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2013 55 Jorge Ortigas Galindo Algebraic and Topological Invariants of Curves and Surfaces with Quotient Singularities (Invariantes topológicos y algebraicos de curvas y superficies con singularidades cociente) Departamento Director/es Matemáticas Cogolludo Agustín, José Ignacio Vallés, Jean Florens, Vincent Director/es Tesis Doctoral Autor Repositorio de la Universidad de Zaragoza – Zaguan http://zaguan.unizar.es UNIVERSIDAD DE ZARAGOZA Departamento Director/es Jorge Ortigas Galindo ALGEBRAIC AND TOPOLOGICAL INVARIANTS OF CURVES AND SURFACES WITH QUOTIENT SINGULARITIES (INVARIANTES TOPOLÓGICOS Y ALGEBRAICOS DE CURVAS Y SUPERFICIES CON SINGULARIDADES COCIENTE) Director/es Matemáticas Cogolludo Agustín, José Ignacio Vallés, Jean Florens, Vincent Tesis Doctoral Autor 2013 Repositorio de la Universidad de Zaragoza – Zaguan http://zaguan.unizar.es UNIVERSIDAD DE ZARAGOZA Departamento Director/es Director/es Tesis Doctoral Autor Repositorio de la Universidad de Zaragoza – Zaguan http://zaguan.unizar.es UNIVERSIDAD DE ZARAGOZA Algebraic and Topological Invariants of Curves and Surfaces with Quotient Singularities DOCTORAL THESIS co-supervised by Professors Jos´e Ignacio Cogolludo Agust´ın from Universidad de Zaragoza-IUMA, Jean Vall`es and Vincent Florens from Universit´e de Pau et des Pays de l’Adour by Jorge Ortigas-Galindo Zaragoza, May 6, 2013 CONTENTS AGRADECIMIENTOS v INTRODUCTION ix RESUMEN (Spanish) xxiii R´ ESUM´ E (French) xxxix Chapter I. Vmanifolds: Quotient Singularities, Embedded QResolutions, Intersection Numbers and Logarithmic Complex 1 I.1 V-manifolds and Quotient Singularities 2 I.1–1 The abelian case: normalized types 3 I.1–2 Orbisheaves 6 I.2 Weighted Blow-ups and Embedded Q-Resolutions 6 I.3 Intersection Theory on Abelian-Quotient V-Surfaces 13 I.3–1 Cartier and Weil Q-Divisors on V-Manifolds 13 I.3–2 Rational Intersection Number and Weighted Blow-ups 15 I.4 Weighted projective plane 16 I.4–1 Weighted B´ezout’s Theorem for Weighted Projective Planes 18 I.5 Logarithmic Complex and residues on V-manifolds 18 I.5–1 Logarithmic forms and residues on P2 w20 Chapter II. Local invariants on quotient singularities 25 II.1 Milnor fibers on quotient singularities 26 II.2 Local invariants on quotient singularities 27 iv CONTENTS II.2–1 Noether’s Formula 27 II.2–2 Definition of the δ-invariant on quotient singularities 28 II.3 The δw-invariant 32 II.3–1 The δw-invariant for function germs 32 II.3–2 The δw-invariant: the general case of local germs 37 Chapter III. Logarithmic Trees 43 III.1 Construction of logarithmic trees: Tnul P43 III.2 Construction of logarithmic trees: ˜ Tδ1δ2 P49 III.2–1 Recursive method to construct ˜ Tδ1δ2 P50 III.3 Some examples 54 III.4 Relation between of ˜ Tδ1,δ2 P(f) and ˜ Tnul P(f) 60 III.5 Number of local conditions 66 Chapter IV. Global Invariants: Adjunction-like Formula on P2 w69 IV.1 A genus formula for weighted projective curves 70 IV.2 Computing the continuous discretely 76 IV.2–1 A preliminary example 76 IV.2–2 Counting points 78 IV.3 Dedekind Sums 80 IV.3–1 Arithmetic properties 81 IV.3–2 Fourier-Dedekind Sums 82 IV.4 An Adjunction-like Formula on P2 w82 Chapter V. Structure of H•(P2 w\R;C) 89 V.1 The spaces Hk(P2 w\D;C) and the residue maps 89 V.2 Logarithmic 1-forms: a basis for H1(P2 w\D;C) 90 V.3 Two examples: ring structure of H•(P2 w\C;C) 93 V.3–1 Ring structure of H•(P2 w\{xyz = 0};C) 93 V.3–2 Ring structure of H•(P2 w\{xyz(xyz+(x3−y2)2)=0};C) 94 V.4 A holomorphic presentation for H2(P2 w\R;C) 98 CONCLUSION AND FUTURE WORK 107 CONCLUSI´ ON Y TRABAJO FUTURO (Spanish) 109 CONCLUSION ET TRAVAUX FUTURS (French) 111 BIBLIOGRAPHY 116 AGRADECIMIENTOS Si alguien me hubiera dicho hace un a˜no que a d´ıa de hoy estar´ıa aqu´ı sentado escribiendo estas l´ıneas no le hubiese cre´ıdo. . . Toda esta aventura, que ahora llega a su fin, comenz´o en 2008. Han sido a˜nos muy duros pero plagados de experiencias inolvidables, buenos momentos y, sobre todo, personas que me han acompa˜nado en este camino, allan´andolo y empuj´andome hasta sacar lo mejor de mi. Es por eso que una parte de esta tesis es suya, queriendo aprovechar este momento para agradecerles todo el apoyo recibido. Me gustar´ıa dar las gracias en primer lugar a las instituciones que, con su soporte econ´omico, han hecho posible el desarrollo de este trabajo. Al IUMA, por su beca destinada a los j´ovenes investigadores al inicio del m´aster. Al Gobierno de Arag´on por su beca de formaci´on del personal investigador y por su contribuci´on a trav´es del grupo “E15 Grupo Consolidado Geometr´ıa”. Al Ministerio de Educaci´on, por la beca FPU recibida y su financiaci´on por medio del proyecto “MTM2010-2010-21740-C02-02”. A las universidades de Zaragoza y Pau junto con sus Departamentos de Matem´aticas por acogerme hasta 2011. Finalmente, al Centro Universitario de la Defensa, mi casa desde 2011, por los medios que pone a nuestra disposici´on y la motivaci´on personal que ha supuesto. Este trabajo no habr´ıa sido posible sin la ayuda de distintos profesores que me han ido acompa˜nando a lo largo de estos a˜nos. Javier Otal, que fue el que meti´o el gusanillo de la Geometr´ıa Algebraica en mi cuerpo. Luis R´andez, por sus numerosos y buenos consejos. Andr´as N´emethi, por su hospitalidad durante mi estancia de 3 meses en Budapest (muchas gracias por todo Elda). Agradecer a Vincent Florens y Jean Vall`es por todo su apoyo, numerosas conversaciones de matem´aticas, su correcci´on de mi franc´es y por su confianza en m´ı al pensar todos estos a˜nos que por “chapurrear” franc´es vi AGRADECIMIENTOS pod´ıan hablar a una velocidad normal y yo enterarme de todo. A Kike por la fuerza que siempre transmite, por todo su conocimiento que nunca deja de sorprenderte e intenta explicarte de forma “f´acil”. Gracias por toda tu ayuda y apoyo durante estos a˜nos. Menci´on especial merece Jos´e Ignacio, mi padre acad´emico, por todo el mal que le he dado este tiempo (sobrevivir a dos hijos nuevos y un becario. . . ). Este trabajo no habr´ıa sido posible sin toda su ayuda, su apoyo an´ımico en los malos momentos, su paciencia a la hora de explicarme su Memoirs (que aunque ´el diga que no. . . de f´acil tiene poco) y su gran implicaci´on en la fase final de esta tesis. De estos a˜nos lo mejor que me llevo son las personas y amigos que han estado (y siguen estando) a mi lado y a las que quiero dar las gracias en estas l´ıneas. Echando la vista atr´as, me acuerdo de c´omo empez´o esta andadura. Me viene a la cabeza el a˜no del m´aster (2008-2009). Aquel mes de octubre donde comenzamos nuestro periplo en Bilbao, la primera lista de la compra con Julia en el supermercado de plaza Zab´alburu (que fuimos incapaces de llevar a la resi solos. . . ), nuestras “euskal” discusiones cari˜nosas con Nahikari y Arantzazu, las aventuras con el “Sunny” y los “Fluvis (de ambos sexos)” junto con Luis, la fabada Litoral con “nuggets” de pollo. . . y todos los maravillosos findes en Donosti que han ido aconteciendo durante estos a˜nos siguientes convirti´endose en una tradici´on m´as (gracias especialmente a ti Nahikari por todo tu ´animo durante estos a˜nos). Recordar a todo el grupo de “becarios aragoneses”, Jos´e Mari, Sergio, Raquel, Michel, Antonio Oller, Carmen, ´ Angeles, “Chavi” y “Chiquillo” por esos numerosos caf´es que tanto ayudan, Antonio Otal (cuantas conversaciones y mutuo ´animo en nuestro anterior despacho), Pooo (formando, la ya famosa dualidad, “Pooo-Chatooo”), la nueva familia alternativa formada por Juan (gracias por t´us ´animos y toda la ayuda burocr´atica francesa. . . ), Adela (por su apoyo en la fase final) y Sim´on (nuestro peque˜no “huevo kinder”) con su hija “la planta” moribunda. Los “becarios for´aneos”, Guadalupe (con su juventud), Eva, Nacho, Irene y Ana; de los que tanto apoyo y cari˜no he recibido y que me han hecho sentirme como en casa cada vez que les visitaba. Agradecer a mis compa˜neros de “Mates III” su apoyo en este ´ultimo a˜no de tesis, Marcos (con sus bromas de pique cari˜noso), Tere y Etel por su ayuda, hacer f´acil el trabajo diario y acompa˜narme dentro del grupo “afterhours” del CUD. Dar las gracias a todos mis compa˜neros de matem´aticas por todos esos caf´es y comidas: Javi, Eva, Mart´ın (gracias por todas esas conversaciones de matem´aticas), ´ Alvaro (y su mascota el “perroala”), Silvia INTRODUCTION xiii pairwise coprime positive integers. There is a natural action of the multiplicative group C∗on C3\{0}given by (x0, x1, x2)7−→ (tw0x0, tw1x1, tw2x2). The set of orbits C3\{0} C∗under this action is denoted by P2 wand it is called the weighted projective plane of type w. Let us recall the definition of one of the most important objects we are dealing with. Definition 2. An embedded Q-resolution of (H, 0) ⊂(M, 0) is a proper analytic map π:X→(M, 0) such that: (1) Xis a V-manifold with abelian quotient singularities, (2) πis an isomorphism over X\π−1(Sing(H)), (3) π−1(H) is a Q-normal crossing hypersurface on X(see Definition (I.2.1)), and (4) the strict transform ˆ H:= π−1(H\{0}) is Q-smooth (see Definition (I.2.1)). Embedded Q-resolutions are a natural generalization of the usual embedded resolutions, for which some of the invariants studied from Chapters II to V can be effectively calculated. In Section I.3 we develop an intersection theory in order to study embedded Q-resolutions in dimension 2 (see [AMO11b] and [Mar11] for further details). One has to deal with two types of divisors on V-manifolds: Weil and Cartier divisors. Weil divisors are locally finite linear combinations with integral coefficients of irreducible subvarieties of codimension 1 and Cartier divisors are global sections of the quotient sheaf of meromorphic functions modulo non-vanishing holomorphic functions. The relationship between Cartier divisors and line bundles provides a useful way to define the intersection multiplicity of two divisors. In the smooth category, both notions coincide but this is not the case for singular varieties. Theorem (I.3.3) ([AMO11a]) allows one to develop a rational intersection theory on Vmanifolds (see [AMO11b]). Definition 3 (Local intersection number on X(d;a, b), [Ort09]).Denote by Xthe cyclic quotient space X(d;a, b) and consider two divisors D1= {f1= 0}and D2={f2= 0}given by f1, f2∈C{x, y}reduced and without common components. Assume that, (d;a, b) is normalized. Then as Cartier divisors D1=1 d{(X, fd 1)}and D2=1 d{(X, fd 2)}. The local number (D1·D2)[P]at a point Pof type (d;a, b) is defined as (D1·D2)[P]=1 d2dimCOP hfd 1, fd 2i xiv INTRODUCTION where OP=C{x, y}Gdis the local ring of functions at P(recall §I.1–2). This local intersection theory developed allows us, for instance, to compute the Weighted B´ezout’s Theorem for weighted projective planes (Proposition (I.4.7)), which will be of particular interest in some forthcoming results. Proposition 1 ([Ort09]).The intersection number of two Q-divisors, D1 and D2on P2 wwithout common components is D1·D2=X P∈D1∩D2 (D1·D2)[P]=1 ¯wdegw(D1) degw(D2)∈Q, where ¯w=w0w1w2and degw(Di) = deg(φ∗(Di)) (see (7)). In §I.5 we develop a De Rham cohomology for projective varieties with quotient singularities. We shall recall some Hodge theoretical results for projective V-manifolds which will be of particular interest for us. All these results, with their respective proofs, can be found in the first chapter of [Ste77]. The definition of logarithmic forms and residues on non-normal crossing Qdivisors on V-surfaces will be provided. Let Dbe a Q-divisor in P2 w. The complement of Dwill be denoted by XD. Let us fix π:XD−→ P2 waQ-resolution of the singularities of Dso that the reduced Q-divisor D= (π∗(D))red is a union of smooth Q-divisors on XDwith Q-normal crossings. Definition 4. AC∞form ϕon XDshall be called logarithmic (with respect to a divisor Dand a Q-resolution π)if π∗ϕis logarithmic on XDwith respect to the Q-normal crossing divisor D(see Definition (I.5.6)). Therefore, one has the corresponding sheaf π∗ΩXD(loghDi). Once Dand πare fixed one can define the residue map Res[∗] π(ϕ)of a logarithmic form ϕas follows π∗Ωk XD(loghDi)Res[k] π −→ H0(D[k];C) ϕ7→ Res[k](π∗ϕ). This previous definition is independent from the Q-resolution. For instance, in the particular case of X(d;a, b) and the Res[2] one has the following. Definition 5. Let hbe an analytic germ on X(d;a, b) written in normalized form (Definition (I.1.9)). Let ϕ=hdx ∧dy xy be a logarithmic 2-form with INTRODUCTION xv poles at the origin. Then Res[2](ϕ) := 1 dh(0,0). Second part: local invariants In Chapter II we extend the concept of Milnor fiber and Milnor number of a curve singularity allowing the ambient space to be a quotient surface singularity (§II.1). A generalization of the local δ-invariant is defined and described in terms of a Q-resolution of the curve singularity (§II.3). In particular, when applied to the classical case (the ambient space is a smooth surface) one obtains a formula for the classical δ-invariant in terms of a Qresolution, which simplifies considerably effective computations. All these tools will finally allow for an explicit description of the genus formula of a curve defined on a weighted projective plane in terms of its degree and the local type of its singularities in Chapter IV. Definition 6 ([CAMO13]).Let C={f= 0} ⊂ X(d;a, b) be a curve germ. The Milnor fiber Fw tof (C,[0]) is defined as follows, Fw t:= {F=t}/Gd. The Milnor number µwof (C, P) is defined as follows, µw:= 1 −χorb(Fw t). Note that alternative generalizations of Milnor numbers can be found, for instance, in [ABFdBLMH10, BLSS02, uT77, STV05]. The one proposed here seems more natural for quotient singularities (see Example (IV.1.18)), but more importantly, it allows for the existence of an explicit formula relating Milnor number, δ-invariant, and genus of a curve on a singular surface (Chapter IV). In §II.3 we present a version of Noether’s formula (see Theorem (II.2.1)) for curves on quotient singularities and Q-resolutions. Theorem 2 (Noether’s Formula, [CAMO13]).Consider Cand Dtwo germs of Q-divisors at [0] without common components in a quotient surface singularity. Then the following formula holds: (C·D)[0] =X Q≺[0] νC,QνD,Q pqd , where Qruns over all the infinitely near points of (CD, [0]) and Qappears after a blow-up of type (p, q)of the origin in X(d;a, b). xvi INTRODUCTION We define the local invariant δwfor curve singularities on X(d;a, b). Definition 7 ([CAMO13]).Let Cbe a reduced curve germ at [0] ∈ X(d;a, b), then we define δwas the number verifying χorb(Fw t) = rw−2δw, where rwis the number of local branches of Cat [0], Fw tdenotes its Milnor fiber, and χorb(Fw t) denotes the orbifold Euler characteristic of Fw t. A recurrent formula for δwbased on a Q-resolution of the singularity is provided in Theorem (II.2.5). Theorem 3 ([CAMO13]).Let (C, [0]) be a curve germ on an abelian quotient surface singularity. Then δw=1 2X Q≺[0] νQ dpq (νQ−p−q+e), where Qruns over all the infinitely near points of a Q-resolution of (C, [0]), Qappears after a (p, q)-blow-up of the origin of X(d;a, b), and e:= gcd(d, aq− bp). In §II.3–1 an interpretation of the δwinvariant as the dimension of a vector space is given. In the classical case this invariant can be interpreted as the dimension of a vector space. Since δwis in general a rational number, a similar result can only be expected in certain cases, namely, when associated with Cartier divisors (see Theorem (II.3.7)). Theorem 4 ([CAMO13]).Let f: (X(d;a, b), P)→(C, P)be a reduced analytic function germ. Assume (d;a, b)is a normalized type. Consider R=OP hfithe local ring associated with fand Rits normalization ring. Then, δw P(f) = dimCR R∈N. In §II.3–2 we will present, in some way, a generalization of this result. To do this some previous definitions are needed. For a given k≥0, one has the module OP(k) (for further details see §I.1–2), OP(k) := {h∈C{x, y}| h(ξa dx, ξb dy) = ξk dh(x, y)}. Let {f= 0}be a germ in P∈X(d;a, b). Note that if f∈ OP(k), then one has the following OP-module, OP(k−a−b) verifying OP(k−a−b) = {h∈C{x, y}| hdx ∧dy fis Gd-invariant}. INTRODUCTION xvii Definition 8. Let D={f= 0}be a germ in P∈X(d;a, b), where f∈ OP(k). Consider πaQ-resolution of (D, P). (1) Let Mlog D,π denote the submodule of OPconsisting of all h∈ OP such that the 2-form ω=hdx ∧dy f∈Ω2 P(a+b−k) is logarithmic at P, with respect to Dand the embedded Q-resolution π(recall Definition 4). (2) Let Mnul D,π denote the submodule of Mlog D,π consisting of all h∈ Mlog D,π such that the 2-form ω=hdx ∧dy f admits a holomorphic extension outside the strict transform b f. This last module will play an important role in the construction of a presentation for the cohomology ring of P2 w\R in Chapter V. Definition 9. Let D={f= 0}be a germ in P∈X(d;a, b). Let us define the following dimension, KP(D) = KP(f) := dimCOP Mnul D,π . The number KP(f) gives us the minimal number of conditions required to a generic germ h∈ OP(s) so that h∈ Mnul D,π(s). Theorem 5. Let f, g ∈ O(k),k∈N, be two germs at P∈X(d;a, b). Then, KP(f)−KP(g) = δw P(f)−δw P(g). In Chapter III we continue defining some other logarithmic modules and sheaves associated with a Q-divisor Dand a Q-resolution πapart from the one in Definition 8. Their global sections will allow for a construction of logarithmic 2-forms on Din Chapter V. We will construct two kinds of trees associated with an analytic germ {f= 0}at P∈X(d;a, b), ˜ Tnul P(f) (see §III.1) and ˜ Tδ1δ2 P(f) (see §III.2) being δ1and δ2two local branches of fat P. These trees will provide a useful description of the logarithmic modules previously defined. Definition 10. Let D={f= 0}be a germ in P∈X(d;a, b), where f∈ OP(k). Consider πaQ-resolution of (D, P). Let us define Mδiδj D,π the submodule of Mlog D,π consisting of all h∈ Mlog D,π such that the 2-form ω=hdx ∧dy f xviii INTRODUCTION has zero residues outside the edges of the path γ(δ1, δ2). As a consequence of the construction of the trees ˜ Tnul P(§III.1) and ˜ Tδ1,δ2 P (§III.2) and Definitions 8 and 10 one has the following characterization: Mnul D,π ={h∈ OP|˜ TP(D, π)|h≥˜ Tnul P(D, π)}. Mδiδj D,π ={h∈ OP|˜ TP(D, π)|h≥˜ Tδiδj P(D, π)}. Consider now the following dimension, Kδiδj P(D) = Kδiδj P(f) := dimCOP Mδiδj D,π . The number Kδiδj P(f) gives us the minimal number of conditions required to a generic germ h∈ OP(s) so that h∈ Mδiδj D,π(s). Definition 11. The degree of a weighted tree Twill be defined as follows deg(T) := X Q∈|T | w(T, Q) 2dpq (w(T, Q) + p+q−e), where w(T, Q) denotes the weight of Tat Q, the vertex Qruns over all the infinitely near points of a Q-resolution of Vf,Qappears after a (p, q)-blowup of the origin of X(d;a, b), and e:= gcd(d, aq −bp). One has the following result (Lemma (III.4.3)) for weighted plane curves in P2 wwhich extends Lemma 2.35 in [CA02] for curves in P2and classical resolutions. Lemma 6. The following result holds, deg( ˜ Tδ1,δ2 P(f)) = deg( ˜ Tnul P(f)) −1. Note that in the case of germs at Pon C2and classical blow-ups, the degree of a tree Tis related with the number of conditions imposed to a germ gso that T|g≥ T . In this situation KP(f) = deg Tnul P(f) = δP(f) (see [CA02]). In our case, deg Tnul P(f) = δw P(f), independent from the Q-resolution, which is in general a rational number. Therefore, KP(f) = deg Tnul P(f) can only be expected when fdefines a function on X(d;a, b). The previous Lemma 6, together with the following Proposition 7 (see Proposition (III.5.6)), will be useful in Chapter V to allow for the proof of Theorem 13. Proposition 7. Let {f= 0}be an analytic germ of curve singularity at [0] on X(d;a, b). Consider δ1, δ2any two different local branches of fat [0], then Kδ1δ2 P(f) = KP(f)−1. INTRODUCTION xix Third part: global invariants In §IV.1 a genus formula for weighted projective curves is provided by means of the δw-invariant (Definition 7). For a given d∈Nand a normalized weight list w∈N3the virtual genus associated with dand wis defined as gd,w := d(d−|w|) 2 ¯w+ 1, having the following result (see Theorem (IV.1.12)). Theorem 8 ([CAMO13]).Let C ⊂ P2 wbe an irreducible curve of degree d > 0, then g(C) = gd,w −X P∈Sing(C) δw P. We also show some practical examples in which the genus of different curves in different weighted projective planes is computed. During the rest of this Chapter IV we focus our efforts on obtaining an Adjunction-like Formula relating the genus of a generic curve of quasihomogeneous degree d, and the dimension of the space of polynomials of degree d+ deg K(note that deg K=−|w|=−(w0+w1+w2)), with Kthe canonical divisor in P2 w(this dimension will be denoted by Dd−|w|,w). One has the following results (see Theorem (IV.4.3) and Corollary (IV.4.4)) which will play an important role in Chapter V. Theorem 9. Let w0, w1, w2be pairwise coprime integers, d∈Nand denote by ¯w=w0w1w2,|w|=w0+w1+w2where w= (w0, w1, w2). Let us consider positive integers pi=wi,qi=−w−1 jwkmod wi∈Nwith j < k (recall that X(wi;wj, wk) = X(pi;−1, qi)), ri=w−1 kdmod wi∈N. Consider Dd−|w|,w = # (x, y, z)∈N3|w0x+w1y+w2z=d−|w|, A(pi,qi) ri= # (x, y)∈N2|pix+qiy≤qiri, x, y ≥1, δ(pi,qi) ri=ri(piri−pi−qi+ 1) 2pi . Then Dd−|w|,w =gd,w + 2 X i=0 δ(pi,qi) ri−A(pi,qi) ri. Let C ⊂ P2 wbe a reduced curve of degree d, the number of global conditions of Cis defined as follows K(C) := X P∈Sing(C) KP(f). xx INTRODUCTION Corollary 10 (Adjunction-like Formula).Let C ⊂ P2 wbe a reduced curve of degree d, then h0(P2 w;O(d−|w|)) = Dd−|w|,w =gd,w −X P∈Sing(C) δw P+K(C). From now on, we will denote by XCthe complement of Cin the complex projective plane P2 w. With all the necessary ingredients previously developed, we will finally focus, in Chapter V, on one of the most important invariants of the pair (P2 w,R), namely, the cohomology ring of XR, where Ris a reduced algebraic (possibly singular) curve in the complex projective plane P2 wwhose irreducible components Riare all rational (g(Ri) = 0). Such curves will be called rational arrangements. The aim of this chapter is to find a presentation for the cohomology ring of XR. Let Dbe a reduced Q-divisor in P2 w. In §V.2, a basis for H1(P2 w\D;C) is given and, in §V.4, a holomorphic presentation for H2(P2 w\R;C) is provided. Consider a system of coordinates [X:Y:Z] in P2 w. If one writes D:= {D= 0},Dcan be expressed as a product C0·C1·. . . ·Cnwhere Ci:= {Ci= 0},Ciare irreducible components of D. One can consider the following differential forms σij := d log Cdj i Cdi j!=djd(log Ci)−did(log Cj). where i, j = 0, ..., n,di:= degw(Ci). Take πaQ-resolution of Dthen, the pull-back π∗σij defines a logarithmic 1-form on XD. The following result holds. Theorem 11. The cohomology classes of B1(D) := {σik}n i=0 i6=k, constitute a basis for H1(XD;C). It is easy to check that, in general, Brieskorn’s Theorem does not hold, that is, ∧2H1(XR;C) does not generate H2(XR;C). In §V.3 some examples of the computation of the ring structure of H2(XD;C) are provided. Finally, in §V.4, a holomorphic presentation for H2(XR;C), for a rational arrangement R, is given. Let us sketch this last result. INTRODUCTION xxi Let Ci,Cj,Ckbe three curves in P2 w(not necessarily different). We will denote by Cijk the union Ci∪Cj∪Ckand consider Cijk a reduced equation for Cijk. We also use dijk := degwCijk. For instance, if i=j=k,Cijk =Ci,Cijk =Ciand dijk = degw(Ci). Using the modules described in terms of logarithmic trees seen in Chapters II and III one can construct the following sheaf M∆ Rijk,π. Definition 12. Let R=SiRibe a rational arrangement and πaQresolution of singularities for R. For every triple (Ri,Rj,Rk), not necessarily i6=j6=k, let us take three points P1∈Sing(Ri∩Rj), P2∈Sing(Rj∩Rk) and P3∈Sing(Ri∩Rk). For every Plchoose two local branches, δil lof Ri and δjl lof Rj. Consider ∆ := h(P1, δi1 1, δj1 1),(P2, δj2 2, δk2 2),(P3, δk3 3, δi3 3)i. Let us construct a sheaf M∆ Rijk,π associated with ∆. Let Q∈ Rijk, one has the following module (M∆ Rijk,π)Q:=            OQif Q /∈Sing(Rijk) (Mnul Rijk,π)Qif Pl6=Q∈Sing(Rijk) (Mδi l,δj l Rijk,π)Qif Q=Plwith δi l6=δj l (Mnul Rijk,π)Qif Q=Plwith δi l=δj l            . This module lead us to the corresponding sheaf M∆ Rijk,π which will be called sheaf of ∆-logarithmic forms along Rijk w.r.t. π. The previous sheaf M∆ Rijk,π does not depend on the choice of the resolution π. For a given Rwe will simply write M∆ Rijk if no ambiguity seems no likely to arise. P1 P2P3 δi1 1 δj1 1 δj2 2 δk2 2 δi3 3 δk3 3 ∆ δj1 1δi1 1 δj2 2 δk2 2 δk3 3 δi3 3 Ci Cj Ck Figure 1. ∆ in H1(¯ Rijk;C). xxii INTRODUCTION With the previous definition, using the Adjunction-like Formula (Corollary 10), Lemma 6 and Proposition 7 one has the following results. Proposition 12. Let Rbe a rational arrangement in P2 was in Definition 12, then dim H0(P2 w,M∆ Rijk (dijk −|w|)) >0. Theorem 13. Let R=SiRibe a rational arrangement in P2 wand π aQ-resolution of singularities for R. Let Hbe a polynomial of quasihomogeneous degree dijk −|w|, such that H∈H0(P2 w,M∆ Rijk (dijk −|w|)). The well-defined global 2-forms ω=HΩ2 Rijk form a holomorphic presentation of H2(P2 w\R,C). The proof in Theorem 13 will provide a method to find the relations among the generators in H2(P2 w\ R;C) by means of the residue operator (Definitions 4 and 5). Most of the results seen from Chapters I to V are illustrated in the particular case of D=V(xyz(xyz + (x3−y2)2)) ⊂P2 wand w= (2,3,7). In Chapter I we study a Q-resolution of its singularities (Example (I.2.8)). In Chapter III we construct different logarithmic trees associated with them (Examples (III.3.2) and (III.3.5)). The local concepts studied in Chapters I and II give us the tools to compute its genus in Chapter IV (Example (IV.1.18)). See also Example (IV.4.5) for an illustrative example of the Adjunction-like Formula. Finally, all these results allow us in Chapter V to study its cohomology ring H•(P2 w\D;C) in §V.3–2 and Example (V.4.9). RESUMEN (Spanish) xxix Definici´on 4. Una forma C∞,ϕen XDse llamar´a logar´ıtmica (con respecto al divisor Dy a la Q-resoluci´on π)si π∗ϕes logar´ıtmica en XDcon respecto al divisor con Q-cruces normales D(v´ease Definici´on (I.5.6)). Entonces, tendremos el correspondiente haz π∗ΩXD(loghDi). Una vez que Dyπest´an fijados, podemos definir la aplicaci´on residuo Res[∗] π(ϕ)de una forma logar´ıtmica ϕde la siguiente manera π∗Ωk XD(loghDi)Res[k] π −→ H0(D[k];C) ϕ7→ Res[k](π∗ϕ). La definici´on anterior es independiente de la Q-resoluci´on. Por ejemplo, en el caso particular de X(d;a, b), para el Res[2] tenemos lo siguiente. Definici´on 5. Sea hun germen anal´ıtico en X(d;a, b) escrito en forma normalizada (Definici´on (I.1.9)). Sea ϕ=hdx ∧dy xy una 2-forma logar´ıtmica con polos en el origen. Entonces Res[2](ϕ) := 1 dh(0,0). Segunda parte: invariantes locales En el Cap´ıtulo II se ampl´ıa el concepto de fibra de Milnor y n´umero de Milnor de una singularidad de curva permitiendo que el espacio ambiente tenga singularidades cociente (§II.1). Definimos una generalizaci´on del invariante δy damos una descripci´on de ´este en t´erminos de una Q-resoluci´on de las singularidades de la curva (§II.3). En particular, aplicando lo visto al caso cl´asico (el espacio de ambiente es una superficie lisa) uno obtiene una f´ormula para el invariante δcl´asico en t´erminos de una Q-resoluci´on, lo que simplifica considerablemente los c´alculos efectivos. Finalmente, todas estas herramientas nos permitir´an dar, en el cap´ıtulo IV, una descripci´on expl´ıcita de la f´ormula de g´enero para una curva definida en un plano proyectivo ponderado en t´erminos de su grado y el tipo de sus singularidades locales. Definici´on 6 ([CAMO13]).Sea C={f= 0} ⊂ X(d;a, b) un germen de curva. La fibra de Milnor Fw tde (C,[0]) se define como, Fw t:= {F=t}/Gd. El n´umero de Milnor µwde (C, P) se define como, µw:= 1 −χorb(Fw t). xxx RESUMEN (Spanish) N´otese que generalizaciones alternativas del n´umero de Milnor se pueden encontrar, por ejemplo, en [ABFdBLMH10, BLSS02, uT77, STV05]. La que aqu´ı se propone parece m´as natural para el caso de singularidades cociente (v´ease el Ejemplo (IV.1.18)), pero m´as importante a´un, permite la existencia de una f´ormula expl´ıcita que relaciona el n´umero de Milnor, el invariante δy el g´enero de una curva en una superficie singular (Cap´ıtulo IV). En §II.3 presentamos una versi´on de la f´ormula de Noether para curvas en espacios con singularidades cociente usando Q-resoluciones (v´ease Teorema (II.2.1)). Teorema 2 (F´ormula de Noether, [CAMO13]).Consideremos CyDdos g´ermenes de Q-divisor en [0] sin componentes comunes en una superficie con singularidades cociente. Se tiene la siguiente f´ormula: (C·D)[0] =X Q≺[0] νC,QνD,Q pqd , donde Qrecorre todos los puntos infinitamente pr´oximos a (CD, [0]) yQ aparece tras una explosi´on ponderada de tipo (p, q)del origen de X(d;a, b). A continuaci´on definimos el invariante δwlocal para singularidades de curvas en X(d;a, b). Definici´on 7 ([CAMO13]).Sea Cun germen reducido en [0] ∈X(d;a, b), definimos δwcomo el n´umero que verifica la siguiente ecuaci´on χorb(Fw t) = rw−2δw, donde rwes el n´umero de ramas locales de Cen [0], Fw tdenota su fibra de Milnor y χorb(Fw t) denota la caracter´ıstica de Euler orbifold de Fw t. En el Teorema (II.2.5) proporcionamos una f´ormula recursiva para δw basada en una Q-resoluci´on de la singularidad. Teorema 3 ([CAMO13]).Sea (C, [0]) un germen de curva en una superficie con singularidades cociente abelianas. Entonces δw=1 2X Q≺[0] νQ dpq (νQ−p−q+e), donde Qrecorre todos los puntos infinitamente pr´oximos de una Q-resoluci´on de (C, [0]),Qaparece tras una explosi´on ponderada de tipo (p, q)del origen de X(d;a, b)ye:= gcd(d, aq −bp). En §II.3–1 damos una interpretaci´on del invariante δwcomo dimensi´on de un espacio vectorial. En el caso cl´asico este invariante se puede interpretar como la dimensi´on de un espacio vectorial, sin embargo, dado que δwes en RESUMEN (Spanish) xxxi general, un n´umero racional, un resultado similar s´olo se puede esperar en ciertos casos, m´as concretamente, cuando se asocie a divisores de Cartier (ver Teorema (II.3.7)). Teorema 4 ([CAMO13]).Sea f: (X(d;a, b), P)→(C, P)un germen reducido de funci´on anal´ıtica. Asumamos que (d;a, b)es de tipo normalizado. Consideremos R=OP hfiel anillo local asociado a fyRsu anillo normalizado. Entonces, δw P(f) = dimCR R∈N. En §II.3–2 presentaremos una generalizaci´on de este resultado. Para ello, necesitaremos algunas de las definiciones anteriormente vistas. Dado k≥0, tenemos el m´odulo OP(k) (para m´as detalles, v´ease §I.1–2), OP(k) := {h∈C{x, y}| h(ξa dx, ξb dy) = ξk dh(x, y)}. Sea {f= 0}un germen en P∈X(d;a, b). Notar que si f∈ OP(k), entonces el siguiente OP-m´odulo, OP(k−a−b), verifica OP(k−a−b) = {h∈C{x, y}| hdx ∧dy fes Gd-invariante}. Definici´on 8. Sea D={f= 0}un germen en P∈X(d;a, b) con f∈ OP(k). Considerar πuna Q-resoluci´on de (D, P). (1) Denotemos Mlog D,π el subm´odulo de OPconsistente en los h∈ OP tales que la 2-forma ω=hdx ∧dy f∈Ω2 P(a+b−k) es logar´ıtmica en P, con respecto a Dy la Q-resoluci´on π(recordar Definici´on 4). (2) Sea (Mnul D,π)Pel subm´odulo de Mlog D,π consistente en todos los h∈ Mlog D,π tales que la 2-forma ω=hdx ∧dy f admite una extensi´on holomorfa fuera de la transformada estricta b f. Este ´ultimo m´odulo jugar´a un papel importante a la hora de construir una presentaci´on para el anillo de cohomolog´ıa de P2 w\R en el Cap´ıtulo V. Definici´on 9. Sea D={f= 0}un germen en P∈X(d;a, b), definimos la siguiente dimensi´on, KP(D) = KP(f) := dimCOP Mnul D,π . xxxii RESUMEN (Spanish) El n´umero KP(f) nos da el m´ınimo n´umero de condiciones que tenemos que pedirle a un germen gen´erico h∈ OP(s) para que h∈ Mnul D,π(s). En el caso particular de que f∈ OP, es decir, (f, [0]) sea un germen de funci´on en X(d;a, b), entonces (v´ease Corolario (II.3.14)) KP(f) = δP(f). Teorema 5. Sean f, g ∈ O(k),k∈N, dos g´ermenes en P∈X(d;a, b). Entonces, KP(f)−KP(g) = δw P(f)−δw P(g). En el Cap´ıtulo III continuamos definiendo otros m´odulos y haces logar´ıtmicos asociados a un Q-divisor Dy a una Q-resoluci´on π. Sus secciones globales nos permitir´an, en el cap´ıtulo V, construir 2-formas logar´ıtmicas sobre D. Construiremos dos tipos de ´arboles asociados a un germen anal´ıtico {f= 0}en P∈X(d;a, b), ˜ Tnul P(f) (v´ease §III.1) y ˜ Tδ1δ2 P(f) (v´ease §III.2) siendo δ1yδ2dos ramas locales de fen P. Estos ´arboles nos permitir´an dar una descripci´on ´util de los m´odulos logar´ıtmicos previamente definidos. Definici´on 10. Sea D={f= 0}un germen en P∈X(d;a, b) con f∈ OP(k). Considerar πuna Q-resoluci´on de (D, P). Definimos Mδiδj D,π el subm´odulo de Mlog D,π consistente en todos los h∈ Mlog D,π tales que la 2-forma ω=hdx ∧dy f tiene residuo nulo fuera de los bordes del camino γ(δ1, δ2). Como consecuencia de la construcci´on de los ´arboles ˜ Tnul P(§III.1) y ˜ Tδ1,δ2 P(§III.2) y las Definiciones 8 y 10, tenemos la siguiente caracterizaci´on: Mnul D,π ={h∈ OP|˜ TP(D, π)|h≥˜ Tnul P(D, π)}. Mδiδj D,π ={h∈ OP|˜ TP(D, π)|h≥˜ Tδiδj P(D, π)}. Consideremos la siguiente dimensi´on, Kδiδj P(D) = Kδiδj P(f) := dimCOP Mδiδj D,π . El n´umero Kδiδj P(f) nos da el menor n´umero de condiciones que tenemos que imponerle a un germen gen´erico h∈ OP(s) para que h∈ Mδiδj D,π(s). Definici´on 11. Definimos el grado de un ´arbol con pesos, T, de la siguiente manera deg(T) := X Q∈|T | w(T, Q) 2dpq (w(T, Q) + p+q−e), RESUMEN (Spanish) xxxiii donde w(T, Q) denota el peso de Ten Q, el v´ertice Qrecorre todos los puntos infinitamente pr´oximos de una Q-resoluci´on de Vf,Qaparece tras una explosi´on de tipo (p, q) del origen de X(d;a, b) y e:= gcd(d, aq −bp). Obtenemos el siguiente resultado (Lema (III.4.3)) para curvas planas ponderadas en P2 wmediante Q-resoluciones que generaliza el Lema 2.35 en [CA02] para curvas en P2usando resoluciones cl´asicas. Lema 6. Tenemos que, deg( ˜ Tδ1,δ2 P(f)) = deg( ˜ Tnul P(f)) −1. N´otese que en el caso de g´ermenes en un punto Pde C2y explosiones cl´asicas, el grado de un ´arbol Test´a relacionado con el n´umero de condiciones que hay que imponer a un germen gpara que T|g≥ T . En esta situaci´on KP(f) = deg Tnul P(f) = δP(f) (v´ease [CA02]). En nuestro caso, deg Tnul P(f) = δw P(f), independiente de la Q-resoluci´on, siendo este grado un n´umero racional. Por tanto, KP(f) = deg Tnul P(f) s´olo se puede esperar cuando fsea una funci´on en X(d;a, b). El Lema 6, junto con la Proposici´on 7 que veremos a continuaci´on (ver Proposici´on (III.5.6)), ser´an ´utiles en el Cap´ıtulo V a la hora de probar el Teorema 13. Proposici´on 7. Sea {f= 0}un germen anal´ıtico de una singularidad de curva en el punto Pde X(d;a, b). Denotemos por δ1, δ2, dos ramas locales cualesquiera de fen P, entonces Kδ1δ2 P(f) = KP(f)−1. Tercera parte: invariantes globales En §IV.1 damos una f´ormula para calcular el g´enero de curvas en el plano proyectivo ponderado por medio del invariante δw(Definici´on 7). Dado d∈Ny una lista de pesos normalizados w∈N3, definimos el g´enero virtual asociado a dywcomo gd,w := d(d−|w|) 2 ¯w+ 1, dando lugar al siguiente resultado (v´ease Teorema (IV.1.12)). Teorema 8 ([CAMO13]).Sea C ⊂ P2 wuna curva irreducible de grado d > 0, entonces g(C) = gd,w −X P∈Sing(C) δw P. xxxiv RESUMEN (Spanish) En el cap´ıtulo tambi´en se muestran algunos ejemplos pr´acticos en los que se calcula el g´enero de diferentes curvas en distintos P2 w. Durante el resto del Cap´ıtulo IV centraremos nuestros esfuerzos en obtener una F´ormula de tipo Adjunci´on que relacione el g´enero de una curva gen´erica de grado cuasihomog´eneo dy la dimensi´on del espacio de polinomios de grado d+ deg K(notar que deg K=−|w|=−(w0+w1+w2)), siendo Kel divisor can´onico de P2 w(esta dimensi´on se denotar´a Dd−|w|,w). Obtenemos los siguientes resultados (v´eanse Teorema (IV.4.3) y Corolario (IV.4.4)) que jugaran un papel importante en el Cap´ıtulo V. Teorema 9. Sean w0, w1, w2enteros primos dos a dos, d∈Ny denotemos por ¯w=w0w1w2,|w|=w0+w1+w2donde w= (w0, w1, w2). Consideremos los siguientes enteros positivos pi=wi,qi=−w−1 jwkm´od wi∈Ncon j < k (notar que X(wi;wj, wk) = X(pi;−1, qi)), ri=w−1 kdm´od wi∈N. Consideremos Dd−|w|,w = # (x, y, z)∈N3|w0x+w1y+w2z=d−|w|, A(pi,qi) ri= # (x, y)∈N2|pix+qiy≤qiri, x, y ≥1, δ(pi,qi) ri=ri(piri−pi−qi+ 1) 2pi . Entonces Dd−|w|,w =gd,w + 2 X i=0 δ(pi,qi) ri−A(pi,qi) ri. Sea C ⊂ P2 wuna curva reducida de grado d, definimos el n´umero de condiciones globales para Cde la siguiente manera K(C) := X P∈Sing(C) KP(f). Corolario 10 (F´ormula de tipo Adjunci´on).Sea C ⊂ P2 wuna curva reducida de grado d, entonces h0(P2 w;O(d−|w|)) = Dd−|w|,w =gd,w −X P∈Sing(C) δw P+K(C). A partir de ahora, denotaremos por XCal complementario de Cen el plano proyectivo ponderado P2 w. Con todos los ingredientes previamente vistos, nos centraremos, a lo largo del Cap´ıtulo V, en uno de los invariantes m´as importantes del par (P2 w,R), el anillo de cohomolg´ıa de XR, donde Res una curva algebraica plana reducida (con o sin puntos singulares) en el plano proyectivo complejo ponderado RESUMEN (Spanish) xxxv P2 wcuyas componentes irreducibles Rison todas racionales (g(Ri) = 0). Tales curvas ser´an llamadas configuraciones racionales. El objetivo del cap´ıtulo ser´a encontrar una presentaci´on para el anillo de cohomolog´ıa de XR. Sea Dun Q-divisor reducido en P2 w. En §V.2, se proporciona una base para H1(P2 w\ D;C) y, en §V.4, damos una presentaci´on holomorfa para H2(P2 w\R;C). Tomemos un sistema de coordenadas [X:Y:Z] en P2 w. Si escribimos D:= {D= 0},Dse puede expresar como producto de C0·C1·. . .·Cndonde Ci:= {Ci= 0}, siendo Cilas componentes irreducibles de D. Consideremos las siguientes formas diferenciales σij := d log Cdj i Cdi j!=djd(log Ci)−did(log Cj). donde i, j = 0, ..., n,di:= degw(Ci). Tomemos πuna Q-resoluci´on de Dentonces, el pull-back π∗σij define una 1-forma logar´ıtmica en XD. Tenemos el siguiente resultado. Teorema 11. Las clases de cohomolog´ıa de B1(D) := {σik}n i=0 i6=k, constituyen una base para H1(XD;C). Es f´acil comprobar que, en general, no podemos esperar el Teorema de Brieskorn, es decir, ∧2H1(XR;C) no genera H2(XR;C). En §V.3 presentamos algunos ejemplos del c´alculo de la estructura de anillo de H2(XD;C). Finalmente, en §V.4, damos una presentaci´on holomorfa para H2(XR;C), con Runa configuraci´on racional. Veamos en detalle este ´ultimo resultado. Sean Ci,Cj,Cktres curvas en P2 w(no necesariamente distintas). Denotaremos por Cijk a la uni´on Ci∪ Cj∪ Ck. Consideremos Cijk una ecuaci´on reducida para Cijk. Tambi´en usaremos dijk := degwCijk. Por ejemplo, en el caso i=j=k, tendremos Cijk =Ci,Cijk =Ciy dijk = degw(Ci). Usando los m´odulos descritos en t´erminos de ´arboles logar´ıtmicos en los Cap´ıtulos II y III podemos construir el siguiente haz M∆ Rijk,π. Definici´on 12. Sea R=SiRiuna configuraci´on racional y πuna Qresoluci´on de singularidades para R. Para cada triple (Ri,Rj,Rk), no necesariamente i6=j6=k, tomemos tres puntos P1∈Sing(Ri∩ Rj), P2∈ xxxvi RESUMEN (Spanish) Sing(Rj∩Rk) y P3∈Sing(Ri∩Rk). Para cada Plelegimos dos ramas, δil l de Riyδjl lde Rj. Consideremos ∆ := h(P1, δi1 1, δj1 1),(P2, δj2 2, δk2 2),(P3, δk3 3, δi3 3)i. Vamos a construir un haz M∆ Rijk,π asociado a ∆. Sea Q∈ Rijk, tenemos el siguiente m´odulo (M∆ Rijk,π)Q:=            OQif Q /∈Sing(Rijk) (Mnul Rijk,π)Qif Pl6=Q∈Sing(Rijk) (Mδi l,δj l Rijk,π)Qif Q=Plwith δi l6=δj l (Mnul Rijk,π)Qif Q=Plwith δi l=δj l            . Este m´odulo nos lleva a la definici´on del haz M∆ Rijk,π que llamaremos haz de formas ∆-logar´ıtmicas sobre Rijk con respecto a π. Este haz M∆ Rijk,π no depende de la elecci´on de la resoluci´on π. Si no hay ambig¨uedad, dado R, simplemente escribiremos M∆ Rijk . P1 P2P3 δi1 1 δj1 1 δj2 2 δk2 2 δi3 3 δk3 3 ∆ δj1 1δi1 1 δj2 2 δk2 2 δk3 3 δi3 3 Ci Cj Ck Figura 2. ∆ en H1(¯ Rijk;C). Con la definici´on previa, usando la F´ormula de tipo Adjunci´on (Corolario 10), el Lema 6 y la Proposici´on 7 obtenemos los siguientes resultados. Proposici´on 12. Sea Runa configuraci´on racional en P2 wcomo en la Definici´on 12, entonces dim H0(P2 w,M∆ Rijk (dijk −|w|)) >0. Teorema 13. Sea R=SiRiuna configuraci´on racional en P2 wyπuna Q-resoluci´on de singularidades para R. Sea Hun polinomio de grado cuasihomog´eneo dijk −|w|, tal que H∈H0(P2 w,M∆ Rijk (dijk −|w|)). RESUMEN (Spanish) xxxvii Las 2-formas ω=HΩ2 Rijk forman una presentaci´on holomorfa para H2(P2 w\ R,C). La demostraci´on del Teorema 13 proporciona un m´etodo para encontrar las relaciones entre los generadores de H2(P2 w\R;C) por medio de las relaciones en H1(¯ R[1];C) y la inyectividad del operador residuo (Definiciones 4 y 5). La mayor parte de los resultados vistos a lo largo de los Cap´ıtulos I a V est´an ilustrados en el caso particular de D=V(xyz(xyz +(x3−y2)2)) ⊂P2 w con w= (2,3,7). En el Cap´ıtulo I estudiamos una Q-resoluci´on de sus singularidades (Ejemplo (I.2.8)). En el Cap´ıtulo III construimos diferentes ´arboles logar´ıtmicos asociados a D(Ejemplos (III.3.2) y (III.3.5)). Los conceptos locales estudiados en los Cap´ıtulos I y II nos dar´an las herramientas necesarias para calcular el g´enero de Den el Cap´ıtulo IV (Ejemplo (IV.1.18)). V´ease tambi´en (IV.4.5) para un ejemplo ilustrativo de la F´ormula de tipo Adjunci´on. Finalmente, todos estos resultados nos permitir´an, en el Cap´ıtulo V, estudiar el anillo de cohomolog´ıa H•(P2 w\ D;C) en §V.3–2 y el Ejemplo (V.4.9). R´ ESUM´ E (French) xlv Soit Dun Q-diviseur dans P2 w. Le compl´ementaire de Dest not´e XD. On fixe π:XD−→ P2 wune Q-r´esolution des singularit´es de Dde telle sorte que le Q-diviseur r´eduit D= (π∗(D))red est une r´eunion de Q-diviseurs lisses dans XD`a Q-croisements normaux. D´efinition 4. Une forme C∞ϕdans XDest dite logarithmique (le long du diviseur Dpar rapport `a la Q-r´esolution π)si π∗ϕest logarithmique en XDpar rapport au diviseur `a Q-croisements normaux D(voir D´efinition (I.5.6)). Alors, on a le faisceau correspondant π∗ΩXD(loghDi). Une fois que Det πsont fix´es, on peut d´efinir l’application r´esidu Res[∗] π(ϕ) d’une forme logarithmique ϕcomme suit π∗Ωk XD(loghDi)Res[k] π −→ H0(D[k];C) ϕ7→ Res[k](π∗ϕ). La d´efinition ci-dessus est ind´ependante de la Q-r´esolution. Par exemple, dans le cas particulier de X(d;a, b), Res[2] s’exprime comme suit. D´efinition 5. Soit hun germe analytique dans X(d;a, b) ´ecrit sous forme standard (D´efinition (I.1.9)). Soit ϕ=hdx ∧dy xy une 2-forme logarithmique avec des pˆoles `a l’origine. Alors Res[2](ϕ) := 1 dh(0,0). Deuxi`eme partie : invariants locaux Dans le Chapitre II on ´etend le concept de fibre de Milnor et de nombre de Milnor d’une singularit´e de courbe dans un espace ambiant ayant des singularit´es quotient (§II.1). On d´efinit une g´en´eralisation du δ-invariant et on donne une description de celui-ci en termes d’une Q-r´esolution des singularit´es de la courbe (§II.3). En particulier, quand on applique ce qu’on a vu dans le cas classique (l’espace ambiant est une surface lisse), on obtient une formule pour l’invariant δclassique en fonction d’une Q-r´esolution, ce qui simplifie consid´erablement les calculs. Enfin, tous ces outils nous permettent de donner, dans le Chapitre IV,une description explicite de la formule du genre d’une courbe d´efinie dans un plan projectif pond´er´e en fonction de son degr´e et du type de singularit´es locales. xlvi R´ ESUM´ E (French) D´efinition 6 ([CAMO13]).Soit C={f= 0} ⊂ X(d;a, b) un germe de courbe. La fibre de Milnor Fw tde (C,[0]) est d´efinie comme Fw t:= {F=t}/Gd. Le nombre de Milnor µwde (C, P) est d´efini comme µw:= 1 −χorb(Fw t). Notez que des g´en´eralisations alternatives pour le nombre de Milnor peuvent ˆetre trouv´ees, par exemple, dans [ABFdBLMH10, BLSS02, uT77, STV05]. La g´en´eralisation propos´ee ici semble plus naturelle dans le cas des singularit´es quotient (voir l’Exemple (IV.1.18)) ; en particulier, elle permet de g´en´eraliser la formule qui r´elie le nombre de Milnor, le δ-invariant et le genre d’une courbe sur une surface singuli`ere (Chapitre IV). Dans §II.3 on pr´esente une version de la formule de Noether pour des courbes dans des espaces avec des singularit´es quotient `a l’aide des Qr´esolutions (voir Th´eor`eme (II.2.1)). Th´eor`eme 2 (Formule de Noether, [CAMO13]).On consid`ere deux germes C, D de Q-diviseurs en [0], sans composantes communes, dans un surface avec des singularit´es quotient et une suite d’´eclatements pond´er´es qui s´eparent les branches de Cet D. Ces germes satisfont la formule suivante : (C·D)[0] =X Q≺[0] νC,QνD,Q pqd , o`u Qparcourt les points infiniment voisins de [0] par la suite d’´eclatements pond´er´es. Ensuite, on d´efinit l’invariant local δwpour les singularit´es de courbes dans X(d;a, b). D´efinition 7 ([CAMO13]).Soit Cun germe r´eduit en [0] ∈X(d;a, b), on d´efinit δwcomme le nombre d´etermin´e par l’´egalit´e suivante : χorb(Fw t) = rw−2δw, o`u rwest le nombre de branches locales Cen [0], Fw td´esigne la fibre de Milnor, et χorb(Fw t) repr´esente la caract´eristique d’Euler de l’orbifold Fw t. Cette d´efinition suit [Mil68, Theorem 10.5]. Dans le Th´eor`eme (II.2.5) on ´enonce une formule r´ecursive pour δwpour les Q-r´esolutions de la singularit´e de surface qui g´en´eralise la formule classique. Th´eor`eme 3 ([CAMO13]).Soit (C, [0]) un germe de courbe dans une surface avec des singularit´es quotient ab´eliennes. Alors δw=1 2X Q≺[0] νQ dpq (νQ−p−q+e), R´ ESUM´ E (French) xlvii o`u Qqui apparaˆıt lors d’un ´eclatement pond´er´e (p, q)d’un point de type X(d;a, b)(normalis´e), parcourt les points infiniment proches d’une Q-r´esolution du (C, [0]) et e:= gcd(d, aq −bp). Dans §II.3–1 on donne une interpr´etation de l’invariant δwcomme la dimension d’un espace vectoriel. Dans le cas classique cet invariant est aussi d´efini comme la codimension de l’anneau du germe dans sa normalisation. δwest en g´en´eral un nombre rationnel ; ce r´esultat est encore vrai dans certains cas, par exemple, lorsqu’il est associ´e `a des diviseurs de Cartier (voir Th´eor`eme (II.3.7)). Th´eor`eme 4 ([CAMO13]).Soit f: (X(d;a, b), P)→(C, P)un germe r´eduit de fonction analytique. On suppose que (d;a, b)est de type normalis´e. On consid`ere R=OP hfil’anneau local associ´e `a fet Rla normalisation de R. Alors, δw P(f) = dimCR R∈N. Dans §II.3–2 on pr´esente une g´en´eralisation de ce r´esultat. Pour ce faire, on a besoin de quelques d´efinitions. On fixe k≥0, et l’on consid`ere le module OP(k) (pour plus de d´etails, voir §I.1–2), OP(k) := {h∈C{x, y}| h(ξa dx, ξb dy) = ξk dh(x, y)}. Soit {f= 0}un germe dans P∈X(d;a, b). Si f∈ OP(k), alors le OPmodule OP(k−a−b) v´erifie : OP(k−a−b) = h∈C{x, y}hdx ∧dy fest Gd-invariant. D´efinition 8. Soit D={f= 0}un germe dans P∈X(d;a, b) avec f∈ OP(k). On consid`ere πune Q-resolution de (D, P). (1) Soit Mlog D,π le sous-module de OPdes h∈ OPtels que la 2-forme ω=hdx ∧dy f∈Ω2 P(a+b−k) est logarithmique en P, par rapport `a Det π(D´efinition 4). (2) Soit Mnul D,π le sous-module de Mlog D,π des h∈ Mlog D,π tels que la 2forme ω=hdx ∧dy f admet une extension holomorphe en dehors de la transform´ee stricte b f. Ce dermier module joue un rˆole important dans la construction de la pr´esentation de l’anneau de cohomologie P2 w\R du Chapitre V. xlviii R´ ESUM´ E (French) D´efinition 9. Soit D={f= 0}un germe dans P∈X(d;a, b). On pose : KP(D) = KP(f) := dimCOP Mnul D,π . Le nombre KP(f) nous donne le nombre minimum de conditions qu’on doit demander `a un germe g´en´erique h∈ OP(s) pour que h∈ Mnul D,π(s). Dans le cas particulier o`u f∈ OP, c’est `a dire, quand (f, [0]) est un germe de fonction dans X(d;a, b), on a (voir Corollaire (II.3.14)) KP(f) = δP(f). Th´eor`eme 5. Soient f, g ∈ O(k),k∈N, deux germes en P∈X(d;a, b). Alors, KP(f)−KP(g) = δw P(f)−δw P(g). Dans le Chapitre III on d´efinit d’autres modules et faisceaux logarithmiques associ´es `a un Q-diviseur Det `a une Q-r´esolution π. Leurs sections globales nous permettent, dans le Chapitre V, de construire des 2-formes logarithmiques sur D. On va construire deux types d’arbres associ´es `a un germe analytique {f= 0}en P∈X(d;a, b), ˜ Tnul P(f) (voir §III.1) et ˜ Tδ1δ2 P(f) (voir §III.2) o`u δ1et δ2sont deux branches locales fau point P. Ces arbres nous permettent de donner une description utile des modules logarithmiques d´efinis pr´ec´edemment. D´efinition 10. Soit D={f= 0}un germe dans P∈X(d;a, b) avec f∈ OP(k) et soit πune Q-resolution de (D, P). On d´efinit Mδiδj D,π comme le sous-module de Mlog D,π des h∈ Mlog D,π tels que la 2-forme ω=hdx ∧dy f a des r´esidus nuls le long du chemin γ(δ1, δ2). ` A la suite de la construction des arbres ˜ Tnul P(§III.1) et ˜ Tδ1,δ2 P(§III.2) et des D´efinitions 8 et 10, on a les caract´erisations suivantes : Mnul D,π ={h∈ OP|˜ TP(D, π)|h≥˜ Tnul P(D, π)}. Mδiδj D,π ={h∈ OP|˜ TP(D, π)|h≥˜ Tδiδj P(D, π)}. On consid`ere la dimension suivante, Kδiδj P(D) = Kδiδj P(f) := dimCOP Mδiδj D,π . Le nombre Kδiδj P(f) donne le plus petit nombre de conditions `a imposer un germe g´en´erique h∈ OP(s) pour que h∈ Mδiδj D,π(s). R´ ESUM´ E (French) xlix D´efinition 11. On d´efinit le degr´e d’un arbre Tavec des poids comme suit : deg(T) := X Q∈|T | w(T, Q) 2dpq (w(T, Q) + p+q−e), o`u w(T, Q) d´esigne le poids de Ten Q, le sommet Qpasse par chaque point infiniment proche d’une Q-r´esolution de Vf,Qapparaˆıt lors d’un ´eclatement de type (p, q) de (X(d;a, b),[0]) et e:= gcd(d, aq −bp). On obtient le r´esultat suivant (Lemme (III.4.3)) pour des courbes dans P2 wen utilisant des Q-r´esolutions ; ce r´esultat g´en´eralise le Lemme 2.35 dans [CA02] pour des courbes dans P2et des r´esolutions classiques. Lemme 6. deg( ˜ Tδ1,δ2 P(f)) = deg( ˜ Tnul P(f)) −1. On remarque que dans le cas des germes (P∈C2et des ´eclatements classiques), le degr´e d’un arbre Test li´e au nombre de conditions que doit satisfaire un germe gpour que T |g≥ T. Dans cette situation, KP(f) = deg Tnul P(f) = δP(f) (voir [CA02]). Dans notre cas, le nombre deg Tnul P(f) = δw P(f)∈Qest ind´ependant de la Q-r´esolution. Ainsi, l’egalit´e KP(f) = deg Tnul P(f) est vraie seulement si fest une fonction `a X(d;a, b). Le Lemme 6, avec la Proposition 7 (voir Proposition (III.5.6)), seront utiles dans le Chapitre V pour d´emontrer le Th´eor`eme 13. Proposition 7. Soit {f= 0}un germe analytique d’une singularit´e de courbe en Pde X(d;a, b). On d´enote δ1, δ2, deux branches locales de fen P. Alors, Kδ1δ2 P(f) = KP(f)−1. Troisi`eme partie : invariants globaux Dans §IV.1 on donne une formule pour le genre de courbes dans le plan projectif pond´er´e qui utilise l’invariant δw(D´efinition 7). Soit d∈Net soit w∈N3une liste de poids normalis´es, on d´efinie le genre virtuel associ´e `a d et wcomme gd,w := d(d−|w|) 2 ¯w+ 1. Il donne lieu au r´esultat suivant (voir Th´eor`eme (IV.1.12)). Th´eor`eme 8 ([CAMO13]).Soit C ⊂ P2 wune courbe irr´eductible de degr´e d > 0, alors g(C) = gd,w −X P∈Sing(C) δw P. lR´ ESUM´ E (French) Dans ce chapitre on montre ´egalement quelques exemples pratiques o`u les genres de diff´erentes courbes dans P2 wsont calcul´es. Pour le reste du Chapitre IV on concentre nos efforts dans l’obtention d’une Formule de type Adjonction concernant le genre d’une courbe g´en´erique de degr´e quasi-homog`ene det la dimension de l’espace des polynˆomes de degr´e d+ deg K(noter que deg K=−|w|=−(w0+w1+w2)), o`u Kle diviseur canonique de P2 w(cette dimension sera not´ee Dd−|w|,w). Les r´esultats suivants (voir Th´eor`eme (IV.4.3) et Corollaire (IV.4.4)) vont jouer un rˆole cl´e dans le Chapitre V. Th´eor`eme 9. Soient w0, w1, w2des entiers deux `a deux premiers entre eux, soit d∈Net on d´enote ¯w=w0w1w2,|w|=w0+w1+w2o`u w= (w0, w1, w2). On consid`ere les entiers positifs suivants pi=wi,qi=−w−1 jwk mod wi∈N,j < k (on note que X(wi;wj, wk) = X(pi;−1, qi)), ri=w−1 kd mod wi∈N). On consid`ere Dd−|w|,w = # (x, y, z)∈N3|w0x+w1y+w2z=d−|w|, A(pi,qi) ri= # (x, y)∈N2|pix+qiy≤qiri, x, y ≥1, δ(pi,qi) ri=ri(piri−pi−qi+ 1) 2pi . Alors Dd−|w|,w =gd,w + 2 X i=0 δ(pi,qi) ri−A(pi,qi) ri. Soit C ⊂ P2 wune courbe r´eduite de degr´e d, on d´efinit le nombre de conditions globales pour Ccomme suit : K(C) := X P∈Sing(C) KP(f). Corollaire 10 (Formule de type Adjonction).Soit C ⊂ P2 wune courbe r´eduite de degr´e d, alors h0(P2 w;O(d−|w|)) = Dd−|w|,w =gd,w −X P∈Sing(C) δw P+K(C). D`es maintenant, on note XCle compl´ementaire de Cdans le plan projectif pond´er´e P2 w. Les invariants calcul´es pr´ec´edemment vont nous servir, dans le Chapitre V, pour calculer l’un des invariants les plus importants de la paire (P2 w,R), l’anneau de cohomologie XR, o`u Rest une courbe alg´ebrique plane r´eduite (avec ou sans point singulier) dans le plan projectif pond´er´e P2 w, dont les composantes irr´eductibles Risont toutes rationnelles (g(Ri) = 0). Ces R´ ESUM´ E (French) li courbes sont appel´ees des arrangements rationnels. Le but du chapitre est de trouver une pr´esentation de l’anneau de cohomologie de XR. Soi Dun Q-diviseur r´eduit dans P2 w. Dans §V.2, on fournit une base pour H1(P2 w\ D;C) et, dans §V.4, on donne une pr´esentation holomorphe de H2(P2 w\R;C). On prend un syst`eme de coordonn´ees [X:Y:Z] en P2 w. Si l’on ´ecrit D:= {D= 0}, la fonction Dpeut s’exprimer comme le produit de C0·C1·. . .·Cn, avec Ci:= {Ci= 0}, o`u Cisont les composantes irr´eductibles de D. On consid`ere les formes diff´erentielles suivantes σij := d log Cdj i Cdi j!=djd(log Ci)−did(log Cj). avec i, j = 0, ..., n,di:= degw(Ci). On prend πune Q-r´esolution de Dalors, le pull-back π∗σij d´efinit une 1-forme logarithmique dans XD. On a le r´esultat suivant. Th´eor`eme 11. Les classes de cohomologie de B1(D) := {σik}n i=0 , i 6=k, fournissent une base pour H1(XD;C). Il est facile de voir que, en g´en´eral, on ne peut pas esp´erer r´ecuperer le Th´eor`eme de Brieskorn, c’est `a dire, V2H1(XR;C) n’engendre pas H2(XR;C). Dans §V.3 on pr´esente quelques exemples de calcul de la structure d’anneau de H2(XD;C). Finalement, dans §V.4, on donne une pr´esentation holomorphe de H2(XR;C), o`u Rest un arrangement rationnel. Voyons en d´etail ce dernier r´esultat. Soient Ci,Cj,Cktrois courbes dans P2 w(pas n´ecessairement diff´erentes). On note Cijk l’union Ci∪Cj∪Cket on consid`ere Cijk une ´equation r´eduite pour Cijk. On utilise ´egalement dijk := degwCijk. Par exemple, dans le cas i=j=k, on a Cijk =Ci,Cijk =Ciet dijk = degw(Ci). En utilisant les modules d´ecrits en termes d’arbres logarithmiques dans les Chapitres II et III on peut construire le faisceau M∆ Rijk,π. D´efinition 12. Soit R=SiRiun arrangement rationnel et πune Qr´esolution des singularit´es pour R. Pour chaque triplet (Ri,Rj,Rk) (les indices ne sont pas n´ecessairement distincts), on prend trois points P1∈ Sing(Ri∩Rj), P2∈Sing(Rj∩Rk) et P3∈Sing(Ri∩Rk). Pour chaque Pl lii R´ ESUM´ E (French) on choisi deux branches, δil lde Riet δjl lde Rj. On consid`ere ∆ := h(P1, δi1 1, δj1 1),(P2, δj2 2, δk2 2),(P3, δk3 3, δi3 3)i. On va construire un faisceau M∆ Rijk,π associ´e `a ∆. Soit Q∈ Rijk ; on consid`ere le module suivant (M∆ Rijk,π)Q:=            OQif Q /∈Sing(Rijk) (Mnul Rijk,π)Qif Pl6=Q∈Sing(Rijk) (Mδi l,δj l Rijk,π)Qif Q=Plwith δi l6=δj l (Mnul Rijk,π)Qif Q=Plwith δi l=δj l            . Ce module conduit `a la d´efinition du faisceau M∆ Rijk,π qu’on appelle le faisceau des formes ∆-logarithmiques sur Rijk par rapport `a π. P1 P2P3 δi1 1 δj1 1 δj2 2 δk2 2 δi3 3 δk3 3 ∆ δj1 1δi1 1 δj2 2 δk2 2 δk3 3 δi3 3 Ci Cj Ck Figure 3. ∆ en H1(¯ Rijk;C). Avec la d´efinition pr´ec´edente, en utilisant la Formule de type Adjonction (Corollaire 10), le Lemme 6 et la Proposition 7 on a les r´esultats suivants. Proposition 12. Soit Run arrangement rationnel dans P2 wcomme dans le D´efinition 12. Alors, dim H0(P2 w,M∆ Rijk (dijk −|w|)) >0. Th´eor`eme 13. Soit R=SiRiun arrangement rationnel dans P2 wet π une Q-r´esolution des singularit´es pour R. Soit Hun polynˆome de degr´e quasi-homog`ene dijk −|w|, tel que H∈H0(P2 w,M∆ Rijk (dijk −|w|)). Les 2-formes ω=HΩ2 Rijk forment une pr´esentation holomorphe pour l’espace H2(P2 w\R,C). R´ ESUM´ E (French) liii La d´emonstration du Th´eor`eme 13 fournit une m´ethode pour trouver la lien entre les g´en´erateurs H2(P2 w\R;C) grˆace aux relations dans H1(¯ R[1];C) et `a l’injectivit´e de l’op´erateur r´esidu (D´efinitions 4 et 5). La plupart des r´esultats vus dans les Chapitres I `a V sont illustr´es dans le cas particulier de D=V(xyz(xyz+(x3−y2)2)) ⊂P2 wavec w= (2,3,7). Dans le Chapitre I on ´etude une Q-r´esolution des singularit´es (Exemple (I.2.8)). Dans le Chapitre III on construit diff´erents arbres logarithmiques associ´es `a D(Exemples (III.3.2) et (III.3.5)). Les concepts locaux ´etudi´es dans les Chapitres I et II nous donnent les outils n´ecessaires pour calculer le genre de Ddans le Chapitre IV (Exemple (IV.1.18)). Voir aussi (IV.4.5) pour un exemple illustratif de la Formule de type Adjonction. Finalement, ces r´esultats nous permettent, dans le Chapitre V, d’´etudier l’anneau de cohomologie H•(P2 w\D;C) dans §V.3–2 et l’Exemple (V.4.9). §I.2. Weighted Blow-ups and Embedded Q-Resolutions 7 We recall that in the class of V-manifolds, the abelian groups of Cartier and Weil divisors are not isomorphic. However the isomorphism can be achieved after tensoring by Q. Such divisors will be referred to as Q-divisors (Section I.3). Definition (I.2.1). Let Xbe a V-manifold with abelian quotient singularities. A hypersurface Don Xis said to be Q-normal crossing if it is locally isomorphic to the quotient of a normal crossing divisor under a group action of type (d;A). That is, for any x∈X, there is an isomorphism of germs (X, x)≃ (X(d;A),[0]) such that (D, x)⊂(X, x) is identified under this morphism with a germ of the form (4) [x]∈X(d;A)|xm1 1···xmk k= 0,[0]. Whenever (D, x) is Q-normal crossing with k= 1 in (4) we say xis a Qsmooth point of D. A Q-divisor with only Q-smooth points will be referred to as a Q-smooth divisor. Let M=Cn+1/Gbe an abelian quotient space not necessarily cyclic or written in normalized form. Consider H⊂Man analytic subvariety of codimension one. Definition (I.2.2). An embedded Q-resolution of (H, 0) ⊂(M, 0) is a proper analytic map π:X→(M, 0) such that: (1) Xis a V-manifold with abelian quotient singularities, (2) πis an isomorphism over X\π−1(Sing(H)), (3) π−1(H) is a Q-normal crossing hypersurface on X, and (4) the strict transform ˆ H:= π−1(H\{0}) is Q-smooth. Remark (I.2.3) ([AMO11b, Mar11]).Let f: (M, 0) →(C,0) be a nonconstant analytic function germ. Consider (H, 0) the hypersurface defined by f. Let π:X→(M, 0) be an embedded Q-resolution of (H, 0) ⊂(M, 0). Then π−1(H)=(f◦π)−1(0) is locally given by a function of the form xm1 1···xmk k:X(d;A)→C. Remark (I.2.4) ([CAMO13]).In some cases, one needs to consider a stronger condition on Q-resolutions, namely, the strict transform of Hdoes not contain any singular points of X. This can always be achieved by blowing up eventually once more the strict preimage of the hypersurface. Such an embedded resolution will be referred to as a strong Q-resolution. In what follows we will use weighted blow-ups of points as a tool for finding embedded Q-resolutions. 8 Chapter I. Vmanifolds: Quotient Singularities, Embedded Q- . . . Let Xbe an analytic surface with abelian quotient singularities. Consider π:b X→Xthe weighted blow-up at a point P∈Xwith respect to w= (p, q), which will be assumed to be coprime. We distinguish three cases. (i) The point Pis smooth. In this case X=C2and π=πw:b C2 w→C2 is the weighted blow-up at the origin with respect to w= (p, q). The new ambient space is covered as b C2 w=U1∪U2=X(p;−1, q)∪X(q;p, −1) and the charts are given by First chart X(p;−1, q)−→ U1, [(x, y)] 7→ ((xp, xqy),[1 : y]w). Second chart X(q;p, −1) −→ U2, [(x, y)] 7→ ((xyp, yq),[x: 1]w). The exceptional divisor E=π−1 w(0) is isomorphic to P1 wwhich is in turn isomorphic to P1under the map [x:y]w7−→ [xq:yp]. The singular points of b C2 ware cyclic quotient singularities located at the exceptional divisor. They actually coincide with the origins of the two charts and they are written in a normalized form. Example (I.2.5) ([AMO11b, Mar11]).Let f:C2→Cbe the function given by f=xp+yqwith gcd(p, q) = 1. Consider π(q,p):b C2 (q,p)→C2the (q, p)-weighted blow-up at the origin. In U1the total transform is given by the function xpq(1 + yq) : X(q;−1, p)−→ C. The equation yq=−1 has just one solution in U1and the local equation of the total transform at this point is of the form xpqy= 0. Hence the proper map π(q,p)is an embedded Q-resolution of the plane curve C={f= 0} ⊂ C2where all spaces are written in a normalized form. m=pq (p;q, −1) (q;−1, p) U1U2 Figure I.1. Embedded Q-resolution of {xp+yq= 0} ⊂ C2. (ii) The point Pis of type (d;p, q).Assume X=X(d;p, q) is written in a normalized form, i.e. gcd(d, p) = gcd(d, q) = 1. Without loss of generality, §I.2. Weighted Blow-ups and Embedded Q-Resolutions 9 pand qcan assumed to be coprime. We will describe π=πw,d :b C2 w,d → X(d;p, q) the weighted blow-up at the origin with respect to w= (p, q). The new ambient space is covered as b C2 w,d =U1∪U2=X(p;−d, q)∪X(q;p, −d) and the charts are given by First chart X(p;−d, q)−→ U1, (xd, y)7→ [(xp, xqy)]d,[1 : y]w. Second chart X(q;p, −d)−→ U2, (x, yd)7→ [(xyp, yq)]d,[x: 1]w. As above, the exceptional divisor E=π−1 w(0) is identified with P1 wwhich is isomorphic to P1under the map [x:y]w7−→ [xq:yp]. The singular points of b C2 w,d are cyclic quotient singularities at the origin of each chart and they are written in a normalized form. Example (I.2.6) ([AMO11b, Mar11]).Assume gcd(p, q) = 1 and p<q. Let f= (xp+yq)(xq+yp) and consider C1={xp+yq= 0}and C2= {xq+yp= 0}the two irreducible components of {f= 0}. Let π1:b C2 (q,p)→C2be the (q, p)-weighted blow-up at the origin. The new space has two singular points of type (q;−1, p) and (p;q, −1) located on the exceptional divisor E1. The local equation of the total transform in the first chart is given by the function xp(p+q)(1 + yq)(xq2−p2+yp) : X(q;−1, p)−→ C. Here x= 0 is the equation of the exceptional divisor and the other factors correspond to the strict transform of C1and C2(denoted again by the same symbol). Hence E1has multiplicity p(p+q); it intersects transversely C1at a smooth point while it intersects C2at a singular point (the origin of the first chart) without Q-normal crossings. E1 (p;q, −1) (q;−1, p) C2 ←− E1 (p;q, −1) (q2−p2) C2 (p;−1, q) C1C1 E2 Figure I.2. Embedded Q-resolution of f= (xp+yq)(xq+yp). 10 Chapter I. Vmanifolds: Quotient Singularities, Embedded Q- . . . Let us consider π2the w= (p, q2−p2)-weighted blow-up at the origin of X(q;−1, p), π2:b C2 w,q −→ X(q;p, q2−p2) = X(q;−1, p). The new space has two singular points of type (p;−q, q2−p2)=(p;−1, q) and (q2−p2;p, −q). In the first chart, the local equation of the total transform of xp(p+q)(xq2−p2+yp) is given by the function xp(p+q)(1 + yp) : X(p;−1, q)−→ C. Thus the new exceptional divisor E2has multiplicity p(p+q) and intersects transversely the strict transform of C2at a smooth point. Hence the composition π=π2◦π1is an embedded Q-resolution of {f= 0} ⊂ C2. Figure I.2 illustrates the whole process. (iii) The point Pis of type (d;a, b).As above, assume that X= X(d;a, b) and the map (5) π=π(d;a,b),w :\ X(d;a, b)w−→ X(d;a, b) is the weighted blow-up at the origin of X(d;a, b) with respect to w= (p, q). The new space is covered as b U1∪b U2=Xp−1q pd a pb −qa ∪Xq p −1 qd qa −pb b . Or equivalently (6) \ X(d;a, b)w=b U1∪b U2=Xpd e; 1,−q+a0pb e∪Xqd e;−p+b0qa e,1 with a0a=b0b≡1 mod (d) and e= gcd(d, pb −qa). The charts are given by First chart Xp−1q pd a pb −qa −→ b U1, (x, y)7→ ((xp, xqy),[1 : y]w)(d;a,b). Second chart Xq p −1 qd qa −pb b −→ b U2, (x, y)7→ ((xyp, yq),[x: 1]w)(d;a,b). §I.2. Weighted Blow-ups and Embedded Q-Resolutions 11 Equivalently, see [Mar11, Remark I.3.14], First chart Xpd e; 1,−q+a0pb e−→ b U1, , (xe, y)7→ ((xp, xqy),[1 : y]w)(d;a,b). Second chart Xqd e;−p+b0qa e,1−→ b U2, (x, ye)7→ ((xyp, yq),[x: 1]w)(d;a,b). The exceptional divisor E=π−1 (d;a,b),w(0) is identified with the quotient space P1 w(d;a, b) := P1 w/Gdwhich is isomorphic to P1under the map P1 w(d;a, b)−→ P1 [x:y]w7→ [xdq/e :ydp/e], where e= gcd(dp, dq, pb −qa). Again the singular points are cyclic and correspond to the origins. They may be not written in normalized form even if gcd(p, q) = 1 and (d;a, b) is normalized. Example (I.2.7) ([AMO11b, Mar11]).Assume gcd(p, q) = gcd(r, s)=1 and p q<r s. Let f= (xp+yq)(xr+ys) and consider C1={xp+yq= 0}, C2={xr+ys= 0} the two irreducible components of f. Working as in Example (I.2.6), one obtains the following picture ((I.2.7)) representing an embedded Q-resolution of {f= 0} ⊂ C2. p(q+s)E1 (p;q, −1) Q C2 (s;−1, r) s(p+r)E2 C1 Q=rq −ps s −q rq −ps −r p  Figure I.3. Embedded Q-resolution of f= (xp+yq)(xr+ys). After writing the quotient spaces in their normalized form one checks that this resolution coincides with the one given in Example (I.2.6) assuming r=qand s=p. Example (I.2.8). Let {f=xy(xy + (x3−y2)2)=0}be a Q-divisor on X(7; 2,3). Let us compute a Q-resolution of {f= 0} ∈ X(7; 2,3). Consider 12 Chapter I. Vmanifolds: Quotient Singularities, Embedded Q- . . . w= (1,5) and let π(7;2,3),w :\ X(7; 2,3)w−→ X(7; 3,2) be the (1,5)-weighted blow-up at the origin. \ X(7; 2,3)w=b U1∪b U2=C2∪X(5; 2,1) First chart C2−→ b U1, , (x7, y)7→ ((x, x5y),[1 : y]w)(7;2,3). Second chart X(5; 2,1) −→ b U2, (x, y7)7→ ((xy, y5),[x: 1]w)(7;2,3). The local equation of the total transform in the first chart is given by the function x12 |{z} E1 ·y |{z} C2 ·y+ (1 −x7y2)2 | {z } C4 , on C2. The local equation of the total transform in the second chart is given by the equation y12 |{z} E1 ·x |{z} C1 ·x+ (x3−y7)2 | {z } C3 , on X(5; 2,1). After a second weighted blow-up with weight w= (2,1) of [(0,0)] ∈X(5; 2,1) one finally has \ X(5; 2,1)w=b U1∪b U2=X(2; 1,1) ∪C2 (1,5) (2,1) C2 C4 C1 C3 E1 C1 C1 C2 C2 C3C3C4 C4 E1 E2 (7; 2,3) (5; 2,1) (2; 1,1) Figure I.4. Embedded Q-resolution of f=xy(xy + (x3−y2)2). §I.3. Intersection Theory on Abelian-Quotient V-Surfaces 13 First chart X(2; 1,1) −→ b U1, , (x5, y)7→ ((x2, xy),[1 : y]w)(5;2,1). Second chart C2−→ b U2, (x, y5)7→ ((xy2, y),[x: 1]w)(5;2,1). Section §I.3 Intersection Theory on Abelian-Quotient V-Surfaces Intersection theory is a powerful tool in complex algebraic (and analytic) geometry, see [Ful98] for a wonderful exposition. It will be frequently used in the successive chapters. The main objects involved in intersection theory on surfaces are divisors, which have two main incarnations, Weil and Cartier. These coincide in the smooth case, but not in general. In the singular case the two concepts are different and a geometric interpretation of intersection theory is yet to be developed. A general definition for normal surfaces was given by Mumford [Mum61] and it was applied by Sakai to study Weil divisors on normal surfaces [Sak84]. I.3–1. Cartier and Weil Q-Divisors on V-Manifolds Let us recall the definitions of Cartier and Weil divisors. The content of this section can be found in detail in [AMO11a]. Let Xbe an irreducible normal complex analytic variety. Denote by OXthe structure sheaf of X and KXthe sheaf of total quotient rings of OX. Denote by K∗ Xthe (multiplicative) sheaf of invertible elements in KX. Similarly O∗ Xis the sheaf of invertible elements in OX. Note that an irreducible subvariety Vcorresponds to a prime ideal in the ring of sections of any local complex model space meeting V. Definition (I.3.1). ACartier divisor on Xis a global section of the sheaf K∗ X/O∗ Xand it can be represented by giving an open covering {Ui}i∈Iof X and, for all i∈I, an element fi∈Γ(Ui,K∗ X) such that fi fj∈Γ(Ui∩Uj,O∗ X),∀i, j ∈I. 14 Chapter I. Vmanifolds: Quotient Singularities, Embedded Q- . . . Two systems {(Ui, fi)}i∈I,{(Vj, gj)}j∈Jrepresent the same Cartier divisor if and only if on Ui∩Vj,fiand gjdiffer by a multiplicative factor in OX(Ui∩Vj)∗. The abelian group of Cartier divisors on Xis denoted by CaDiv(X). If D:= {(Ui, fi)}i∈Iand E:= {(Vj, gj)}j∈Jthen D+E={(Ui∩Vj, figj)}i∈I,j∈J. The functions fiabove are called local equations of the divisor on Ui. A Cartier divisor on Xis effective if it can be represented by {(Ui, fi)}i with all local equations fi∈Γ(Ui,OX). Any global section f∈Γ(X, K∗ X) determines a principal Cartier divisor (f)X:= {(X, f)}by taking all local equations equal to f. Definition (I.3.2). AWeil divisor on Xis a locally finite linear combination with integral coefficients of irreducible subvarieties of codimension one. The abelian group of Weil divisors on Xis denoted by WeDiv(X). If all coefficients appearing in the sum are non-negative, the Weil divisor is called effective. The following theorem allows us to identify both notions on V-manifolds after tensorizing by Q. Theorem (I.3.3) ([AMO11a, Mar11]).Let Xbe a V-manifold. Then the notion of Cartier and Weil divisor coincide over Q. More precisely, the linear map TX⊗1 : CaDiv(X)⊗ZQ−→ WeDiv(X)⊗ZQ is an isomorphism of Q-vector spaces. In particular, for a given Weil divisor Don Xthere always exists k∈Zsuch that kD ∈CaDiv(X). Definition (I.3.4). Let Xbe a V-manifold. The vector space of Q-Cartier divisors is identified under TXwith the vector space of Q-Weil divisors. A Q-divisor on Xis an element in CaDiv(X)⊗ZQ= WeDiv(X)⊗ZQ.The set of all Q-divisors on Xis denoted by Q-Div(X). In [AMO11a], we give a way to construct the inverse of TX⊗1. Here we summarize how to write a Weil divisor as a Q-Cartier divisor where X is an algebraic V-manifold. (1) Write D=Pi∈Iai[Vi]∈WeDiv(X), where ai∈Zand Vi⊂X irreducible. Also choose {Uj}j∈Jan open covering of Xsuch that Uj=Bj/Gjwhere Bj⊂Cnis an open ball and Gjis a small finite subgroup of GL(n, C). (2) For each (i, j)∈I×Jchoose a reduced polynomial fi,j :Uj→C such that Vi∩Uj={fi,j = 0}, then [Vi|Uj] = 1 |Gj|{(Uj, f|Gj| i,j )}. §I.3. Intersection Theory on Abelian-Quotient V-Surfaces 15 (3) Identifying {(Uj, f|Gj| i,j )}with its image CaDiv(Uj),→CaDiv(X), one finally writes Das a sum of locally principal Cartier divisors over Q, D=X (i,j)∈I×J ai |Gj|{(Uj, f|Gj| i,j )}. See [AMO11a] for a detailed explanation. I.3–2. Rational Intersection Number and Weighted Blow-ups Now we have all the necessary ingredients to develop a rational intersection theory on surfaces with quotient singularities. The content of this section can be found in detail in [AMO11b]. Let Xbe an algebraic V-manifold of dimension 2. Consider D1and D2 two effective Q-divisors on X, and P∈Xa point. The divisor Diis locally given in a neighborhood of Pby a reduced polynomial fi,i= 1,2. On the other hand the point Pcan be assumed to be a normalized type of the form (d;a, b). Hence the computation of (D1·D2)Pis reduced to the following particular case. Definition (I.3.5) (Local intersection number on X(d;a, b), [AMO11b, Mar11, Ort09]).Denote by Xthe cyclic quotient space X(d;a, b) and consider two divisors D1={f1= 0}and D2={f2= 0}given by f1, f2∈ C{x, y}reduced and without common components. Assume that, (d;a, b) is normalized. Then as Cartier divisors D1=1 d{(X, fd 1)}and D2=1 d{(X, fd 2)}. The local number (D1·D2)[P]at a point Pof type (d;a, b) is defined as (D1·D2)[P]=1 d2dimCOP hfd 1, fd 2i where OP=C{x, y}Gdis the local ring of functions at P(recall §I.1–2). Example (I.3.6) ([AMO11b, Mar11, Ort09]).Let x1and x2be the local coordinates of the axes in M:= X(d;a, b). Consider Xi:= {(M, xi)} the Q-divisors associated with them. Then (X1·X2)0=1 d. See [AMO11b] for further details and [Ort09] for a more direct approach. In the preceeding sections, weighted blow-ups were introduced as a tool for computing embedded Q-resolutions. Here we calculate self-intersection 16 Chapter I. Vmanifolds: Quotient Singularities, Embedded Q- . . . numbers of exceptional divisors of weighted blow-ups on analytic surfaces with abelian quotient singularities. Proposition (I.3.7) ([AMO11b, Mar11]).Let Xbe an analytic surface with abelian quotient singularities and let π:b X→Xbe the (p, q)-weighted blow-up at a point P∈Xof type (d;a, b). Assume gcd(p, q)=1and (d;a, b)is a normalized type, i.e. gcd(d, a) = gcd(d, b)=1. Also write e= gcd(d, pb −qa). Consider two Q-divisors Cand Don X. As usual, denote by Ethe exceptional divisor of π, and by b C(resp. b D) the strict transform of C(resp. D). Let νand µbe the (p, q)-multiplicities of Cand Dat P, i.e. x(resp. y) has (p, q)-multiplicity p(resp. q). Then the following equalities hold: (1) π∗(C) = b C+ν eE. (2) E·b C=eν dpq. (3) E2=−e2 dpq. (4) b C·b D=C·D−νµ dpq. In addition, if Dhas compact support then b D2=D2−µ2 dpq. Example (I.3.8) ([AMO11b]).Let us compute now the self-intersection of the divisors in Example (I.2.6). After the first blow-up (of type (q, p) over a smooth point) the divisor E1has self-intersection −1 pq . Let us consider the second blow-up, of type (p, q2−p2) at a point of type (q;p, q2−p2); the exceptional divisor is E2and its self-intersection is −q p(q2−p2). The strict transform of E1has multiplicity pand hence its self-intersection is −1 pq − p q(q2−p2)=−q p(q2−p2). For a detailed example of this intersection theory see [AMO12,§6]. Section §I.4 Weighted projective plane The main reference that has been used in this section is [Dol82], see also [Ort09] for a more detailed and down to earth exposition. Here we concentrate our attention on describing the analytic structure and singularities. Let w:= (w0, w1, w2) be a weight vector, that is, a triple of pairwise coprime positive integers. There is a natural action of the multiplicative group C∗on C3\{0}given by (x0, x1, x2)7−→ (tw0x0, tw1x1, tw2x2). §I.5. Logarithmic Complex and residues on V-manifolds 23 Example (I.5.17). Consider now the 2-form τ:= Ω2 xyz =w2zdx ∧dy xyz +w0xdy ∧dz xyz +w1ydz ∧dx xyz in P2 (w0,w1,w2)with (9) Ω2:= w2zdx ∧dy +w0xdy ∧dz +w1ydz ∧dx. This form (9) will be called from now on the weighted volume form. Denote by P0:= [1 : 0 : 0]w,P1:= [0 : 1 : 0]wand P2:= [0 : 0 : 1]wthe three vertices of P2 w. Let us compute the residues of τat the three origins of the weighted projective space. One has Res[2] P0(τ) = Res[2]w2 dx ∧dy xy = (−1)σ(3,1,2) = (−1)2= 1, Res[2] P1(τ) = Res[2]w1 dz ∧dx zx = (−1)σ(2,3,1) = (−1)2= 1, Res[2] P2(τ) = Res[2]w0 dy ∧dz yz = (−1)σ(1,2,3) = (−1)0= 1. Definition (I.5.18). In general we define the weighted volume form on Pn w with w= (w0, . . . , wn) as (10) Ωn:= n X i=0 (−1)iwixidxˇı, where dxˇı=dx1∧···∧dxi−1∧dxi+1 ∧···∧dxn. II Local invariants on quotient singularities In this chapter we define and investigate some of the properties of local invariants of curve germs on quotient singular surfaces. In particular, we define a Milnor fiber, Milnor number, and a δ-invariant which generalize their analogues over smooth surfaces. In §II.1 and §II.3 we show that such invariants can be effectively computed in terms of embedded Q-resolutions. These results will allow us in Chapter IV to obtain a formula for the genus of a curve in the weighted projective plane in terms of its degree and the previous invariants. Part of the results of this chapter can be found in [CAMO13]. Let Xbe a surface quotient singularity and C={f= 0}aQ-divisor on X. By means of the cyclic action, one can canonically obtain a function Fon Xand thus define the Milnor fiber and Milnor number µwof Fin a standard way. Note that alternative generalizations of Milnor numbers can be found, for instance, in [ABFdBLMH10, BLSS02, uT77, STV05]. The approach proposed in the present work seems more natural for quotient singularities (see Example (IV.1.18)), but more importantly, it allows for the existence of an explicit formula relating Milnor number, δ-invariant, and genus of a curve on a singular surface. The new local invariant δw 0 can be given in terms of the Milnor fiber of (C,0) by means of the formula µw= 2δw−rw+ 1, where µwis the Milnor number of (C,0) and rwis the number of local branches of Cat 0. In the classical case (X=C2) the invariant δcan be obtained from a resolution of the local singularity (C,0) in (C2,0) as (11) δ=X Q≺0 νQ(νQ−1) 2, 25 26 Chapter II. Local invariants on quotient singularities where Qruns over all infinitely near points of 0 of the σ-process in a resolution of (C,0) and νQdenotes the multiplicity of the strict transform of C at Q. A two-fold generalization of this result is shown here: first by allowing the resolution to be an embedded Q-resolution and second by allowing X to be a quotient surface singularity (see Theorem (II.2.5)). In the classical case, the δ-invariant can be interpreted as the dimension of a vector space. Since, in general, δwis a rational number, a similar result can only be expected in certain cases, namely, when associated with Cartier divisors. However, we can associate a natural number f δwto δwwhich can also be understood as a difference of two dimensions of vector spaces. Some of the techniques used in the study of f δwwill be used in Chapter II to obtain a numerical version of the Adjunction Formula to compute the degree of the canonical divisor of (non-smooth) curves in P2 w. Section §II.1 Milnor fibers on quotient singularities Our purpose in this section is to provide a definition for the Milnor fiber of the germ of an isolated curve singularity (f, [0]) defined on an abelian V-surface X(d;a, b). For the sake of completeness we include this chapter which can be found in [CAMO13]. In the classical case, if (C={f= 0},0) ⊂(C2,0) defines a local singularity, the Milnor fiber is defined as Ft={f=t}and it satisfies χ(Ft) = r−2δ, where ris the number of local branches of (C,0). Note that this cannot be extended directly to the case C ⊂ X(d;a, b) because, in general, the germ (f, [0]) does not define a function on X(d;a, b). However, F:= Qg∈Gfg=ufd(ua unit) is a well-defined function on X(d;a, b) and hence the set {F=t}is also well defined and invariant under the action of the cyclic group Gd. One can offer the following alternative definition for the Milnor fiber of C. Definition (II.1.1) ([CAMO13]).Let C={f= 0} ⊂ X(d;a, b) be a curve germ. The Milnor fiber Fw tof (C,[0]) is defined as follows, Fw t:= {F=t}/Gd. The Milnor number µwof (C, P) is defined as follows, µw:= 1 −χorb(Fw t). The symbol χorb(M) denotes the orbifold Euler characteristic of M⊂ X=C2/Gdas a subvariety of a quotient space which carries an orbifold structure. Note that one can also consider Cas a germ in (C2,0) in which §II.2. Local invariants on quotient singularities 27 case, χorb(Fw t) = 1 dχ(Ft), where Ft={F=t} ⊂ C2. Therefore, 1 −µw= 1 d−µ d, which implies (12) µw=d−1 d+µ d. Note the difference between this definition and [ABFdBLMH10, Definition 2.10]. Also note that µw(x) = µw(y) = d−1 d, which extends immediately to all local curves that can become an axis after an action-preserving change of coordinates. This motivates the following. Definition (II.1.2) ([CAMO13]).The curve {f= 0} ⊂ X(d;a, b) is called aQ-smooth curve if there exists g∈C{x, y}such that C{f, g}G=C{x, y}G. Corollary (II.1.3) ([CAMO13]). {f= 0}is a Q-smooth curve ⇐⇒ µw(f) = d−1 d. Section §II.2 Local invariants on quotient singularities II.2–1. Noether’s Formula In this section we present a version of Noether’s formula for curves on quotient singularities with respect to Q-resolutions. During this section we will use the notation introduced in Chapter I. As an immediate consequence of Proposition (I.3.7)(4) one has the following formula, (13) (C·D)[0] =νCνD pqd +X Q≺[0] (b C·b D)Q, where Qruns over all infinitely near points to [0] ∈X(d;a, b) after a weighted (p, q)-blow-up. By induction, using formula (13), one can prove the following generalization of Noether’s formula for Q-divisors on quotient surface singularities. Theorem (II.2.1) (Noether’s Formula, [CAMO13]).Consider Cand D two germs of Q-divisors at [0] without common components in a quotient surface singularity. Then the following formula holds: (C·D)[0] =X Q≺[0] νC,QνD,Q pqd , where Qruns over all the infinitely near points of (C·D, [0]) and Qappears after a blow-up of type (p, q)of the origin in X(d;a, b). 28 Chapter II. Local invariants on quotient singularities Remark (II.2.2) ([CAMO13]).Note that p, q, d, a, b in Theorem (II.2.1) depend on Qand its predecessor. II.2–2. Definition of the δ-invariant on quotient singularities In this section the local invariant δwfor curve singularities on X(d;a, b) is defined. Definition (II.2.3) ([CAMO13]).Let Cbe a reduced curve germ at [0] ∈ X(d;a, b). We define δwas the number verifying (14) χorb(Fw t) = rw−2δw, where rwis the number of local branches of Cat [0], Fw tdenotes its Milnor fiber, and χorb(Fw t) denotes the orbifold Euler characteristic of Fw t. Using the same argument as in (12), one can check that (15) δw=1 dδ+1 2rw−r d where δdenotes the classical δ-invariant of Cas a germ in (C2,0) and ris the number of local branches of Cin (C2,0). Remark (II.2.4) ([CAMO13]).At this point it is worth mentioning that rw and rdo not necessarily coincide. For instance, x2−y4defines an irreducible curve germ in X(2; 1,1) (hence rw= 1), but it is not irreducible in C2(where r= 2). One can check that δw= 1, δ= 2 (see Example (II.2.7)), which verify (15). The purpose of this section is to give a recurrent formula for δwbased on a Q-resolution of the singularity. For technical reasons it seems more natural to use strong Q-resolutions (see Remark (I.2.4)) for the statement, but this is not a restriction (see Remark (II.2.8)). Before we state the result, let us introduce some notation. Assume (f, 0) ⊂X(d;a, b) and consider a (p, q) blow-up πat the origin. Denote by ν0(f) the (p, q)-multiplicity of fat 0. We will use the following notation: (16) δw 0,π(f) := ν0(f) 2dpq (ν0(f)−p−q+e), The result now can be stated as. Theorem (II.2.5) ([CAMO13]).Let (C, [0]) be a curve germ on an abelian quotient surface singularity. Then (17) δw(C) = X Q≺[0] δw Q,π(p,q)(f) §II.2. Local invariants on quotient singularities 29 where Qruns over all the infinitely near points of a strong Q-resolution of (C, [0]),π(p,q)is a (p, q)-blow-up of Q, the origin of X(d;a, b), and e:= gcd(d, aq −bp). Proof. Since we want to proceed by induction, let us assume [0] ∈ X(d;a, b). After a (p, q)-blow-up of [0] there are three types of infinitely near points to [0], namely, P1(resp. P2) a point on the surface of local type X(pd e; 1,−q+a0pb e) (resp. X(qd e;−p+b0qa e,1)) and P3, . . . , Pnsmooth points on the surface (see (6)). Outside the neighborhoods Biof the points Pi, the preimage of the Milnor fiber is a covering of degree ν eover E. Therefore χorb(Fw t) = χorb(E\{P1, . . . , Pn})ν e+X i χorb(Bi∩{xdν e˜ Fi=t}), since the intersection is glued over puncture disks, whose Euler characteristic is zero. Here ˜ Fidenotes the strict preimage of Fat Pi. In order to compute χorb(Bi∩{xdν e˜ Fi=t}) we have to distinguish three cases: P1, P2, and Pi(i= 3, . . . , n). Assume first that P=Pi,i= 3, . . . , n, then one can push ˜ Fiinto ˜ F0 iin a direction transversal to Eas in Figure II.1, then χorb(Bi∩{xdν e˜ Fi=t}) = χ(( ˜ F0 i)t)−(E, ˜ Fi)Pi. ˜ Fi˜ F0 i E Figure II.1. Pushing ˜ Fi In case P=P1∈X(pd e; 1,−q+a0pb e), one has χorb(B1∩{xdν e˜ F1=t}) = e pd χ(B1∩{xdν e 1 d˜ f1=t}), where ˜ f1is the preimage of ˜ F1in C2. Therefore, after applying the pushing strategy, χ(B1∩{xν e˜ f1=t}) = ν e(1 −(E, f1)P1), and hence χorb(B1∩{xdν e˜ F1=t}) = ν ee pd −(E, ˜ F1)P1. 30 Chapter II. Local invariants on quotient singularities One obtains an analogous formula for P2. Adding up all the terms and applying Proposition (I.3.7)(1) one obtains: χorb(Fw t) = ν dp +ν dq −eν dpq(1 + ν e) + X i χorb(˜ Fi) =−ν dpq(ν−p−q+e) + X i χorb(˜ Fi). The formula follows by induction since, after a strong Q-resolution, Piχorb(˜ Fi) = rwand hence χorb(Fw t) = rw−Xν dpq(ν−p−q+e) = rw−2δw.  As an immediate consequence of Theorem (II.2.5), one can obtain a recursive formula for δw 0(C) in terms of δw Q(ˆ C) for Q≺0 which can be useful for our purposes. Corollary (II.2.6). Let Cbe a curve singularity at 0∈X(d;a, b),πa weighted (p, q)blow-up and Q1, . . . , Qrinfinitely near points of Cafter π. Then δw 0(C) = δw 0,π(C) + r X j=1 δw Qj(ˆ C), with the notation used above. A similar comment to Remark (II.2.2) applies to Theorem (II.2.5). Example (II.2.7) ([CAMO13]).Assume xp−yq= 0, for p,qcoprime, defines a curve on a surface singularity of type X(d;a, b). Note that a simple (q, p)-blow-up will be a (strong) Q-resolution of the singular point. Therefore, using Theorem (II.2.5) one obtains δw=ν 2dqp(ν−q−p+e). Note that ν=pq. Also, since xp−yq= 0 defines a set of zeros in X(d;a, b) by hypothesis, this implies ap ≡bq mod dand hence e:= gcd(d, ap −bq) = d. Thus (18) δw=(pq −p−q+d) 2d. Note that this provides a direct proof, for the classical case (d= 1), that δ=(p−1)(q−1) 2 for a singularity of type xp−yqin (C2,0). §II.2. Local invariants on quotient singularities 31 Another direct consequence of (18) is that (19) δw(x) = δw(y) = d−1 2d and the same formula holds for any Q-smooth curve (see Definition (II.1.2)). Remark (II.2.8). By (19), a Q-resolution is enough to obtain δw. Note that a Q-resolution ends when the branches are separated and each strict transform is a Q-smooth curve. Therefore if (17) is applied to a (not necessarily strong) Q-resolution, one needs to add di−1 2difor each local branch γi⊂X(di;ai, bi), i= 1, . . . , rw. Corollary (II.2.9) ([CAMO13]).Let Cand Dbe two reduced Q-divisors at [0] ∈X(d;a, b)without common components. Then δw(C·D) = δw(C) + δw(D)+(C·D)[0]. Proof. One has, (20) νC·D(νC·D−p−q+e)=(νC+νD)(νC+νD−p−q+e) =νC(νC−p−q+e) + νD(νD−p−q+e)+2νCνD. Dividing (20) by 2dpq and adding over all the infinitely near points to Pone obtains, δw(C·D) = δw(C) + δw(D) + X Q≺[0] νC,QνD,Q pqd =δw(C) + δw(D) + (C·D)[0].  As an inmediate consequence of Corollary (II.2.9) after applying induction one has the following result. Lemma (II.2.10). Let Ci,i= 1 ...,n ∈Nbe reduced Q-divisors at [0] ∈ X(d;a, b)without common components. Then δw(C1·. . . ·Cn) = n X i=1 δw(Ci) + X i6=j (Ci·Cj)[0]. Example (II.2.11). Let x= 0 and y= 0 on X(d;a, b), by (19) and Corollary (II.2.9) one has δw(xy) = δw(x) + δw(y)+(x·y)[0] = 2d−1 2d+1 d= 1. Definition (II.2.12). Let Ebe a reduced Q-divisor. We define δw E:= X P∈E δw P(E). Note that the sum is finite since δw P(E) = 0 for non-singular points. 32 Chapter II. Local invariants on quotient singularities Example (II.2.13). Recall that after a weighted blow-up at the origin of X(d;a, b) with respect to w= (p, q) (see 6) one has (21) \ X(d;a, b)w=Xpd e; 1,−q+a0pb e∪Xqd e;−p+b0qa e,1. Let us compute δw E, where E=π−1 (d;a,b),w(0) is the exceptional divisor after a weighted blow-up of type wat the origin of X(d;a, b). Using (19), (21), and Definition (II.2.12), one has δw E= qd e−1 2qd e + pd e−1 2pd e = 1 −(p+q)e 2pqd . Using Corollary (II.2.6), note that one can obtain δwfrom a sequence of weighted blow-ups such that the strict preimage of Cis Q-smooth, which is weaker than asking for a (strong) Q-resolution as follows: (22) δw=1 2X Q≺[0] νQ dpq (νQ−p−q+e) + δw ˆ C, where δw ˆ Cis a finite sum of terms of the form di−1 2di, for Qi∈X(di;ai, bi). Section §II.3 The δw-invariant In the classical case this invariant can be interpreted as the dimension of a vector space. Since, in general, δwis a rational number, a similar result cannot be expected for any germ. The following section is devoted to proving that when the germ is defined by a function, the number δwis a non-negative integer and in fact it can also be interpreted as the dimension of a vector space. After that, in the second part we present a discussion on the δw-invariant for general germs. II.3–1. The δw-invariant for function germs Let us start with the following constructive result which allows one to see any singularity on the quotient X(d;a, b) as the strict transform of some {g= 0} ⊂ C2after performing a certain weighted blow-up. Remark (II.3.1). The Weierstrass division theorem states that given f, g ∈ C{x, y}with f y-general of order k, there exist q∈C{x, y}and r∈C{x}[y] §II.3. The δw-invariant 39 Finally one gets, A(p,q) pa =a(pqa −p−q+ 1) 2=δ(p,q) pa . 2) Proving equation (27) is simple and direct computation. To prove equation (28), let us describe A(p,q) r1,A(p,q) rand A(p,q) r1−r: A(p,q) r1= #{(i, j)∈N2|pi +qj ≤qr +pqa;i, j ≥1}. pa +r 0 1 r qa A(p,q) r1−rA(p,q) r1 A(p,q) r aqr Figure II.3. A(p,q) r= #{(i, j)∈N2|pi +qj ≤qr;i, j ≥1} = #{(i, j)∈N2|pi +qj ≤qr +apq −apq;i, j ≥1} = #{(i, j)∈N2|p(i+aq) + qj ≤qr1;i≥1, j ≥1} = #{(i, j)∈N2|pi +qj ≤qr1;i≥aq + 1, j ≥1}. A(p,q) r1−r= #{(i, j)∈N2|pi +qj ≤qr1−qr;i, j ≥1} = #{(i, j)∈N2|pi +q(j+r)≤qr1;i, j ≥1} = #{(i, j)∈N2|p+qj ≤qr1;i≥1, j ≥r+ 1}. To conclude the proof it is enough to apply Pick’s Theorem again and look at Figure II.3 which represent the situation one has.  40 Chapter II. Local invariants on quotient singularities As a result of this Proposition (II.3.12) one has the following result. Theorem (II.3.13). Let f1, f2∈ O(k)be two germs at [0] ∈X(d;a, b). Then, K0(f1)−K0(f2) = δw 0(f1)−δw 0(f2). Proof. By Remark (II.3.1) and the discussion after it, we can assume that f`(x, y) = yr`+X i>0≤j<r` aijxiyj∈C{x}[y]. in X(p;−1, q) (p=d, q ≡ −ba−1mod d). Consider g1∈C{x, y}the reduced germ obtained after applying Lemma (II.3.2) to f1. Denote by π(p,q)the blowing-up at the origin. Note that νp,q(g1) = qr1, and thus δ(p,q) r1=δw π(p,q)(g1) (see (26) and (16)). Consider the form ω:= φdx∧dy g1,φ∈C{x, y}and let us calculate the local equations for the pull-back of ωafter blowing-up the origin on C2, (29) φdx ∧dy g1 π(p,q) ←− xνφ+p+q−1−qr1hdx ∧dy f1 . Using the definitions of Mnul g1and Mnul f1(see Definition (II.3.8)) this implies that φ∈ Mnul g1⇔h∈ Mnul f1(νφ),and νφ+p+q−1−qr1≥0. Therefore φ(x, y)7→ φ(xp, xqy) induces an isomorphism Mnul g1∼ =A(p,q) r1∩Mnul f1, where A(p,q) r1:= {h∈C{x, y} | ordh+p+q−1−qr1≥0}and ordhis the order of h. Since dimCC{x,y} A(p,q) r1 =A(p,q) r1, one obtains (30) K0(g1) = A(p,q) r1+K0(f1) On the other hand (see Remark (II.3.11) and Corollary (II.2.6)), (31) K0(g1) = δ0(g1) = δw π(p,q)(f) + δw 0(f1) = δ(p,q) r1+δw 0(f1). Therefore, from (30) and (31), (32) K0(f1) = δw 0(f1) + δ(p,q) r1−A(p,q) r1. Following a similar procedure we get, (33) K0(f2) = δw 0(f2) + δ(p,q) r2−A(p,q) r2. Notice that k≡qr1≡qr2mod p, which implies r1≡r2mod psince pand qare coprime. Therefore by Proposition (II.3.12) A(p,q) r1−δ(p,q) r1=A(p,q) r2−δ(p,q) r2, §II.3. The δw-invariant 41 and finally from (32) and (33) it can be concluded that K0(f1)−K0(f2) = δw 0(f1)−δw 0(f2).  From Proposition (II.3.12)(1) and (32) one has the following result which generalizes Remark (II.3.11). Corollary (II.3.14). If (f, [0]) is a function germ on X(d;a, b), then K0(f) = δw 0(f). The result in Theorem (II.3.13) motivates the following definition. Definition (II.3.15). Let {f= 0}be a germ in 0 ∈X(d;a, b), where f∈ O0(k). Consider g∈ O0(k) generic. The ∆w 0-invariant is defined as follows ∆w 0(f) := δw 0(f)−δw 0(g). Remark (II.3.16). Notice that ∆w P(f) is always a well-defined integer. Note that if gis a generic holomorphic germ in (C2, P) (and thus smooth), then δw P(g) = 0. Hence, if fis a holomorphic germ in (C2, P) then ∆w P(f) = δP(f). III Logarithmic Trees Some basic definitions of how to construct logarithmic modules (Definition (II.3.8)) and trees (Definition (III.1.10)) associated with a Q-resolution will be given in the following sections. They extend those given in [CA02, §2.5 and §2.6]. In this chapter we also define some new logarithmic modules associated with a germ f∈(X(d;a, b), P) and a Q-resolution π, recall the ones already defined in Chapter II (Definition (II.3.8)). A useful description of these logarithmic modules will result from the use of multiplicity trees. Their global sections will allow for the construction of logarithmic 2-forms on D ⊂ P2 wwith respect to π. All these results will allow us in Chapter V to provide a (rational) presentation for the cohomology ring of P2 w\R, where Ris a reduced algebraic curve (possibly singular) in the complex projective plane P2 wwhose irreducible components are all rational. Section §III.1 Construction of logarithmic trees: Tnul P Let us start this section with some technical definitions about the construction of multiplicity trees. We will end this section describing a basic example of a logarithmic tree. Remark (III.1.1). For the sake of simplicity and if no ambiguity seems no likely to arise, when refering to a (pP, qP)-weighted blow-up of a point P, the subindex Pwill be omitted. Analogously, the type of surface singularity appearing after weighted blow-ups X(d;a, b) will be omitted when possible. Notation (III.1.2). Let fbe an analytic germ at P∈X0=X(d;a, b) whose set of zeros is a germ of curve Vf⊂X0. Consider the sequence of 43 44 Chapter III. Logarithmic Trees weighted blow-ups X0ε1 ←− X1ε2 ←− X2ε3 ←− ... εm ←− Xm=ˆ X in a Q-resolution of X0at P. •Denote by πkthe composition of the first kweighted blow-ups πk=εk◦... ◦ε1. •The germ of curve ˆ Vf,k =π−1 k(Vf\{P}) shall be called the strict transform of Vfin Xkand its equation denoted by ˆ fk. •The reduced divisor π∗ k(Vf)red shall be denoted by Vf,k and called the total transform of Vfin Xk. For simplicity let us write ˆ Vf:= ˆ Vf,m and Vf:= Vf,m. •The exceptional divisor in Xkresulting from the weighted blow-up of a point in Xk−1shall be denoted by Ekand the points P1 k, ..., PNk k in Ek∩ˆ Vf,k will be called the infinitely near points to Pin Ek. For convenience, the point Pis also considered to be infinitely near to itself. •For the sake of simplicity, if a (p, q)-weighted blow-up occurs at a point Qwe shall denote by νQ(f) its (p, q)-weighted multiplicity, omitting the weights if no ambiguity seems no likely to arise. Definition (III.1.3) (Multiplicity tree of πat P).Let πbe a Q-resolution of singularities. Let us construct the multiplicity tree of πat Pwhich will be denoted by TP(π, f), or simply by TP(f) if the Q-resolution πof X0is fixed. TP(f) is a labeled tree with triples (w, `1, `2) at each vertex and is defined as follows. T1. The vertices of TP(f) are in bijection with the infinitely near points to P(for simplicity we shall denote the vertices of the tree by their corresponding infinitely near points). T2. Two vertices of TP(f), say Qand Q0, are joined by an edge if and only if: one of the points, say Q0, belongs to Xk; the other point Q belongs to Xk−1and Q0∈ε−1 k(Q) = Ek. T3. For convenience, this tree is considered to simply be a vertex if X0 is smooth and Pis not a singular point of f. If f(P)6= 0, then TP(f) := ∅. T4. The weight wof the label at a vertex Qwill be defined as νQ(f) and denoted by w(TP(f), Q) = νQ(f). If a (p, q)-weighted blow-up occurs at a point Qof type X(d;a, b), its labels will be `1= (p, q), `2= (d;a, b). For technical reasons, if no blow-up occurs at a point Qof type X(d;a, b), then Qmust be a Q-smooth point and its label will be (νQ(f),(1,1),(d; 1,1)) or simply (νQ(f),(1,1),(d)). §III.1. Construction of logarithmic trees: Tnul P45 Notice that both multQ(ˆ Vf,k, Ek) = 1 dand δQ(Ek) = δQ(ˆ Vf,k) = d−1 2ddo not depend on (p, q) and (a, b). T5. Depending on the context, for the sake of simplicity and if no ambiguity seems likely to arise, we might only write in each vertex Q of the tree the weight w(TP(f), Q). Let us define an extension of these multiplicity trees. Definition (III.1.4) (Extended multiplicity tree at P).The extended multiplicity tree at Pwill be denoted by ˜ TP(f). It contains the multiplicity tree TP(f) as a subtree and it can be constructed as follows. ˜ T1. The vertices of ˜ TP(f) correspond to the points of Sing( ˆ X). Note that each extra vertex can be of two types: (a) Those corresponding to a non-infinitely near point in Sing( ˆ X) in the intersection of two exceptional divisors, say Eiand Ej, which shall be denoted by eij with the convention i<j. (b) Those corresponding to a non-infinitely near point in Sing( ˆ X) which only belong to one exceptional divisor, say Eiwhich shall be denoted by ein, where n∈ {1,2}. Note that there are at most two of these points on each exceptional divisor Ei. The corresponding vertices will be shown in the tree as ◦ instead of the usual solid vertices •. Note that the singular points of type (b) are necessarily Q-smooth in ¯ Vf. ˜ T2. A vertex of type (a), say eij, will be joined to a vertex Qof TP(f) if Qbelongs to Xjand eij ∈ε−1 j(Q). A vertex of type (b), say einis joined to a vertex Qof TP(f) if Qbelongs to Xiand ein∈ε−1 i(Q). These new edges will be shown on the extended tree as dashed segments. ˜ T3. The corresponding label at each eij or eijshall be (0,(1,1),(d; 1,1)) or simply (0,(1,1),(d)). The triples at the remaining vertices coincide with those as vertices of TP(f). Remark (III.1.5). Notice that existence of points of type ˜ T1.(b) in Definition (III.1.4) constitutes a remarkable difference with the classical construction in [CA02,§2.5]. Example (III.1.6). Assume gcd(p, q) = 1 and p<q. Let f= (xp+yq)(xq+ yp) be a germ of curve singularity at 0 ∈C2. Recall Examples (I.2.6) and (I.3.8). Figure III.1 is a possible graph for ˜ T0(f). 46 Chapter III. Logarithmic Trees A B C D E F G Figure III.1. Extended multiplicity tree ˜ T0(f). A= (p(p+q),(q, p),(1; 1,1)). B= (0,(1,1),(p; 1,1)). C= (1,(1,1),(1; 1,1)). D= (p(p+q),(p, q2−p2),(q;−1, p)). E= (0,(1,1),(p; 1,1)). F= (1,(1,1),(1; 1,1)). G= (0,(1,1),(q2−p2; 1,1)). Let us see now some basic definitions coming from graph theory. Definition (III.1.7). The set of vertices |˜ TP(f)|of an extended multiplicity tree ˜ TP(f) is endowed with a partial order as follows. Consider the preferred point (root), Pand direct the edges of the tree towards P. In this directed tree, a point Qis said to be greater than Q0(denoted Q≥Q0) if there is a directed path from Qto Q0. That means that all arrows are pointing in the same direction. In graph theory this situation is commonly described by calling Qan ancestor of Q0, or Q0adescendant of Q. Given a set of points {P1, ..., Pn} ⊂ ˜ TP(f) one can define Asc(P1, ..., Pn) = {Q∈ TP(f)|Q≥Pii= 1, ..., n}, Desc(P1, ..., Pn) = {Q∈ TP(f)|Q≤Pii= 1, ..., n}. Remark (III.1.8). Multiplicity trees are quasi-strongly connected trees, which means that the set of common descendants Desc(P1, ..., Pn) is non vacuous and inherits a linear order from TP(f). The maximal element in §III.1. Construction of logarithmic trees: Tnul P47 Desc(P1, ..., Pn) is called the greatest common descendant and it is denoted by gcd(P1, ..., Pn), for a reference see [Ber70]. Notation (III.1.9). In order to simplify, we shall write T∼ =T0for two weighted trees that are isomorphic as trees and `i=`0 i,i= 1,2. We will say T=T0(resp. ≥,≤,<or >) if T∼ =T0and w(T, Q) = w(T0, Q) (resp. ≥, ≤,<or >) for any Q∈ |T | =|T0|(note that we are using the isomorphism of trees to identify the vertices). Sometimes it will be necessary to compare empty trees. In this case, the conditions =,≤,≥are vacuous and hence always satisfied. In what follows we are going to define some logarithmic trees and tree invariants which will be of particular interest. Definition (III.1.10). Let (f, P) be a germ and TP(f) its multiplicity tree. A labeled tree Tis said to be a logarithmic tree for (f, P) if it satisfies the following properties: • T ∼ =TP(f), and •the OP-module MT:= {h∈ OP| TP(f)|h≥ T } is logarithmic (recall Definition (II.3.8)). Remark (III.1.11). Note that if T1≥ T2, then MT2⊂ MT1. Definition (III.1.12). Consider a labeled tree Twhose labels are triples of type (w, (p, q),(d;a, b)) as above. The degree of Tshall be defined as (34) deg(T) := X Q∈|T | w(T, Q) 2dpq (w(T, Q) + p+q−e), where w(T, Q) denotes the weight of Tat Qand e:= gcd(d, aq −bp). Also, one can define the null tree associated with T(denoted by Tnul) as a labeled tree isomorphic to Twhose labels are triples of type (w0,(p, q),(d;a, b)) where (35) w(Tnul, Q) = w(T, Q)−p−q+e. Remark (III.1.13). Note that Tnul P(f) is a logarithmic tree for (f, P)∈ (X(d;a, b), P). Definition (III.1.14). Let f, g ∈ OP(k) be germs at. One can define the restriction of gto ˜ TP(f) or ˜ TP(f)|gas a labeled tree isomorphic to ˜ TP(f) where w(˜ TP(f)|g, Q) = νQ(g) is the (p, q)-multiplicity of gat Qa point of label (w, (p, q),(d;a, b)). Example (III.1.15). Recall Example (III.1.6). Assume gcd(p, q) = 1 and p<q. Let g= (xq−8yp) be a germ of curve singularity at 0 ∈C2. The 48 Chapter III. Logarithmic Trees restriction ˜ TP(f)|gis given by the tree in Figure III.1 with the following labels: A= (p2,(q, p),(1; 1,1)). B= (0,(1,1),(p; 1,1)). C= (0,(1,1),(1; 1,1)). D= (p(p+q),(p, q2−p2),(q;−1, p)). E= (0,(1,1),(p; 1,1)). F= (0,(1,1),(1; 1,1)). G= (0,(1,1),(q2−p2; 1,1)). Assume gcd(p, q) = 1 and p<q. Let f= (xp+yq)(xq+yp) be a germ of curve singularity at 0 ∈C2. Recall Examples (I.2.6) and (I.3.8). Figure III.1 is a possible graph for ˜ T0(f). Remark (III.1.16). Notice that from (34), Theorem (II.2.5) and Corollary (II.2.6) one has deg( ˜ Tnul P(f)) = deg(Tnul P(f)) = δw P(f). Remark (III.1.17). Note that according to (III.1.4), in the case of germs in (C2, P) the degree of a tree Tis related with the number of conditions imposed to a germ gso that T |g≥ T. In this situation KP(f) = deg Tnul P(f) = δP(f) (see also [CA02, Lemma 2.35]). In our case, deg Tnul P(f) = δw P(f), which is in general a rational number. However, when fdefines a function on X(d;a, b) one still obtains the equality KP(f) = deg Tnul P(f) = δw P(f) (see Corollary (II.3.14)). Let us see a lemma which will be of particular interest through this chapter. It gives local conditions under which meromorphic forms are logarithmic. Such conditions can be expressed in terms of multiplicity trees. Lemma (III.1.18). Let fand hbe analytic germs where xand yare local equations of Pon a surface X(d;a, b). Let ψ:= hdx ∧dy f be a invariant 2-form on X(d;a, b). Let πbe a Q-resolution of Vf⊂X0 such that (36) ˜ TP(f)|h≥˜ Tnul P(f) Then the following results hold: §III.3. Some examples 55 w(˜ Ti+1 j, Q) = (w(˜ Ti ∗, Q) + νQ(Eki) if Q∈ |γj|, j = 1,2 w(˜ Ti ∗, Q) if Q∈ |˜ Ti+1 j|\|γj|, j = 1,2 w(Q) = (w(˜ Ti ∗, Q) + νQ(Eki)−p−q+eif Q∈ |γj|, j = 3, ..., n2 w(˜ Ti ∗, Q)−p−q+eif Q∈ |˜ Ti+1 j|\|γj|, j = 3, ..., n2. Example (III.3.2). Let f=xy be a curve germ singularity on X(d;a, b). It has only two local branches δ1=xand δ2=yat P= [(0,0)] on X(d;a, b). After one (a, b)-weighted blow-up (recall (6)) one has (see Figure III.4) (45) \ X(d;a, b)w=X(a; 1, α)∪X(b;β, 1) = X(a; 1, α)∪Xb; 1, β0. (a+b, (a, b),(d;a, b)) δ2 (a; 1, α)(b;β, 1) (1,(1,1),(b; 1,1)) (1,(1,1)(a; 1,1)) δ1 δ1 δ2 Figure III.4. Q-Resolution of Cand ˜ T0(xy). One can observe the construction of ˜ Tnul 0(xy) and ˜ Tδ1δ2 0(xy) in Figure III.5. ˜ Tnul 0˜ Tδ1δ2 0 (d, (a, b),(d;a, b)) δ2 b−1a−1 δ1 (0,(a, b),(d;a, b)) δ2 00 δ1 Figure III.5. ˜ Tnul 0(xy) and ˜ Tδ1δ2 0(xy). Let us compute deg( ˜ Tδ1,δ2 0(xy)) −deg( ˜ Tnul 0(xy)), recalling the definition of degree seen in (34). 56 Chapter III. Logarithmic Trees On the one hand, deg( ˜ Tnul 0(xy)) = d(d+a+b−d) 2dab +(a−1)(a−1+1+1−a) 2a +(b−1)(b−1+1+1−b) 2b=a+b 2ab +a−1 2a+b−1 2b = 1, Recall Example (II.2.11) and Remark (III.1.16), as we already know deg( ˜ Tnul 0(xy)) = δw 0(xy). On the other hand one has deg( ˜ Tδ1,δ2 0(xy)) = 0. Finally one gets deg( ˜ Tδ1,δ2 0(xy)) = deg( ˜ Tnul 0(xy)) −1. Example (III.3.3). Let f=xy(xb+ya). with a,bcoprime, be a germ of curve singularity on X(d;a, b). It has three local branches δ1=xand δ2=yand δ3=xb+yaat P= [(0,0)] on X(d;a, b). After one (a, b)-weighted blow-up (recall (6)) one has (see Figure III.6) (46) \ X(d;a, b)w=X(a; 1, α)∪X(b;β, 1) = X(a; 1, α)∪Xb; 1, β0. (a+b+ab, (a, b),(d;a, b)) δ2 (a; 1, α)(b;β, 1) (1,(1,1),(b)) (1,(1,1)(a)) δ1 δ1 δ2δ3 (1,(1,1),(1)) δ3 Figure III.6. Q-Resolution of Cand ˜ T0(xy). One can observe the construction of ˜ Tnul 0(xy(xb+ya)) and ˜ Tδiδj 0(xy(xb+ ya)) in Figure III.7. Let us compute deg( ˜ Tδ1,δ2 0(xy(xb+ya))) −deg( ˜ Tnul 0(xy(xb+ya))). Recall the definition of degree seen in (34). On the one hand, deg( ˜ Tnul 0(xy(xb+ya))) = (d+ab)(d+ab +a+b−d) 2dab +(a−1) 2a+(b−1) 2b = 1 + 1 2+(ab +a+b) 2d. §III.3. Some examples 57 ˜ Tnul 0 δ2 δ1δ3 d+ab b−10a−1 δ2 δ1δ3 ab 010 ˜ Tδ1δ2 0 δ2 δ1δ3 ab 0 ˜ Tδ1δ3 0 δ2 δ1δ3 ab 0 ˜ Tδ2δ3 0 00 b a Figure III.7. ˜ Tnul 0(xy(xb+ya)) and ˜ Tδiδj 0(xy(xb+ya)). On the other hand, deg( ˜ Tδ1δ2 0(xy(xb+ya))) = (ab)(ab +a+b−d) 2dab +1(1 + 1 + 1 −1) 2 =(ab +a+b) 2d−1 2+ 1. deg( ˜ Tδ1δ3 0(xy(xb+ya))) = (ab)(ab +a+b−d) 2dab +a(a+ 1 + 1 −a) 2a =(ab +a+b) 2d−1 2+ 1. deg( ˜ Tδ2δ3 0(xy(xb+ya))) = (ab)(ab +a+b−d) 2dab +b(b+ 1 + 1 −b) 2b =(ab +a+b) 2d−1 2+ 1. Finally one gets deg( ˜ Tδ1,δ2 0(xy)) = deg( ˜ Tnul 0(xy)) −1. Example (III.3.4). Assume xp+yq= 0, with p,qcoprime, defines a curve on a surface singularity of type X(d;a, b). Note that ν=pq. Also, since xp+yq= 0 defines a set of zeros in X(d;a, b) by hypothesis, this implies 58 Chapter III. Logarithmic Trees ap ≡bq mod dand hence e:= gcd(d, ap −bq) = d. Note that a simple (q, p)-blow-up will be a (strong) Q-resolution of the singular point. Let f= (xq+yp)(xq−yp) be a germ of curve singularity at [(0,0)] on X(d;a, b). After one (p, q)-weighted blow-up one has (recall (6)) 0000 2pq 2pq −p−q+e2pq −p−q ˜ T0(f)˜ Tnul 0(f)˜ Tδ1δ2 0(f) 11 (qd e)(pd e) 00qd e−1pd e−1 qd e−2pd e−2 Figure III.8. Strong Q-resolution of f,˜ T0(f), ˜ Tnul 0(f) and ˜ Tδ1δ2 0(f). Recall the notation used in Example (II.2.13). Let us compute deg(Tδ1,δ2 0(f)) −deg(Tnul 0(f)) = (2pq −p−q)(2pq −e) 2dpq + qd e−1 2qd e + pd e−1 2pd e−(2pq −p−q+e)(2pq) 2dpq = 1 −(p+q) 2pq +e(p+q) 2dpq −2e d e=d = 1 −(p+q) 2pq +(p+q) 2pq −2 = −1. Finally one has deg( ˜ Tδ1,δ2 0(f)) = deg( ˜ Tnul 0(f)) −1. Example (III.3.5). Let f=xy(xy + (x3−y2)2) be a germ of curve singularity at [(0,0)] on X(7; 2,3), recall §V.3–2. Figure III.9 is a possible graph for ˜ T0(f). Let us denote by δ1=x,δ2=yand δ3, δ4the two local branches of xy + (x3−y2)2at the origin (see Figure V.1). δ2 δ4 δ1 δ3 δ1δ3 δ2δ4 12 11 01 4 1 δ2δ4 δ1 δ3 (1,5),(2,1) Figure III.9. Q-Resolution of Dat [0 : 0 : 1] and ˜ T0(f). §III.3. Some examples 59 One can observe the construction of ˜ Tnul 0(f) in Figure III.10, the one of ˜ Tδ1δ2 0(f) in two steps, ˜ Tδ1δ4 0(f), ˜ Tδ2δ3 0(f) and ˜ Tδ3δ4 0(f) in Figure III.11,the construction of ˜ Tδ1δ3 0(f) in Figure III.12 and finally the one of ˜ Tδ2δ4 0(f) in Figure III.13. 13 6 (p, q, e) = (1,5,7) (p, q, e) = (2,1,5) 00 00 0 Figure III.10. ˜ Tnul 0(f). 6 201 01 0˜ Tδ1δ2 0 6 501 11 1 6 210 0 1 0˜ Tδ1δ4 0 6 201 00 1 6 210 0 0 1 ˜ Tδ2δ3 0˜ Tδ3δ4 0 Figure III.11. Construction of ˜ Tδ1δ2 0(f) in two steps, ˜ Tδ1δ4 0(f), ˜ Tδ2δ3 0(f) and ˜ Tδ3δ4 0(f). Let us compute deg( ˜ Tδi,δj 0(f)) −deg( ˜ Tnul 0(f)). One has deg( ˜ Tnul 0(f)) = 13(13 + 1 + 5 −7) 2·7·1·5+6(6 + 2 + 1 −5) 2·5·2·1=24 7, deg( ˜ Tδ1δ2 0(f)) = deg( ˜ Tδ1δ4 0(f)) = deg( ˜ Tδ2δ3 0(f)) = deg( ˜ Tδ3δ4 0(f)) =6(6 + 1 + 5 −7) 2·7·1·5+ 1 + 1 = 17 7. 60 Chapter III. Logarithmic Trees 13 100 1 0 0 Figure III.12. Construction of ˜ Tδ1δ3 0(f). 6 700 10 0 Figure III.13. Construction of ˜ Tδ2δ4 0(f). deg( ˜ Tδ1δ3 0(f)) = 13(13 + 1 + 5 −7) 2·7·1·5+1(1 + 2 + 1 −5) 2·5·2·1+1 4=17 7, deg( ˜ Tδ2δ4 0(f)) = 6(6 + 1 + 5 −7) 2·7·1·5+7(7 + 2 + 1 −5) 2·5·2·1+1 4=17 7, Finally one gets deg( ˜ Tδi,δj 0(f)) = deg( ˜ Tnul 0(f)) −1,∀1≤i<j≤4. Section §III.4 Relation between of ˜ Tδ1,δ2 P(f)and ˜ Tnul P(f) The following result (Lemma (III.4.3)) shows the relation between the degree of ˜ Tnul P(f) and the degree of ˜ Tδi,δj P(f) constructed above. Remark (III.4.1). This Lemma generalizes [CA02, Lemma 2.35]. Notice that in our case the singular points of the exceptional divisors which appear will play an important role along all the proof and so the δw-invariant defined in Section II.3. §III.4. Relation between of ˜ Tδ1,δ2 P(f)and ˜ Tnul P(f)61 We have seen in subsection §III.2–1 a way to construct the weights of ˜ Tδ1,δ2using a recursive process. In order to simplify the proof of the main result some notation will be used. Notation (III.4.2). A vertex Qof a tree ˜ Twill be said to have a defect f, denoted by f(˜ T, Q) if w(˜ T, Q) = w(˜ TP(f), Q) + f. Also f(Q) will denote f(˜ Tδ1,δ2, Q). Let A⊂V[2] fbe a set of maximal points of ˜ T. A function χAshall be defined as follows: χA= #A∩{δ1, δ2}. Note that, if Aand Bare disjoint, then χA+χB=χA∪B. Also note that X •distinguished point χ{•} = 2. Lemma (III.4.3). One has the following result, (47) deg( ˜ Tδ1,δ2 P(f)) = deg( ˜ Tnul P(f)) −1. Proof. For the sake of simplicity we will write ˜ Tδ1,δ2for ˜ Tδ1,δ2 P(f) and ˜ Tnul for ˜ Tnul P(f). The result is equivalent to deg( ˜ Tδ1,δ2)−deg( ˜ Tnul) = −1, since the addition of non-infinitely near points does not affect the degree of either tree. An inductive method will be used in order to prove equation (47). The idea consists in computing first the difference deg( ˜ T)−deg( ˜ Tnul|˜ T) for a tree that only decomposes into simple subtrees (base case of induction). Secondly, the same difference in any other subtree of type ˜ Ti(inductive step) will be studied. Finally, applying induction hypothesis the desired result will be concluded. (A) Consequences of the construction of ˜ Tδ1,δ2. Let us see a few remarks about simple subtrees of type i≥1. Some obvious consequences of the construction of ˜ Tδ1,δ2are: (A.1) Simple subtrees have either one or no distinguished vertices. (A.2) Before a vertex belongs to a simple tree, it can never belong to any distinguished path. (A.3) Before a vertex belongs to a simple tree, it always has to belong to trees of type 1. 62 Chapter III. Logarithmic Trees (A.4) Any point of the tree ˜ Tδ1,δ2can only belong to at most one distinguished path and be at most once a central point. (IB) Base case of induction: simple trees. Recall that a simple subtree is a tree in the induction process which has no further decomposition. (IB.0) If ˜ T=◦. It is easy to check by construction that if we are in a point of type ◦then, (48) deg( ˜ T)−deg( ˜ Tnul|˜ T) = δw ◦(Ek). From now on we will only focus on points of type •. (IB.1) Tree of type 1. (1) A subtree ˜ Tof type 1 is simple if and only if it is a single vertex. (2) In that case, (49) deg( ˜ T)−deg( ˜ Tnul|˜ T) = mult•(ˆ Vf,k, Ek) + δw •(Ek)−1 Proof. For the first part (i), the “if” direction is obvious. There is only one direction left to prove, i.e. that any simple subtree of type 1 cannot have more than one vertex. Let ˜ Tbe a subtree of type 1. Let us denote by m∈Ek⊂Xkthe root of ˜ T. If ˜ Thas more than one vertex, then Ekis not transversal to ˆ Vf,k at mand hence, according to step 3, the vertex ek,∗in ˜ Thas to be distinguished. Therefore, ˜ Thas two distinguished points and hence it is not simple. For the second part (ii) is a direct consequence (43), if •=δi, i = 1,2 then deg( ˜ T)−deg( ˜ Tnul|˜ T) = (w(˜ Tδ1,δ2,•))(w(˜ Tδ1,δ2,•) + p+q−e) 2dpq −(w(˜ Tnul,•))(w(˜ Tnul,•) + p+q−e) 2dpq =0(0 + 1 + 1 −d) 2d−(d−1)(d−1+1+1−d) 2d = 0 −d−1 2d=1 d+d−1 2d−1 = mult•(ˆ Vf,k, Ek) + δw •(Ek)−1.  (IB.2) Tree of type 2with no distinguished points. A simple subtree ˜ Tof type 2 with no distinguished points must be a single point. By (43) such vertices must have defect 0. Moreover, (50) deg( ˜ T)−deg( ˜ Tnul|˜ T) = mult•(ˆ Vf,k, Ek) + δw •(Ek). §III.4. Relation between of ˜ Tδ1,δ2 P(f)and ˜ Tnul P(f)63 Proof. Applying (43), one has deg( ˜ T)−deg( ˜ Tnul|˜ T) = (w(˜ Tδ1,δ2,•))(w(˜ Tδ1,δ2,•) + p+q−e) 2dpq −(w(˜ Tnul,•))(w(˜ Tnul,•) + p+q−e) 2dpq =d(d+ 1 + 1 −d) 2d−(d−1)(d−1+1+1−d) 2d = 1 −d−1 2d=1 d+d−1 2d= mult•(ˆ Vf,k, Ek) + δw •(Ek).  (IB.3) Tree of type 2with one distinguished point. Let ˜ Tbe a simple subtree of type 2 with one (and hence unique) distinguished vertex, say Q. In this case, for vertices of type •one has: (51) f(˜ T, Q) = νQ(Ek)−p−q+eif Q∈ |γ| −p−q+eif Q∈ |˜ T|\|γ|, where γis the path joining Qand m= gcd( ˜ T). Otherwise, note that f(◦) = d−1 Moreover, (52) deg( ˜ T)−deg( ˜ Tnul P(f)|˜ T) = multm(ˆ Vf,k, Ek)+ X •∈|γ| ν•(Ek)(ν•(Ek)−p−q+e) 2dpq . where ˜ Tnul|˜ Tdenotes the corresponding subtree of ˜ Tnul isomorphic to ˜ T and where m∈Ek. Proof. Combining (A.2) with (44, one has (51). Moreover, using Proposition (I.3.7) and Theorem (II.2.1) deg( ˜ T)−deg( ˜ Tnul|˜ T) = X •∈|γ| (ν•(f) + ν•(Ek))(ν•(f) + ν•(Ek)−p−q+e) 2dpq −ν•(f)(ν•(f)−p−q+e) 2dpq =X •∈|γ| ν•(f)ν•(Ek) dpq +X •∈|γ| ν•(Ek)(ν•(Ek)−p−q+e) 2dpq = multm(ˆ Vf,k, Ek) + X •∈|γ| ν•(Ek)(ν•(Ek)−p−q+e) 2dpq .  64 Chapter III. Logarithmic Trees Note that the difference of degrees between two isomorphic weighted trees can be calculated by decomposing each one into similar subtrees and then adding the differences of degrees for each pair of similar subtrees. Let ˜ Tibe a subtree of ˜ Tδ1,δ2whose decomposition produces only simple subtrees ˜ Ti+1 1,˜ Ti+1 2,˜ Ti+1 3, ... , ˜ Ti+1 ni. Items (IB.0), (IB.1), (IB.2) and (IB.3) allow for the calculation of the difference of degrees of ˜ Tiand ˜ Tnul|˜ Ti. Since (53) deg( ˜ Tδ1,δ2 P|˜ Ti+1 0) = deg( ˜ Tnul P|˜ Ti+1 0), one has deg( ˜ Ti)−deg( ˜ Tnul|˜ Ti) = (54) f(Pi+1) + X •∈Ek mult•(ˆ Vf,k, Ek)−χAi+X m∈Ek δw m(Ek) (55) = f(Pi+1) + νPi+1 (f)e dpq Pi+1 −χAi+ 1 −(p+q)e 2pqd Pi+1 . where Pi+1 is the central point of ˜ Ti,Ekis the exceptional divisor resulting after blowing up Pi+1, and Ai= ni+1 [ j=0 |˜ Ti+1 j|. Notice that one has, (56) f(Pi+1) = (νPi+1 −p−q)(νPi+1 −e) 2dpq −νPi+1 (νPi+1 −p−q+e) 2dpq = =(p+q)e 2pqd Pi+1 −νPi+1 (f)e dpq Pi+1 . In particular we obtain deg( ˜ Ti) = deg( ˜ Tnul|˜ Ti)−1. (IS) Inductive step. An obvious consequence of A.4 is that all central points Pi+1 belong to exactly one distinguished path, except for P2, which belongs to none. Hence f(˜ Tδ1,δ2, Pi+1) = νPi+1 (Eki)−p−qif i≥2 −p−qif i= 1. §IV.1. A genus formula for weighted projective curves 71 Definition (IV.1.2) ([CAMO13]).For a given d∈Nand a normalized weight list w∈N3the virtual genus associated with dand wis defined as gd,w := d(d−|w|) 2 ¯w+ 1. Remark (IV.1.3). Note that this number, gd,w, is a generalization of the combinatorial number gd=d−1 2for w= (1,1,1). Let us see a lemma which will be useful in future results to understand the genus in the non-irreducible case. Lemma (IV.1.4). Let di, n ∈Nwith i= 1, . . . , n and a normalized weight list w∈N3. Consider d=d1+···+dnthen, gd,w = n X i=1 gdi,w +X i6=j didj ¯w−(n−1). Proof. If d=d1+d2, after straightforward computation one has gd,w =gd1,w +gd2,w +d1d2 ¯w−1. It is enough to proceed by induction to conclude the desired result.  Remark (IV.1.5) ([CAMO13]).Note that gd,w is always defined regardless of whether or not there actually exist smooth curves of degree din P2 w. For instance, it is easy to see that there are no smooth curves of degree 5 in P2 (2,3,5) (see Remark (IV.1.9)). In general, by the discussion above, if gd,w is not a positive integer, then no smooth curves in P2 wof degree dtransversal w.r.t. the axes can exist. However, this is not a sufficient condition, since g40,(2,3,5) = 20, but all curves of degree 40 need to pass through at least one vertex. The characterization is given by the following. Lemma (IV.1.6) ([CAMO13]).Given dand was above, then the space of smooth curves of degree dtransversal w.r.t. the axes in P2 wis non-empty if and only if ¯w|d. Moreover, any smooth curve can be deformed into a smooth curve of the same degree and transversal w.r.t. the axes. Proof. Let Fbe a weighted homogeneous polynomial of degree dwhose set of zeros defines C. The condition Pi/∈ C implies that Fcontains a monomial of type λiXdi i,λi6= 0, i= 0,1,2. Therefore widi=d, which implies the result since gcd(wi, wj) = 1 by hypothesis on the weights w. For the converse, assume ¯w|d, then X d w0 0+X d w1 1+X d w2 2is a smooth curve of degree dtransversal w.r.t. the axes in P2 w. 72 Chapter IV. Global Invariants: Adjunction-like Formula on P2 w The moreover part is a consequence of the fact that if Cis smooth then ¯w|dand hence C+λ0X d w0 0+λ1X d w1 1+λ2X d w2 2is transversal w.r.t. the axes for an appropriate generic choice of λi. Remark (IV.1.7). A generic curve Cof degree din P2 wis not smooth in general (recall the definition of Sing(C) seen in Definition (IV.1.1)). Proof. It is enough to apply Lemma (IV.1.6).  Corollary (IV.1.8) ([CAMO13]).If Cis a smooth weighted curve in P2 w of degree d, then g(C) = gd,w. Proof. By Lemma (IV.1.6) one can assume that Cis transversal w.r.t. the axes. The result follows immediately from the discussion above.  Remark (IV.1.9) ([CAMO13]).Note that Corollary (IV.1.8) does not apply to quasi-smooth weighted curves, that is, curves whose equation is a weighted homogeneous polynomial whose only singularity in C3is {0}. For instance, the curve C:= {X0X1=X2} ⊂ P2 (2,3,5) of degree 5 can be parametrized by the map P1→P2 (2,3,5), given by [t:s]7→ [t2:s3:t2s3]w. Hence it is rational and g(C) = g(P1) = 0. However, g5,(2,3,5) =7 12. As a consequence of Corollary (IV.1.8), there are no smooth curves of degree 5 in P2 (2,3,5) (see Remark (IV.1.5)). Remark (IV.1.10). Let V(F) be a curve of degree din P2 w. Then d6≡ 0 mod wi⇒F(Pi)=0. Moreover, if V(F) is generic then d6≡ 0 mod wi⇔F(Pi)=0. Proof. For the first part notice that if F(Pi)6= 0 then F=Xa i+. . ., a∈Nwith awi=dand so d≡0 mod wi. For the second one it is enough to prove that F(Pi) = 0 ⇒d6≡ 0 mod wi. Since V(F) is generic, one can write F=X w0i+w1j+w2k=d αijkXi 0Xj 1Xk 2, with αijk ∈C. If F(Pi) = 0, then the monomial Xa i,a∈Nwith awi=d cannot belong to F, and thus d6≡ 0 mod wi. Example (IV.1.11) ([CAMO13]).Consider Fermat curves of type C:= {Xaw1w2 0+Xaw0w2 1+Xaw0w1 2= 0} ⊂ P2 w, §IV.1. A genus formula for weighted projective curves 73 w= (w0, w1, w2). Note that d=a¯wand hence applying Corollary (IV.1.8) one obtains g(C) = gd,w =a 2(a¯w−|w|)+1, (see [Ker07] for the special case w0= 1). Now we are in conditions to prove the main theorem of this section. Theorem (IV.1.12) ([CAMO13]).Let C ⊂ P2 wbe an irreducible curve of degree d > 0, then (66) g(C) = gd,w −X P∈Sing(C) δw P. Proof. Let F∈ C[X0, X1, X2] be a defining equation for C. Note that F¯wdefines a function. Also, from the proof of Lemma (IV.1.6) one can obtain an algebraic family of smooth curves Ct={Ft= 0},t∈(0,1] of degree d¯wwhose defining polynomials Ftdegenerate to F¯w=F0such that Ctand Cintersect transversally and outside the axes. Therefore, by Corollary (IV.1.8), g(Ct) = gd¯w,w =d¯w 2 ¯w(d¯w−|w|) + 1 = d 2(d¯w−|w|)+1. On the other hand, define It:= C ∩Ct. Using B´ezout’s Theorem (I.4.7), (C ·Ct) = X Pt∈It (C ·Ct)Pt=1 ¯wd2¯w=d2. Since Ctand Cintersect transversally outside the axes at smooth points one has (C ·Ct)Pt= 1 (see Example (I.3.6)) and hence #It=d2. For each P∈Sing(C), consider BS Pa regular neighborhood of Cat P and for each Pt∈Itconsider BI Pta regular neighborhood of C ∪Ctat Pt. Note that, outside B:= SP∈Sing(C)BS P∪SPt∈ItBI Pt,Ctprovides a ¯w: 1 covering of C, that is, χ(Ct\B∩Ct) = ¯w·χ(C \B∩C). Also, in each BS P, the curve Ctis the disjoint union of ¯wMilnor fibers of (C, P). Therefore (67) χ(Ct) = χ(Ct\B∩Ct) + PP∈Sing(C)χ(BS P∩Ct) + PPt∈Itχ(BI Pt∩C) = ¯w·χ(C \B∩C) + ¯w·hPP∈Sing(C)2δw P+PP∈Sing(C)rPi+d2. On the other hand (68) χ(Ct)=2−2d 2(d¯w−|w|)+1=d|w|−d2¯w 74 Chapter IV. Global Invariants: Adjunction-like Formula on P2 w and χ(C) = (2−2g(C)−PP∈Sing(C)(rP−1) χ(C \B∩C) + d2+ # Sing(C). Therefore (69) χ(C \B∩C)=2−2g(C)−d2−X P∈Sing(C) rP. Substituting (68) and (69) in (67) one obtains d|w|−d2¯w= ¯w· 2−2g(C)−d2−X P∈Sing(C) rP+X P∈Sing(C) 2δw P+X P∈Sing(C) rP +d2, which after simplification becomes 2 ¯wg(C) = d2−d|w|+ 2 ¯w−2 ¯w· X P∈Sing(C) δw P  and results into the desired formula.  Remark (IV.1.13). From Remark (III.1.16) one can rewrite formula (66) as follows (70) g(C) = gd,w −X P∈Sing C δw P=gd,w −X P∈Sing C deg(Tnul P(C)), where Cis an irreducible curve on P2 wof quasi-homogeneous degree d. Remark (IV.1.14). Let C=∪iCi⊂P2 w, with Ciirreducible curves, then g(C) = X i g(Ci). Remark (IV.1.15). Notice that g(C) = h0(ˆ C; Ω1). Example (IV.1.16). Let us consider the curve C={X0X1−X2} ⊂ P2 w, with w= (a, b, a +b). Note that P1→P2 wgiven by [t:s]7→ [ta:sb: tasb] is an isomorphism and hence g(C) = g(P1) = 0 (see discussion in Remark (IV.1.9)). In order to use Theorem (IV.1.12) one needs to compute the virtual genus of Cgd,w =2ab −a−b 2ab . On the other hand Sing(C) = {P0, P1}. Note that both singularities P0 and P1are of type xp−yqwith (p, q) = (1,1) in their respective quotientsingularity charts (P0∈X(a;b, a +b) and P1∈X(b;a, a +b)) and thus, formula (18) implies: δw P0=1 2 (1 −1−1 + a) a=a−1 2a §IV.1. A genus formula for weighted projective curves 75 and δw P1=1 2 (1 −1−1 + b) b=b−1 2b. Therefore, according to Theorem (IV.1.12) g(C) = gd,w −δw P0−δw P1=2ab −a−b 2ab −a−1 2a−b−1 2b= 0. Example (IV.1.17). Let us consider now the curve C={X0X1−X2 2} ⊂ P2 w, with w= (2k−1,2k+ 1,2k). In order to use Theorem (IV.1.12) one needs to compute the virtual genus of C g4k,w =4k(4k−6k) 2(4k2−1)2k+ 1 = 1 −2k (4k2−1). On the other hand Sing(C) = {P0, P1},P0∈X(2k−1; 2k+ 1,2k) and P1∈X(2k+ 1; 2k−1,2k). Using Example (II.2.7) one has, δw P0=2−1−2 + (2k−1) 2(2k−1) =k−1 2k−1 and δw P1=2−1−2 + (2k+ 1) 2(2k+ 1) =k 2k+ 1. Therefore, according to Theorem (IV.1.12) g(C) = 1 −2k (4k2−1) −k−1 2k−1−k 2k+ 1 = 0. Example (IV.1.18) ([CAMO13]).In the following example (recall Example (I.2.8)), the curve Cis tangent to one of the axes. From the point of view of this work, such points are not special and do not contribute to the genus of Csince they are smooth in C. Note that this is one of the main differences with the approach shown in [ABFdBLMH10]. Let us consider the curve C={X0X1X2+ (X3 0−X2 1)2} ⊂ P2 wof quasihomogeneous degree d= 12, with w= (2,3,7). Note that g12,w =12(12 −12) 2 ¯w+ 1 = 1. On the other hand, Sing(C) = {P2}, which is a quotient singularity of local type xy + (x2−y3)2in X(7; 2,3). In order to obtain δw P2one can perform, for instance, a blow-up of type (1,5). The multiplicity of the exceptional divisor is 6 and hence (71) ν(ν−p−q+e) 2dpq =6(6 −1−5 + 7) 2·7·1·5=3 5. After this first blow-up, the two branches separate and the strict preimage becomes a smooth branch (at a smooth point of the surface) and a 76 Chapter IV. Global Invariants: Adjunction-like Formula on P2 w singularity of type xp−yq, where (p, q) = (2,1), in X(5; 2,1). Using formula (18) one obtains: (72) ν(ν−p−q+e) 2dpq =pq −p−q+d 2d=2−2−1+5 2·5=2 5. Combining (71) and (72) one obtains δw P2=3 5+2 5= 1. Therefore g(C) = 1 −1 = 0 according to Theorem (IV.1.12). Section §IV.2 Computing the continuous discretely Consider an irreducible curve C ⊂ P2 wof quasi-homogeneous degree d. During the rest of this chapter we will focus our efforts on obtaining an Adjunction-like Formula relating the genus of a generic curve of quasihomogeneous degree d, and the dimension of the space of polynomials of degree d+ deg K, with Kthe canonical divisor in P2 w(this dimension will be denoted by Dd−|w|,w). Notation (IV.2.1). Consider w0, w1, w2, k ∈N. We will use the following notation, Dk,w := # (x, y, z)∈N3|w0x+w1y+w2z=k. Remark (IV.2.2). Notice that, with the previous notation, one has Dk−|w|,w =h0(P2 w;O(k−|w|)). IV.2–1. A preliminary example Let us start this section with one basic illustrative example. Let us compute the number of solutions (a, b, c)∈N3of the equation aw0+bw1+cw2=k¯w with w0, w1, w2, k ∈Nfixed, or equivalently, the number of monomials in OP2 wof quasi-homogeneous degree k¯w(recall the notation used in §IV.1). This number will be denoted by Dk¯w,w. Notice that this is equivalent to computing the number of natural solutions (a, b, c) to aw0+bw1= (kw01 − c)w2(recall the notation used in the beginning of Section IV.1). To do this, consider the following sets: ˜ A:= (a, b)∈N2, a > 1, b > 1|aw0+bw1=αw2, α = 0, . . . , kw01, §IV.2. Computing the continuous discretely 77 Aα α kw2−1 kw1−1 kw0−1 y=−n0w2 w1 x kw01 P P= (kw01,−n0w2w0k)Q Q= (kw01,−n1w2w1k) y=n1w2 w0 x Pm= (mw1,−n0w2m), m = 0, . . . , kw0 Qm= (mw0, n1w2m), m = 0, . . . , kw1 Qm Pm Figure IV.1. ˜ B:= (a, 0) ∈N2, a > 1 or (0, b)∈N2, b > 1|aw0=αw2, α = 0, . . . , kw01. Denote by A= # ˜ Aand B= # ˜ B. With the previous notation one has Dk¯w,w =A+B+ 1. To compute the cardinal of Atake two integers n0, n1 such that n0w0+n1w1= 1 with n1>0 and n0≤0 (it can always be done because the weights are pairwise coprime). Consider Aα:= # λ∈N|− n0αw2 w1 <λ<n1αw2 w0 Note that by virtue of Pick’s theorem the area of the triangle is equal to the number of natural points in its interior Iplus one half the number of points in the boundary plus one. The area of the triangle equals k2¯ω 2, thus k2¯ω 2=I+k|w| 2+ 1, therefore A=I+(kw2−1) = k2¯w 2−k|w| 2+ 1+(kw2−1) = 1 2(k2¯w−k|w|)+kw2. 78 Chapter IV. Global Invariants: Adjunction-like Formula on P2 w It is easy to check that B=kw0+kw1, then we have Dk¯w,w =1 2k(k¯w+|w|)+1 It is known that the genus of a smooth curve on P2 wof degree dtransversal w.r.t. the axes is (Definition (IV.1.2)) gd,w =d(d−|w|) 2 ¯w+ 1. We want to find dsuch that Dk¯w,w =gd,w. To do that it is enough to solve the equation 1 2k(k¯w+|w|) + 1 = d(d−|w|) 2 ¯w+ 1. One finally gets that Dk¯w,w =g|w|+k¯w,w. An important by-product of this section is that only to compute Dk¯w,w, the computations have been, in some way, a bit ”tricky”. The natural question now is, how can Dd,w be computed for an arbitrary degree d? In the following section this question will be solved. IV.2–2. Counting points The main reference used in this section is [BR07]. Let a, b, c, t be positive integers with gcd(a, b) = gcd(a, c) = gcd(b, c) = 1. The aim of this section is to give a way to compute the cardinal of the following two sets, ∆1:= {(x, y)∈N2|ax +by ≤t}, ∆1 t/a t/b 0 Note that #∆1cannot be computed by means of Pick’s Theorem unless t is divisible by aand b. ∆2:= {(x, y, z)∈N3|ax +by +cz =t}, §IV.2. Computing the continuous discretely 79 x y z ∆2 t/a t/b t/c gcd(a, b) = gcd( b, c ) = gcd( a, c )=1 Denote by L∆i(t) the cardinal of ∆i. Let us consider the following notation, Notation (IV.2.3). If we denote by ξa:= e2iπ a, consider (73) p{a,b,c}(t) := poly{a,b,c}(t) + 1 aPa−1 k=1 1 (1−ξkb a)(1−ξkc a)ξkt a +1 bPb−1 k=1 1 (1−ξka b)(1−ξkc b)ξkt b +1 cPc−1 k=1 1 (1−ξka c)(1−ξkb c)ξkt c. with (74) poly{a,b,c}(t) := t2 2abc +t 21 ab +1 ac +1 bc+3(ab +ac +bc) + a2+b2+c2 12abc . Remark (IV.2.4). Notice that in particular, one has (75) p{a,b,1}(t) = poly{a,b,1}(t)+1 a a−1 X k=1 1 (1 −ξkb a)(1 −ξk a)ξkt a +1 b b−1 X k=1 1 (1 −ξka b)(1 −ξk b)ξkt b . with (76) poly{a,b,1}(t) = t2 2ab +t 21 ab +1 a+1 b+3(ab +a+b) + a2+b2+ 1 12ab . 80 Chapter IV. Global Invariants: Adjunction-like Formula on P2 w Theorem (IV.2.5) ([BR07]).One has the following result, L∆1(t) = p{a,b,1}(t)and L∆2(t) = p{a,b,c}(t). Section §IV.3 Dedekind Sums In this section we are going to define the Dedekind sums giving some properties which will be particularly useful for future results. See [RG72] and [BR07] for a more detailed exposition. Definition (IV.3.1). Let a, b be integers, gcd(a, b) = 1, b≥1. The Dedekind sum s(a, b) is defined as follows (77) s(a, b) := b−1 X j=1 ja bj b, where the symbol ((x)) denotes ((x)) = x−[x]−1 2if xis not an integer, 0 if xis an integer, with [x] the greatest integer not exceeding x. This is the well-known sawtooth function of period 1, 1/2 −1/2 12 0 −1 −2 y x which at the points of discontinuity takes the mean value between the limits from the right and from the left. Let us see some properties of this kind of sums (see [BR07, Corolary 8.5] or [RG72, Theorem 2.1] for further details). §IV.4. An Adjunction-like Formula on P2 w87 Finally, it is straightforward to check that D12−|w|,w =D0,w = 1 and therefore one has the result previously seen in Corollary (IV.4.4), 1 = D0,w =g12,w −δw [0.0.1] +K(C) = 1 −1+1. V Structure of H•(P2 w\R;C) In [CAM12], Cogolludo-Agust´ın and Matei determine an explicit presentation by generators and relations of the cohomology algebra H•(P2\C,C) of the complement of an algebraic curve Cin the complex projective plane P2, via the study of log-resolution logarithmic forms on P2. Our aim in this Chapter is to extend this result for rational arrangements in P2 w(see Definition (V.1.1) below). As a first approach , in §V.2, we present a basis for H1(P2 w\C;C). In §V.3 some examples of the computation of the ring structure of H2(P2 w\C;C) are provided. Finally, in §V.4, a holomorphic presentation for H2(P2 w\R;C), for a rational arrangement R, is given. Section §V.1 The spaces Hk(P2 w\D;C)and the residue maps Let us start with some basic definitions and useful notation. Definition (V.1.1). A reduced Q-divisor in P2 wwill be called an arrangement. If all irreducible components C0, ..., Cnin an arrangement Dare rational curves (g(Ci) = 0), we shall say Dis a rational arrangement. Notation (V.1.2). From now on, Dwill denote an arrangement in P2 w, and Ra rational arrangement. The complement of D(resp. R) will be sometimes denoted by XD(resp. XR) for simplicity. Let us fix π:XD−→ P2 waQ-resolution of the singularities of an arrangement Dso that the reduced Q-divisor D= (π∗(D))red is a union of Q-smooth divisors on XDwith Q-normal crossings as described in Chapter I. 90 Chapter V. Structure of H•(P2 w\ R;C) Let us see two technical results which will be used in the subsequent sections. Proofs are omitted because they are similar to the correspondent results in P2, see [CA02, Propositions 2.2 and 2.5] for further details. Proposition (V.1.3). Let Dbe an arrangement in P2 w, then H2(X;C)∼ =H1(D;C),and H1(XD;C)∼ =H2(D;C)/C=Cr+1/C. (V.1.4). Let Ybe a topological space. In what follows we will denote by hi(Y) (resp. hi(Y)) the dimension of the vector space Hi(Y;C) (resp. Hi(Y;C)). Note that, by the Universal Coefficient Theorem, hi(Y) = hi(Y). In the following proposition the residue maps are applied to the case of rational arrangements. Proposition (V.1.5). Let Dbe an arrangement in P2 w. Then there is an injection H1(XD;C)Res[1] ,→H0(D[1];C) and a map H2(XD;C)Res[2] →H0(D[2];C) via the Poincar´e residue operator (recall Definition (I.5.7)). Moreover, Res[2] is injective if and only if Dis a rational arrangement. Section §V.2 Logarithmic 1-forms: a basis for H1(P2 w\D;C) The aim of this section is to compute a basis for H1(P2 w\D;C) generalizing [CA02, Theorem 2.11]. Notation (V.2.1). In what follows we shall consider a system of coordinates [X:Y:Z] in P2 w. If one writes D:= {D= 0},Dcan be expressed as a product C0·C1·. . .·Cnwhere Ci:= {Ci= 0},Ciare irreducible components of D. Denote also Cij := {CiCj= 0}. Definition (V.2.2). One can consider the following differential forms (89) σij := d log Cdj i Cdi j!=djd(log Ci)−did(log Cj). where i, j = 0, ..., n,di:= degw(Ci). §V.2. Logarithmic 1-forms: a basis for H1(P2 w\ D;C)91 Note that, since any two determinations of log Cdj i Cdi j differ by a constant, their differential is well defined. Lemma (V.2.3). The holomorphic 1-forms σij are well defined on P2 w\ Cij. Proof. This is a consequence of the following two facts: (1) Each σij is invariant under the C∗-action λ·(X, Y, Z)=(λw0X, λw1Y, λw2Z). (2) They vanish on the space tangent to the fibers of the natural projection C3\{Cj= 0}p −→ P2 w\Cj (X, Y, Z)7→ [X:Y:Z]w at any point. Part (1) is straightforward. Let us prove (2). The vector E(X, Y, Z) = w0X∂ ∂X +w1Y∂ ∂Y +w2Z∂ ∂Z generates the space tangent to the fibers of jat (X, Y, Z). Hence one has to check σij(E(X, Y, Z)) = 0. Note that σij(E(X, Y, Z)) = djw0XCi,X Ci +w1YCi,Y Ci +w2ZCi,Z Ci −diw0XCj,X Cj +w1YCj,Y Cj +w2ZCj,Z Cj, where Ci,X,Ci,Y and Ci,Z are the derivatives of Ciwith respect to X,Y and Zrespectively. Finally, by the Euler identity X i wiXi ∂F ∂Xi = degwF, w0XCi,X +w1Y Ci,Y +w2ZCi,Z =diCi, and therefore σij(E(X, Y, Z)) = djdi Ci Ci−didj Cj Cj =djdi−didj= 0.  Definition (V.2.4). From the discussion above, σij defines a global differential 1-form on XDfor any i, j = 0, ..., n. Remark (V.2.5). With the previous notation, the following equalities hold: (1) σij =−σji. 92 Chapter V. Structure of H•(P2 w\ R;C) (2) dkσij +diσjk +djσki = 0 Proof. Straightforward computation. Direct consequences of the 1form σij definition (89).  Proposition (V.2.6). The pull-back π∗σij defines a logarithmic 1-form on XD. Proof. Since the 1-forms σij are C∞on XDthe statement is trivial on XD. Hence, it is enough to check the statement locally at the points of D=XD\XD. Let P∈ D be a point of type X(d;a, b) on the inverse image of D. Hence, by Remark (I.2.3) the pull-back of σij by πcan be written locally at Pas a multiple of d(xnym) xnym=ndx x+mdy y which is logarithmic at P. Finally, the injectivity of the residue map Res[1] (Proposition (V.1.5)) will prove that, fixing k∈ {0, . . . , n}, the forms σik,i6=kdefine a basis for H1(XD;C). From Corollary (I.5.12), if no ambiguity seems likely to arise, we will use σik instead of π∗σik. Theorem (V.2.7). The cohomology classes of B1(D) := {σik}i6=k for a fixed k, constitute a basis for H1(XD;C). Proof. By the de Rham theorem, the classes [σik] with respect to dcohomology are elements of H1(XD;C). Moreover, their residues can be obtained as follows Res[1] πσikˆ Cj =   dkif i=j6=k 0 if i6=j6=k −diif j=k where ˆ Cj∈ D[1] is the strict transform of Cj. Since the images are linearly independent in H0(D[1];C), the forms σik are also linearly independent in H1(XD;C). Finally, by Proposition (V.1.3), B1(D) has maximal cardinality.  §V.3. Two examples: ring structure of H•(P2 w\ C;C)93 Section §V.3 Two examples: ring structure of H•(P2 w\C;C) For line arrangements in P2, the 2-forms d`i `i∧d`j `jgenerate H2(XL;C), but in general it is no longer true for rational arrangements in P2. For instance, let C=C0∪C1∪C2be a plane quartic, where C1is a conic, C2is a line tangent to C1, and C0is a transversal line. One has h1(XC) = h2(XC) = 2 and hence, by dimension reasons, H2(XC;C) cannot be generated by ∧2H1(XC;C). As a first approach to the general problem of curves in P2 wwe will describe the ring structure of H•(XC;C) for some particular examples. V.3–1. Ring structure of H•(P2 w\{xyz = 0};C) As a first example, let us compute the ring structure of H•(P2 w\D;C) being D=V(xyz). First of all the Euler characteristic χ(P2 w\D) is computed, 3 = χ(P2 w) = χ(P2 w\D)+(χ(V(z)) −2) + (χ(V(y)) −2) + (χ(V(x)) −2) + 3, then χ(P2 w\D) = 0. On the other hand h1(P2 w\D) = h2(D)−1 = 2 and one has χ(P2 w\D) = dim H0(P2 w,P2 w\D)−h1(P2 w\D) + h2(P2 w\D). We conclude that h1(P2 w\D) = 2 and h2(P2 w\D) = 1 so we need to find two logarithmic 1-forms and one logarithmic 2-form invariant under the Euler operator to generate our space. Consider the following global logarithmic 1-forms (see Lemma (V.2.3)): σ01 =w1 dx x−w0 dy y, σ02 =w2 dx x−w0 dz z, σ12 =w2 dy y−w1 dz z. The following relation is easily checked (see Remark (V.2.5)). R0:= w2σ01 +w0σ12 +w1σ20 = 0. By Theorem (V.2.7), σ01 and σ02 are generators of H1(P2 w\D). Consider now the 2-form τ:= Ω2 xyz =w2 dx ∧dy xy +w0 dy ∧dz yz +w1 dz ∧dx zx , 94 Chapter V. Structure of H•(P2 w\ R;C) which is clearly logarithmic (recall the definition of the weighted volume form in (9) Ω2:= w2zdx ∧dy +w0xdy ∧dz +w1ydz ∧dx). By direct computation, the following relations hold: R1:= σ01 ∧σ12 =w1τ, (90) R2:= σ01 ∧σ02 =w0τ, R3:= σ02 ∧σ12 =w2τ. Summarizing all the results we get the ring structure of H•(P2 w\ {xyz = 0};C). The generators are: H•(P2 w\{xyz = 0};C) = H0(P2 w\D;C)⊕H1(P2 w\D;C)⊕H2(P2 w\D;C) =h1i⊕hσ01, σ02i⊕hτi, with the relation σ01 ∧σ02 =w0τ. Note that H•(P2 w\{xyz = 0};C) is independent of w. V.3–2. Ring structure of H•(P2 w\{xyz(xyz + (x3−y2)2) = 0};C) As a second example (recall Example (I.2.8) and Figure I.4), let us see how to compute the ring structure of H•(P2 w\D;C) being D=V(xyz(xyz + (x3−y2)2)) and w= (2,3,7). Denote by C0={x= 0},C1={y= 0}, C2={z= 0}and C3=V(F) = {xyz + (x3−y2)2= 0}. Under the previous notation D=C0∪C1∪C2∪C3. We have already seen that g(C3) = 0 (see Example (IV.1.18)). Also note that # Sing(D)∩C0= # Sing(D)∩C1= # Sing(D)∩C3= 2, # Sing(D)∩C2= 3. and χ(C3\ {[0 : 0 : 1],[1 : 1 : 0]}) = −1 since C3has two branches at [0 : 0 : 1]. Finally we know that 3 = χ(P2 w) = χ(P2 w\D)+(χ(C0)−2) + (χ(C1)−2) + (χ(C2)−3) + (χ(C3)−2) + 4, then χ(P2 w\D) = 1. One also has that dim H0(P2 w,P2 w\D) = 1 and h1(P2 w\ D) = h2(D)−1 = 3, therefore χ(P2 w\D) = dim H0(P2 w,P2 w\D)−h1(P2 w\D)+h2(P2 w\D)⇒h2(P2 w\D) = 3. We need to find three global logarithmic 1-forms and three independent 2-forms to generate our space. Consider the following three global 1-forms, σ01 = 3dx x−2dy y, σ02 = 7dx x−2dz z, §V.3. Two examples: ring structure of H•(P2 w\ C;C)95 x= 0 y= 0 z= 0 C3 [1 : 1 : 0]wP1 P2 P3 P4P5 P6 P7 P8 P9 (1,5),(2,1) (1,2) 1 7 23 4 5 6 Figure V.1. Q-Resolution of D. σ03 = 12dx x−2dF F. By Theorem (V.2.7), they constitute a basis for H1(P2 w\D;C). Consider the following three well-defined global 2-forms, τ0:= Ω2 xyz , τ1:= (x3−y2)x2 yzF Ω2, τ2:= (x3−y2)y xzF Ω2. The following relations hold, (91) σ01 ∧σ02 (90) = 2τ0, (92) σ01 ∧σ03 = 2τ= 2(τ0−τ1+τ2), (93) σ02 ∧σ03 = 8τ2−2τ= 2(−τ0+τ1+ 3τ2), so in particular, by construction, the forms τ0,τ1and τ2are logarithmic. 96 Chapter V. Structure of H•(P2 w\ R;C) Consider also the well defined 2-form τ:= Ω2 F, noticing that τ=τ0−τ1+τ2. The forms τiare independent. To check that it is enough to observe their residues (recall Definition (I.5.14)) at different points in Table 1. Let us compute three particular illustrative examples of the computation of the residues at different points. First of all we have to establish an order for the divisors in the Q-resolution (see Figure V.1). Let us compute Res[2] P6(τ0). After two weighted blow-ups (recall Example (I.2.8)) one has 7dx ∧dy xy x=u1¯v1 y= ¯v15, v1=¯v7 1 ←− 5du1∧dv1 u1v1 u1=¯u2 2, u2=¯u5 2 v1=¯u2v2 ←− 2du2∧dv2 u2v2 . Therefore, locally at P6∈X(2; 1,1) the form can be written as τ0= 2du2∧dv2 u2v2 . Then, Res[2] P6(τ0) = Res[2] 02du2∧dv2 u2v2 =2 2(−1)σ(1,4,5,6,7,3,2) = (−1)9=−1. Let us compute now Res[2] P7(τ1). After a weighted blow-up of type (1,5) one has 7(x3−y2)x2dx ∧dy y(xy + (x3−y2)2) x=¯u, u=¯u7 y=¯u5v ←− (1 −uv2)du ∧dv uv(v+ (1 −uv2)2). Hence, locally at P7(smooth point of ¯ XD) the form τ1can be written τ1=(1 −uv2)du ∧dv uv(v+ (1 −uv2)2) Res[2] P7(τ1) = Res[2] (0,−1) (1 −uv2)du ∧dv uv(v+ (1 −uv2)2) = Res[2] (0,0) (1 −u(v−1)2)du ∧d(v−2uv2+u2v4) u(v−1)(v−2uv2+u2v4) = (−1)(−1)σ(1,2,4,5,6,3,7) = (−1)(−1)3= 1. As a final example, let us compute Res[2] P2(τ2). In the first chart (X(2; 1,1)) the form τ2can be written in the following way τ2= 2 (1 −y2)ydy ∧dz z(yz + (1 −y2)2).