ACKNOWLEDGEMENT

The completion of this thesis would not have been possible without the help and the support of some dear people for whom I shall dedicate the following words.

First and foremost, I would like to express my sincere gratitude to my advisor Alexandru DIMCA. Thank you for the continuous support you gave me, for your patience, motivation and immense knowledge. Your guidance helped me in all the time of research and writing of this thesis. It is and it will always be an honor to work with you.

My deepest appreciation goes to my reading committee members, Lucian BADESCU and Michel GRANGER for spending time reading, reviewing and commenting my work. Michel GRANGER, thank you for the interest you had in my research, and I wish that one day I will have the honor to work with you.

I would like to thank my thesis jury members who accepted to judge my work. It is to my honor and privilege to present my work to brilliant specialist like yourselves. I gratefully acknowledge the support of the Lebanese National Council for Scientific Research (CNRS-L), without which the present study could not have been completed. Charles, thank you for your support and for helping me to have the financial support.

Thank you for the municipality of Cheikh Taba, my beloved village, for giving me a complement financial support. I owe my deepest gratitude again to my advisor, to Adam PARUSINSKI, Michel MERLE, Pauline BAILET and Jean-Baptiste CAMPESATO for listenning to my presentations, commenting my work, giving me advice. This work group was very important to me. It has increased myself confidence, and took my fear away when I speak in front of an audience. Special thanks also to André for being there for me whenever I needed him.

I could not find words to express my gratitude to you. Louli, you made my dream come true, the dream of entering the world of research. I am grateful for you for the rest of my life.

The last years of Ph.D. studies weren't that easy for me, and the most difficult part was to know how to take the right decisions, especially when it comes to my future. You always give me faith whenever I feel down. Bina, I am grateful for you for helping me get through all of these difficulties.

I am indebted to all my colleagues and friends for providing a stimulating and fun environment in which to learn and grow. I am especially grateful to Ahed, Ben, Bienvenu, Brice and Giovanni for the special moments we have spent together. Pauline, my thesis becomes more fun and less difficult with you! Thank you for helping me get through the difficult times, and for all the emotional support, entertainment, and caring you have provided. Ghina, we went through this long journey together, and now we finish together, we shared the same difficult moments, and most importantly, we shared lovely moments with "3ab2a"! Maher I appreciate you being there for me, helping me in all my problems, especially the administrative issues. Thank you for the fun times we spent together.

From Beirut to Paris to Angola, I want to thank you all my dear friends. Sadi2eeee! I did it! I am a doctor like you now, a different kind of doctors but who cares! We both did it! Che, not only I would like to thank you for what you have done to me, but I have to tell you that you are one of the few who was able to draw a sincere smile on my face, even during the most difficult times I passed through. Bébé, maybe you are not aware of what you have done to me, but living away from my country, family, and friends was like impossible for me. Yet, you knew how to encourage me and make it easier on me. I am lucky to have a friend like you! Hita and Risha, words could never express how much I appreciate you standing by me. You were more like sisters to me! I would never imagine having friends as supportive, helpful, caring and fun as you. "If a friendship lasts longer than 7 years, psychologist say that it will last a lifetime", and now I am sure that our friendship will last more than 7 years.

Maybe we haven't spent too much time together, but what we have spent was so special. Thank you Hu for being there for me when I needed you, for bringing me up when I felt down, for making me laugh when I felt sad, for listening to me when I needed to talk, and for everything you have done to me. It is not easy to tolerate the nagging of a woman, but bravo you did it! Lastly and most importantly, I wish to thank all my family. 'Cheikh El Chabeb', you always bring joy to my life ever since you were a child. I am truly blessed to have you in my life. Thank you dear for being the best brother I could ever have. "A sister is a gift to the heart, a friend to the spirit, a golden thread to the meaning of life." Nisso, I do not want to thank you for always being there for me, I want to thank you for being my sister. You were always special for all the family, but to me you are the most precious gift I could ever have. Mom, Dad, I am speechless when it comes to you. Dad, you always trusted me and you called me "batale". I am "batale" because of the faith and strength you gave me. Without you I would not be strong enough to get through all the difficult times and lonely moments. Mom, you believed in me, and you believed that I am able to reach my goals. If I succeeded in my life, it is because of you. I know that you are proud of me now, but you would rather be proud of yourself because you knew how to play your role, the role of being the perfect mother. I am grateful for each day you gave me, my lovely family, you bore me, raised me, supported me, taught me, and loved me. To you I dedicate this thesis.

Abstract (Français)

Soit S = C[x 0 , • • • , c n ] l'anneau gradué des polynômes en x 0 , • • • , x n à coefficient complexes, S = r≥0 S r où S r désigne l'espace vectoriel des polynômes homogènes de degré r. Pour un polynôme homogène, f ∈ S N , de degré N , on définit l'algèbre de Milnor (ou du Jacobien) par M (f ) = S/J f , où J f est l'idéal Jacobien de f , i.e. l'idéal engendré par les dérivées partielles f 0 = ∂f ∂x 0 , • • • , f n = ∂f ∂xn . M (f ) est une C-algèbre graduée, dont la graduation est induite par celle de S.

L'étude de l'algèbre de Milnor est liée aux singularités de l'hypersurface H définie par f = 0 dans l'espace projectif P n , ainsi qu'à la structure de Hodge mixte sur la cohomologie de H et de son complémentaire U = P n \ H. En effet, l'algèbre de Milnor de f est égale, avec un décalage de graduation, au groupe d'homologie d'ordre 0 ou le groupe de cohomologie d'ordre n+1 du complexe de Koszul des dérivées partielles de f , donc c'est normal d'étudier les autres groupes de (co)homologie de ce complexe.

Dans la première partie de cette thèse, on étudie la relation entre la théorie de Hodge mixte du complémentaire U de H et les singularités de H. L'importance de la théorie de Hodge est qu'il existe une structure de Hodge mixte sur les groupes de cohomologie de toute variété algébrique X, compatible avec les morphismes induits par les applications régulières u : X → Y . Cette structure consiste essentiellement de deux filtrations, une filtration décroissante F s , la filtration de Hodge, et une filtration croissante W m , la filtration par le poids. On s'intéresse au calcul des dimensions des groupes gradués associés à la filtration de Hodge dans le cas où H est une courbe dans P 2 , qu'on notera C dans la suite, qui admet des points doubles et triples ordinaires comme singularités. En particulier, on a obtenu le résultat suivant.

Theorem 0.1. Soit C ⊂ P 2 une courbe de degré N . Supposons que C n'admet que n noeuds et t points triples comme singularités. Notons U = P 2 \C. Soient C = j=1,r C j la décomposition de C en union de composantes irréductibles, ν : Cj → C j les normalisations, et g j = g( Cj ), le genre de Cj . On a alors

dim Gr 1 F H 2 (U, C) = r j=1 g j et dim Gr 2 F H 2 (U, C) = (N -1)(N -2) 2 -t.
Ce théorème nous permet de calculer tous les nombres de Hodge mixtes du groupe de cohomologie H 2 (U ) de U , le complémentaire de C, et par conséquent, les nombres de Betti correspondants qui sont des invariants topologiques importants. Un cas spécial est celui des courbes rationnelles, où g i = 0 pour tout j. Dans ce cas, H 2 (U ) est pure de type [START_REF] Bott | Differential Forms in Algebraic Topology[END_REF][START_REF] Bott | Differential Forms in Algebraic Topology[END_REF]. Ceci est une propriété connue dans le cas des complémentaires des arrangements de droites.

On étudie ensuite le cas où C admet des singularités isolées quelconques, et on trouve que le même résultat pour la dimension dim Gr 1 F H 2 (U, C). Le cas où C admet des singularités de multiplicités atteignant l'ordre 4 montre qu'on ne peut pas s'attendre à trouver des formules simples pour dim Gr 2 F H 2 (U, C).

Dans la deuxième partie de cette thèse, on s'intéresse à trouver les dimension de M (f ) r qui sont des invariants projectifs de l'hypersurface H : f = 0. Le cas lisse est déjà étudié, et la série de Poincaré définie par HP (M (f ))(t) = r dim M (f ) r t r est complètement déterminée. Explicitement on a,

HP (M (f ))(t) = (1 -t N -1 ) n+1 (1 -t) n+1 .
L'étude du cas où H admet des singularités isolées, a D'après cette étude, on peut se poser la question: Que se passe-t-il si C admet d'autres singularités que les noeuds? On restreint l'étude dans ce qui suit sur le cas des courbes ayant des points doubles et triples comme singularités, par exemple la courbe définie par (x 2 -y 2 )(y 2 -z 2 )(x 2 -z 2 ) = 0, qui est l'union de 6 droites se coupant en 4 points triples et 3 points doubles. Contrairement au cas nodale, dans ce cas l'inclusion F s ⊂ P s dûe à Deligne-Dimca, où P s est la filtration par l'ordre du pôle, peut être stricte, ce qui rend l'étude plus difficile.

dim M (f ) r = dim M (f s ) r pour tout r ≤ N -2.
On généralise dans ce cas l'équation 0.1. Plus précisément, on obtient le résultat suivant. Theorem 0.2. Soit C ⊂ P 2 une courbe de degré N . Supposons que C admet n noeuds (A 1 ) et t points triples (D 4 ). Soit C = j=1,r C j la décomposition de C en union de composantes irréductibles, ν : Cj → C j les normalisations et notons g j = g( Cj ) les genres de Cj . Soit τ = n + 4t le nombre de Tjurina global de C. On a,

0 ≤ dim M (f ) 2N -3 -τ ≤ r j=1 g j . En particulier, (i) Si g i = 0 pour tout i, on a dim M (f ) 2N -3 = τ . (ii) L'égalité dim M (f ) 2N -3 -τ = r j=1 g j est vérifiée si et seulement si H 2 (U ) vérifie F 2 H 2 (U ) = P 2 H 2 (U ).
On considère le sous-module gradué de S, AR(f ) ⊂ S n+1 , de toutes les relations entre les dérivées partielles f j du polynôme f , soit

a = (a 0 , ..., a n ) ∈ AR(f ) m si et seulement si a 0 f 0 + a 1 f 1 + ... + a n f n = 0.
Dans le module AR(f ) il y a un sous-module de S des relations de Koszul, nommé aussi le sous-module des relations triviales, engendré par les relations t ij ∈ AR(f ) d-1 pour 0 ≤ i < j ≤ n, où la i ième coordonné de t ij est égale à f j , sa j ième coordonné égale à -f i et les autres sont nulles.

On appelle le module quotient ER(f ) = AR(f )/KR(f ), le module des relations essentielles, ou les relations non triviales, car c'est le module des relations qu'on doit ajouter aux relations de Koszul pour avoir toutes les relations, ou les syzygies, entre les f j .

On décrit dans cette thèse les dimensions de l'espace de syzygies de l'idéal Jacobien de degré N -2. Plus précisément, on a obtenu le résultat suivant. Theorem 0.3. Avec les mêmes hypothèses du théorème précédent, on a

max(r -1 + t - r j=1 g j , r -1) ≤ dim ER(f ) N -2 ≤ r -1 + t.
En particulier, dim ER(f ) N -2 = r -1 + t si g j = 0 pour tout j.

Exemple 5.6 montre que les courbes avec points doubles et triples sont plus compliquées que les courbes nodales. En particulier, contrairement au cas nodal rationnel, dans le cas des courbes rationnelles à points doubles et triples ordinaires, la série de Poincaré HP (M (f )) n'est pas complètement déterminée par N , le nombre des composantes irréductibles et le nombre de points doubles et triples. Il montre aussi que c'est difficile de contrôler les composantes homogènes M (f ) r pour r = 2N -3.

On voudrait continuer au futur l'étude de ces questions intéressantes et difficiles. Les relations entre les syzygies de l'idéal Jacobian et la cohomologie de Rham de la fibre de Milnor définie par F : f = 1 est aussi un autre sujet d'investigation.

Abstract (English)

Let S = C[x 0 , • • • , c n ] =
r≥0 S r be the graded ring of polynomial functions in x 0 , • • • , x n with complex coefficients, where S r denotes the vector space of homogeneous polynomials of degree r. Let f ∈ S N be a homogeneous polynomial of degree N , and define M (f ) = S/J f to be the Milnor (or Jacobian) algebra of f , where J f is the Jacobian ideal of f , i.e. the ideal generated by the first order partial derivatives f j = ∂f ∂x j for j = 0, 1, ..., n of f .

The study of such Milnor algebras is related to the singularities of the hypersurface H ⊂ P n defined by f = 0 in the complex projective space P n , as well as to the mixed Hodge structure on the cohomology of H, and of its complement U = P n \ H. In fact the Milnor algebra of f can be seen up to a twist of grading as the first (respectively the last) homology (respectively cohomology) group of the Koszul complex of the partial derivatives of f , so it is natural to study the other (co)homology groups of this complex as well.

In the first part of this thesis, we study the relation between the mixed Hodge theory of the complement of the hypersurface H and the singularities of H. The importance of Hodge theory is the existence of a mixed Hodge structure on the cohomology groups of each algebraic variety X, compatible with the morphisms induced by regular mappings u : X → Y . This structure consists essentially of two filtrations, the decreasing Hodge filtration F s and the increasing weight filtration W m . We are interested in computing the dimensions of the associated graded groups of the former one in the case where H is a curve in P 2 , that we will denote in the sequel by C, having only ordinary double and triple points as singularities. In particular we have obtained the following result.

Theorem 0.4. Let C ⊂ P 2 be a curve of degree N and set U = P 2 \C. Suppose that C has only n nodes (A 1 ) and t triple points (D 4 ) as singularities. Let C = j=1,r C j be the decomposition of C as a union of irreducible components, let ν : Cj → C j be the normalization mappings and let g j = g( Cj ) be the corresponding genera. Then one has

dim Gr 1 F H 2 (U, C) = r j=1 g j and dim Gr 2 F H 2 (U, C) = (N -1)(N -2) 2 -t.
This theorem allows us to compute all the mixed Hodge numbers of the second cohomology group H 2 (U ) of the complement U of C, and consequently the correspondant Betti numbers which are important topological invariants. A special case is the case of rational curves, where g j = 0 for all j. In this case H 2 (U ) is pure of type (2, 2), a well known property in the case of line arrangement complements.

Then we study the case where C has more general isolated singularities, and we find the same result for dim Gr 1 F H 2 (U, C). The case where C has singularities of multiplicities up to 4 shows that we cannot expect simple formulas for dim Gr 2 F H 2 (U, C).

In the second part of this thesis, we are interested in finding the dimensions of M (f ) r which are projective invariants of the hypersurface H : f = 0. The case where H is smooth is already known, and the Hilbert-Poincaré series, defined by

HP (M (f ))(t) = r dim M (f ) r t r , is all determined. More explicitely, HP (M (f ))(t) = (1 -t N -1 ) n+1 (1 -t) n+1 .
The study of the case where H has isolated singularities, say at the points a 1 , • • • a p , has begun by A. Dimca and A.D.R. Choudary in [START_REF] Choudary | Dimca: Koszul Complexes and Hypersurface Singularities[END_REF], who proved that dim M (f ) r stabilizes for r > (n + 1)(N -2). In this case dim M (f ) r = τ (H), where τ (H) is the sum of all Tjurina numbers τ (H, a j ) for j = 1, • • • , p. Then A. Dimca and G. Sticlaru proved in [START_REF] Dimca | Koszul Complexes and Pole Order Filtrations[END_REF] that dim M (f ) r coincides with the dimensions of M (f s ) r of a homogeneous polynomial f s defining a smooth hypersurface of same degree of H for small r. Indeed, they have noticed that for

r ≤ N -2, dim M (f ) r = dim M (f s ) r .
When n = 2, i.e. H is a curve C in P 2 , the study becomes simpler, and we have more interesting results. In particular, if C is a nodal curve , dim M (f ) r = dim M (f s ) r for all r ≤ 2N -4, and the next dimension is given by dim

M (f ) 2N -3 = n(C) + r j=1 g j , (0.2)
where n(C) is the number of nodes of C. If C is a rational nodal curve, then dim M (f ) 2N -3 = n(C) = τ (C), and therefore the Poincaré series is all determined in terms of the number of nodes and the degree of C.

This study gives rise to the open question: What will happen if C has singularities other than nodes? We restrict our studies to the curves having only ordinary double and triple points as singularities, for instance, the curve defined by (x 2 -y 2 )(y 2 -z 2 )(x 2 -z 2 ) = 0, which is the union of 6 lines intersecting in 4 triple points and 3 nodes. This case is more subtle since, unlike the nodal case, the well-known inclusion due to Deligne-Dimca, F s ⊂ P s , may be strict, where P s is the pole order filtration.

We give in this case a generalization of equation (0.2). More precisely, we have obtained the following result.

Theorem 0.5. Let C ∈ P 2 be a curve of degree N . Suppose C has n nodes (A 1 ) and t triple points (D 4 ). Let C = j=1,r C j be the decomposition of C as a union of irreducible components, let ν : Cj → C j be the normalization mappings and set g j = g( Cj ). Then we have the following.

0 ≤ dim M (f ) 2N -3 -τ ≤ r j=1 g j .
In particular, (i) If all g i = 0, one has dim M (f ) 2N -3 = τ .

(ii) One has equality, i.e. dim M (f ) 2N -3 -τ = r j=1 g j if and only if

H 2 (U ) satisfies F 2 H 2 (U ) = P 2 H 2 (U ).
One can consider the graded S-submodule AR(f ) ⊂ S n+1 of all relations involving the partial derivatives f j 's of the polynomial f , namely

a = (a 0 , ..., a n ) ∈ AR(f ) m if and only if a 0 f 0 + a 1 f 1 + ... + a n f n = 0.
Inside the module AR(f ) there is the S-submodule of Koszul relations KR(f ), called also the submodule of trivial relations, spanned by the relations t ij ∈ AR(f ) d-1 for 0 ≤ i < j ≤ n, where t ij has the i-th coordinate equal to f j , the j-th coordinate equal to -f i and the other coordinates zero.

The quotient module ER(f ) = AR(f )/KR(f ) may be called the module of essential relations, or non trivial relations, since it tells us which are the relations which we should add to the Koszul relations in order to get all the relations, or syzygies, involving the f j 's.

We describe in this thesis the dimension of the space of syzigies of the Jacobian ideal of degree N -2. More precisely, we have obtained the following result.

Theorem 0.6. Under the same hypothesis of the previous theorem, we have

max(r -1 + t - r j=1 g j , r -1) ≤ dim ER(f ) N -2 ≤ r -1 + t.
In particular, dim ER(f ) N -2 = r -1 + t if g j = 0 for all j.

Example 5.6 shows that the curves with ordinary nodes and triple points are much more subtle than the nodal curves. In particular, for rational curves with ordinary nodes and triple points the Poincaré series HP (M (f )) is not determined by N , the number of irreducible components, the number of double and triple points (as was the case for rational nodal curves). It also shows that it is rather difficult to control the dimensions of the homogeneous components M (f ) r for r = 2N -3.

We plan to continue the study of these difficult and interesting questions in the future. The relations between syzygies of the Jacobian ideal and the de Rham cohomology of Milnor fibers given by F : f = 1 is also a subject of further investigations.

Chapter 1 Introduction 1.1 Introduction (version Française)

Les variétés algébriques sont un objet central de la géométrie algébrique. Une des plus importantes problématiques dans ce domaine est l'étude des singularités des hypersurfaces, qui s'étaient remarquées dans le cas des courbes, comme le point double ordinaire de la courbe définie par y 2 = x 2 -x 3 , le point de rebroussement de la courbe y 2 = x 3 et le point triple ordinaire de la courbe y 3 = x 3 -x 4 . Soit S = C[x 0 , • • • , x n ] l'anneau de polynômes à n + 1 variables à coefficients dans C. S est un anneau gradué dont les éléments homogènes de degré r sont les polynômes homogènes de degré r. On note S = r≥0 S r . Pour un polynôme f ∈ S N , on définit l'algèbre de Milnor par M (f ) = S/J f , où J f est l'idéal Jacobian de f , c.à.d. l'idéal engendré par les dérivées partielles

f 0 = ∂f ∂x 0 , • • • , f n = ∂f ∂xn . M (
f ) est une C-algèbre graduée, dont la graduation est induite par celle de S.

Dans cette thèse, on étudie l'algèbre de Milnor M (f ) d'un polynôme homogène f et sa relation avec l'hypersurface projective correspondante, V (f ) : f = 0. En particulier, on s'intéresse à trouver les dimensions des composantes homogènes M (f ) r de cette algèbre de Milnor qui sont des invariants projectifs des hypersurfaces. Le cas lisse étant complètement étudié, on s'interesse au cas où V (f ) admet des singularités isolées, plus précisément aux courbes qui ont des points doubles et triples ordinaires. De telles ques-tions ont récemment attiré beaucoup d'intérêt, voir [START_REF] Choudary | Dimca: Koszul Complexes and Hypersurface Singularities[END_REF], [START_REF] Dimca | Syzigies of Jacobian Ideals and Defects of Linear Systems[END_REF], [START_REF] Dimca | Graded Koszul cohomology and spectrum of certain homogeneous polynomials[END_REF], [START_REF] Dimca | Syzygies and logarithmic vector fields along plane curves[END_REF], [START_REF] Dimca | Sticlaru: Chebyshev Curves, Free Resolutions and Rational Curve Arrangements[END_REF], [START_REF] Dimca | Koszul Complexes and Pole Order Filtrations[END_REF], [START_REF] Eissydieux | Sur l'application des periodes d'une variation de structure de Hodge attachee aux familles de hypersurfaces a singularites simples[END_REF], [START_REF] Sernesi | The Local Cohomology of the Jacobian Ring[END_REF], [START_REF] Sticlaru | Invariants and rigidity of projective hypersurfaces[END_REF], et [START_REF] Sticlaru | Log-concavity of Milnor algebras for projective hypersurfaces[END_REF]. D'autre part, la théorie de Hodge joue un rôle important dans la théorie de singularités. D'après P. Deligne, il existe une structure de Hodge mixte sur les groupes de cohomologie de toute variété algébrique, voir [START_REF] Deligne | Theorie de Hodge, II and III[END_REF]. Cette structure définie une filtration (la filtration de Hodge) dont les dimensions des groupes gradués associés seront calculées pour certains cas. Plus précisement, on étudie dans cette thèse la structure de Hodge mixte de la cohomologie du complémentaire des courbes singulières dans P 2 , dont les singularités sont des points doubles et triples, en particulier la relation entre la filtration de Hodge et la filtration par l'ordre du pôle.

On commence par le chapitre 2 où on rappelle les notions de base des germes et des singularités isolées des hypersurfaces, les suites régulières et les intersections complètes. Notons que les dérivées partielles f 0 , • • • , f n forment une suite régulière si et seulement si V (f ) est une hypersurface lisse, ce qui explique le fait que l'étude de l'algèbre de Milnor M (f ) est simple dans ce cas. Puis on rappelle le théorème de Bézout, le théorème de Cayley-Bacharach, et les défauts des systèmes linéaires par rapport aux sous-ensembles finis (ou sous-schémas de dimension 0) dans P n . Ceux-ci jouent un rôle important dans la compréhension de l'algèbre de Milnor M (f ) quand V (f ) admet des singularités isolées, voir [START_REF] Dimca | Syzigies of Jacobian Ideals and Defects of Linear Systems[END_REF].

Dans le chapitre 3, on introduit le complexe de Koszul K * (f ) des dérivées partielles d'un polynôme homogène f , dont le groupe de cohomologie d'ordre n + 1 est égale à l'algèbre de Milnor avec un décalage de graduation. Quand V (f ) admet des singularités isolées, le seul groupe de cohomologie non nul de K * (f ) distinct de H n+1 décrit les syzygies de l'idéal Jacobian J f . Cette relation nous donne des résultats importants sur les composantes homogènes de M (f ). Le cas des courbes nodales a été étudié par A. Dimca et G. Sticlaru dans [START_REF] Dimca | Koszul Complexes and Pole Order Filtrations[END_REF]. En particulier, dans le cas des courbes nodales rationnelles, où chaque composante irréductible C i de C = V (f ) est rationnelle, la série de Hilbert-Poincaré de M (f ) est donnée explicitement en fonction du degré de f et le nombre de noeuds, voir Corollaire 3.3. On généralise partiellement ce résultat au cas des courbes dans P 2 à points doubles et triples ordinaires, voir Théorème 5.5, et on montre dans l'exemple 5.6 qu'on ne peut pas espérer à le généraliser complètement.

Dans le chapitre 4, on introduit les structures de Hodge mixtes et leurs relations avec les singularités des hypersurfaces. Dans la première partie de ce chapitre, on rappelle quelques définitions et propriétés introduites par P. Deligne [START_REF] Deligne | Theorie de Hodge, II and III[END_REF]. Le premier résultat de la thèse, énoncé dans la deuxième partie de ce chapitre, relie la théorie de Hodge du complémentaire de la courbe plane C = V (f ), admettant des points doubles et triples ordinaires comme singularités, à la topologie des composantes irréductibles C i de C ainsi qu'au nombre de points triples. Avec ce résultat on peut calculer les nombres de Hodge mixtes du groupe de cohomologie d'ordre 2 du complémentaire des courbes à points doubles et triples, et par conséquent les nombres de Betti correspondants. Puis on considère le cas où C est une courbe plane à singularités isolées quelconques, où on calcul le polynôme de Hodge-Deligne de C et de son complémentaire U . On généralise ainsi le théorème 4.2 aux arrangements des courbes ayant des singularités ordinaires et se coupant transversalement. Dans la dernière section, on montre que le cas des courbes planes à singularités ordinaires dont les multiplicités atteignent l'ordre 4 (sans l'hypthèse que les courbes se coupent transversalement) est beaucoup plus compliqué.

On commence le chapitre 5 par donner des exemples de syzygies de l'idéal Jacobian J f et les relier au calcul de la série de Hilbert-Poincaré de l'algèbre de Milnor M (f ) en utilisant le logiciel Singular. Puis on donne une brève présentation des suites spectrales associées à un complexe filtré, et on l'applique au complexe de Rham du complémentaire d'une hypersurface afin de définir la filtration par l'ordre du pôle. Cette filtration, qu'on note P s , et la filtration de Hodge F s vérifient la relation F s ⊂ P s démontrée par Deligne-Dimca dans [START_REF] Deligne | Filtrations de Hodge et par l'ordre du pole pour les hypersurfaces singulières[END_REF]. Dans le cas des courbes nodales, cette relation devient une égalité, ce qui explique pourquoi la théorie est plus simple dans ce cas. Pour les courbes à points doubles et triples ordinaires, l'inclusion peut être stricte, voir Exemple 5.5.

L'idée principale est la description de la suite spectrale associée à la filtration par l'ordre du pôle en fonction des composantes homogènes de la cohomologie du complexe de Koszul K * (f ), voir la section (5.4) et le fait que cette suite spectrale dégénère au terme E 2 dans le cas des courbes planes aux singularités quasi-homogènes, voir [START_REF] Dimca | Koszul Complexes and Pole Order Filtrations[END_REF]. Une approche différente de ces résultats est donnée dans le nouvel article de A. Dimca et M. Saito, voir [START_REF] Dimca | Graded Koszul cohomology and spectrum of certain homogeneous polynomials[END_REF]. Dans les Propositions 5.4 et 5.5, on montre comment on peut trouver des représentants pour des classes de cohomologie de Rham dans la cohomologie de la fibre de Milnor pour deux arrangements classiques de droites en utilisant les syzygies de l'idéal Jacobian. Notons qu'au contraire de la cohomologie du complémentaire d'un arrangement d'hyperplans, où la description en fonction des formes différentielles est due à Arnold, Brieskorn, Orlik et Solomon, voir [START_REF] Orlik | Arrangements of Hyperplanes[END_REF], nos propositions 5.4 et 5.5 sont les premiers résultats de ce type concernant la cohomologie de la fibre de Milnor.

Le deuxième résultat de cette thèse, voir Théorème 5.5 décrit la dimension de M (f ) 2N -3 et celle de l'espace de syzygies

EF (f ) N -2 = {(a, b, c) ∈ S 3 N -2 : af x + bf y + cf z = 0}
de degré N -2 d'une courbe dans P 2 de degré N à points doubles et triples. Contrairement au cas nodal, on trouve dans notre cas un encadrement pour ces invariants, et le fait que ces inégalités deviennent des égalités dépend de l'égalité entre la filtration par l'ordre du pôle P s et la filtration de Hodge F s .

On montre que ces égalités sont vraies quand les composantes irréductibles C i de C = V (f ) sont rationnelles, et cela entraîne une nouvelle situation où on a une égalité F s = P s . Cependant, même dans ce cas, Exemple 5.6 montre qu'on ne peut pas avoir des formules simples pour dim M (f ) s comme dans le cas des courbes nodales décrites dans le Corollaire 3.3.

Introduction (English version)

Algebraic varieties are a fundamental object of study in Algebraic Geometry. One of the most important problems in this field is the study of the singularities of hypersurfaces, which were first noticed in the case of curves, for instance, the ordinary double point of the curve defined by y 2 = x 2 -x 3 , the cusp of the curve given by y 2 = x 3 and the ordinary triple point

y 3 = x 3 -x 4 . Denote by S = C[x 0 , • • • , x n ]
the polynomial ring over C in n + 1 variables, with its natural grading, S = ⊕ r≥0 S r , where S r is the vector space of homogeneous polynomials in S of degree r. For a polynomial f ∈ S N , we define the Milnor algebra by M (f ) = S/J f , where J f is the ideal of S spanned by the partial derivatives

f 0 = ∂f ∂x 0 , • • • , f n = ∂f ∂xn .
Note that M (f ) is a graded C-algebra, whose grading is the one induced from the grading of S.

In this thesis, we study the Milnor algebra M (f ) of a homogeneous polynomial f and its relation to the corresponding hypersurface V (f ) : f = 0 in P n . In particular, we are interested in finding the dimensions of the graded pieces of this Milnor algebra, which are projective invariants of hypersurfaces. The smooth case being completely studied, we are only interested in the case when V (f ) has isolated singularities, specifically the case of curves in P 2 with ordinary double and triple points. Such questions have attracted a lot of interest recently, see [START_REF] Choudary | Dimca: Koszul Complexes and Hypersurface Singularities[END_REF], [START_REF] Dimca | Syzigies of Jacobian Ideals and Defects of Linear Systems[END_REF], [START_REF] Dimca | Graded Koszul cohomology and spectrum of certain homogeneous polynomials[END_REF], [START_REF] Dimca | Syzygies and logarithmic vector fields along plane curves[END_REF], [START_REF] Dimca | Sticlaru: Chebyshev Curves, Free Resolutions and Rational Curve Arrangements[END_REF], [START_REF] Dimca | Koszul Complexes and Pole Order Filtrations[END_REF], [START_REF] Eissydieux | Sur l'application des periodes d'une variation de structure de Hodge attachee aux familles de hypersurfaces a singularites simples[END_REF], [START_REF] Sernesi | The Local Cohomology of the Jacobian Ring[END_REF], [START_REF] Sticlaru | Invariants and rigidity of projective hypersurfaces[END_REF], [START_REF] Sticlaru | Log-concavity of Milnor algebras for projective hypersurfaces[END_REF].

On the other hand, Hodge theory plays an important role in Singularity Theory. Following P. Deligne, there exists a mixed Hodge structure on the cohomology groups of any algebraic variety, see [START_REF] Deligne | Theorie de Hodge, II and III[END_REF]. As part of such a structure, there are Hodge filtrations on these cohomology groups whose dimensions are computed thereafter in some cases. More precisely, we study in this thesis the mixed Hodge structures on the cohomology of the complements of singular curves in P 2 with ordinary double and triple points, in particular the relation between the Hodge filtration and the pole order filtration.

In chapter 2 we recall the basic facts on germs and isolated hypersurface singularities, regular sequences and complete intersections. Note that f 0 , ..., f n is a regular sequence if and only if V (f ) is a smooth hypersurface, which explains why the study on the Milnor algebra M (f ) is easy in this case. Then we discuss the Bézout Theorem, the Cayley-Bacharach Theorem and the defects of linear systems with respect to finite subsets (or 0-dimensional subschemes) in P n . Such defects play a key role in understanding the Milnor algebra M (f ) when V (f ) has isolated singularities, see [START_REF] Dimca | Syzigies of Jacobian Ideals and Defects of Linear Systems[END_REF].

In chapter 3, we introduce the Koszul complex K * (f ) of the partial derivative of a homogeneous polynomial f , whose top cohomology group is up to a shift in grading the Milnor algebra M (f ) of f . When V (f ) has isolated singularities, the only other nonzero cohomology group of K * (f ) describes the syzygies of the Jacobian ideal J f , i.e. the (homogeneous) syzygies among the derivatives f 0 , ..., f n . This relation gives interested results about the dimensions of the homogeneous components of M (f ). An important case is the one of nodal curves studied by A. Dimca and G. Sticlaru in [START_REF] Dimca | Koszul Complexes and Pole Order Filtrations[END_REF]. In particular, in the case of rational nodal curves, i.e. each irreducible component C i of C = V (f ) is rational, the Hilbert-Poincaré series is given explicitely in terms of the degree of f and the number of nodes, see Corollary 3.3. We partially generalize this result in the case of curves in P 2 with ordinary double and triple points, see Theorem 5.5 and show in Example 5.6 that a complete generalization cannot be hoped for.

In chapter 4 we give a brief introduction about Hodge structures and their relation to hypersurfaces singularities. In the first part of this chapter, we recall some definitions and properties introduced by P. Deligne [START_REF] Deligne | Theorie de Hodge, II and III[END_REF]. In the second part, we state and prove our first result, see Theorem 4.2, which relates the mixed Hodge theory of the complement of a plane curve C = V (f ) with double and triple points and the topology of the irreducible components C i of the curve C and the number of triple points. Using this result, we can compute all mixed Hodge numbers of the second cohomology group of the complement of curves with double and triple points, and therefore we can find the corresponding Betti numbers.We consider then the case where C has more general isolated singularities, where we compute the Hodge-Deligne polynomial of C and of its complement U . We generalize Theorem 4.2 to the case of arrangements of curves having ordinary singularities and intersecting transversely at smooth points. In the last section we show that the case of plane curves with ordinary singularities of multiplicity up to 4 (without assuming transverse intersection) is definitely more complicated.

In chapter 5 we start by giving some examples of syzygies of the Jacobian ideal J f , and relate them to the computation of the Poincaré series of the Milnor algebra M (f ) via the SINGULAR software. Then we give a quick presentation of the spectral sequence associated to a filtered complex, and apply this to the global algebraic de Rham complex of a hypersurface complement to define the pole order filtration. The pole order filtration P s and the Hodge filtration F s satisfies the inclusion F s ⊂ P s , as shown by Deligne-Dimca in [START_REF] Deligne | Filtrations de Hodge et par l'ordre du pole pour les hypersurfaces singulières[END_REF]. In the case of nodal curves, this inclusion becomes an equality and this explains why the theory is much simpler in this case. For curves with double and triple points, this inequality can be strict, and the first such examples are given in Example 5.5.

The main technical ingredient is the description of the pole order spec-tral sequence refered to above in terms of the homogeneous components of the cohomology of the Koszul complex K * (f ), see section (5.4) and the fact that this spectral sequence degenerates at the E 2 -term in the case of plane curves having only weighted homogeneous singularities, see [START_REF] Dimca | Koszul Complexes and Pole Order Filtrations[END_REF]. A different approach for these results can be found in the recent paper by A. Dimca and M. Saito, see [START_REF] Dimca | Graded Koszul cohomology and spectrum of certain homogeneous polynomials[END_REF].

In Propositions 5.4 and 5.5 we show for two classical line arrangements how to get de Rham representatives in the cohomology of the Milnor fiber using the syzygies of the Jacobian ideal. Note that unlike the cohomology of a hyperplane arrangement complement, where the description in terms of differential forms is due to Arnold, Brieskorn, Orlik and Solomon see [START_REF] Orlik | Arrangements of Hyperplanes[END_REF], Propositions 5.4 and 5.5 are the first such results for the Milnor fiber cohomology.

The second main result of this thesis is Theorem 5.5 in which we describe the dimension of M (f ) 2N -3 and the dimension of the space of syzygies of degree N -2 for a degree N curve in P 2 with double and triple points. Unlike the nodal case, here we obtain just some bounds for these invariants, and the equalities between the invariants and the bounds depend on the equality between the pole order filtration P s and the Hodge filtration F s .

We prove that such equalities hold when all the irreducible components C i of C = V (f ) are rational, and this yields a new situation when one has an equality F s = P s . However, even in this case, Example 5.6 show that simple formulas for all the dimensions dim M (f ) s as in the case of nodal curves described in Corollary 3.3 cannot be expected.

1 , f 1 ) ∼ (U 2 , f 2 ) if and only if f 1 | U 0 = f 2 | U 0 for some neighborhood U 0 of the origin with U 0 ⊂ U 1 ∩ U 2 .
An equivalence class of this relation is called a germ of analytic map from C n to C p at the origin and is denoted by (U, f ), or simply f . The set of all these germs is denoted by E n,p . When p = 1, we simply write

E n . Proposition 2.1. E n is a local C-algebra with unique maximal ideal m n = {f ∈ E n : f (0) = 0}. Let E 0 n,p be the set of germs f ∈ E n,p with f (0) = 0. An element of E 0 n,p
will also be denoted by f : (C n , 0) → (C p , 0).

Contact Equivalence

Denote by D n the group of complex map germs g : (C n , 0) → (C n , 0) which are analytic isomorphisms, the group operation being the composition of map germs, and by M n,p the group Gl(p, E n ), i.e. the group of invertible matrices of order p with entries in E n . Definition 2.1. Let G = M n,p D n be the semi-direct product given by the multiplication rule:

(A 1 , h 1 ).(A 2 , h 2 ) = (A 1 (A 2 • h -1 1 ), h 1 h 2 )
where composition with h -1 1 refers to all the entries of the matrix A 2 . Define an action on G by

G × E 0 n,p → E 0 n,p ((A, h), f ) → (A, h).f = A(f • h -1 )
where the germs f and f • h -1 are considered as column vectors with entries in m n ⊂ E n . An equivalence relation associated to this action is given by f 1 ∼ f 2 if and only if there exists (A, h) ∈ G such that f 2 = (A, h).f 1 and is called contact equivalence or K-equivalence. When p = 1, M n,p = E n , two germs f 1 and f 2 are K-equivalent if and only if there exist a map germ u ∈ E n and h :

(C n , 0) → (C n , 0) an analytic isomorphism, such that f 2 = u.(f 1 • h -1 ). Example 2.1. Let f ∈ m 2 n , and assume that V ( ∂f ∂x 1 , • • • , ∂f ∂xn ) = {x ∈ C n ; ∂f ∂x i = 0 i = 1, • • • , n} ⊆ {0}, i.e. f has at most isolated singularities. (i) If in addition corankf = n -rank( ∂ 2 f ∂x i ∂x j ) i,j=1,••• ,n ≤ 1,then f is K- equivalent to the normal form A k : f k = x k+1 1 +x 2 2 +• • • x 2 n . In particular, if corankf = 0, i.e. rank( ∂ 2 f ∂x i ∂x j ) i,j=1,••• ,n = n,
we have a nondegenerate singularity, and by Morse lemma f is K-equivalent to the normal form

A 1 : f 1 = x 2 1 + x 2 2 + • • • + x 2
n which corresponds to a nodal hypersurface.

(ii) The simplest singularities of corank 2 are the polynomials f K-equivalent to the normal form D 4 :

x 3 1 + x 1 x 2 2 + x 2 3 + • • • + x 2 n .
When n = 2, this corresponds to a triple point of a plane curve.

One can define in a similar way germs of analytic sets and one has the following, see [8, p.23],

Theorem 2.1. f 1 ∼ f 2 if and only if (f -1 1 (0), 0) (f -1 2 (0), 0).

Regular Sequences and Complete Intersections

Definition 2.2. Let X = f -1 (0) be the analytic set germ defined by f , where f :

(C n , 0) → (C p , 0) is an analytic map germ. X is called a complete intersection if dim X = n -p. Example 2.2. If p = 1, then X is a hypersurface of dimension n -1.
Therefore, X is a complete intersection called a hypersurface singularity.

Definition 2.3. Let E be a ring, a 1 , • • • , a p non-invertible elements in E. The sequence a 1 , • • • , a p is called a regular sequence in E if a j is not a zero divisor in the quotient ring E/(a 1 , • • • , a j-1 ) for j = 1, • • • , p. A maximal regular sequence (a 1 , • • • , a p )
, • • • , f n n homogeneous polynomials in S = C[x 1 , • • • , x n ]. Then {f 1 , • • • , f p } forms a regular sequence in S if and only if the algebraic variety V = V (f 1 , • • • , f n ) = {x ∈ C n ; f 1 (x) = • • • = f n (x) = 0} is a complete intersection, i.e. dim(V, x) = n -p for every x ∈ V .

Milnor Number and Tjurina Number

Let f : (C n , 0) → (C, 0) be a function germ. Denote by J f the Jacobian ideal of f , i.e. the ideal generated by ∂f ∂x

i , i = 1, • • • , n in the local ring E n . Definition 2.4. (i) The analytic Milnor algebra of the germ f is defined by M (f ) = E n /J f . Its dimension µ(f ) = dim C M (f ) is called the
Milnor number of the germ f.

(ii) The Tjurina algebra of the germ f is defined by

T (f ) = E n /(f, J f ). Its dimension τ (f ) = dim C T (f ) is called the Tjurina number of the germ f . Example 2.3. (i) µ(A k ) = τ (A k ) = k. (ii) µ(D 4 ) = τ (D 4 ) = 4. Let S = C[x 0 , • • • , x n ]
the polynomial ring over C in n + 1 variables, with its natural grading, S = ⊕ d≥0 S d , where S d is the vector space of homogeneous polynomials in S of degree d. For a polynomial f ∈ S d , we define the algebraic Milnor algebra by M (f ) a = S/J a f , where J a f is the ideal of S spanned by the partial derivatives ∂f ∂x 0 , • • • , ∂f ∂xn . One has the following result, see [8, p.111] Theorem 2.3. If f has an isolated singularity at the origin, then the nat-

ural morphism S -→ E n+1 induces an isomorphism of local C -algebras M (f ) a -→ M (f ).
We will use in the sequel the same notation for the algebraic and the analytic setting, namely M (f ) = S/J f . Note that M (f ) is a graded Calgebra, whose grading is the one induced from the grading of S. Definition 2.5. For any graded module M = ⊕ s≥s 0 M s over a graded Calgebra of finite type, define the Poincaré Series by

P M (t) = s≥s 0 (dim C M s )t s .
P M (t) = (1 -t d 1 ) • • • (1 -t dp ) (1 -t) n+1 .

Bézout's Theorem, Cayley-Bacharach Theorem and defects of linear systems

Let C and D be distinct curves in P 2 defined by F (x, y, z) = 0 and G(x, y, z) = 0 respectively. Suppose that C and D have no common components, and let P ∈ C ∩ D such that z P = 0. Regard P as a point in the affine plane with coordinates (x, y, 1). Let f and g be the dehomogenization of F and G respectively with respect to z. We define the intersection multiplicity of C and D at the point P by

i(C, D; P ) = dim O P (f, g) P ,
where O P is the local ring of germs of regular functions at P , and (f, g) P is the ideal generated by the germs of f and g in O P .

Remark 2.1. If z P = 0, then either x P or y P must be nonzero. In this case we dehomogenize F and G with respect to the nonzero coordinate, and we define the intersection multiplicity in an analogous way as above.

Example 2.4. Suppose we want to find the intersection multiplicity of each point of intersection of the curves C : F (x, y, z) = xz -y 2 = 0 and D : G(x, y, z) = y -z = 0. C and D have two points of intersection, namely, P 1 (1 : 1 : 1) and P 2 (1 : 0 : 0). Since z P 1 = 0, let f 1 (x, y) = x -y 2 and g 1 (x, y) = y -1 be the dehomogenizations of F and G respectively. Then,

O P 1 (f 1 , g 1 ) P 1 C[x, y] (x-1,y-1) (x -y 2 , y -1) (x-1,y-1) C[x, y] (x -y 2 , y -1) (x-1,y-1) C[x] (x -1) (x-1)
C.

Therefore, i(C, D; P 1 ) = 1. Now z P 2 = 0, then P 2 can be considered as the point at infinity. In this case, one dehomogenizes with respect to x = 0. Let f 2 (y, z) = z -y 2 and g 2 (x, z) = y -z be the corresponding dehomogenizations. Then,

O P 2 (f 2 , g 2 ) P 2 C[y, z] (y;z) (z -y 2 , y -z) (y,z) C[y, z] (z -y 2 , y -z) (y,z) C[y] (y(1 -y)) (y) C.
Hence, i(C, D; P 2 ) = 1 Suppose that Γ is a set of γ distinct points in C n . One may ask about "the failure of Γ to impose independent conditions on polynomials of degree ≤ k" for some positive integer k. The classical Cayley-Bacharach theorem gives an answer in the case of homogeneous polynomials of degree 3 in P 2 . The extended and thought-provoking history of the result starts with a prominent result by Papus of Alexandria proved in the fourth century A.D., and then it developed to have nine versions of the Cayley-Bacharach Theorem from which we will state only one, see [START_REF] Eisenbud | Cayley-Bacharach theorems and conjectures[END_REF]CB7].

Defects of Ideals and Defects of Linear Systems

Let I be a homogeneous ideal of S, we define the saturation I of I to be the set of elements s ∈ S such that for each i there exists a positive integer m i such that x m i i s ∈ I. I is a homogeneous ideal of S. Definition 2.6. Let Y be a 0-dimensional subscheme in P n defined by a homogeneous ideal I. We introduce the corresponding sequence of defects

def k Y = dim H 0 (Y, O Y ) -dim S k I k .
When Y is a subscheme defined by the Jacobian ideal of a homogeneous polynomial f , then

def k Y = τ (V ) -dim S k J k ,
where V is the hypersurface defined by f = 0, J = J f , and τ (V ) is the sum of all the Tjurina numbers of the singularities of V .

This positive integer is called the failure of Y to impose independent conditions on homogeneous polynomials of degree k.

Cayley-Bacharach Theorem

Before we state the Cayley-Bacharach theorem, we shall define the notion of residual subscheme Γ to a subscheme Γ of a zero-dimensional scheme Γ. We want this definition to have the following two properties: (α) the sum of the degrees of Γ and Γ should be equal to that of Γ, (β) the residual subscheme to Γ should be again Γ . We begin first by defining the Gorenstein local ring. Definition 2.7. Let A be a local Artinian ring, and m ⊂ A its maximal ideal. We say that A is Gorenstein if the annihilator of m has dimension one as a vector space over K = A/m.

Definition 2.8. A local ring R is Gorenstein if for every maximal regular sequence (F 0 , • • • , F k ) of elements of R the quotient A = R/(F 0 , • • • , F k ) is a Gorenstein Artinian ring.
Proposition 2.2. [START_REF] Eisenbud | Cayley-Bacharach theorems and conjectures[END_REF] The local rings of a zero-dimensional complete intersection scheme are Gorenstein. Definition 2.9. Let Γ be a zero-dimensional scheme with coordinate ring A(Γ) = S/I(Γ). Let Γ ⊂ Γ be a closed subscheme and I Γ ⊂ A(Γ) its ideal. By the subscheme of Γ residual to Γ we mean the subscheme defined by the ideal

I Γ = Ann(I Γ ) ⊂ A(Γ).
Remark 2.3. If Γ is a zero-dimensional complete intersection scheme, then Γ and Γ verify the properties (α) and (β) cited above. More generally, if Γ is a zero-dimensional Gorenstein scheme, then Γ and Γ are residual to each other, since I Γ = (I Γ ) ⊥ , where the orthogonal complement ⊥ is taken with respect to the natural paring

Q : A(Γ) × A(Γ) -→ C, (u, v) → p(u, v), with p : A(Γ) -→ C is a linear map inducing isomorphisms p x : m ⊥ x ←→ C for any point x ∈ Γ. Theorem 2.6. (Cayley-Bacharach) Let X 1 , • • • , X n be hypersurfaces in P n of degrees d 1 , • • • , d n , and suppose that the intersection Γ = X 1 ∩ • • • ∩ X n is zero-dimensional.
Let Γ and Γ be subschemes of Γ residual to one another in Γ, and set s = d i -n -1. If k ≤ s is a nonnegative integer, then the dimension of the family of hypersurfaces in P n of degree k containing Γ , which coincides with ( I Γ ) k , (modulo those containing all of Γ, which are exactly the elements of ( I Γ ) k ) is equal to the failure of Γ to impose independant conditions on hypersurfaces of complementary degree s -k.

Example 2.6. Let C be the degree 3 cuspidal curve defined by f = x 3 -y 2 z = 0. The Jacobian ideal of f is J f = (x 2 , y 2 , yz), therefore J f = (x 2 , y). Using Definition 2.6, we get def 0 C = 1 and def k C = 0 for k ≥ 1. Therefore, by Cayley-Bacharach Theorem, the dimension of the family of hypersurfaces of degree 1 (respectively 0) which contains J f is 1 (respectively 0).

Chapter 3 Koszul Complexes

In this chapter, we recall the definitions of the Koszul complexes of the partial derivatives of a homogeneous polynomial f in S. The importance of these complexes is that they are related to the Milnor algebra of f , M (f ), and they are useful tools in the computation of the dimensions of the graded pieces of M (f ). Note that a homology type complex can be seen as a cohomology type one. Indeed, if we consider the complex

K * : 0 → K n → K n-1 → • • • → K 0 → 0,
and if we set L s = K -s , we get the equivalent cohomology type complex L * : 0 → L -n → L -n+1 → • • • → L 0 → 0. Therefore, all definitions given below about the homology type complexes remain true for cohomology type complexes, and vice versa.

Homogeneous Koszul Complexes

Let f be a homogeneous polynomial in S of degree e. Define a complex

K * (f ) : 0 → Se f d → S → 0
where d(e f ) = f and d is S-linear. Here we regard e f as a homogeneous vector of degree e. Then d is a homogeneous morphism of degree 0, and can be seen as the multiplication by f . K * (f ) is called the Koszul complex of f . Note that (Se f ) k = S k-e , in other words, Se f = S[-e] as graded S-modules. Let f 0 , • • • , f n be homogeneous polynomials of the same degree e, then the tensor product

K * (f) = K * (f 0 , • • • , f n ) = K * (f 0 ) ⊗ • • • ⊗ K * (f n ) is the Koszul complex of f, where f = (f 0 , • • • , f n ). Using the shift introduced above, we have K m (f) = S km [-me], with k m = n+1 m , or explicitly 0 → S[-(n + 1)e] → S n+1 [-ne] → • • • → S n+1 [-e] → S → 0. Let K : 0 → K n+1 → • • • → K 0 →
0 be a (homology type) Koszul complex of graded finite type S-modules with a degree zero (i.e. homogeneous) differential. And let K s : 0 → K s,n+1 → • • • → K s,0 → 0 for every s ∈ N be the complex obtained by taking the homogeneous components of degree s. Define the Euler characteristic of K s by:

χ(K s ) = n+1 j=0 (-1) j dim(K s,j ).
Then Poincaré series associated to K is defined by:

P (K)(t) = s≥0 χ(K s )t s . Example 3.1. Let K : 0 → • • • → 0 → K 0 → 0, then P (K)(t) = s≥0 χ(K s )t s = P (K 0 )(t) as in Definition 2.5. Lemma 3.1. Let K : 0 → K n+1 d - → . . . d - → K 0 →
0 be a complex of graded finite type S-modules, then χ(K s )(t) = χ(H * (K s ))(t), and hence, P (K)(t) = P (H * (K))(t), where H * can be viewed as a complex

H * : 0 → H n+1 d - → . . . d - → H 0 → 0, with d = 0.
Proof. It is known that for any homomorphism d j : K j → K j-1 , K j = imd j + ker d j , therefore the Euler characteristic χ(K s )(t) can be given by

χ(K s )(t) = dim K s,0 -dim imd 1 -dim ker d 1 + dim imd 2 -dim ker d 2 + • • • + (-1) n+1 dim imd n+1 + (-1) n+1 dim ker d n+1 = dim cokerd 1 -dim ker d 1 imd 2 + • • • + (-1) n+1 dim ker d n+1 For j = 1, • • • , n, H j (K s ) = ker d j imd j+1 , H 0 = ker d 0 imd 1 = cokerd 1
, and H n+1 = ker d n+1 . This proves the first equality. Taking the sum with respect to s, we get the second equality.

A regular differential p-form on C n+1 is a differential form

ω = I c I dx i 1 ∧ • • • ∧ dx ip , where I = (i 1 , • • • , i p ) and c I ∈ S = C[x 0 , • • • , x n ].
Denote by Ω p be the C-vector space of all the p-forms on C n+1 . Ω p can be made into a graded S-module. Indeed, if f is a homogeneous polynomial of degree N , we set deg(f

dx i 1 ∧ • • • ∧ dx ip ) = N + p.
Let f 0 , • • • , f n be homogeneous polynomials in S e . Let K * (f 0 , • • • , f n ) be the (cohomology type) Koszul complex of f j which can be represented as follows:

K * (f) = K * (f 0 , • • • , f n ) : 0 ω∧ -→ Ω 0 ω∧ -→ Ω 1 → • • • → Ω n+1 → 0 where ω = f 0 dx 0 + f 1 dx 1 + • • • + f n dx n .
Note that |ω| = e + 1, so in order to have homogeneous differentials we have to consider the complex K * (f):

K * (f) : 0 → Ω 0 [-(n + 1)(e + 1)] → • • • → Ω n [-e -1] ω∧ -→ Ω n+1 → 0 Theorem 3.1. Let f 0 , • • • , f n be
homogeneous polynomials of degree e, and let K * (f 0 , . . . , f n ) and K * (f 0 , . . . , f n ) be the Koszul complexes defined as above, then

H p (K * (f)) s H n+1-p (K * (f)) s+n+1-p(e+1) = H n+1-p ( K * (f)) s+n+1 .
Proof. Let e i be a basis for the free S-module Se i , where e i is a homogeneous vector of degree e, for i = 0, • • • , n. Then {e I } |I|=p where

I = {i 1 , • • • i p } with i 1 < i 2 < • • • < i p
, and e I = e i 1 ⊗ • • • ⊗ e ip , form a basis for K p . On the other hand, a basis for Ω n+1-p is given by {dx J } |J|=n+1-p , where

J = {j 1 , • • • j n+1-p } with j 1 < j 2 < • • • < j n+1-p , and dx J = dx j 1 ∧ • • • ∧ dx j n+1-p .
Define the isomorphisms α p : K p -→ Ω n+1-p such that α p (e I ) = sign(σ)dx J where J = {0, • • • , n}\I and sign(σ) is the sign of the permutation that transforms (I, J) into (0, • • • , n). These isomorphisms α p induce isomorphisms between the complexes K * (f) and K * (f) (or K * (f) up to a sign, and they give the isomorphisms between H p (K * ) s and H n+1-p (K * ) s+n+1-p(e+1) . The grading on K * (f) is simpler to define and it will be used in the sequel of this thesis. The grading on K * (f) is useful in some questions since this makes the differential homogeneous of degree 0.

Remark 3.1. For p = 0, H n+1 (K * ) H 0 (K * ) = M (f). In particular, H n+1 (K * ) s+n+1 H 0 (K * ) s M (f) s .

Poincaré Series associated to the Jacobian ideal

In this section, we consider the hypersurface H ⊂ P n defined by f = 0 and we study the relation between the homology (repectively cohomology) groups of the Koszul complex of the partial derivatives

f 0 , • • • , f n of f , denoted by K * (f ) (respectivly K * (f ))
, and the singularities of H.

Remark 3.2. Note that H 0 (K * (f )) is the Milnor algebra of f given by M (f ) = S/J f .

For any hypersurface H, we have the following due to Saito [START_REF] Saito | On a Generalization of De Rham Lemma[END_REF].

Proposition 3.1. Let Σ be the singular locus of the hypersurface H. Then

H k (K) = 0 f or k > dim(Σ) + 1.
The case when H is smooth is already known and we have the following, see [8, p.109], as well as Proposition 3.1, Theorem 2.2 and Theorem 2.4. Proposition 3.2. The following are equivalent:

(i) H k (K * (f )) = 0 for k > 0 and P (M (f ))(t) = (1 -t N -1 ) n+1 /(1 -t) n+1 .
(ii) The hypersurface H is smooth. Proposition 3.3. For a hypersurface H with isolated singularities, we have:

P (H 0 (K * ))(t) -P (H 1 (K * ))(t) = (1 -t N -1 ) n+1 /(1 -t) n+1 .
Proof. Since the Poincaré series of a complex does not depend on differentials, then consider a homogeneous polynomial f s of degree N , such that D s : f s = 0 is a smooth hypersurface in P n , and let s K * be the Koszul complex associated to its partial derivatives, therefore we have by Lemma 3.1

P (H • (K * ))(t) = P (K * )(t) = P ( s K * )(t) = P (H • ( s K * ))(t) Since f s is smooth, then by (ii) ⇒ (i) of Proposition 3.2, P (H • ( s K * ))(t) = P (H 0 ( s K * ))(t) = (1 -t N -1 ) n+1 /(1 -t) n+1 .
On the other hand, we have by definition

P (H • (K * ))(t) = P (H 0 (K * ))(t) -P (H 1 (K * ))(t).
And the result follows.

If H is a hypersurface with only isolated singularities, a 1 , • • • , a p , then its singular locus Σ has dimension 0, therefore by Proposition 3.1 and Proposition 3.3, it is enough to study H 0 (K * ), so we are only left to compute P (M (f ))(t). A. D. R. Choudary and A. Dimca proved in [START_REF] Choudary | Dimca: Koszul Complexes and Hypersurface Singularities[END_REF] that: (β) The pairs (P n , H) and (P n , H) are homeomorphic.

Then neither (α) ⇒ (β) nor (β) ⇒ (α) holds in general.

And we have the following results, making the claim (A) more precise:

Corollary 3.1. [START_REF] Choudary | Dimca: Koszul Complexes and Hypersurface Singularities[END_REF] The morphism M (f ) k-1 → M (f ) k given by the multiplication by a generic linear form l ∈ S 1 , is an epimorphism for k > n(N -2) and an isomorphism for k > (n + 1)(N -2) + 1.

Corollary 3.2. [3] dim M (f ) k = τ (H) for all k > (n + 1)(N -2)
, where τ (H) is the sum of all Tjurina numbers τ (H, a j ).

Proposition 3.4. [9, p.162] Let H be a smooth hypersurface in P n having the same degree as H. Then

χ(H) = χ( H) + (-1) n µ(H),
where µ(H) is the sum of all Milnor numbers µ(H, a i ).

Proposition 3.5. Let H be a hypersurface in P n of degree N with isolated singularities and K * the homogeneous Koszul complex associated to its partial derivatives defined in section 3.1, then

P (H n ( K * (f )))(t) = P (H n+1 ( K * (f ))) -P (H n+1 ( K * (f s ))),
where f s is a homogeneous polynomial of degree N defining a smooth hypersurface in P n .

Proof. The equation follows from the isomorphism between the homology and the cohomology of the Koszul complexes given in Theorem 3.1, as well as Proposition 3.3.

Proposition 3.6. [START_REF] Dimca | Koszul Complexes and Pole Order Filtrations[END_REF] Under the same hypothesis of the previous proposition, we have:

dim H n (K * (f )) j = dim M (f ) j+N -n-1 -dim M (f s ) j+N -n-1 .
For a hypersurface H : f = 0 in P n with isolated singularities, A. Dimca and G. Sticlaru introduced three integers, as follows, see [START_REF] Dimca | Koszul Complexes and Pole Order Filtrations[END_REF]:

Definition 3.1. (i) the coincidence threshold ct(H) defined as ct(H) = max{q : dim M (f ) k = dim M (f s ) k f or all k ≤ q},
with f s a homogeneous polynomial in S of degree N such that H s : f s = 0 is a smooth hypersurface in P n .

(ii) the stability threshold st(H) defined as st(H) = min{q : dim M (f ) k = τ (H) f or all k ≥ q} where τ (H) is the total number of H, i.e. the sum of all the Tjurina numbers of the singularities of H.

(iii) the minimal degree of nontrivial syzygy mdr(H) defined as

mdr(H) = min{q : H n (K * (f )) q+n = 0}.
Using Proposition 3.6, we can prove that ct(H) = mdr(H) + N -2, and since by Proposition 3.

2 dim M (f s ) k = 0 for k > (n + 1)(N -2), then N -2 ≤ ct(H) ≤ (n + 1)(N -2)
. Also by Corollary 3.2, we have st(H) ≤ (n + 1)(N -2) + 1. We have the following results, see [START_REF] Dimca | Koszul Complexes and Pole Order Filtrations[END_REF]. Proposition 3.7. Let C : f = 0 be nodal curve of degree N in P 2 . Then one has ct(C) ≥ 2N -4.

Therefore, the dimensions of M (f ) k are determined for all k < 2N -3, and the next dimension is given by dim

M (f ) 2N -3 = n(C) + r j=1 g j = g + r -1 (3.1)
where n(C) = τ (C) is the total number of nodes of C and g j are the genera of the normalizations of the irreducible components C j of C whose number is r, and

g = (N -1)(N -2) 2 .
Corollary 3.3. If C is a rational nodal curve, then the Hilbert-Poincaré series P (M (f )) is completely determined by the degree N and the number of nodes n(C). In particular, st(C) = 2N -3 unless C is a generic line arrangement and then st(C) = 2N -4.

Proof. For a rational curve, g j = 0 for j = 1, • • • , r, then by Equation 3.

1 dim M (f ) 2N -3 = n(C) = τ (C). On the other hand, by Corollary 3.1 dim M (f ) k is decreasing for k ≥ 2N -3 and constant for k > 3N -5, this proves that st(C) = 2N -3. If C is a generic line arrangement, then dim M (f ) 2N -3 = n(C) = N 2 , dim M (f ) 2N -4 = dim M (f s) 2N -4 = dim M (f s ) N -2 = N 2 , and dim M (f ) 2N -5 = dim M (f s ) 2N -5 > N 2 . Example 3.2.
Let C be the degree 4 curve defined by f = x(x 3 + y 3 + z 3 ). Then C has 3 collinear nodes. st(C) ≤ 3N -5 = 7 and ct(C) ≥ 2N -4 = 4. Indeed a computation using Singular [START_REF] Decker | Sch¨onemann: Singular 3-1-3 -A computer algebra system for polynomial computations[END_REF] yields the following Hilbert-Poincaré series

HP (M (f ))(t) = 1 + 3t + 6t 2 + 7t 3 + 6t 4 + 4t 5 + 3(t 6 + t 7 + • • • )
and hence ct(C) = 4 and st(C) = 6.

HP (M (f ))(t) = 1 + 3t + 6t 2 + 7t 3 + 6(t 4 + t 5 + • • • ), which implies ct(C) = 4.
Theorem 3.3. (nodal case) Let H : f = 0 be a degree N nodal hypersurface in P n and N denotes the set of its nodes. Then

dim H n (K * (f )) nN -n-1-k = def k (N ) for 0 ≤ k ≤ nN -2n-1 and dim H n (K * (f )) j = τ (H) = |N | for j ≥ n(N -1). In other words, dim M (f ) T -k = dim M (f s ) k + def k (N ) for 0 ≤ k ≤ nN -2n -1, where T = (n, N ) = (n + 1)(N -2). In particular dim M (f ) T = τ (H), i.e. st(H) ≤ T.
For the general case of H with isolated singularities, see Theorem 1 in [START_REF] Dimca | Syzigies of Jacobian Ideals and Defects of Linear Systems[END_REF].

P (M (f ))(t) = 1 + 3t + 6t 2 + 7t 3 + 6t 4 + 4t 5 + 3(t 6 + t 7 + • • • ).
Example 3.5. Every cubic cuspidal curve is K-equivalent to the normal form f = x 3 -zy 2 whose Tjurina number is τ (C) = 2. We have st(C) ≤ 4, and ctC) ≥ 3 -2 = 1. Using Theorem 1 in [START_REF] Dimca | Syzigies of Jacobian Ideals and Defects of Linear Systems[END_REF] and Example 2.6, we get M (f ) 2 = 3 and M (f ) 3 = 2. Hence, the corresponding Poincaré series is

P (M (f ))(t) = 1 + 3t + 3t 2 + 2(t 4 + t 5 + • • • ). Example 3.6. Let C : f = x p y q + z N = 0 where p > 0, q > 0 and p + q = N > 2. st(C) ≤ 3(N -2) + 1 = 3N -5 and since qxf x -pyf y = 0, then mdr(C) = 1. Therefore ct(C) = N -1. By Proposition 1 in [11], one can compute dim M (f ) 3N -6-k for 0 ≤ k ≤ 2N -5, or dim M (f ) j for d -1 ≤ j ≤ 3d -6.
Hence, the corresponding Poincaré series is known for every N > 2.

In the case of plane curves with ordinary double and triple points we have the following results, see [START_REF] Dimca | Syzygies and logarithmic vector fields along plane curves[END_REF], Example 2.2 (iii) and Example 2.8 (iii).

Chapter 4 Hodge Structures 4.1 Basic Facts about Mixed Hodge Theory

A (pure) Hodge structure is an algebraic structure introduced by W. V. D. Hodge that applies to smooth and compact Kähler manifolds. In 1970, P. Deligne defined the mixed Hodge structure which is a generalization of the pure Hodge structure, that applies to all complex algebraic varieties. In this section, we give some basic definitions and properties. For more details, we refer to Deligne [START_REF] Deligne | Theorie de Hodge, II and III[END_REF], Voisin [START_REF] Voisin | Théorie de Hodge et Géométrie algébrique complexe[END_REF] and Appendix C [START_REF] Dimca | Singularities and Topology Hypersurface[END_REF]. (i) F is a finite filtration, i.e., there exist s, t ∈ Z with F s H C = H C and

F t H C = 0; (ii) H C = F p H C ⊕ F m-p+1
for all p ∈ Z, where the conjugation on H C is induced from the complex conjugation on C.

Or, equivalently, if we set H p,q = F p H C ∩ (F q H C ) for any pair (p, q) with p + q = m, then we have the following relations:

(α) H C = p+q=m H p,q .
(β) H p,q = H q,p . Conversly, given (α) and (β), H has a Hodge Structure where the Hodge filtration is defined by the formula

F p H C = s≥p H s,m-s .
Definition 4.2. The Hodge numbers of H C are defined by the formula

h p,q (H) = dim C H p,q
Remark 4.1. Since H p,q = H q,p , then h p,q = h q,p . Example 4.1. If X is a smooth projective variety, then the cohomology group H m (X, Q) has a pure Hodge structure of weight m, for any m ≥ 0. In particular the Hodge numbers h p,q (H) = h p,q (H p+q (X, C)) are well-defined.

We recall now the definition and some properties about mixed Hodge structure. (i) H is a finite dimensional Q-vector space;

(ii) W is a finite increasing filtration on H called the weight filtration;

(iii) F is a finite decreasing filtration on H C called the Hodge filtration, such that (Gr W k H, F ) is a Hodge structure of weight k for all k.

where

Gr W k H = W k H/W k-1
H, and the induced filtration is given by

F p (Gr W k H) C = (F p H C ∩ W k H C + W k-1 H C )/W k-1 H C .
When (H, W, F ) is a MHS we can define the mixed Hodge numbers by

h p,q (H) = dim Gr p F Gr W p+q H C .
The importance of the MHS is that the cohomology of any algebraic variety has such a structure. More precisely, we have the following theorem.

Theorem 4.1. (Deligne)

There is a functorial MHS on H * (X; Q) for any quasi-projective variety X such that the following properties hold for all m ≥ 0:

(i) the weight filtration W on H m (X; Q) satisfies 0 = W -1 ⊂ W 0 ⊂ • • • ⊂ W 2m = H m (X; Q); for m ≥ n = dim X, we also have W 2n = • • • = W 2m ; (ii) the Hodge filtration F on H m (X; C) satisfies H m (X; C) = F 0 ⊃ • • • ⊃ F m+1 = 0. For n = dim X, we also have F n+1 = 0; (iii) if X is a smooth variety, then W m-1 H m (X, Q) = 0 (i.e., all weights on H m (X; Q) are ≥ m) and W m H m (X, Q) = j * H m (X, Q) for any compactification j : X → X; (iv) if X is a projective variety, then W m H m (X, Q) = H m (X, Q) (i.
e., all weights on H m (X; Q) are ≤ m) and W m-1 = kerp * for any proper map p : X → X with X smooth. For a projective variety X ⊂ P n , let j m : H m (P n ) → H m (X) be the morphism induced by the inclusion j : X → P n . Define the primitive m-th cohomology group of X to be H m 0 (X) = cokerj m . The mixed Hodge numbers of the complement U of X and those of X are related by

h p,q (H m (U )) = h n-p,n-q (H 2n-m-1 0 (X)), (4.1)
see [9, p.246].

Let X be a variety, and let H * c (X; C) be the cohomology groups of X with compact support. These groups also admit MHS by the work of P. Deligne. If X is smooth then H m c (X; C) is dual to H 2n-m (X; C), where n = dim C X, (see Theorem 6.23 in [19, p.155]), and we have the following. Proposition 4.1. With the above notations, we have

h p,q (H m (X, C)) = h n-q,n-p (H 2n-m c (X, C)),
where h p,q are the mixed Hodge numbers for every p, q ≤ n.

We are only interested in the case m = 2, where the Hodge filtration on H 2 (U ) is given by

H 2 (U ) = F 0 = F 1 ⊃ F 2 ⊃ F 3 = 0, (4.2)
and the weight filration by

0 = W 1 ⊂ W 2 ⊂ W 3 ⊂ W 4 = H 2 (U ), (4.3)
Note that for k > 2 = dim U H k (U ) = 0 since U is an affine variety.

A. Dimca and G. Sticlaru described the dimensions of Gr s F H 2 (U ) for nodal curves in the following proposition. Proposition 4.3. Let C ⊂ P 2 be a nodal curve of degree N and set U = P 2 \C. Let C = j=1,r C j be the decomposition of C as a union of irreducible components, let ν : Cj → C j be the normalization mappings and set g j = g( Cj ). Then one has

dim Gr 1 F H 2 (U, C) = r j=1 g j and dim Gr 2 F H 2 (U, C) = (N -1)(N -2) 2 .
Example 4.2. Let C ⊂ P 2 be the nodal curve defined by f = x(x 3 +y 3 +z 3 ) = 0. C is the union of two irreducible components, a line C 1 : f 1 = x = 0 and a smooth curve C 2 : f 2 = x 3 +y 3 +z 3 = 0, see the figure below. Since C 1 and C 2 are smooth, the genus of each one is equal to the arithmetic genus given by the formula p a (C j ) =

(N j -1)(N j -2) 2
where N j is the degree of f j for j = 1, 2. Then we have g 1 = 0 and g 2 = 1. By Theorem 4.1, the Hodge filtration is given by 0

⊂ F 2 ⊂ F 1 = H 2 (U, C), and hence dim Gr 1 F H 2 (U, C) = dim F 1 F 2 = 1, and dim Gr 2 F H 2 (U, C) = dim F 2 = 3.
One of the main results of this thesis is a generalization of Proposition 4.3 to curves with ordinary double and triple points as singularities.

Theorem 4.2. Let C ⊂ P 2 be a curve of degree N . Suppose that C has only n nodes and t triple points. Set U = P 2 \C. Let C = j=1,r C j be the decomposition of C as a union of irreducible components,let ν : Cj → C j be the normalization mappings and set g j = g( Cj ). Then one has

dim Gr 1 F H 2 (U, C) = r j=1 g j and dim Gr 2 F H 2 (U, C) = (N -1)(N -2) 2 -t.
Proof. Suppose that the curve C j : f j = 0 has degree N j , and has n j nodes and t j triple points.

Using the normalization maps ν j and the additivity of the Hodge-Deligne polynomial, it follows that, P (C j ) = P (C j \(C j ) sing ) + P ((C j ) sing ) = P ( Cj \{(2n j + 3t j ) points}) + n j + t j = P ( Cj ) -P ({2n j + 3t j points}) + n j + t j = uv -g j u -

g j v + 1 -n j -2t j .
Indeed, it is known that for a smooth curve C, the genus g(C) is exactly the Hodge number h 1,0 (C) = h 0,1 (C). Now,

P (C) = P (C 1 ∪ • • • ∪ C r ) = r j=1 P (C j ) - 1≤i<j≤r P (C i ∩ C j ) + 1≤i<j<k≤r P (C i ∩ C j ∩ C k ) = ruv - r j=1 g j u - r j=1 g j v + r - r j=1 (n j + 2t j ) -( 1≤i<j≤r
N i N j -s) + t ,
where s (respectively t ) denotes the total number of triple points which are intersection of only two (respectively three) curves. The term 1≤i<j≤r N i N j -s is a result of Bézout's Theorem. Indeed, the total number of intersection points counted with multiplicities is 1≤i<j≤r N i N j , and it is easy to see that the intersection multiplicity of each of the s triple points is 2, i.e., each one is counted twice, whereas the intersection multiplicity of each of the t triple points is 1, in other words 3t contribution to the sum 1≤i<j≤r N i N j -s, each such being the intersection of two locally smooth curves in three ways. Next we have by the additivity P (U ) = P (P 2 )-P (C) where P (P 2 ) = u 2 v 2 +uv+1. Let's look at the cohomology of the smooth surface U . The group

H 4 c (U, C) is dual to the group H 0 (U, C), which is 1-dimensional of type (0,0). It follows that H 4 c (U, C) is 1-dimensional of type (2,
2) and its contribution to P (U ) is exactly the term u 2 v 2 . The group H 3 c (U, C) is dual to the group H 1 (U, C), which is (r-1)-dimensional of type (1,1). Indeed, by Theorem 4.1, the only nonzero weights on H 1 (U, C) are m = 1 and m = 2. But W 1 is zero since W 1 = j * (H 1 (P 2 )) = 0 where j : U -→ P 2 is the compactification of U and we apply again Theorem 4.1 (iii). It follows that the contribution of H 3 c (U, C) to P (U ) is exactly the term -(r -1)uv. The remaining terms come from group H 2 c (U, C), which is dual to the group H 2 (U, C). By theorem 4.1, the only nonzero weights on H 2 (U, Q) are m = 2, 3 and 4. On the other hand, W 2 = 0, since W 2 = j * H 2 (P 2 ). But H 2 (P 2 ) = αQ, where α = c 1 (L), c 1 denotes the Chern class of the line bundle L = O(1). Therefore, by the naturality property of Chern classes (see [START_REF] Bott | Differential Forms in Algebraic Topology[END_REF]), N α = j * (c 1 (L ⊗N )) = c 1 (j * (L ⊗N )) = c 1 (L ⊗N |U ) which is equal to zero since the line bundle L ⊗N |U has a nowhere vanishing section given by f , the defining equation of C. This implies that H 2 (U, C) has only classes of type (2, 1), (1, 2), and (2, 2). Therefore the dimension dim Gr 1 F H 2 (U, C) is the number of independent classes of type [START_REF] Atiyah | Introduction to Commutative Algebra[END_REF][START_REF] Bott | Differential Forms in Algebraic Topology[END_REF], which correspond to classes of type (1, 0) in H 2 c (U ) according to Proposition 4.1, and hence to the terms in u in P (U ). This gives the first equality. Now the dimension dim Gr 2 F H 2 (U, C) is the number of independent classes of type (2,1) or (2,2), which correspond respectively to the terms in v or the constant terms in the polynomial P (U ). This yields

dim Gr 2 F H 2 (U, C) = r j=1 (g j + n j + 3t j -1)+ 1≤i<j≤r d i d j +1-( r j=1 t j +s+t ).
Recall the formula

g j + n j + 3t j = p a (C j ) = (N j -1)(N j -2) 2 ,
where p a denotes the arithmetic genus, see [20, p.298 and p.54]. Recall also that the δ-invariant of a node A 1 is 1 and the δ-invariant of a triple point D 4 is 3. Let t := r j=1 t j + s + t , using the fact that N = r j=1 N j , we get:

dim Gr 2 F H 2 (U, C) = r j=1 (N j -1)(N j -2) 2 -1 + 1≤i<j≤r N i N j + 1 -t = r j=1 N 2 j -3N j 2 + 1≤i<j≤r N i N j + 1 -t = ( r j=1 N 2 j 2 + 1≤i<j≤r N i N j ) - r j=1 3N j 2 + 1 -t = 1 2 ( r j=1 N j ) 2 - 3N 2 + 1 -t = N 2 -3N + 2 2 -t = (N -1)(N -2) 2 -t.
Remark 4.3. As shown in the proof above, we have W 2 H 2 (U, C) = 0, and this implies

h 2,0 (H 2 (U )) = h 1,1 (H 2 (U )) = h 0,2 (H 2 (U )) = 0.
On the other hand, since F 0 = F 1 , then h 0,j = h j,0 = 0, for all j. We have also h 1,3 = h 3,1 = 0, since F 3 = 0.

This last remark and Theorem 4.2 allow us to compute all mixed Hodge numbers h p,q (H 2 (U )). Indeed, dim Gr 1 F H 2 (U ) = h 1,2 , and dim Gr 2 F H 2 (U ) = h 2,1 + h 2,2 . We conclude, Corollary 4.1. With the above notation and assumptions, we have the following.

(i) h 2,1 (H 2 (U )) = h 1,2 (H 2 (U )) = r j=1 g j . (ii) h 2,2 (H 2 (U )) = (N -1)(N -2) 2 -r j=1 g j -t. (iii) b 2 (U ) = (N -1)(N -2)

2

+ r j=1 g j -t, where b 2 (U ) denotes the second Betti number of the complement U .

In particular, it follows that H 2 (U ) is pure of type (2, 2) when g j = 0 for all j, a well known property in the case of line arrangements. In addition, in the case of rational curves with double and triple points, the Hodge filtration on H 2 (U ) is given by

H 2 (U ) = F 0 = F 1 = F 2 ⊃ F 3 = 0, since Gr 1 F (H 2 (U )) = F 1 F 2 = 0. Example 4.3. Let C : (x 2 -y 2 )(y 2 -z 2 )(x 2 -z 2 ) = 0.
C is the union of 6 lines in P 2 . It has 4 triple points ((1, 1, 1), (1, -1, 1), (-1, 1, 1), (-1, -1, 1)) and 3 nodes ((0, 1, 0),(1, 0, 0),(0, 0, 1)). We have g i = 0 for every i = 1, • • • 6, N = 6, and t = 4. Then, dim H 1 = 5, and according to the aforementioned theorem (4.2) we get dim

Gr 1 F H 2 (U, C) = dim F 1 F 2 = 0 and dim Gr 2 F H 2 (U, C) = dim F 2 = 6. Hence b 2 (U ) = 6.
Example 4.4. Let C : xyz(x 2 y + x 2 z + y 2 x + y 2 z + z 2 x + z 2 y) = 0. C is the union of 3 lines giving rise to a triangle and a smooth cubic curve. It has 3 triple points (the vertices of the triangle) and 3 nodes. We have

g 1 = g 2 = g 3 = 0, g 4 = 1, d = 6, and t = 3. Then dim H 1 (U ) = 3, dim Gr 1 F H 2 (U, C) = 1 and dim Gr 2 F H 2 (U, C) = 7, and b 2 (U ) = 1 + 7 = 8.
Example 4.5. Let C : (x 3 -y 3 )(y 3 -z 3 )(x 3 -z 3 ) = 0. C is the union of 9 lines. It has 12 triple points and no nodes. We have g i = 0 for every i = 1, • • • 6, N = 9, and t = 12. Then dim

H 1 = 8, dim Gr 1 F H 2 (U, C) = 0 and dim Gr 2 F H 2 (U, C) = 16 = b 2 (U ).

On Hodge Theory of Singular Plane Curves

In this section, we generalize the results of the previous section to curves with more general singularities. Suppose first C is an irreducible curve of degree N . Denote by a k , k = 1, ..., p the singular points of C, and let r(C, a k ) be the number of irreducible branches of the germ (C, a k ). Let ν : C → C be the normalization mapping. Using the normalization map ν and the additivity of the Hodge-Deligne polynomial, it follows that,

P (C) = P (C\(C) sing ) + P ((C) sing ) = P ( C\(∪ k ν -1 (a k )) + p = P ( C) - k P (ν -1 (a k )) + p = uv -gu -gv + 1 - k (r(C, a k ) -1).
It is known that one has the formula

g = (N -1)(N -2) 2 - k δ(C, a k ), (4.4)
relating the genus, the degree and the local singularities of C, and the δinvariants can be computed using the formula

2δ(C, a k ) = µ(C, a k ) + r(C, a k ) -1, (4.5)
where µ(C, a k ) is the Milnor number of the singularity (C, a k ). For both formulas above, see [21, p.85]. This proves the following result. (i) The Hodge-Deligne polynomial of C is given by

P (C)(u, v) = uv -gu -gv + 1 - k (r(C, a k ) -1),
with g given by the formula (4.4).

(ii) H 0 (C) = C is pure of type (0, 0).

(iii) H 2 (C) = C is pure of type (1 , 1).
(iv) The mixed Hodge numbers of the MHS on H 1 (C) are given by

h 0,0 (H 1 (C)) = k (r(C, a k ) -1), h 1,0 (H 1 (C)) = h 0,1 (H 1 (C)) = g.
In particular, one has the following formulas for the first Betti number of C.

b 1 (C) = k (r(C, a k ) -1) + 2g = (N -1)(N -2) - k µ(C, a k ).
Now we consider the case of a curve C having several irreducible components. More precisely, let C = j=1,r C j be the decomposition of C as a union of irreducible components C j , let ν j : Cj → C j be the normalization mappings and set g j = g( Cj ). Suppose that the curve C j has degree N j , denote by a j k for k = 1, ..., p j be the singular points of C j and let r(C j , a j k ) be the number of branches of the germ (C j , a j k ). Then the formulas (4.4) and (4.5) can be applied to each irreducible curve C j , as well as Proposition 4.4.

Let A be the union of the singular sets of the curves C j . Let B be the set of points in C sitting on at least two distinct components C i and C j . For b ∈ B, let n(b) be the number of irreducible components C j passing through b. By definition, n(b) ≥ 2. Moreover, note that the sets A and B are not disjoint in general, and their union is precisely the singular set of C.

Using the additivity of Hodge-Deligne polynomials we get

P (C) = P (C 1 ∪ • • • ∪ C r ) = r j=1 P (C j ) + (-1) l-1 0≤i 1 <•••<i l ≤r P (C i 1 ∩ • • • ∩ C i l ).
The first sum is easy to determine using Proposition 4.4. (i) The Hodge-Deligne polynomial of C is given by

P (C)(u, v) = ruv - r j=1 g j u - r j=1 g j v + r - j,k ((r(C j , a j k ) -1) - b∈B (n(b) -1).
with g j given by the formula (4.4).

(ii) H 0 (C) = C is pure of type (0, 0).

(iii) H 2 (C) = C r is pure of type (1, 1).

(iv) The mixed Hodge numbers of the MHS on H 1 (C) are given by

h 0,0 (H 1 (C)) = j,k ((r(C j , a j k ) -1) + b∈B (n(b) -1) -r + 1, h 1,0 (H 1 (C)) = h 0,1 (H 1 (C)) = j g j .
In particular, one has the following formula for the first Betti number of C.

b 1 (C) = j,k ((r(C j , a j k ) -1) + b∈B (n(b) -1) -r + 1 + 2 j g j .
Note that a point in the intersection A ∩ B will give a contribution to the last two sums in the above formula for P (C).

Example 4.6. Suppose C is a nodal curve. Then for each singularity a j k ∈ A one has a j k / ∈ B (otherwise we get worse singularities than nodes) and r(a j k ) = 2. Moreover, each point b ∈ B satisfies n(b) = 2. It follows that in this case we get

P (C)(u, v) = ruv - r j=1 g j u - r j=1 g j v + r -n 2 ,
with n 2 the number of nodes of C. More precisely, in this case we have n 2 = n 2 + n 2 , where n 2 (resp. n 2 ) is the number of nodes of C in A (resp. in B) and one clearly has

n 2 = S 1 := j,k ((r(C j , a j k ) -1), n 2 = S 2 := b∈B (n(b) -1).
Example 4.7. Suppose C has only nodes and ordinary triple points as singularities. Then let n 3 be the number of triple points and note that we can write as above n 3 = n 3 + n 3 , where n 3 (resp. n 3 ) is the number of triple points of C in A 0 = A \ B (resp. in B). For a point a ∈ A 0 , the contribution to the sum S 1 is 2, while the contribution to the sum S 2 is 0.

A point b ∈ B can be of two types. The first type, corresponding to the partition 3 = 1 + 1 + 1, is when b is the intersection of three components C j , all smooth at b. The contribution of such a point b is 0 to the sum S 1 and 2 to the sum S 2 . The second type, corresponding to the partition 3 = 2 + 1, is when b is the intersection of two components, say C i and C j , such that C i has a node at b, and C j is smooth at b. The contribution of such a point b is 1 to the sum S 1 and 1 to the sum S 2 .

It follows that the contribution of any triple point to the sum S 1 + S 2 is equal to 2. Since the double points in C can be treated exactly as in Example 4.6, this yields the following.

P (C)(u, v) = ruv - r j=1 g j u - r j=1 g j v + r -n 2 -2n 3 .
When there are only triple points in B of the first type, then we obviously have the following additional relations It follows that the contribution of any point of multiplicity 4 to the sum S 1 + S 2 is equal to 3. Since the double and triple points in C can be treated exactly as in Example 4.7, this yields the following.

S 1 = n 2 + 2n 3 , S 2 = n 2 + 2n 3 .
P (C)(u, v) = ruv - r j=1 g j u - r j=1 g j v + r -n 2 -2n 3 -3n 4 .
When there are only points of multiplicity 4 in B of the first type, then we obviously have the following additional relations

S 1 = n 2 + 2n 3 + 3n 4 , S 2 = n 2 + 2n 3 + 3n 4 .

Arrangements of transversely intersecting curves

Recall that C = j=1,r C j is the decomposition of C as a union of irreducible components C j , and the curve C j has degree N j . In this section we assume that any curve C j has only ordinary multiple points as singularities and let n k (C j ) denote the number of ordinary points on C j of multiplicity k. We also assume that the intersection of any two distinct components C i and C j is transverse, i.e. the points in C i ∩ C j are nodes of the curve C i ∪ C j . This implies in particular that A ∩ B = ∅. The formulas (4.4) and (4.5) yield the equality.

g j = (N j -1)(N j -2) 2 - 1 2 k µ(C j , a j k ) + r(C, a j k ) -1 , (4.6)
Using this, Theorem 4.3 gives the formula

dim Gr 2 F H 2 (U, C) = r j=1 (N j -1)(N j -2) 2 - 1 2 j,k µ(C j , a j k ) -r(C, a j k ) + 1 + + b∈B (n(b) -1) -r + 1.
If a j k is an ordinary m-multiple point on the curve C j , one has µ(C j , a j k ) = (m -1) 2 and hence

µ(C j , a j k ) -r(C, a j k ) + 1 = (m -1)(m -2).
If we denote by n m (resp. n m ) the number of m-multiple points of C coming from just one component C j (resp. from the intersection of several components C j ), we see that we have

j,k µ(C j , a j k ) -r(C, a j k ) + 1 = m (m -1)(m -2)n m .
This equality explains the contribution of the points in A. These formulas give the following result.

Theorem 4.4. With the above assumptions and notation, one has

dim Gr 2 F H 2 (U, C) = (N -1)(N -2) 2 - m m -1 2 n m ,
with n m = n m + n m the number of ordinary m-tuple points of C.

Curves with ordinary singularities of multiplicity ≤ 4

Let C ⊂ P 2 be a curve of degree N having only ordinary singular points of multiplicity at most 4. Set U = P 2 \ C, and let C = ∪ r j=1 C j be the decomposition of C in irreducible components. Then,

P (C) = r j=1 P (C j ) - 0≤i<j≤r P (C i ∩ C j ) + 0≤i<j<k≤r P (C i ∩ C j ∩ C k ) - 0≤i<j<k<l≤r P (C i ∩ C j ∩ C k ∩ C l ).
Let a j m denote the number of singular points of multiplicity m that belong to the component C j (note that a point can be singular on two components, being a node on each of them). Chapter 5

Syzygies of Jacobian Ideals for curves with double and triple points

Let f ∈ S be a homogeneous polynomial of degree N , f 0 , • • • , f n the corresponding partial derivatives, and H the hypersurface defined by f = 0. Denote by AR(f ) the graded S-submodule in S n+1 of all the relations involving the f j s, namely a = (a 0 , • • • , a n ) ∈ AR(f ) q if and only if R(q) : a 0 f 0 +• • •+a n f n = 0. Let KR(f ) ⊂ AR(f ) be the S-submodule of Koszul relations or the trivial relations. Note that a trivial relation is an element of the S-submodule in AR(f ) spanned by the relations R N -1 with a i equal to f j and a j equal to -f i , and the other coordinates are zero, i.e. (f i )f j + (-f j )f i = 0. The quotient module ER(f ) = AR(f )/KR(f ), called the module of essential relations, is the graded S-submodule of nontrivial syzygies. By Theorem 3.1 in [START_REF] Dimca | Syzigies of Jacobian Ideals and Defects of Linear Systems[END_REF],

dim ER(f ) nN -2n-1-k = dim H n (K * (f )) nN -n-1-k = def k Σ f (5.1) and def k Σ f = dim M (f ) T -k -dim M (f s ) k (5.2) for 0 ≤ k ≤ nN -2n-
P (M (f ))(t) = 1 + 3t + 6t 2 + 10t 3 + 15t 4 + 21t 5 + 28t 6 + 36t 7 + 42t 8 + 46t 9 +48(t 10 + t 11 + t 12 • • • .
By the same argument of the previous example, we find that the first nontrivial syzygies have degree 4 and doing a similar computation we find that a basis of ER(f ) 4 can be given by {r

1 : y 2 z 2 f x + x 2 z 2 f y + x 2 y 2 f z = 0, r 2 : (x 4 -2xy 3 -2xz 3 )f x + (y 4 -2x 3 y - 2yz 3 )f y + (z 4 -2x 3 z -2y 3 z)f z = 0}.
To find a basis of ER(f ) 5 whose dimension is 6, we simply multiply r 1 and r 2 by x, y, and z respectively, i.e. {xr 1 , yr 1 , zr 1 , xr 2 , yr 2 , zr 2 }. Indeed, these vectors are clearly linearly independant modulo the trivial relations since they are homogeneous of degree 4 < N -1 = 8.

Remark 5.1. The two example above give an equality in the inequality mdr(C) ≥ 2N/3 -2 from Proposition 3.8. This shows that this inequality is the best possible in general for curves with ordinary double and triple points.

Spectral sequences

Spectral sequences are algebraic tools introduced by Jean Leray in 1946, that are useful in the computation of homology and cohomology groups. These latters are usually not easy to be computed. The idea behind spectral sequences is to construct a sequence of bigraded abelian groups that converges to the homology or cohomology group we wish to obtain. Definition 5.1. A (first quadrant, cohomological) spectral sequence is a sequence {E r , d r } of bigraded groups

E r = p,q≥0 E p,q r ,
together with differentials

d r : E p,q r → E p+r,q-r+1 r , d 2 r = 0, such that H * (E r , d r ) = E r+1 .
The E p,q r term and the differential d r are shown in the figure below.

Definition 5.2. A spectral sequence {E r } is said to be convergent, if E r = E r+1 = • • • for r ≥ r 0 . The limit of E r is denoted by E ∞ .
A very common type of spectral sequences is the one associated to a filtered complex. Let K * : K

0 d → K 1 d → K 2 → • • • be a complex of abelian groups. A subcomplex K of K * is a subgroup such that dK ⊂ K * . A sequence of subcomplexes K * = F 0 K * ⊃ F 1 K * ⊃ • • •
is called a decreasing filtration on K * . The associated graded complex to K * is defined by

Gr p K * = F p K * F p+1 K *
with differential the one induced by d. The filtration on K * induces a filtration on its cohomology H * (K * ) = p≥0 H p (K * ), given by

F p H q = im(H q (F p K * ) → H q (K * ))
The associated graded cohomology is then

Gr p H * = F p H * F p+1 H * .
Proposition 5.1. Let K * be a filtered complex. There exists a spectral sequence {E r } with E p,q 0 = Gr p K p+q , E p,q 1 = H p+q (Gr p K * ), and E p,q ∞ = Gr p H p+q (K * ), for any p and q. We say that the spectral sequence converges to H * (K * ).

The differential d 0 is defined by d 0 : E p,q 0 -→ E p,q+1 0 . By the above isomorphism, d 0 : K p,q -→ K p,q+1 . Then the differential d 0 is exactly the differential δ, and the complex (E p,q 0 , d 0 ) is then the complex

• • • → K p,q-1 δ - → K p,q δ - → K p,q+1 → • • • . Consequently E p,q 1 = H p+q δ (K p, * ).
The differential d 1 is defined by

d 1 : E p,q 1 → E p+1,q 1
. The E p,q 2 term is the cohomology of the complex (E p,q 1 , d 1 ) given by

• • • → H p+q δ (K p-1, * ) d 1 -→ H p+q δ (K p, * ) d 1 -→ H p+q δ (K p+1, * ) → • • •
where the differential d 1 is the one induced by d. Hence

E p,q 2 = H d H δ (K * , * ).
Remark 5.2. There exists another filtration on K * given by

F q K n = q ≥q p+q=n K p,q .
By symmetry of p and q, one can construct another spectral sequence { E r , d r } converging to H * (K * ) such that E p,q 0 = K p,q , d 0 = d, E p,q 1 = H d (K * ,q ), d 1 induced by δ, and E p,q 2 = H δ H d (K * , * ). Definition 5.3. Given two spectral sequences {E * , * r , d r } and { E * , * r , d r }, we define a morphism of spectral sequences to be a sequence of homomorphisms of bigraded groups, f r : E * , * r -→ E * , * r of bidegree (0, 0), such that f r commutes with the differential d r for every r ≥ 0, i.e. f r • d r = d r • f r , and each f r+1 is induced by f r on cohomology, that is,

f r+1 : E * , * r+1 ∼ = H(E * , * r , d r ) -→ H( E * , * r , d r ) ∼ = E * , * r .

Rational differential forms and pole order filtrations

A regular differential p-form on C n+1 is a differential form

ω = I c I dx i 1 ∧ • • • ∧ dx ip , where I = (i 1 , • • • , i p ) and c I ∈ S = C[x 0 , • • • , x n ].
Denote by Ω p be the C-vector space of all the p-forms on C n+1 . Ω p can be made into a graded S-module. Indeed, if f is a homogeneous polynomial of degree N , we set deg(f

dx i 1 ∧ • • • ∧ dx ip ) = N + p.
Suppose f as above, and let U (f ) = {x ∈ P n ; f (x) = 0}. The differential p-forms on U (f ) can be regarded as elements of H 0 (U (f ), Ω p P n ). Every differential p-form ω on U (f ) can be written as ω = ∆(γ) f s , where γ ∈ Ω p+1 sN and ∆ is the contraction with the Euler vector field E = n j=0 x j ∂ ∂x j , see [9, chapter 6]. Definition 5.4. Let ω be a differential p-form on U (f ), the minimal value of s such that one can write ω = ∆(γ) f s is called the order of the pole of the rational differential form ω along the hypersurface H : f = 0, and is denoted by ord H (ω).

Let (A * 0 , d) = (H 0 (U (f ), Ω * ), d) be the de Rham complex associated to the hypersurface H : f = 0, i.e. the complex given by 0

→ A 0 0 d - → A 1 0 d - → • • • → A n 0 → 0. The differential d is given by d(ω) = d( ∆(γ) f s ) = - d f (γ)) f s+1 , where d f (γ) = f dγ -|γ| N df ∧ γ. Theorem 5.1. [18, p.453] (Grothendieck) H * (U (f )) = H * (A * 0 , d).
Proposition 5.2. (i) For any a ∈ Z/N Z there exists a spectral sequence

(E r (f ) a , d r ) on A * a with E s,t 1 (f ) a = H s+t ( P s A * a / P s+1 A * a ),
which converges to the Milnor fiber cohomology eigenspace H * (F ) a .

(ii) There exists a spectral sequence

(E r (f ), d r ) on A * with E s,t 1 (f ) = H s+t ( P s A * / P s+1 A * ),
which converges to the Milnor fiber cohomology H * (F ).

Spectral Sequences Associated to the Koszul Complex

The spectral sequences associated to the de Rham cohomology defined in section 5.2 can be compared to some algebraic spectral sequences of the Milnor fiber F : f = 1 associated to the Koszul Complex K * (f ) defined in section 3.1.

E r (f ), d r ) with E s,t 1 (f ) a = H s+t (B s, * a )
and converging to the cohomology H * (B * a ).

Remark 5.3. The cohomology group H s+t (B s, * a ) is in fact the cohomology of the complex

• • • → B s,t-1 a d 1 -→ B s,t a d 1 -→ B s,t+1 a → • • •
which is exactly, Denote by Ẽr (f ) a the reduced spectral sequence of E r (f ) a obtained by replacing E 0,0 r (f ) 0 by E 0,0 r (f ) 0 /C for a = 0, and Ẽr (f ) a = E r (f ) a for a = 0. Similarly, we define the reduced spectral sequence Ẽr (f ) a to be the spectral sequence given by replacing Ẽ0,0 r (f ) 0 by E 0,0 r (f ) 0 /C for a = 0, and Ẽr (f ) a = E r (f ) a for a = 0. Let δr : ( Ẽr (f ) a , d r ) -→ ( Ẽr (f ) a , d r ) be the induced morphisms. 3 ). Theorem 1 in [START_REF] Dimca | Monodromy of Triple Point Line Arrangements[END_REF] says that H 0,1 (F ) λ = 2 in this case. Hence, since the two 2-forms ω 1 and ω 2 survives at E 0,1 ∞,a = H 0,1 (F ) λ , and their corresponding forms [j * ∆(ω 1 )] and [j * ∆(ω 2 )] form a basis for H 0,1 (F ) λ .

• • • → Ω s+t-1 tN +a d 1 -→ Ω s+t (
We conclude that finding explicit Jacobian syzygies allows us to describe explicit bases for the eigenvalues of the monodromy acting on the cohomology of the Milnor fibers of homogeneous polynomials. This type of results is new even in the case of line arrangements, where the cohomology of the complement U was described in terms of differential forms since a long time, see [START_REF] Orlik | Arrangements of Hyperplanes[END_REF].

Consider now the case when a = 0. We will denote in the sequel E r (f ) 0 and E r (f ) 0 by E r (f ) and E r (f ) respectively. Suppose that we have a curve C ∈ P 2 : f = 0 with isolated singularities. we know that in this case the only nonzero cohomology groups of the Koszul complex K * (f ) are H 2 (K * (f )) and H 3 (K * (f )). Therefore, the nontrivial terms of the E 1 -spectral sequence belong to the lines (L) : s + t = 3 and (L ) : s + t = 2.

Consider first the terms on the line (L). By Theorem 5.4 and Remark 3.1, we have where f s = 0 defines a smooth curve in P 2 of degree N .

A study by A. Dimca and G. Sticlaru using the spectral sequences defined above, showed that for a nodal curve dim M (f ) 2N -3 = n(C)+ r j=1 g j , where n(C) is the number of nodes of C. We extend now this result to the case when C has nodes and ordinary triple points as singularities.

Theorem 5.5. Let C ∈ P 2 be a curve of degree N . Suppose C has n nodes (A 1 ) and t triple points (D 4 ). Let C = j=1,r C j be the decomposition of C as a union of irreducible components, let ν : Cj → C j be the normalization mappings and set g j = g( Cj ). Then we have the following.

(A) 0 ≤ dim M (f ) 2N -3 -τ ≤ r j=1 g j . In particular, (i) If all g i = 0, one has dim M (f ) 2N -3 = τ , i.e. st(C) ≤ 2N -3.

(ii) One has equality, i.e. dim M (f ) 2N -3 -τ = r j=1 g j if and only if H 2 (U ) satisfies F 2 H 2 (U ) = P 2 H 2 (U ).

(B) max(r-1+t-r j=1 g j , r-1) ≤ dim ER(f ) N -2 ≤ r-1+t. In particular, dim ER(f ) N -2 = r -1 + t if g j = 0 for all j.

Proof. (A) Consider the spectral sequence E p,q 1 (f ) = H p+q (K * (f )) (q+1)N that converges to H p+q-1 (U ), where U = P 2 \ C. By Theorem 2.4 (ii) in [START_REF] Dimca | Koszul Complexes and Pole Order Filtrations[END_REF], the differential d t 1 : E 2-t,t 1 → E 3-t,t 1 is bijective for t ≥ 2 and injective for t = 1. Consider first the case when t = 1. Since

d 1 : E 1,1 1 → E 2,1 1 is injective, then dim E 1,1 ≤ dim E 2,1 1 . dim E 1,1 1
= dim H 2 (K * (f )) 2N = dim ER(f ) 2N -2 which is equal to τ (C) by Equation 5.1. On the other hand, dim E 2,1 1 (f ) = dim H 3 (K * (f )) 3N = dim M (f ) 2N -3 . This proves the left hand side of the inequality in (A). To prove the right hand side of the inequality, consider the limit term E Cette thèse met en oeuvre le lien entre la théorie de Hodge et les singularités des hypersurfaces, en particulier des courbes planes.

Dans la première partie, on donne une étude détaillée des groupes de cohomologie du complémentaire U = P 2 \ C d'une courbe plane C. Le premier résultat principal montre que les dimensions des groupes gradués associés à ces groupes de cohomologie peuvent être calculées en fonction des invariants gémétriques simples dans le cas où C admet des points doubles et triples ordinaires . Une étude plus approfondie montre que le cas des singularités isolées quelconques est plus difficile à étudier. Ainsi, lorsque les singularités atteignent l'ordre 4, on ne peut pas s'attendre à trouver des formules simples.

Dans la deuxième partie, on trouve d'abord pour certains cas particuliers des représentants pour des classes de cohomologie de Rham dans la cohomologie de la fibre de Milnor d'un polynôme homogène f qui définit une courbe C à des singularités doubles et triples. Ces résultats sont les premiers résultats de ce type concernant la cohomologie de la fibre de Milnor et donnent lieu à de nouvelles questions intéressantes et difficiles.

On s'intéresse également dans cette partie, à trouver les dimensions des groupes gradués de l'algèbre de Milnor, M (f ) r , qui sont des invariants projectifs de l'hypersurface H définie par f = 0. Le deuxième résultat prin-cipal et les exemples qui suivent montrent que contrairement au cas des courbes nodales, dans le cas des courbes à points doubles et triples ordinaires, l'inclusion F s ⊂ P s dûe à Deligne-Dimca entre la filtration de Hodge F s et la filtration par l'ordre de pôle P s peut être stricte et rend l'étude plus difficile.

Question

Conclusion(English version)

This thesis studies the relation between the Hodge theory and singularities of hypersurfaces, in particular plane curves.

In the first part, we give a detailed study of the cohomology group of the complement U = P 2 \ C of a plane curve C. The first main result shows that the dimensions of the graded quotient groups of these cohomology groups with respect to the Hodge filtration can be computed in terms of simple geometric invariants in the case where C has only ordinary double and triple points as singularities. For more complicated singularities, the study becomes more subtle. In particular, we show that for curves singularities of multiplicity 4, we can not expect simple formulas as in the previous case.

In the second part, first we find in some particular cases, de Rham representatives in the cohomology of the Milnor fiber of a homogeneous polynomial f that defines a curve with ordinary double and triple points, using the syzygies of the Jacobian ideal. These results are the first such results for the Milnor fiber cohomology, and give rise to new interesting and difficult questions.

Pour n = 2

 2 , c.à.d. quand H est une courbe C ⊂ P 2 , l'étude devient plus simple et on a des résultats plus intéressants. Quand C est une courbe nodale, A. Dimca et G. Sticlaru ont trouvé que dim M (f ) r = dim M (f s ) r pour tout r ≤ 2N -4, c.à.d. les dimensions de M (f ) r sont connus jusqu'au rang 2N -4, et pour le rang suivant on a, dim M (f ) 2N -3 = n(C) + r j=1 g j , (0.1) où n(C) est le nombre des noeuds de C. Si C est une courbe nodale rationnelle, alors dim M (f ) 2N -3 = n(C) = τ (C), et par conséquent la série de Poincaré est complètement déterminée en fonction du nombre des noeuds et du degré de C.

  Germs, equivalence relations for germs and singularitiesGerms of Complex Analytic MappingsLet M = {(U, f ) where U is an open neighborhood of the origin in C n and f : U → C p an analytic map}. Define on M an equivalence relation given by, (U

Theorem 2 . 4 .

 24 [8, p.108] Let f 1 , • • • , f p ∈ S be homogeneous polynomials of degree d 1 , • • • , d p respectively, and assume that {f 1 , • • • , f p } form a regular sequence in S. If I = (f 1 , • • • , f p ) then M = S/I is a graded S-module and

Theorem 2 . 5 .Example 2 . 5 .

 2525 (Bézout's Theorem, [20, p.54]) Let C and D be distinct curves in P 2 , having degrees d and e respectively. Let C ∩ D = {P 1 , • • • , P s }. Then s j=1 i(C, D; P j ) = de. If C and D are the curves in example 2.4. The degree of C is 2 and that of D is 1. Then by Bézout's theorem, i(C, D; P 1 )+i(C, D; P 2 ) = 2, which is the result we got in the previous example.

Remark 2 . 2 .

 22 For a nodal hypersurface D with N nodes, h ∈ J k if and only if h vanishes on the set N and we get def k (N ) = |N | -codim{h ∈ S k |h(a) = 0 f or any a ∈ N }.

Theorem 3 . 2 .

 32 (A) The Euler characteristic χ(H) and dim M (f ) k for k > (n+1)(N -2) depend only on local invariants of the singularities (H, a j ), for example their Milnor numbers µ(H, a j ) and their Tjurina numbers τ (H, a j ). (B) The Betti numbers b m (H) and the Poincaré series P (M (f )) depend in general not only on local invariants but also on the position of the singularities of the hypersurface H. (C) Let H : f = 0 and H : f = 0 be two hypersurfaces as above. Consider the statements (α) P (M (f )) = P (M ( f )).

Example 3 . 3 .

 33 Let C be a generic line arrangement defined by f = xyz(x + y + z) = 0. Then C has 6 nodes, by Corollory 3.3, HP (M (f )) is all determined and we have st(C) = 2N -4 = 4 and ct(C) ≥ 4. Therefore

Example 3 . 4 . 4 .

 344 Let C : f = x(x 3 + y 3 + z 3 ). C has 3 collinear nodes. We have by Corollary 3.2, st(C) ≤ 7, and by Proposition 3.7, ct(C) ≥ The remaining dimensions, dim M (f ) 5 and dim M (f ) 6 , can be computed by Theorem 3.3. dim M (f ) 6 = dim M (f s ) 0 + def 0 (N ) = 1 + 2 = 3 using Definition 2.6. Similarly, one gets dim M (f ) 5 = 4. Hence, the corresponding Poincaré series is

Proposition 3 . 8 .

 38 Let C : f = 0 be a curve of degree N in P 2 with ordinary double and triple points. Then one has mdr(C) ≥ 2N/3 -2 and ct(C) ≥ 5N/3 -4.

Definition 4 . 1 .

 41 A (pure) Hodge Structure of weight m is a pair (H, F ), where H is a finite dimensional Q-vector space and F is a decreasing filtration on H C = H ⊗ C (called the Hodge filtration) such that:

Definition 4 . 3 .

 43 A mixed Hodge structure (MHS) is a triple (H, W, F ) where:

Remark 4 . 2 .

 42 By (iii) and (iv) in Theorem 4.1, if X is a smooth projective variety, then H m (X, Q) has a pure Hodge structure of weight m, as in Example 4.1.

Proposition 4 . 4 .

 44 With the above notation and assumptions, we have the following for an irreducible plane curve C ⊂ P 2 .

3 - 1 . 4 . 5 .

 3145 (C j , a j k ) -1).Consider now the alternated sum, where l ≥ 2. The only points of C that give a contribution to this sum are the points in B. Now, for a point b ∈ B, its contribution to the alternated sum is clearly given byc(b) = -... + (-1) n(b)-1 n(b) n(b) = -n(b) +Proposition With the above notation and assumptions, we have the following for a reducible plane curve C = j=1,r C j .

Example 4 . 8 .

 48 Suppose C has only ordinary points of multiplicity 2, 3 and 4 as singularities. Then let n 4 be the number of points of multiplicity 4 and note that we can write as above n 4 = n 4 + n 4 , where n 4 (resp. n 4 ) is the number of points of multiplicity 4 of C in A 0 = A \ B (resp. in B). For a point a ∈ A 0 of multiplicity 4, the contribution to the sum S 1 is 3, while the contribution to the sum S 2 is 0.A point b ∈ B can be of 4 types. The first type, corresponding to the partition 4 = 1 + 1 + 1 + 1, is when b is the intersection of 4 components C j , all smooth at b. The contribution of such a point b is 0 to the sum S 1 and 3 to the sum S 2 .The second type, corresponding to the partition 4 = 2 + 1 + 1, is when b is the intersection of 3 components, say C i , C j and C k , such that C i has a node at b, and C j and C k are smooth at b. The contribution of such a point b is 1 to the sum S 1 and 2 to the sum S 2 .The third type, corresponding to the partition 4 = 2 + 2, is when b is the intersection of 2 components, say C i and C k , such that C i and C k have a node at b. The contribution of such a point b is 2 to the sum S 1 and 1 to the sum S 2 .The fourth type, corresponding to the partition 4 = 3 + 1, is when b is the intersection of 2 components, say C i and C k , such that C i has a triple point at b, and C k is smooth at b. The contribution of such a point b is 2 to the sum S 1 and 1 to the sum S 2 .

2 times

 2 Now let b ∈ B such that n(b) = m. The number of such points is precisely n m . It follows that b∈B (n(b) -1) = m (m -1)n m .Let 1 ≤ i < j ≤ r and consider the intersection C i ∩ C j . It contains exactly N i N j points, since C i and C j intersects transversely. The sum S = 1≤i<j≤r N i N j represents the number of all such intersection points. Note that a point b ∈ B is counted in this sum exactly n(b) -1)n m .

Denote by b k 3 (respectively b k 4 ) 2 F H 2 (n m + b 2 4 ,

 34224 the number of triple points (respectively points of multiplicity 4) of C that are intersection of exactly k components, for k = 2, 3 (respectively k = 3, 4). Let b 2 4 (respectively b2 4 ) be the number of singular points p of multiplicity 4 in C representing the intersection of exactly Theorem 4.5. Let C ⊂ P 2 be a curve of degree N having only ordinary singular points of multiplicity at most 4. If U = P 2 \ C, then one has dim Gr with n m the number of ordinary m-tuple points of C and b 2 4 the number of singular points p of C which are smooth on one component C i of C and have multiplicity 3 on the other component C j of C passing through p.

Proposition 5 . 3 .

 53 Consider the double complex (B, d , d ) defined by B s,t a = Ω s+t (t+1)N +a s, t ∈ Z, d = d, and d (ω) = -|ω|N -1 df ∧ ω for a homogeneous differential form ω. Let (B * a , D f ) be the associated total complex, namely, B k a = s+t=k B s,t a , and D f = d + d with d d + d d = 0, and F p B k a = s≥p s+t=k B s,t a the decreasing filtration on B * a as in Example 5.3. By Proposition 5.1 and Example 5.3, we have: There exists a spectral sequence (

t+1)N +a d 1 -

 1 → Ω s+t+1 (t+2)N +a → • • • for any a ∈ {-N + 1, • • • , 1, 0}. By Example 5.3 d 1 is induced by d = -|ω| N df ∧ω.Therefore, this is exactly the Koszul complex defined in section 2.2 with differential a multiple of the differential of the Koszul complex. Hence, we conclude, Theorem 5.3. The E 1 term of the spectral sequence of Proposition 5.3 is given byE s,t 1 (f ) a = H s+t (K * f ) (t+1)N +a .The complexes B * a (respectively B * = a B * a ) and A * a (respectively A * ) defined in Section 1.9 are related by the following morphisms of complexes δ :B * a → A * -1 a δ : B * → A * -1 δ(ω) = ∆(ω)f -(t+1) , f or ω ∈ B s,t .These morphisms δ induce morphisms of spectral sequences, δ r : ( E r (f ) a , d r ) -→ (E r (f ) a , d r ).

Theorem 5 . 4 .

 54 [9, p.192] δr are isomorphisms for all r ≥ 1 and hence, by equation 5.3 they induce isomorphismsH k (B * a , D f ) = H k-1 (F ) a .Proof. In Example 5.2, the relations r 1 and r 2 correspond to the forms ω 1 and ω 2 ∈ H 2 (K * f ) 6 respectively, given byω 1 = y 2 z 2 dy ∧ dz -x 2 z 2 dx ∧ dz + x 2 y 2 dx ∧ dyand ω 2 = (x 4 -2xy 3 -2xz 3 )dy∧dz-(y 4 -2x 3 y-2yz 3 )dx∧dz+(z 4 -2x 3 z-2y 3 z)dx∧dy, and we have dω 1 = dω 2 = 0. Here we choose a = -3 to get N + a = 9 + a = 6, and therefore λ = exp( 2πia N ) = exp( -2πi

E 3 -t,t 1 (

 31 f ) = H 3 (K * (f )) (t+1)N = M (f ) (t+1)N -3 . By Proposition 3.6, the terms E 2-t,t 1 (f ) = H 2 (K * (f )) (t+1)N on the line (L ) have dimensions dim E 2-t,t 1 (f ) = dim M (f ) (t+2)N -3 -dim M (f s ) (t+2)N -3 ,

  OuverteDans la partie (B) du Théorème 5.5, on sait que dim ER(f) N -2 = r-1+t si g j = 0 pour tout j = 1, • • • ,r. En d'autres termes, chaque composante irréductible C j et chaque point triple p ∈ C entraîne une relation et il n'y a qu'une seule relation de dépendance parmi les r + t relations. Est-ce possible d'écrire les relations associées à la composante C j et le point associé à C j et au point p de la même manière que dans le cas nodal traité dans Théorèm 4.1 in [15]?

  [START_REF] Atiyah | Introduction to Commutative Algebra[END_REF], where T = (n+1)(N -2), and dim ER(f ) nN -2n+k = τ (H) for k ≥ 0. These formulas allow us to compute dim ER(f ) s for all s if we know the corresponding Poincaré series HP (M (f )), e.g. by computing it using the Singular software. We begin this chapter by some examples that Example 5.2. In this example we consider the curve C of Example 4.3(ii) which has only 12 triple points. Then dim ER(f ) 14+j = τ (C) = 12 × 4 = 48 for j ≥ 0. Using Singular again, we find:P (M (f s ))(t) = 1 + 3t + 6t 2 + 10t3 + 15t 4 + 21t 5 + 28t 6 + 36t 7 + 42t 8 + 46t 9 + 48t 10 +48t 11 +46t 12 +42t 13 +36t 14 +28t 15 +21t 16 +15t 17 +10t 18 +6t 19 +3t 20 +t 21 , and

  2,1 ∞ = P 1 H 2 (U ) P 2 H 2 (U ).By Theorem 5.2, P s H m (U ) ⊃ F s H m (U ). For s = 1 we have in addition F 1 H 2 (U ) = H 2 (U ), then P 1 H 2 = F 1 H 2 (U ) = H 2 (U ). For s = 2, P 2 H 2 ⊃ F 2 H 2 (U ) ⊃ H 2 (U ). It follows that the map Gr 1 F (H 2 (U )) -→ E 2,1

	Chapter 6
	Conclusion
	6.1 Conclusion(version Française)
	∞

Definition 4.4. Let X be a quasi-projective complex variety. The Hodge-Deligne polynomial of X is defined by P (X)(u, v) = p,q E p,q (X)u p v q where E p,q = s (-1) s dim Gr p F Gr W p+q H s c (X, C) = s (-1) s h p,q (H s c (X, C)).

Proposition 4.2. Let Y be a closed subvariety of X, then P (X) = P (X\Y )+ P (Y ).

Proof. This follows immediatley from the long exact sequence of the compactly supported cohomology, namely,

which is strictly compatible with the filtrations F * and W * , i.e. the sequence remains exact after application of Gr p F Gr W p+q .

Hodge theory for plane curves with double and triple points

In this section, we take X = C to be a curve in P 2 , and U = P 2 \ C. By Theorem 4.1, H * (U, C) has a mixed Hodge structure. In particular, for the Hodge filtration on H m (U ), one has

where the equality F 0 = F 1 follows from the fact that dim Gr 0 F H m (U ) = dim F 0 F 1 = h 0,m + h 0,m+1 + h 0,2m = 0 since h 0,q (H m (U )) = h 2,2-q (H 3-m 0 (C)) = 0, for all q since 2 > dim C = 1 (see Equation 4.1).

The case m = 1 is not interesting to be studied because the Hodge filtration on H 1 (U ) is trivial, i.e. is given by

Let's look now at the cohomology of the smooth surface U = P 2 \ C. By the additivity we get P (U ) = P (P 2 ) -P (C) where P (P 2 ) = u 2 v 2 + uv + 1. This yields the following consequence. Corollary 4.2.

As in the proof of Theorem 4.2, the contribution of H 4 c (U, C) to P (U ) is the term u 2 v 2 , and that of

) is the number of independent classes of type (1,2), which correspond to classes of type (1, 0) in H 2 c (U ), and hence to the terms in u in P (U ). This proves the following result.

In particular, all the components C j of the curve C are rational if and only if H 2 (U ) is pure of type (2, 2).

Example 4.9. Suppose C has only ordinary points of multiplicity 2, 3 and 4 as singularities. Then let n k be the number of points of multiplicity k, for k = 2, 3, 4 Then using Example 4.8, we get the formula

2 components, such that one of which has a triple point at p (respectively each one has a node at p). Then one has 0≤i<j≤r

Indeed, a point of type b

Hence, by Proposition 4.4, we get the following.

. Therefore, as above, we obtain

Finally we get

with n m the number of ordinary m-tuple points of C.

will be used later to give us an idea about the geometric interpretation of these syzygies.

Examples of syzygies

Using the program Singular, we can find the Poincaré series P (M (f ))(t) and P (M (f s ))(t). We have:

and

Using these two Poincaré series and equation ( 5.2), one can find dim ER(f

Therefore, the first nontrivial relations have degree 2.

Let ω = dQ 1 ∧ dQ 2 , then w ∧ df = 0. On the other hand, dQ 1 = 2(xdx -ydy) and dQ 2 = 2(ydy -zdz). Therefore ω = 4(xydx∧dy -xzdx∧dz +yzdy ∧dz).

Hence, ω ∧ df = 4(yzf x + xzf y + xyf z )dx ∧ dy ∧ dz = 0, which gives the first relation r : yzf x + xzf y + xyf z = 0, and a basis for ER(f ) 2 as a C-vector space can be given then by {r}. Multiplying r by x, y, and z respectively, we can find 3 elements of AR(f ) 3 . Doing some computation we find

) are clearly linearly independant, modulo the trivial relations which have degree ≥ N -1 = 5, they belong to ER(f ) 3 which has dimension 4, hence, they form a basis for ER(f ) 3 .

For the proof, refers to [START_REF] Ph | Principles of Algebraic Geometry[END_REF].

An important example of spectral sequences is the spectral sequence of a double complex.

Example 5.3. A double complex K * , * = K p,q is a complex with two differential operators, the vertical operator δ and the horizontal operator d, such that the rows and the columns of the diagram below are complexes of abelian groups, i.e. d 2 = δ 2 = 0, with the condition dδ + δd = 0. . . . . . . . . .

Out of a double complex we can form the associated single complex (K * , D) defined by

By Proposition 5.1, there exists a spectral sequence {E r , d r } converging to H * (K * ), such that

Definition 5.5. The polar filtration on the complex (A * 0 , d) is the decreasing filtration P defined by

The polar filtration on (A * 0 , d) induces a polar filtration on the cohomology groups H * (U ) given by

}, and we have the following, Theorem 5.2. [9, p.185] (Deligne-Dimca). For any hypersurface complement U = P n \H we have

for any integers s and m.

with differential d given by d( ω f s ) = dω f s -(s + a N ) df ∧ω f s+1 . This complex has a polar filtration given by

for m -s + 1 ≥ 0 and P s A m a = 0 for m -s + 1 < 0. Denote by A * the sum complex of A * a , i.e. A * = a A * a .

Let F be the Milnor fiber of f , i.e. the smooth hypersurface given by f = 1. There is a monodromy isomorphism h : F -→ F given by the multiplication by λ = exp( 2πi N ) and an induced monodromy operator h * : H * (F ) -→ H * (F ). The eigenvalues of h * are exactly the N th -roots of unity. We denote by H * (F ) a the eigenspace corresponding to λ a , we have (see [9, p.189]),

Cohomology of Milnor Fibers and of Plane Curves Complements

For a curve C ⊂ P 2 with isolated singularities, the only nontrivial groups in the spectral sequence are

0 can be written as ω ∧ df = 0 where df = f x dx + f y dy + f z dz and ω = ady ∧ dz -bdx ∧ dz + cdx ∧ dy. Therefore, it gives rise to a 2-form ω ∈ H 2 (K * f ) |ω|+2 . For the next two results, set λ = exp( -2πi

3 ) and let j : F → C 3 be the inclusion. Proposition 5.4. Let C be the curve defined by f : (x 2 -y 2 )(y 2 -z 2 )(x 2z 2 ) = 0, and let F : f -1 = 0 be the Milnor fiber. There exists a basis for H 0,1 (F ) λ given by [j * ∆(ω)], where

Proof. In Example 5.1, the degree 2 relation r gives rise to ω ∈ H 2 (K * (f )) 4 , therefore we should choose a = -2 to get N + a = 4. It follows that λ = exp( 2πia N ) = exp( -2πi 3 ). Note that dω = (a x + b y + c z )dx ∧ dy ∧ dz = 0 in this case. Since d 1 is induced by d, then d 1 (ω) = 0 and ω survives to E 0,1 ∞ (f ) a = H 0,1 (F ) λ whose dimension is 1, see Theorem 1 in [START_REF] Dimca | Monodromy of Triple Point Line Arrangements[END_REF], and the corresponding form [j * ∆(ω)] yields a basis for H 0,1 (F ) λ .

Proposition 5.5. Let C be the curve defined by f : (x 3 -y 3 )(y 3 -z 3 )(x 3z 3 ) = 0, and let F : f -1 = 0 be the Milnor fiber. There exists a basis for H 0,1 (F ) λ given by {[j * ∆(ω 1 )], [j * ∆(ω 2 ]}, where

and

is an epimorphism, and hence, dim E 2,1 ∞ ≤ dim Gr 1 F (H 2 (U )). By Proposition 4.2 dim Gr 1 F (H 2 (U )) = r j=1 g j . On the other hand, by Proposition 2.4 (iii) in [START_REF] Dimca | Koszul Complexes and Pole Order Filtrations[END_REF] the spectral sequence degenerate at the E 2 terms, i.e. E 2,1

and this proves the inequality (A).

For the proof of part (i), we refer to the proof of Corollary 3.3. (B) To prove the second inequality, we consider the differential d 0 1 :

by Theorem 4.2, and

On the other hand, Theorem 1 in [START_REF] Dimca | Monodromy of Triple Point Line Arrangements[END_REF] implies dim Er(f

The last two formulas implies that b 2 (U ) = 2g -τ + r -1. The first formula for b 2 (U ) now implies that r j=1 g j -t = 2g -τ + r -1. If we apply part A of the theorem we get Example 5.5. Let C be the degree 9 curve defined by f = (x 3 + y 3 + z 3 ) 3 + (x 3 + 2y 3 + 3z 3 ) 3 = 0. C is the union of 3 smooth curves, and have 9 triple points as singularities. Using Singular we can find dim M (f ) 16 = τ = 36.

Hence, one has strict inequality in (A)

Moreover, the inequalities in this case are

Example 5.6. Consider the following two distinct realizations of the Pappus configuration 9 3 , see [START_REF] Cohen | On Milnor fibrations of arrangements[END_REF] and [START_REF] Dimca | Singularities and Topology Hypersurface[END_REF], Example (6.4.16), p. 213. The first one is the line arrangement

A Singular computation yields HP (M (f ))(t) = 1+3t+6t 2 +10t 3 +15t 4 +21t 5 +28t 6 +36t 7 +42t 8 +46t 9 +48t 10 + +48t 11 + 47t 12 + 45(t 13 + ...

The second one is the line arrangement A 2 given by A 2 : g = xyz(x + y)(x + 3z)(y + z)(x + 2y + z)(x + 2y + 3z)(4x + 6y + 6z) = 0.

Using again the Singular software, we get HP (M (g))(t) = 1+3t+6t 2 +10t 3 +15t 4 +21t 5 +28t 6 +36t 7 +42t 8 +46t 9 +48t 10 + +48t 11 + 46t 12 + 45(t 13 + ...

Both arrangements have N = n = t = 9 and HP (M (f )(t) -HP (M (g)(t) = t 12 = 0. This shows once again that the curves with ordinary nodes and triple points are much more subtle than the nodal curves. In particular, for rational curves with ordinary nodes and triple points the Poincaré series HP (M (f )) is not determined by N , the number of irreducible components, the number of double and triple points, compare with Corollary 3.3. Not even the weaker invariant ct(C) is determined by these numbers, since ct(V (f )) = 11 and ct(V (g)) = 12. It also shows that it is rather difficult to control the dimension of the homogeneous components M (f ) r for r = 2N -3.

Furthermore, we are interesting in this section, in finding the dimensions of the graded pieces of the Milnor algebra, M (f ) r , which are projective invariants of the hypersurfaces H given by f = 0. The second main result and the following example show that unlike the case of nodal curves, in the case of curves with ordinary double and triple points, the inclusion F s ⊂ P s due to Deligne-Dimca between the Hodge filtration F s and the pole order filtration P s can be strict and and make the study more subtle.

Open Question

By Theorem 5.5, part (B), we know that dim ER(f ) N -2 = r -1 + t if all g j = 0. In other words, each irreducible component C j and each triple point p ∈ C yield a relation and there is just one dependence relation among these r + t relations. Is it possible to write the relations associated to the component C j and to the point p in a similar way to the case of nodal curves treated in Theorem 4.1 in [START_REF] Dimca | Koszul Complexes and Pole Order Filtrations[END_REF]?

Cohomology of Algebraic Plane Curves

Abstract: We describe in this thesis the dimensions of the graded quotients of the cohomology of a plane complement curve with respect to the Hodge filtration in terms of simple geometric invariants. The case of curves with ordinary singularities is discussed in details. In particular, we find the Hodge-Deligne polynomial of any curve C with isolated singularities and that of its complement, from which we can compute the mixed Hodge numbers of the second cohomology group of the complement of the curve, and consequently the correspondant Betti numbers. Furthermore, in the case of curves with ordinary double and triple points, we give relations to the Milnor algebra of the homogeneous polynomial f defining C, to the syzygies of the Jacobian ideal of f and pole order filtration on the second cohomology group of the curve complement.

---------------------------------Key Words: Plane Curves, Milnor algebras, Syzygies, Hodge and pole ordre Filtration.

Cohomologie des Courbes Planes Algébriques

Résumé: On décrit dans cette thèse les dimensions des groupes quotients gradués associés à la cohomologie du complémentaire d'une courbe plane par rapport à la filtration de Hodge en fonction de certains invariants géométriques. Le cas des courbes à singularités ordinaires est détaillé. En particulier, on trouve le polynôme de Hodge-Deligne d'une courbe C quelconque à singularités isolées et celui de son complémentaire duquel on déduit les nombres de Hodge mixtes ainsi que les nombres de Betti correspondants. Dans le cas des courbes dont les singularités sont des noeuds et des points triples ordinaires, on donne des relations importantes avec l'algèbre de Milnor du polynôme homogène f qui définit C, les syzygies de l'idéal Jacobien de f et la filtration par l'ordre de pôle du groupe cohomologique d'ordre 2 du complémentaire de la courbe.

---------------------------------Mots Clés: Courbes planes, Algèbres de Milnor, Syzygies, Filtration de Hodge et par l'ordre du pôle.