f ( j<>l1 and j<>l2 and j<>l3 and j<>l4 and j<>l5 ) and (({ a [ j ]=0 ,b [ j ]=0} subset S [ i ] ) or ({a [ j ]=0 , c [ j ]=0} subset S [ i ] ) or ({b [ j ]=0 , c [ j ]=0} subset S [ i ] ) ) then > good:= f a l s e : > end i f ; > end do ; >i f good then > R[ t ]:=S [ i ] : > t:=t +1: > end i f ; > end do ; >for ifrom 1 to t−1 do > pr in tf ( ’ Next_realization ’ ) ; >print( evalm (R[ i ] ) ) ; > od ; > RETURN( evalm (R)) end : real i zati o n :=proc (C) > loc al EQ, EQ1, eq , v , V, S , K, i , R, l , n : 129 APPENDIX A. CODE > n:=lines_number (C) : > EQ:= equations (C) : > l := r i g i d i f i c a t i o n (C, 10); > EQ1:= fix ed_lin es ( l [ 1 ] , l [ 2 ] , l [ 3 ] , l [ 4 ] , l [ 5 ] ) : > eq:=EQ union EQ1: >for ifrom 1 to n do > v[3∗i−2]:=a [ i ] : > v[3∗i−1]:=b [ i ] : > v[3∗i ]:= c [ i ] : > od : > V:= convert (v , set ) : > S:= solve (eq ,V) : > R:=reduc ( [ S ] , l [ 1 ] , l [ 2 ] , l [ 3 ] , l [ 4 ] , l [ 5 ] , n ) : > RETURN( eval (R)) end : 130 Bibliography [1] Donu Arapura, Geometry of cohomology support loci for local systems. I, J. Algebraic Geom. 6(1997), no. 3, 563–597. MR 1487227 (98m:32043) [2] Vladimir I. Arnold, The cohomology ring of the group of dyed braids, Mat. Zametki 5(1969), 227–231. MR 0242196 (39 #3529) [3] Enrique Artal, José I. Cogolludo-Agustín, and Jorge Ortigas-Galindo, Kummer covers and braid monodromy, Preprint available at arXiv:1205.5427 [math.AG], accepted in J. Inst. Math. Jussieu. [4] Enrique Artal-Bartolo, Sur les couples de Zariski, J. Algebraic Geom. 3(1994), no. 2, 223–247. MR 1257321 (94m:14033) [5] , Topology of arrangements and position of singularities, Annales de la fac. des sciences de Toulouse, to appear. (2013). [6] Enrique Artal-Bartolo, Jorge Carmona-Ruber, José I. Cogolludo-Agustín, and Miguel A. Marco-Buzunáriz, Topology and combinatorics of real line arrangements, Compos. Math. 141 (2005), no. 6, 1578–1588. MR 2188450 (2006k:32055) [7] , Invariants of combinatorial line arrangements and Rybnikov’s example, Singularity theory and its applications (Tokyo), Adv. Stud. Pure Math., vol. 43, Math. Soc. Japan, 2006, pp. 1–34. MR 2313406 (2008g:32042) [8] Enrique Artal-Bartolo, José I. Cogolludo-Agustín, and Daniel Matei, Characteristic varieties of quasi-projective manifolds and orbifolds, Geom. Topol. 17 (2013), no. 1, 273–309. MR 3035328 [9] William A. Arvola, The fundamental group of the complement of an arrangement of complex hyperplanes, Topology 31 (1992), 757–765. [10] David Bessis, Variations on Van Kampen’s method, J. Math. Sci. (N. Y.) 128 (2005), no. 4, 3142–3150, Geometry. MR 2171593 (2006g:55014) BIBLIOGRAPHY [11] Egbert V. Brieskorn, Die Fundamentalgruppe des Raumes der regulären Orbits einer endlichen komplexen Spiegelungsgruppe, Inventiones Mathematicae 12 (1971), 57– 61. [12] , Sur les groupes de tresses [d’après v.i. arnold], Séminaire Bourbaki, 24ème année (1971/1972), Exp. No. 401, Springer, Berlin, 1973, pp. 21–44. Lecture Notes in Math., Vol. 317. [13] Egbert V. Brieskorn and Horst Knörrer, Plane algebraic curves, Birkhäuser Verlag, Basel, 1986, Translated from the German by John Stillwell. MR 886476 (88a:14001) [14] Nero Budur, Unitary local systems, multiplier ideals, and polynomial periodicity of Hodge numbers, Adv. Math. 221 (2009), no. 1, 217–250. MR 2509325 (2010j:32046) [15] Jorge Carmona-Ruber, Monodromía de trenzas de curvas algebraicas planas, Ph.D. thesis, Universidad de Zaragoza, 2003. [16] Giovanni Ceva, De lineis rectis se invicem secantibus statica constructio, Lud. Montiae, 1678. [17] Denis Cheniot, Une démonstration du théorème de Zariski sur les sections hyperplanes d’une hypersurface projective et du théorème de Van Kampen sur le groupe fondamental du complémentaire d’une courbe projective plane, Compositio Math. 27 (1973), 141–158. MR 0366922 (51 #3168) [18] Oscar Chisini, Una suggestiva rappresentazione reale per le curve algebriche piane, Ist. Lombardo, Rend., II. Ser. 66 (1933), 1141–1155. [19] Jose Ignacio Cogolludo, Topological invariants of the complement to arrangements of rational plane curves, Memoirs of the American Mathematical Society 159 (2002), no. 756. [20] Daniel C. Cohen, Graham Denham, Hal Schenck, Alexander I. Suciu, Hiroaki Terao, and Sergey Yuzvinsky, Complex arrangements: Algebra, geometry, topology, 2009. [21] Daniel C. Cohen and Alexander I. Suciu, The braid monodromy of plane algebraic curves and hyperplane arrangements, COMMENT. MATH. HELVETICI 72 (1997), no. 2, 285–315. [22] , Alexander invariants of complex hyperplane arrangements, Trans. Amer. Math. Soc. 351 (1999), no. 10, 4043–4067. MR 1475679 (99m:52019) [23] , Characteristic varieties of arrangements, Math. Proc. Cambridge Philos. Soc. 127 (1999), no. 1, 33–53. MR 1692519 (2000m:32036) [24] , The boundary manifold of a complex line arrangement, Groups, homotopy and configuration spaces, Geom. Topol. Monogr., vol. 13, Geom. Topol. Publ., Coventry, 2008, pp. 105–146. MR 2508203 (2010c:32051) 132 BIBLIOGRAPHY [25] Alex Degtyarev, On deformations of singular plane sextics, J. Algebraic Geom. 17 (2008), no. 1, 101–135. MR 2357681 (2008j:14061) [26] Pierre Deligne, Théorie de Hodge. II, Inst. Hautes Études Sci. Publ. Math. (1971), no. 40, 5–57. MR 0498551 (58 #16653a) [27] , Les immeubles des groupes de tresses généralisés, Invent. Math. 17 (1972), 273–302. MR 0422673 (54 #10659) [28] Angelo Di Pasquale, Links and complements of arrangements of complex projective plane algebraic curves, Ph.D. thesis, University of Melbourne, Department of Mathematics and Statistics, 1999. [29] Alexandru Dimca, On admissible rank one local systems, J. Algebra 321 (2009), no. 11, 3145–3157. MR 2510044 (2010e:14015) [30] Alexandru Dimca, Denis Ibadula, and Daniela Anca Macinic, Pencil type line arrangements of low degree: classification and monodromy, 2013. [31] Alan H. Durfee, Neighborhoods of algebraic sets, Trans. Amer. Math. Soc. 276 (1983), no. 2, 517–530. MR 688959 (84m:32013) [32] William G. Dwyer and David Fried, Homology of free abelian covers. I, Bull. London Math. Soc. 19 (1987), no. 4, 350–352. MR 887774 (88g:55011) [33] Michael Falk, Homotopy types of line arrangements, Invent. Math. 111 (1993), no. 1, 139–150. MR 1193601 (93j:52020) [34] Vincent Florens, Benoît Guerville, and Miguel A. Marco-Buzunuáriz, On complex line arrangements and their boundary manifolds, 2013. [35] M. Green and R. Lazarsfeld, Deformation theory, generic vanishing theorems, and some conjectures of Enriques, Catanese and Beauville, Invent. Math. 90 (1987), no. 2, 389–407. [36] , Higher obstructions to deforming cohomology groups of line bundles, J. Amer. Math. Soc. 4(1991), no. 1, 87–103. [37] Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Wiley Classics Library, John Wiley & Sons Inc., New York, 1994, Reprint of the 1978 original. MR 1288523 (95d:14001) [38] Phillip Griffiths and Wilfried Schmid, Recent developments in Hodge theory: a discussion of techniques and results, Discrete subgroups of Lie groups and applicatons to moduli (Internat. Colloq., Bombay, 1973), Oxford Univ. Press, Bombay, 1975, pp. 31–127. MR 0419850 (54 #7868) [39] Eriko Hironaka, Alexander stratifications of character varieties, Ann. Inst. Fourier (Grenoble) 47 (1997), no. 2, 555–583. MR 1450425 (98e:14020) 133 BIBLIOGRAPHY [40] , Boundary manifolds of line arrangements, Mathematische Annalen 319 (2001), 17–32. [41] Viktor S. Kulikov and Mina Ta˘ıkher, Braid monodromy factorizations and diffeomorphism types, Izv. Ross. Akad. Nauk Ser. Mat. 64 (2000), no. 2, 89–120. MR 1770673 (2001f:14030) [42] Anatoly Libgober, Alexander polynomial of plane algebraic curves and cyclic multiple planes, Duke Math. J. 49 (1982), no. 4, 833–851. MR 683005 (84g:14030) [43] , On the homotopy of the complement to plane algebraic curves., Journal für die reine und angewandte Mathematik 367 (1986), 103–114 (eng). [44] , On the homology of finite abelian coverings, Topology Appl. 43 (1992), no. 2, 157–166. MR 1152316 (93e:57003) [45] , Characteristic varieties of algebraic curves, Applications of algebraic geometry to coding theory, physics and computation (Eilat, 2001), NATO Sci. Ser. II Math. Phys. Chem., vol. 36, Kluwer Acad. Publ., Dordrecht, 2001, pp. 215–254. MR 1866902 (2003e:14008) [46] , Lectures on topology of complements and fundamental groups, Singularity theory, World Sci. Publ., Hackensack, NJ, 2007, pp. 71–137. MR 2342909 (2008i:14043) [47] , Non vanishing loci of Hodge numbers of local systems, Manuscripta Math. 128 (2009), no. 1, 1–31. MR 2470184 (2010g:32026) [48] , On combinatorial invariance of the cohomology of the Milnor fiber of arrangements and the Catalan equation over function fields, Arrangements of hyperplanes—Sapporo 2009, Adv. Stud. Pure Math., vol. 62, Math. Soc. Japan, Tokyo, 2012, pp. 175–187. MR 2933797 [49] Anatoly Libgober and Sergey Yuzvinsky, Cohomology of the Orlik-Solomon algebras and local systems, Compositio Math. 121 (2000), no. 3, 337–361. MR 1761630 (2001j:52032) [50] Saunders Maclane, Some interpretations of abstract linear independence in terms of projective geometry. [51] Boris G. Moishezon, Stable branch curves and braid monodromies, Algebraic geometry (Chicago, Ill., 1980), Lecture Notes in Math., vol. 862, Springer, Berlin, 1981, pp. 107–192. MR 644819 (83c:14008) [52] Shaheen Nazir and Masahiko Yoshinaga, On the connectivity of the realization spaces of line arrangements, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 11 (2012), no. 4, 921–937. MR 3060685 134