Cette thèse est le point d’intersection entre deux facettes de l’étude des arrangements de droites : la combinatoire et la topologie. Dans une première partie nous avons étudié l’inclusion de la variété bord dans le complémentaire d’un arrangement. Nous avons ainsi généralisé le résultat d’E. Hironaka au cas de tous les arrangements complexes. Pour contourner les problèmes provenant des arrangements non réels, nous avons étudié le diagramme de câblage, dit wiring diagram, qui code la monodromie de tresses sous forme de tresse singulière. Pour pouvoir l'utiliser, nous avons implémenté un programme sur Sage permettant de calculer ce diagramme en fonction des équations de l’arrangement. Cela nous a permis de d’obtenir deux descriptions explicites de l’application induite par l’inclusion de la variété bord dans le complémentaire sur les groupes fondamentaux. Nous obtenons ainsi deux nouvelles présentations du groupe fondamental du complémentaire d’un arrangement. L’une d’entre elle généralise le théorème de R. Randell au cas des arrangements complexes. Pour continuer ces travaux, nous avons étudié l’application induite par l’inclusion sur le premier groupe d’homologie. Nous obtenons deux descriptions simples de cette application. En s’inspirant des travaux de J.I. Cogolludo, nous décrivons une décomposition canonique du premier groupe d’homologie de la variété bord comme produit de la 1-homologie et de la 2-cohomologie du complémentaire, ainsi qu'un isomorphisme entre la 2-cohomologie du complémentaire et la 1-homologie du graphe d’incidence. Dans la seconde partie de notre travail nous nous sommes intéressés à l’étude des caractères du groupe fondamental du complémentaire. Nous partons des résultats obtenus par E. Artal sur le calcul de la profondeur d’un caractère. Cette profondeur peut être décomposée en un terme projectif et un terme quasi-projectif. Un algorithme pour calculer la partie projective a été donné par A. Libgober. Les travaux de E. Artal concernent la partie quasi-projective. Il a obtenu une méthode pour la calculer en fonction de l’image de certains cycles particuliers du complémentaire par le caractère. En utilisant les résultats obtenus dans la première partie, nous avons obtenu un algorithme complet permettant le calcul de la profondeur quasi-projective d’un caractère. A travers l’étude de cet algorithme, nous avons obtenu une condition combinatoire pour admettre une profondeur quasi-projective potentiellement non combinatoire. Nous avons ainsi défini la notion de caractère inner-cyclic . Cette notion nous a permis de formuler des conditions fortes sur la combinatoire pour qu’un arrangement n’ait que des caractères de profondeur quasi-projective nulle. Enfin pour diminuer le nombre d’exemples à considérer nous avons introduit la notion de combinatoire première. Si une combinatoire ne l’est pas, alors les variétés caractéristiques de ses réalisations sont définies par celles d’un arrangement avec moins de droites. En parallèle à cette étude, nous avons observé que la composition de l’application induite par l’inclusion sur le premier groupe d’homologie avec un caractère nous fournit un invariant topologique de l'arrangement obtenu en désingularisant les points multiples (blow-up). De plus, nous montrons que cet invariant n’est pas de nature combinatoire. Il nous a ainsi permis de découvrir deux nouvelles nc-paires de Zariski. / This thesis is the intersection point between the two facets of the study of line arrangements: combinatorics and topology. In the first part, we study the inclusion of the boundary manifold in the complement of an arrangement. We generalize the results of E. Hironaka to the case of any complex line arrangement. To get around the problems due to the case of non complexified real arrangement, we study the braided wiring diagram. We 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 induced by the inclusion on the fundamental groups. From theses descriptions, we obtain two new presentations of the fundamental group of the complement. One of them is a generalization of the R. Randell Theorem to any complex line arrangement. In the next step of this work, we study the map induced by the inclusion on the first homology group. Then we obtain two simple descriptions of this map. 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. Finally, we obtain an isomorphism between the 2-cohomology of the complement with the 1-homology of the incidence graph of the arrangement. In the second part, we are interested by the study of character on the group of the complement. We start from the results of E. Artal 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. 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. From the study of this algorithm, we obtain a strong combinatorial condition on characters to admit a quasi-projective 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. 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 less line. 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. We show that the invariant captures more than combinatorial information. Thereby, we detect two new examples of nc-Zariski pairs.

Neste trabalho, estudamos famílias de curvas genericamente reduzidas. Estendemos para o caso genericamente reduzido alguns resultados conhecidos para famílias de curvas reduzidas como a equivalência entre a Whitney equisingularidade e a resolução simultânea forte da família e a equivalência entre a Whitney equisingularidade e a constância do número de Milnor e da multiplicidade de cada curva Xt da família. Estudamos também a equisingularidade topológica e a Whitney equisingularidade de famílias de superfícies em C3 parametrizadas por germes de aplicações A-finitamente determinados. Em ([51]), Ruas apresentou uma conjectura cujo enunciado diz que se f : (C2, 0) r→ (C3, 0) é um germe de aplicação finitamente determinado, então um desdobramento F a 1-parâmetro de f é topologicamente trivial se, e somente se F é Whitney equisingular se, e somente se o número de Milnor μ(D(ft)) de D(ft) é constante, onde D(ft) é a curva de pontos duplos de ft. Apresentamos contra-exemplos que mostram como esta conjectura pode falhar. Mostramos também uma classe de famílias de germes aplicações ft : (C2, 0) → (C3, 0) em que a conjectura é verdadeira. No caso em que f é homogênea e de coposto 1, mostramos também algumas fórmulas para a multiplicidade da imagem da curva de pontos duplos f(D(f)), o número de Milnor da seção transversal μ1(f(C2)) e o invariante J(f) em termos dos graus de f. Em [44], Nuño-Ballesteros e Jorge Pérez apresentam alguns resultados sobre germes de aplicações f : (Cn, 0) → (C2n-1, 0) com n ≥ 3. Quando f é finitamente determinado, a curva dos pontos duplos D(f) de f tem uma estrutura de curva genericamente reduzida. Apresentamos uma outra forma de abordar alguns problemas descritos em [44] usando resultados sobre curvas genericamente reduzidas. / In this work, we study families of generically reduced curves. We extend to the generically reduced case some results known for families of reduced curves as the equivalence between Whitney equisingularity and strong simultaneous resolution of the family and the equivalence between Whitney equisingularity and the constancy of the Milnor number and the multiplicity of each curve Xt of the family. We also study the topological triviality and the Whitney equisingularity of families of surfaces in C3 parametrized by A-finitely determined map germs. In [51], Ruas presented a conjecture whose statement says that if f : (C2, 0) → (C3, 0) is a finitely determined map germ, then an 1-parameter unfolding F = (ft, t) of f is topological trivial if and only if it is Whitney equisingular if and only if the Milnor number μ(D(ft)) is constant, where D(ft) is the double point curve of ft. We present counter-examples that show how the conjecture can fail. We also show a class of families of map germs ft : (C2, 0) → (C3, 0) in which the conjecture is true. We also give formulas for the multiplicity of the image of the double point curve f(D(f)), the Milnor number of the transversal generic section μ 1f(C2)) and the invariant J(f) in terms of degrees of f in the case in which f is homogeneous and has corank 1. In [44], Nuño-Ballesteros and Jorge Pérez give some results in the case of families of map germs f : (Cn, 0) → (C2n-1, 0) with n ≥ 3. When f is finitely determined, the double point. curve D(f) of f is a generically reduced curve. We present another way of approaching some problems in [44] using results on generically reduced curves.

在本碩士論文中, 我們闡述了投射有限群表現, 以及其形變理論。 我們亦特別研究這些表示在 GL_1 和 GL_2 之形變, 並且給了可表示化 的判定準則。 最後, 我們解釋相對應的泛形變環之扎里斯基切空間與 群餘調之關連, 並計算了 GL_1 的泛形變表現。 / In this master thesis, we give an exposition of the deformation theory of representations for GL_1 and GL_2, respectively, of certain profinite groups. We give rigidity conditions of the fixed representation and verify several conditions for the representability. Finally, we interpret the Zariski tangent spaces of respective universal deformation rings as certain group cohomology and calculate the universal deformation for GL_1.