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2011 14 Jorge Martín Morales Embedded QResolutions and Yomdin-Lê Surface Singularities Departamento Director/es Matemáticas Artal Bartolo, Enrique Cogolludo Agustín, José Ignacio Director/es Tesis Doctoral Autor Repositorio de la Universidad de Zaragoza – Zaguan http://zaguan.unizar.es UNIVERSIDAD DE ZARAGOZA Departamento Director/es Jorge Martín Morales EMBEDDED Q-RESOLUTIONS AND YOMDIN-LÊ SURFACE SINGULARITIES Director/es Matemáticas Artal Bartolo, Enrique Cogolludo Agustín, José Ignacio Tesis Doctoral Autor 2011 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 Embedded Q-Resolutions and Yomdin-Lˆe Surface Singularities DOCTORAL THESIS Supervised by Professors Enrique Artal Bartolo and Jos´e Ignacio Cogolludo Agust´ın from University of Zaragoza by Jorge Mart ´ ın-Morales & Zaragoza, October 10, 2011 CONTENTS ACKNOWLEDGMENTS vii RESUMEN (Spanish) ix INTRODUCTION xxi Chapter I. Quotient Singularities and Embedded Q-Resolutions 1 I.1 V-manifolds and Quotient Singularities 1 I.1–1 The abelian case: normalized types 2 I.1–2 Dimension 1, 2, 3 and the cyclic case 5 I.1–3 Working with local equations 7 I.2 Weighted Projective Spaces 8 I.3 Weighted Blow-ups and Embedded Q-Resolutions 11 I.3–1 Dimension 2 15 I.3–2 Dimension 3 22 I.3–3 Higher dimension 26 Chapter II. Cartier and Weil Divisors on V-Manifolds: Pull-Back of a Q-Divisor 29 II.1 Divisors on Complex Analytic Varieties 29 II.2 Divisors on V-Manifolds: Q-Divisor 33 II.2–1 Writing a Weil divisor as a Q-Cartier divisor 35 II.3 Holomorphic Line Bundles and their Sections 40 II.3–1 Line bundle associated with a Cartier divisor 40 II.3–2 Meromorphic sections of a line bundle 42 II.4 Pull-Back of a Q-Divisor 44 iv CONTENTS Chapter III. Intersection Theory on Surfaces with Quotient Singularities 49 III.1 Intersection Numbers: Generalities 49 III.2 Computing Local Intersection Numbers 54 III.3 Intersection Numbers and Weighted Blow-ups 56 III.4 B´ezout’s Theorem for Weighted Projective Planes 65 Chapter IV. Monodromy Zeta Function and Lefschetz Numbers 69 IV.1 Toward A’Campo’s Formula 71 IV.2 Partial Statement and Examples 73 IV.3 Proof of the Theorem 81 IV.3–1 A result by Dimca 81 IV.3–2 Zeta function of a normal crossing divisor 84 IV.3–3 A’Campo’s formula for embedded Q-resolutions 87 IV.4 Zeta Function of Not-Well-Defined Functions 90 IV.5 Why Abelian? D4as a Quotient Singularity 94 Chapter V. Mixed Hodge Structure on the Cohomology of the Milnor Fiber 97 V.1 The Semistable Reduction 97 V.2 Monodromy Filtration 104 V.3 The Spectral Sequence by Steenbrink 106 V.4 Example of a Plane Curve 112 Chapter VI. An Embedded Q-Resolution for Superisolated Singularities 117 VI.1 Preparations for the Q-Resolution 118 VI.2 Construction of the Embedded Q-Resolution 120 VI.3 The Characteristic Polynomial of the Monodromy 133 VI.4 Higher Dimension 138 Chapter VII. Yomdin-Lˆe Surface Singularities 141 VII.1 An Embedded Q-Resolution for YS 143 VII.2 The Characteristic Polynomial of the Monodromy 150 VII.3 Weighted Yomdin-Lˆe Surface Singularities 152 CONTENTS v Chapter VIII. Algorithms for Checking Rational Roots of b-Functions and their Applications 155 VIII.1 Introduction 156 VIII.2 The checkRoot Family of Algorithms 157 VIII.2–1 Multiplicities 160 VIII.2–2 Local versus global b-functions 163 VIII.2–3 b-functions with respect to weights and checkRoot 167 VIII.3 Computing b-Functions via Upper Bounds 169 VIII.3–1 Embedded resolutions 169 VIII.3–2 Topologically equivalent singularities 172 VIII.3–3 A’Campo’s formula 173 VIII.4 Integral Roots of b-Functions 175 VIII.4–1 Upper bounds from different ideals 175 VIII.4–2 Minimal integral root of bf(s) and LCT 175 VIII.4–3 Intersection homology D-module 177 VIII.5 Stratification Associated with Local b-Functions 179 VIII.6 Other Applications 182 VIII.6–1 Bernstein-Sato polynomials for varieties 182 VIII.6–2 A remark in Narv´aez’s paper 183 CONCLUSION AND FUTURE WORK 185 CONCLUSI´ ON (Spanish) 189 BIBLIOGRAPHY 200 RESUMEN (Spanish) xiii Este n´umero de intersecci´on racional fue primero introducido por Mumford para superficies normales, ver [Mum61, Pag. 17]. Nuestra definici´on coincide con la de Mumford gracias al buen comportamiento con respecto al pull-back, ver Theorem (III.1.5). La principal ventaja es que la nuestra no involucra una resoluci´on del espacio ambiente y, por ejemplo, esto nos permite encontrar f´acilmente f´ormulas para la auto-intersecci´on de los divisores excepcionales de explosiones ponderadas sin calcular ninguna resoluci´on. De hecho, esto es el segundo resultado importante de este trabajo. Proposici´on 2. Sea π:= π(d;a,b),ω el morfismo definido en (2). Consideramos dos Q-divisores CyDen X(d;a, b). Entonces, (1) E·π∗(C)=0,(4) E2=−e2 dpq, (2) π∗(C) = b C+ν eE, (5) b C·b D=C·D−νµ dpq, (3) E·b C=eν dpq,(6) b D2=D2−µ2 dpq (Dcompacto), donde νyµdenotan la (p, q)-multiplicidad de CyDen P, es decir, x (resp. y) tienen (p, q)-multiplicidad p(resp. q). Nuestro tercer resultado importante es una versi´on del teorema de B´ezout para cocientes de planos proyectivos ponderados. Proposici´on 3. Sean m1,m2,m3los determinantes de los tres menores de orden 2de la matriz p q r a b c . Supongamos que gcd(p, q, r)=1y escribamos e= gcd(d, m1, m2, m3). Si ω= (p, q, r), entonces el n´umero de intersecci´on de dos Q-divisores en P2 ω(d;a, b, c) := P2 ω/µdes D1·D2=e dpqr degω(D1) degω(D2)∈Q. N´otese que el divisor excepcional de la explosi´on (p, q, r)-ponderada de un punto de tipo (d;a, b, c) es isomorfo a P2 ω(d;a, b, c), ver §I.3–2. As´ı este resultado nos ayudar´a a describir Q-resoluciones encajadas de superficies en C3, ver el cap´ıtulo VI donde se trata con detalle el caso superaislado. Ahora ya tenemos todos los ingredientes necesarios para estudiar los dos invariantes mencionados en t´erminos de una Q-resoluci´on encajada de la singularidad y la normalizaci´on semiestable asociada. Ambos resultados dependen de una estratificaci´on de un Q-divisor con cruces normales. As´ı, necesitamos introducir algo de notaci´on. Notaci´on. Sea f: (M, 0) →(C,0) el germen de una funci´on anal´ıtica y sea (H, 0) la hipersuperficie definida por f. Dada una Q-resoluci´on encajada de (H, 0), π:X→(M, 0), consideramos E1, . . . , Eslas componentes irreducibles del divisor excepcional y b Hla transformada estricta. xiv RESUMEN (Spanish) Normalmente se escribe E0=b HyS={0,1, . . . , s}para que la estratificaci´on de Xasociada al Q-divisor con cruces normales π−1(H) = Si∈SEi est´e definida por E◦ I:= ∩i∈IEi\∪i/∈IEi, para I⊆Sposiblemente vac´ıo. Sea tambi´en X=Fj∈JQjuna estratificaci´on de Xdada por los puntos singulares cocientes de manera que la ecuaci´on local de g:= f◦πen P∈ E◦ I∩Qjsea de la forma xm0 0·. . . ·xmk k:X(d;A)−→ C,(0 ≤k≤n) y las multiplicidades miy la acci´on µdson la misma a lo largo de cada estrato E◦ I∩Qj. En este contexto m(E◦ I∩Qj) est´a definido por m(E◦ I∩Qj) := gcd m0, . . . , mk,Pk j=0 a0jmj d0 ,...,Pk j=0 arjmj dr. A veces tambi´en lo denotamos por m(E, P) o incluso m(P), P∈E◦ I∩Qj, si no hay lugar a confusi´on, ver (IV.3.12) y (V.1.4). El cuarto resultado importante de este trabajo es la generalizaci´on de la f´ormula de A’Campo para Q-resoluciones encajadas, ver (IV.3.14) para un enunciado m´as completo. Su demostraci´on est´a basada en [Dim04, Th. 6.1.14.] y as´ı necesitamos trabajar con complejos constructibles de haces con respecto a una estratificaci´on y el “nearby cycles” de f. Teorema 4. Z(f;t) = Y i=1,...,s, j∈J1−tm(E◦ {i}∩Qj)χ(E◦ {i}∩Qj). N´otese que solo los estratos E◦ {i}∩Qjque provienen del divisor excepcional contribuyen a Z(f;t). Esto refleja el buen comportamiento de las singularidades cocientes abelianas con respecto a los cruces normales. Por el contrario, las no abelianas parecen funcionar de otra manera, ver §IV.5 donde se muestra que los “puntos dobles” pueden contribuir a Z(f;t). Nota. Si la ecuaci´on de gen P∈E◦ {i}∩Qjes de la forma xm:X(d;a, b)→C y el tipo (d;a, b) est´a normalizado, entonces m(P) = m d. As´ı esta f´ormula ya ha sido estudiada en [Vey97] para singularidades de curvas planas. Vamos a describir la normalizaci´on semiestable de g:X→D2 η. Sea e el m´ınimo com´un m´ultiplo de todas las multiplicidades que aparecen en el divisor E:= g−1(0) = E0∪ · · · ∪ Esy consideremos σ:D2 η1/e →D2 ηla cubierta ramificada definida por σ(t) = te. Denotamos por (X1, g1, σ1) el pull-back de gyσ. Finalmente, sea ν:e X→X1la normalizaci´on de X1y denotemos por eg:= g1◦νy%:= σ1◦νlos morfismos naturales. Tambi´en pongamos Di=%−1(Ei) para i= 0, . . . , s yD=Ss i=0 Di. RESUMEN (Spanish) xv Este diagrama conmutativo representa el proceso completo de la normalizaci´on semiestable. Di// % e Xν// %  X1 g1// σ1  D2 η1/e σ  Ei//X X g//D2 η En esta situaci´on, m(g∗(0), P) con P∈g−1(0) se puede interpretar como el cardinal de la fibra sobre Pde la cubierta %:e X→X. Nuestro quinto resultado importante es una descripci´on detallada de esta cubierta. Su demostraci´on est´a basada en el c´alculo expl´ıcito de la normalizaci´on de te−xm0 0· · · xmk kvisto como elemento de C[x0, . . . , xn]µd⊗CC[t], ver (V.1.7). Proposici´on 5. La variedad e Xsolo tiene singularidades cocientes abelianas situadas en eg−1(0) = D, el cual es un divisor reducido con cruces normales en e X. Adem´as, %:e X→Xes una cubierta c´ıclica de ehoja no ramificada sobre X\g−1(0). Para ∅ 6=I⊆S:= {0,1, . . . , s}yj∈J, se tiene: (1) La restricci´on %|:%−1(E◦ I∩Qj)→E◦ I∩Qjes una cubierta c´ıclica ramificada de m(E◦ I∩Qj)hojas no ramificada sobre E◦ I∩Qj. (2) El espacio %−1(E◦ I∩Qj)es una V-variedad con singularidades cocientes abelianas con gcd({m(P)|P∈E◦ I∩Qj})componentes conexas. (3) Sea ϕ:e X→e Xel generador can´onico de la monodrom´ıa de la cubierta %. Entonces, su restricci´on a %−1(E◦ I∩Qj)es un generador de la monodrom´ıa de %|:%−1(E◦ I∩Qj)→E◦ I∩Qj. La idea principal que hay detr´as de esta construcci´on es que en el caso cl´asico despu´es de considerar la normalizaci´on semiestable, el espacio ambiente contiene singularidades cocientes. La proposici´on anterior prueba que lo mismo es cierto para Q-resoluciones encajadas y as´ı la construcci´on de Steenbrink con la sucesi´on espectral se puede adaptar para proporcionar una EHM sobre los grupos de cohomolog´ıa (V.3.4). El prop´ositos del cap´ıtulo V es la descripci´on expl´ıcita de una sucesi´on espectral que converge a la cohomolog´ıa de la fibra de Milnor a partir de una Q-resoluci´on encajada §V.3. Puesto que la Q-resoluci´on encajada se puede elegir para que “casi todo” divisor excepcional contribuya a la monodrom´ıa, nuestra sucesi´on espectral es mejor en el sentido de que menos divisores aparecer´an en la normalizaci´on semiestable y por tanto la combinatoria ser´a m´as sencilla. Vamos a ver con un ejemplo c´omo se aplican todos los resultados anteriormente presentados. xvi RESUMEN (Spanish) Ejemplo. Supongamos gcd(p, q) = gcd(r, s) = 1 y p q<r s. Sea f= (xp+ yq)(xr+ys) y consideremos C1={xp+yq= 0}yC2={xr+ys= 0}. Una Q-resoluci´on encajada de {f= 0} ⊂ C2se puede calcular con la (q, p)- explosi´on del origen de C2, seguida de la (s, qr −ps)-explosi´on de un punto de tipo (q;−1, p), comparar con (2), ver figura 1. 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  Figura 1. Q-resoluci´on encajada de f= (xp+yq)(xr+ys). La auto-intersecciones se calculan con la proposici´on 2 y la matriz de intersecci´on es A=1 rq−ps −r/p 1 1−q/s . Por el teorema 4, el polinomio caracter´ıstico es ∆(t) = t−1tp(q+s)−1ts(p+r)−1 tq+s−1tp+r−1. Estudiamos la normalizaci´on semiestable con la proposici´on 5. Su grafo dual ponderado se muestra en la figura 2. g1=(p−1)(q+s)−gcd(p, s)+1 2 D1D2 m2= 1 g2=(s−1)(p+r)−gcd(p, s)+1 2 C2 C1 m1= 1 . . . Q0 gcd(p, s) Figura 2. Grafo dual de la normalizaci´on semiestable de f. La estructura de Hodge mixta de la cohomolog´ıa de la fibra de Milnor H1(F, C) se obtiene de la sucesi´on espectral de Steenbrink: H1(F, C) = H0,0 |{z} GrW 0H1(F,C) ⊕H0,1⊕H1,0 | {z } GrW 1H1(F,C) ⊕H1,1 |{z} GrW 2H1(F,C) , donde H0,0=Cgcd(p,s)−1, H0,1=Cg1⊕Cg2, H1,1=Cgcd(p,s). Los g´eneros g1yg2se han calculado en la figura 2. La acci´on de la monodrom´ıa sobre GrW 0H1(F, C) est´a dada por el polinomio tgcd(p,s)−1 t−1. N´otese que esto proporciona los autovalores de la monodrom´ıa con bloques de Jordan de tama˜no 2, ver (V.4.3) para m´as detalles. RESUMEN (Spanish) xvii Segunda Parte (Cap´ıtulos VI y VII) En estos dos cap´ıtulos aplicamos parcialmente las nuevas t´ecnicas desarrolladas anteriormente para el estudio de singularidades superaisladas de hipersuperficies y singularidades de Yomdin-Lˆe (ponderadas) de superficies. Estas singularidades han sido estudiadas ampliamente por muchos autores, ver el “survey” [ALM06] donde se repasa parte de la teor´ıa de estas singularidades y sus aplicaciones incluyendo algunos desarrollos recientes y novedosos. Fueron introducidas por Luengo y tambi´en aparecen en un art´ıculo de Stevens, donde se considera el estrado µ-constante, ver [Lue87] y [Ste89]. Despu´es, Artal describi´o en su tesis doctoral [Art94b] una resoluci´on encajada de tales singularidades usando explosiones de puntos y curvas racionales. Aqu´ı, en el cap´ıtulo VI, presentamos una descripci´on de una Q-resoluci´on encajada de singularidades superaisladas de superficies en t´erminos de una Q-resoluci´on encajada (global) de su cono tangente. Probamos que solamente se necesitan explosiones ponderadas de puntos. Por el contrario, el espacio total que aparece tiene singularidades cocientes abelianas. M´as concretamente, sea f=fm+fm+1 +· · · la descomposici´on de f en componentes homog´eneas. Denotamos por C:= V(fm)⊂P2el cono tangente y supongamos que V:= V(f) es superaislada, es decir, Sing(C)∩ V(fm+1) = ∅. El principal resultado de esta parte es una colecci´on de resultados que se pueden resumir como sigue, ver (VI.2.2), (VI.2.10), (VI.2.13). Teorema 6. Sea %P:YP→(C, P)una Q-resoluci´on encajada del cono tangente para P∈Sing(C). Supongamos que (%P)∗(C, P) = b C+X a∈S(ΓP +) mP aEP a es la transformada total de (C, P), donde EP aes el divisor excepcional de la (pP a, qP a)-explosi´on de un punto Paque pertenece al lugar de no transversalidad. Denotemos por νP ala (pP a, qP a)-multiplicidad de Cen Pa. Entonces, se puede construir una Q-resoluci´on encajada ρ:X→(V, 0) de la singularidad de superficie tal que la transformada total es ρ∗(V, 0) = b V+mE0+X P∈Sing(C) a∈S(ΓP +) (m+ 1)mP aEP a, yEP aaparece despu´es de la (pP a, qP a, νP a)-explosi´on del punto Pa(n´otese que el lugar de no transversalidad en dimensi´on 2y3se identifican). xviii RESUMEN (Spanish) La principal ventaja comparada con la de Artal [Art94b] es que en ´esta ´ultima se necesitan en cada paso νP a(en lugar de solo una) explosi´on de puntos y curvas racionales para llegar a una situaci´on parecida. En el cap´ıtulo VI, aplicamos el teorema 4 (f´ormula de A’Campo’s generalizada) para calcular el polinomio caracter´ıstico y el n´umero de Milnor, ver teorema (VI.3.5) y corolario (VI.3.7). En particular, las f´ormulas de [Sie90] y [Ste89] se pueden obtener de esta manera. En el futuro estudiaremos otros invariantes m´as sofisticados como la estructura de Hodge mixta de la cohomolog´ıa de la fibra de Milnor. Como consecuencia probamos que un divisor excepcional de la Q-resoluci´on encontrada para (V, 0) contribuye a la monodrom´ıa compleja si y solo si lo hace el correspondiente divisor en el cono tangente, ver (VI.3.3). As´ı los pesos se pueden elegir para que todo divisor excepcional de la Q-resoluci´on encajada de (V, 0) contribuya a la monodrom´ıa. Estas t´ecnicas se pueden aplicar para estudiar singularidades superaisladas en dimensi´on superior, ver §VI.4, y lo mismo ocurre para singularidades de Yomdin-Lˆe (ponderadas) de superficies, ver cap´ıtulo VII. Tercera Parte (Cap´ıtulo VIII) El ´ultimo cap´ıtulo trata sobre el algoritmo checkRoot y sus aplicaciones para calcular el polinomio de Bernstein-Sato con bases de Gr¨obner. Para dar una descripci´on m´as detallada de los problemas que estamos interesados y los resultados que hemos obtenido, pasamos a recordar algunas definiciones b´asicas en campo de los D-m´odulos. Dado un polinomio f∈C[x] en varias variable el polinomio de BernsteinSato (tambi´en llamado b-funci´on global) de fse define como el polinomio m´onico no nulo bf(s) de menor grado que verifica P(s)fs+1 =bf(s)fs∈C[x, s, 1/f]·fs para P(s)∈Dn[s] := Dn⊗CC[s], donde Dndenota la n-´esima ´algebra de Weyl. La existencia de tal polinomio no nulo est´a garantizada por [Ber72]. An´alogamente se define el polinomio de Bernstein-Sato local (tambi´en llamado b-funci´on local) de fen p∈Cny se denota por bf,p(s). Se conocen varios algoritmos para calcular la b-funci´on de un polinomio, ver por ejemplo [Oak97c], [SST00], [BM02], [Nor02], [Sch04a], [LM08]. Sin embargo, desde el punto de vista computacional, es muy complejo obtener este polinomio en general. A pesar de recientes progresos, en la pr´actica solo se pueden tratar un n´umero limitado de ejemplos. RESUMEN (Spanish) xix Motivados por esto y para mejorar el c´alculo del polinomio de BernsteinSato con bases de Gr¨obner, estudiamos los siguientes problemas computacionales: (1) Encontrar B(s) = Qd i=1(s−αi)mi∈C[s] tal que bf(s)|B(s). (2) Comprobar si αies ra´ız de la b-funci´on. (3) Calcular la multiplicidad de αicomo ra´ız de bf(s). Existen algunos m´etodos conocidos para obtener una cota superior para el polinomio de Bernstein-Sato de una hipersuperficie singular, una vez conocida, por ejemplo, una resoluci´on encajada de tal singularidad [Kas77], ver secci´on VIII.3. Sin embargo, no conocemos ning´un algoritmo para calcular la b-funci´on a partir de esta cota superior. El resultado m´as importante de esta parte final es el teorema (VIII.2.1), que tiene varias consecuencias, ver (VIII.2.6) y (VIII.5.1). En particular, se tiene lo siguiente que resuelve los problemas (2) y (3) anteriores. Corolario 7. Sea mα(resp. mα(p)) la multiplicidad de αcomo ra´ız de bf(−s)(resp. bf,p(−s)). Sean los ideales I= AnnDn[s](fs) + Dn[s]hfie Iα,i =I: (s+α)i+D[s]hs+αi. Entonces, (1) mα> i ⇐⇒ Iα,i 6=Dn[s], (2) mα(p)> i ⇐⇒ p∈V(Iα,i ∩C[x]). El correspondiente algoritmo se llama checkRoot y en general es mucho m´as r´apido que el c´alculo de todo el polinomio de Bernstein, debido a que en (1) no hace falta usar ´ordenes de eliminaci´on para calcular una base de Gr¨obner de Iα,i. Adem´as, el elemento (s+α)i, a˜nadido como generador, parece simplificar tremendamente los c´alculos, comparar con [Nak09]. Como primera aplicaci´on, despu´es de calcular una resoluci´on encajada, hemos hallado bf(s) de la singularidad no aislada f= (xz+y)(x4+y5+xy4) en unos 30 segundos, ver (VIII.3.3). Este ejemplo (que apareci´o primero en [CU05]) era intratable con cualquier sistema de ´algebra computacional. Este algoritmo tiene varias aplicaciones como el c´alculo de la b-funci´on cuando se puede encontrar una cota superior (mediante resoluci´on encajada, para singularidades topol´ogicamente equivalentes o usando la f´ormula de A’Campo y los n´umero espectrales), las ra´ıces enteras de bf(s) (importantes por ejemplo en el problema de comparaci´on logar´ıtmico) y una estratificaci´on de Cncon la b-funci´on local constante en cada estrato (el algoritmo propuesto no emplea descomposici´on primaria, comparar con [NN10]). Los m´etodos de este cap´ıtulo han sido implementados en Singular en las librer´ıas dmod.lib ybfun.lib. Todos los ejemplos que se presentan aqu´ı han sido calculados con esta implementaci´on. INTRODUCTION First Part (Chapters I–V) One of the main invariant of a given hypersurface singularity is the mixed Hodge structure (MHS) on the cohomology of the Milnor fiber. In the isolated case, Steenbrink gave a method for computing this Hodge structure using a spectral sequence that is constructed from the divisors associated with the semistable reduction of an embedded resolution, see [Ste77]. However, in practice the combinatorics of the exceptional divisor of the resolution is often so complicated that the study of the spectral sequence becomes very hard, see e.g. [Art94b] where an embedded resolution and its associated semistable reduction for superisolated surface singularities is computed using blow-ups at points and rational curves. After the semistable reduction process the new ambient space contains normal singularities which are obtained as the quotient of a ball in Cnby the linear action of a finite group. Spaces admitting only such singularities are called V-manifolds. They were introduced in [Sat56] and have the same homological properties over Qas manifolds, e.g. they admit a Poincar´e duality if they are compact and carry a pure Hodge structure if they are compact and K¨ahler, see [Bai56]. Moreover, a natural notion of normal crossing divisor can be defined on V-manifolds, see [Ste77]. Motivated by this fact and in order to try to simplify the combinatorics of the exceptional divisor mentioned above, we introduce the notion of embedded Q-resolution. The idea is as follows. Classically an embedded resolution of {f= 0} ⊂ Cn+1 is a proper map π:X→(Cn+1,0) from a smooth variety Xsatisfying, among other conditions, that π∗({f= 0}) is a normal crossing divisor. To weaken the condition on the preimage of the singularity we allow the new ambient space Xto contain abelian quotient singularities and the divisor π∗({f= 0}) to have “normal crossings” on X. xxii INTRODUCTION More precisely, here is the formal definition of one of the main objects of our study. Definition. Let Mbe an abelian quotient space. Consider H⊂Man analytic subvariety of codimension one. 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∗(H) is a hypersurface with Q-normal crossings on X. The present work is devoted to the study of invariants of a hypersurface singularity (H, 0) ⊂(Cn+1,0) by looking at either an embedded Q-resolution or its associated semistable reduction. We will focus on two important invariants of H, namely the characteristic polynomial of the complex monodromy (Chapter IV) and the mixed Hodge structure on the cohomology of the Milnor fiber (Chapter V). As mentioned above, the motivation for using embedded Q-resolutions rather than standard ones is twofold. On the one hand, they are natural generalization of the usual embedded resolutions, for which the invariant above are expected to be calculated effectively. On the other hand, the combinatorial and computational complexity of embedded Q-resolutions is much simpler, but they keep as much information as needed for the comprehension of the topology of the singularity. Notation. To deal with these resolutions, some notations need to be introduced. Let G:= µd0× · · · × µdrbe an arbitrary finite abelian group written as a product of finite cyclic groups, that is, µdiis the cyclic group of di-th roots of unity. Consider a matrix of weight vectors A:= (aij)i,j = [a0| · · · | an]∈Mat((r+ 1) ×(n+ 1),Z) and the action (3) (µd0× · · · × µdr)×Cn+1 −→ Cn+1, ξd,x7→ (ξa00 d0· · · ξar0 drx0, . . . , ξa0n d0· · · ξarn drxn). The set of all orbits Cn+1/G is called (cyclic)quotient space of type (d;A) and it is denoted by X(d;A) := X   d0a00 · · · a0n . . .. . ..... . . drar0· · · arn   . The orbit of an element (x0, . . . , xn) under this action is denoted by [(x0, . . . , xn)](d;A)and the subindex is omitted if no ambiguity seems likely to arise. INTRODUCTION xxix Second Part (Chapters VI and VII) In these two chapters the new techniques developed above are partially applied to the study of superisolated hypersurface singularities and (weighted) Yomdin-Lˆe surface singularities. These singularities have been extensively studied by many authors, see the survey [ALM06] where part of the theory of these singularities and their applications including some new and recent developments are reviewed. They were introduced by Luengo and also appear in a paper by Stevens, where the µ-constant stratum is considered, see [Lue87] and [Ste89]. Afterward Artal described in his PhD thesis [Art94b] an embedded resolution of such singularities using blow-ups at points and rational curves. Here, in chapter VI, we present a detailed description of an embedded Q-resolution for superisolated surface singularities in terms of a (global) embedded Q-resolution of its tangent cone. It is proven that only weighted blow-ups at points are needed. By contrast, the final total space produced has abelian quotient singularities. More precisely, let f=fm+fm+1 +· · · be the decomposition of f into its homogeneous parts. Denote by C:= V(fm)⊂P2its tangent cone and assume that V:= V(f) is superisolated, i.e. Sing(C)∩V(fm+1) = ∅. The main result of this part is a collection of several results that can be summarized as follows, cf. (VI.2.2), (VI.2.10), (VI.2.13). Theorem 6. Let %P:YP→(C, P)be an embedded Q-resolution of the tangent cone for P∈Sing(C). Suppose that (%P)∗(C, P) = b C+X a∈S(ΓP +) mP aEP a is the total transform of (C, P), where EP ais the exceptional divisor of the (pP a, qP a)-blow-up at a point Pabelonging to the locus of non-transversality. Denote by νP athe (pP a, qP a)-multiplicity of Cat Pa. Then, one can construct an embedded Q-resolution ρ:X→(V, 0) of the superisolated singularity such that the total transform is ρ∗(V, 0) = b V+mE0+X P∈Sing(C) a∈S(ΓP +) (m+ 1)mP aEP a, and EP aappears after the (pP a, qP a, νP a)-blow-up at the point Pa(note that the locus of non-transversality in dimension 2 and 3 are identified). xxx INTRODUCTION The main advantage compared with Artal’s resolution [Art94b] is that in the latter νP a(rather than just one) blow-ups at points and rational curves at each step are needed to achieve a similar situation. The generalized A’Campo’s formula, Theorem 4, is applied and the characteristic polynomial and the Milnor number are calculated as an application, see Theorem (VI.3.5) and Corollary (VI.3.7). In particular, the formulas in [Sie90] and [Ste89] can be obtained in this way. Other more sophisticated invariants, including mixed Hodge structure of the cohomology of the Milnor fiber, are the subjects of our study for the future. As a consequence, we prove that an exceptional divisor in the Q-resolution obtained for (V, 0) contributes to the complex monodromy if and only if so does the corresponding divisor in the tangent cone, see (VI.3.3). Thus the weights can be chosen so that every exceptional divisor in the embedded Q-resolution of (V, 0) contributes to its monodromy. This techniques can be applied to study superisolated singularities in higher dimension, see §VI.4, and the same applies to (weighted) Yomdin-Lˆe surface singularities, see Chapter VII. Third Part (Chapter VIII) The last chapter is about the checkRoot algorithm and its applications for the computation of the Bernstein-Sato polynomial by means of noncommutative Gr¨obner bases. In order to give a more precise description of the problems we are interested in and the results we obtain, let us recall some basic definitions from the realm of D-modules. Given a polynomial f∈C[x] in several variables, the Bernstein-Sato polynomial (also called global b-function) of fis defined as the (non-zero) monic polynomial bf(s)∈C[s] of minimal degree satisfying P(s)fs+1 =bf(s)fs∈C[x, s, 1/f]·fs for some P(s)∈Dn[s] := Dn⊗CC[s], where Dndenotes the n-th Weyl algebra. The existence of such a non-zero polynomial is guaranteed by [Ber72]. Analogously, it is defined the local Bernstein-Sato polynomial (also called local b-function) of fat p∈Cn, and it is denoted by bf,p(s). Several algorithms for computing the b-function associated with a polynomial are known, see for instance [Oak97c], [SST00], [BM02], [Nor02], [Sch04a], [LM08]. However, from the computational point of view it is very hard to obtain this polynomial in general. Despite significant recent progress, only restricted number of examples can be actually treated. INTRODUCTION xxxi Motived by this fact and in order to enhance the computation of the Bernstein-Sato polynomial via Gr¨obner bases, we study the following computational problems: (1) Find B(s) = Qd i=1(s−αi)mi∈C[s] such that bf(s) divides B(s). (2) Check whether αiis a root of the b-function. (3) Compute the multiplicity of αias a root of bf(s). There exist some well-known methods to obtain an upper bound for the Bernstein-Sato polynomial of a hypersurface singularity once we know, for instance, an embedded resolution of such singularity [Kas77], see Section VIII.3. However, as far as we know, there is no algorithm for computing the b-function from this upper bound. The main result of this final part is Theorem (VIII.2.1), which has several consequences, see e.g. (VIII.2.6) and (VIII.5.1). In particular, one obtains the following result solving problems (2) and (3) above. Corollary 7. Let mα(resp. mα(p)) be the multiplicity of αas a root of bf(−s)(resp. bf,p(−s)). Consider the ideals I= AnnDn[s](fs) + Dn[s]hfi and Iα,i =I: (s+α)i+D[s]hs+αi. Then, (1) mα> i ⇐⇒ Iα,i 6=Dn[s], (2) mα(p)> i ⇐⇒ p∈V(Iα,i ∩C[x]). The corresponding algorithm is called checkRoot and in general is much faster than the computation of the whole Bernstein polynomial because no elimination ordering is needed in (1) for computing a Gr¨obner basis of Iα,i. Also, the element (s+α)i, added as a generator, seems to simplify tremendously such a computation, cf. [Nak09]. As a first application, after computing an embedded resolution, we could obtain bf(s) for the non-isolated singularities f= (xz +y)(x4+y5+xy4) in about 30 seconds, see Example (VIII.3.3). This example (first appeared in [CU05]) was intractable by any computer algebra system. Applications of this algorithm includes the computation of the b-function where there is a possibility to compute an upper (it can be achieved by means of embedded resolution, for topologically equivalent singularities or using the formula by A’Campo and spectral numbers), the integral roots of bf(s) (important e.g. for the logarithmic comparison problem), and a stratification of Cnwith the local b-function being constant on each stratum (the algorithm we propose does not employ primary decomposition, cf. [NN10]). The methods from this chapter have been implemented in Singular as libraries dmod.lib and bfun.lib. All the examples presented here have been computed with this implementation. I Quotient Singularities and Embedded Q-Resolutions The purpose of this chapter is to fix the notation and provide several tools to calculate a special kind of embedded resolutions allowing the ambient space to contain abelian quotient singularities. These resolutions are called embedded Q-resolutions, see Definition (I.3.2) below. To do this, we study weighted blow-ups with smooth center. Special attention is paid to the case of dimension 2 and 3 and blow-ups at points. In Chapter III, we develop an intersection theory on this natural context of varieties with abelian quotient singularities. This theory was first introduced by Mumford over normal surfaces, see [Mum61]. The tools presented in this chapter will permit computing the self-intersection numbers of the exceptional divisors of weighted blow-ups in dimension two, see Proposition (III.3.2). All these techniques are applied in Chapters VI and VII, and they are essential for our study of (weighted) Yomdin-Lˆe singularities. We do not pretend to be exhaustive and though objects presented here have many interesting properties, we focus on those that are used later. As for notation through this work we often use (i1, . . . , ik) instead of gcd(i1, . . . , ik) in case of complicated and long formulas if no ambiguity seems likely to arise. Section §I.1 V-manifolds and Quotient Singularities Definition (I.1.1). AV-manifold of dimension nis a complex analytic space which admits an open covering {Ui}such that Uiis analytically isomorphic to Bi/Giwhere Bi⊂Cnis an open ball and Giis a finite subgroup of GL(n, C). 2 Chapter I. Quotient Singularities and Embedded Q-Resolutions V-manifolds were introduced in [Sat56] and have the same homological properties over Qas manifolds. For instance, they admit a Poincar´e duality if they are compact and carry a pure Hodge structure if they are compact and K¨ahler, see [Bai56]. They have been classified locally by Prill [Pri67]. To state this local result we need the following. Definition (I.1.2). A finite subgroup Gof GL(n, C) is called small if no element of Ghas 1 as an eigenvalue of multiplicity precisely n−1, that is, Gdoes not contain rotations around hyperplanes other than the identity. (I.1.3). For every finite subgroup Gof GL(n, C) denote by Gbig the normal subgroup of Ggenerated by all rotations around hyperplanes. Then, the Gbig-invariant polynomials form a polynomial algebra and hence Cn/Gbig is isomorphic to Cn. The group G/Gbig maps isomorphically to a small subgroup of GL(n, C), once a basis of invariant polynomials has been chosen. Hence the local classification of V-manifolds reduces to the classification of actions of small subgroups of GL(n, C). Theorem (I.1.4) ([Pri67]).Let G1and G2be small subgroups of GL(n, C). Then Cn/G1is isomorphic to Cn/G2if and only if G1and G2are conjugate subgroups.  I.1–1. The abelian case: normalized types We are interested in V-manifolds where the quotient spaces Bi/Giare given by (finite) abelian groups. In this case the following notation is used. (I.1.5). Let G:= µd1×· · ·×µdrbe an arbitrary finite abelian group written as a product of finite cyclic groups, that is, µdiis the cyclic group of di-th roots of unity. Consider a matrix of weight vectors A:= (aij)i,j = [a1| · · · | an]∈Mat(r×n, Z) and the action (5) (µd1× · · · × µdr)×Cn−→ Cn, ξd,x7→ (ξa11 d1· · · ξar1 drx1, . . . , ξa1n d1· · · ξarn drxn). Note that the i-th row of the matrix Acan be considered modulo di. The set of all orbits Cn/G is called (cyclic)quotient space of type (d;A) and it is denoted by X(d;A) := X   d1a11 · · · a1n . . .. . ..... . . drar1· · · arn   . §I.1. V-manifolds and Quotient Singularities 3 The orbit of an element (x1, . . . , xn) under this action is denoted by [(x1, . . . , xn)](d;A)and the subindex is omitted if no ambiguity seems likely to arise. Sometimes we use multi-index notation d= (d1, . . . , dr), aj= (a1j, . . . , arj), ξd= (ξd1, . . . , ξdr),x= (x1, . . . , xn), µd=µd1× · · · × µdr, so that the action takes the simple form µd×Cn−→ Cn,(ξd,x)7→ (ξa1 dx1,...,ξan dxn). The following result shows that the family of varieties which can locally be written like X(d;A) is exactly the same as the family of V-manifolds with abelian quotient singularities. Lemma (I.1.6). Let Gbe a finite abelian subgroup of GL(n, C). Then, Cn/G is isomorphic to some quotient space of type (d;A). Proof. Let us write G=Cd1× · · · × Cdras a product of cyclic groups. Let M1, . . . , Mrbe generators of these cyclic groups so that G={Mi1 1· · · Mir r|ik= 0, . . . , dk−1}. Each of these matrices Mi,i= 1, . . . , r, is conjugated to a diagonal matrix of the form Mi∼   ζai1 di... ζain di   , where ζdiis a primitive di-th root of unity. Moreover, they are simultaneously diagonalizable because they commute. This proves that Cn/G ≃X((d1, . . . , dr); (aij)i,j).  Different types (d;A) can give rise to isomorphic quotient spaces, see Remark (I.1.7). We shall prove that they can always be represented by an upper triangular matrix of dimension (n−1)×n, see Lemma (I.1.8). Finding a simpler type (d;A) to represent a quotient space will lead us to the notion of normalized type, see Definition (I.1.10). Remark (I.1.7). Assume just for a while that n= 3. The simple group automorphism on µd×µdgiven by (ξ, η)7→ (ξη−1, η) shows that the following two spaces are isomorphic under the identity map. Xd a11 a12 a13 d a21 a22 a23 =Xd a11 a12 a13 d a21 −a11 a22 −a12 a23 −a13  Note that the determinants of the minors of order 2 are the same in both side of the previous equation. Analogous considerations hold for higher dimension. 4 Chapter I. Quotient Singularities and Embedded Q-Resolutions Lemma (I.1.8). The space X(d;A) = Cn/µdcan always be represented by an upper triangular matrix of dimension (n−1) ×n. More precisely, there exist a vector e= (e1, . . . , en−1), a matrix B= (bi,j)i,j, and an isomorphism [(x1, . . . , xn)] 7→ [(x1, . . . , xk n)] for some k∈Nsuch that X(d;A)∼ =   e1b1,1· · · b1,n−1b1,n . . .. . ..... . .. . . en−10· · · bn−1,n−1bn−1,n   =X(e;B). Proof. To keep the proof as simple as possible, consider only the case n= 3. The general case is analogous. Let (d1;a11, a12, a13) and (d2;a21, a22, a23) be the first two rows of the matrix defining the quotient space. Multiplying conveniently, one can assume d1=d2. Choose α, β satisfying B´ezout’s identity αa11 +βa21 = gcd(a11, a21). Using repeatedly Remark (I.1.7), one finds an isomorphism induced by the identity map between our space Xd;a11 a12 a13 d;a21 a22 a23 and X dgcd(a11, a21)αa12 +βa22 αa13 +βa23 d0a11a22−a21a12 gcd(a11,a21) a11a23−a21a13 gcd(a11,a21)!. This process allows one to reduce the claim to the case n= 1. The proof is complete after Example (I.1.12).  (I.1.9). The action shown in (5) is free on (C∗)n, that is, x∈(C∗)n,ξd·x=x=⇒ξd=1, if and only if the group homomorphism µd→GL(n, C) given by (6) ξd= (ξd1, . . . , ξdr)7−→    ξa1 d... ξar d    is injective. If this is not the case, let Hbe the kernel of this group homomorphism. Then Cn/H ≡Cnand the group µd/H acts freely on (C∗)n under the previous identification. Thus one can always assume that the free (as well as the small) condition is satisfied. This motivates the following definition. Definition (I.1.10). The type (d;A) is said to be normalized if the following two conditions hold. (1) The action is free on (C∗)n. (2) The group µdis identified with a small subgroup of GL(n, C) under the group homomorphism given in (6). By abuse of language we often say the space X(d;A) is written in a normalized form when we actually mean the type (d;A) is normalized. §I.1. V-manifolds and Quotient Singularities 5 Proposition (I.1.11). The space X(d;A)is written in a normalized form if and only if the stabilizer subgroup of Pis trivial for all P∈Cnwith exactly n−1coordinates different from zero. In the cyclic case the stabilizer of a point as above (with exactly n−1 coordinates different from zero) has order gcd(d, a1,...,bai, . . . , an). The procedures described in (I.1.9) and (I.1.3) can be used to convert general types (d;A) into their normalized form. Theorem (I.1.4) allows one to decide whether two quotient spaces are isomorphic. In particular, one can use this result to compute the singular points of the space X(d;A). This method is specially simple in the cyclic case, see (I.1.15) below. I.1–2. Dimension 1, 2, 3 and the cyclic case Now, in the following examples, we discuss the previous normalization process in dimension one, two, and three separately. Also a paragraph is devoted to the cyclic case. Example (I.1.12). (Dimension 1). When n= 1 all spaces X(d;A) are isomorphic to C. Note that X((d1, . . . , dr); (a11, . . . , ar1)t) is the same space as X((d0 1, . . . , d0 r); (a0 11, . . . , a0 r1)t) where d0 i=di gcd(di,ai1)and a0 i1=ai1 gcd(di,ai1). Therefore we can assume that gcd(di, ai1) = 1. The map [x]7→ xd1gives an isomorphism between X(d1;a11) and C. For r= 2 one has that (we write the symbol “=” when the isomorphism is induced by the identity map) C µd1×µd2 =C/µd1 µd2 ∼ = −→ C/µd2 (∗) =X(d2;a21d1)∼ = −→ C, [x]7→ xd1,[x]7→ x d2 gcd(d1,d2). To see the equality (∗) observe that ξd2·xd1≡ξd2·[x] = [ξa21 d2x]≡ξa21d1 d2xd1. It follows that the corresponding quotient space is isomorphic to Cunder the map [x]7→ xlcm(d1,d2). In higher dimension (without assuming gcd(di, ai1) = 1) the isomorphism takes the form X((d1, . . . , dr); (a11, . . . , ar1)t)−→ C: [x]7→ x`, `= lcm d1 gcd(d1, a11),..., dr gcd(dr, ar1). This integer `is closely related to our notion of multiplicity (at a point) of a normal crossing divisor, see (IV.3.12) and (V.1.4). 6 Chapter I. Quotient Singularities and Embedded Q-Resolutions Example (I.1.13). (Dimension 2). Following Lemma (I.1.8), all quotient spaces for n= 2 are cyclic. The space X(d;a, b) is written in a normalized form if and only if gcd(d, a) = gcd(d, b) = 1. If this is not the case, one uses the isomorphism1(assuming gcd(d, a, b) = 1) X(d;a, b)−→ Xd (d,a)(d,b);a (d,a),b (d,b), (x, y)7→ (x(d,b), y(d,a)) to convert it into a normalized one. On the other hand, one can have spaces like Xd;a b e;r s also written in a normalized form. In fact, the previous quotient space is written in a normalized form if and only if so are both rows and gcd(d, e) = 1. Example (I.1.14). (Dimension 3). The space X(d;a, b, c) is written in a normalized form if and only if gcd(d, a, b) = gcd(d, a, c) = gcd(d, b, c) = 1. As above, isomorphisms of the form [(x, y, z)] 7→ [(x, y, zk)] can be used to convert types (d;a, b, c) into their normalized form. For n= 3 there exists non-cyclic quotient spaces written in a normalized form. As an example we give X2;110 2;101. In fact, the general space Xd;a b c e;r s t is written a in normalized form if and only if so are both rows and (d, e, m1)=(d, e, m2)=(d, e, m3) = 1, where m1,m2,m3are the determinants of the three minors of order 2. (I.1.15). (Cyclic case). In the cyclic case the order of the stabilizer subgroup is specially easy to compute and hence the normalized form can be described explicitly. In fact, X(d;a1, . . . , an) is written in a normalized form if and only if gcd(d, a1,...,bai, . . . , an) = 1, ∀i= 1, . . . , n. Here we summarize how to convert types (d;a1, . . . , an) into their normalized form. (1) X(d;a1, . . . , an)≃X(d;aσ(1), . . . , aσ(n)), ∀σ∈Σn. (2) X(d; 0, a2, . . . , an) = C×X(d;a2, . . . , an). (3) X(d;a1, . . . , an) = X(d k;a1 k,...,an k) if kdivides dand all ai’s. (4) X(d;a1, . . . , an) = X(d;ka1, . . . , kan) if gcd(d, k) = 1. (5) X(d;a1, . . . , an)≃X(d k;a1,a2 k,...,an k), the isomorphism is given by [(x1, x2, . . . , xn)] 7→ [(xk 1, x2, . . . , xn)]. In [Fuj75], the author computes resolutions of these cyclic quotient singularities and also studies, among others, the properties shown above. 1Recall the notation (i1,...,ik) = gcd(i1,...,ik) in case of complicated or long formulas. §I.3. Weighted Blow-ups and Embedded Q-Resolutions 13 Again b Cn+1 L(ω) = U0∪· · ·∪Ukcan be covered by k+1 charts. However, the map ϕ0:Cn+1 →U0given by Cn+1 ϕ0 −→ U0={u06= 0} ⊂ b Cn+1 L(ω), x7→ (xp0 0, xp1 0x1, . . . , xpk 0xk, xk+1, . . . , xn),[1 : x1:. . . :xk]ω, is surjective but not injective. In fact, ϕ0(x) = ϕ0(y) if and only if ∃ξ∈µp0:       y0=ξ−1x0, yi=ξpixi, i = 1, . . . , k, yi=xi, i =k+ 1, . . . , n. Hence the previous map ϕ0induces the isomorphism X(p0;−1, p1, . . . , pk)×Cn−k−→ U0. Note that these charts are compatible with the ones described in (I.2.3) for the weighted projective space. In U0the exceptional divisor is {x0= 0} and the first chart of Pk ωis the quotient space X(p0;p1, . . . , pk). (I.3.6). ((p0, . . . , pk)-blow-up of X(d;A) with smooth center). Assume the center is L:{x0=· · · =xk= 0}. Let ω= (p0, . . . , pk) be a weight vector. The action µdon Cn+1 extends naturally to an action on b Cn+1 L(ω) as follows, ξd·x,[u]ωµd 7−→ (ξa0 dx0,...,ξan dxn),[ξa0 du0:. . . :ξak duk]ω. Let \ X(d;A)L(ω) := b Cn+1 L(ω)µddenote the quotient space under this action. Then the induced projection π:\ X(d;A)L(ω)−→ X(d;A),(x,[u]ω)(d;A)7→ [x](d;A) is an isomorphism over \ X(d;A)L(ω)\π−1(L) and the exceptional divisor E:= π−1(L) is identified with the variety (Pk ω×Cn−k)/µd. The action µdabove respects the charts of b Cn+1 L(ω) so that the new ambient space can be covered as \ X(d;A)L(ω) = b U0∪ · · · ∪ b Uk, where b Ui:= Ui/µd={ui6= 0}. Let us study, for instance, the first chart. By using ϕ0one identifies U0 with X(p0;−1, p1, . . . , pk)×Cn−k and µd=µd1× · · · × µdrwith the group µp0d µp0×(r) . . . ×µp0 . 14 Chapter I. Quotient Singularities and Embedded Q-Resolutions Finally, one has the following action µp0d/(µp0×(k) · · · ×µp0)×X(p0;−1, p1, . . . , pk)×Cn−k defined by (ξa0x0,ξp0a1−p1a0x1,...,ξp0ak−pka0xk),ξp0ak+1 xk+1,...,ξp0anxn. This shows that Xp0−1p1· · · pk0· · · 0 p0d a0p0a1−p1a0· · · p0ak−pka0p0ak+1 · · · p0an is isomorphic to b U0and the isomorphism is defined by [x]bϕ0 7−→ (xp0 0, xp1 0x1, . . . , xpk 0xk, xk+1, . . . , xn),[1 : x1:. . . :xk]ω. For i= 1, . . . , k, one proceeds analogously. As for the the exceptional divisor E=π−1(L) = (Pk ω×Cn−k)/µd, it is usually written as E=b V0∪ · · · ∪ b Vk so that these charts are compatible with the ones of \ X(d;A)L(ω) in the sense that b Vi=b Ui|{xi=0},i= 0, . . . , k. Hence, for example, b V0∼ =Xp0p1· · · pk0· · · 0 p0dp0a1−p1a0· · · p0ak−pka0p0ak+1 · · · p0an. Remark (I.3.7). Let ω= (p0, . . . , pk) be a weight vector and write e= gcd(p0, . . . , pk). Denote p0 i=pi/e for i= 0, . . . , k and ω0= (p0 0, . . . , p0 k). Using the previous notation there is an isomorphism F:\ X(d;A)L(ω)−→ \ X(d;A)L(ω0) of blowing-ups (i.e. F◦πω0=πω) induced by the identity map. Hence one can always assume that gcd(p0, . . . , pk) = 1. For instance, in the first chart F:b Uω,0→b Uω0,0takes the form F0: [(x0, x1, . . . , xn)] 7−→ [(xe 0, x1, . . . , xn)] , p0−1p1· · · pk0· · · 0 p0d a0p0a1−p1a0· · · p0ak−pka0p0ak+1 · · · p0an # bϕω,0 // F0  b Uω,0 F  p0 0−1p0 1· · · p0 k0· · · 0 p0 0d a0p0 0a1−p0 1a0· · · p0 0ak−p0 ka0p0 0ak+1 · · · p0 0anbϕω0,0 //b Uω0,0. §I.3. Weighted Blow-ups and Embedded Q-Resolutions 15 Definition (I.3.8). Let π:\ X(d;A)L(ω)→X(d;A) be the ω-blow-up with smooth center L:{x0=· · · =xk= 0}. Then the total transform π∗(H) decomposes as π∗(H) = b H+mE, where E:= π−1(L) is the exceptional divisor of π,b H:= π−1(H\L) is the strict transform of H, and mis the multiplicity of Eat a smooth point. I.3–1. Dimension 2 Let Xbe an analytic surface with abelian quotient singularities. Consider π:b X→Xthe weighted blow-up at a point P∈Xwith respect to ω= (p, q). We distinguish three different situations. (i) The point Pis smooth. Without lost of generality one can assume that X=C2and π=πω:b C2 ω→C2is the weighted blow-up at the origin with respect to ω= (p, q). The new ambient space is covered as b C2 ω=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]ω). Second chart X(q;p, −1) −→ U2, [(x, y)] 7→ ((xyp, yq),[x: 1]ω). The exceptional divisor E=π−1 ω(0) is isomorphic to P1 ωwhich is in turn isomorphic to P1under the map [x:y]ω7−→ [xq1:yp1], p1=p gcd(p, q), q1=q gcd(p, q). The singular points of b C2 ωare cyclic quotient singularities located at the exceptional divisor. They actually coincide with the origins of the two charts; in the case gcd(p, q) = 1 they are written in a normalized form. Example (I.3.9). Let f:C2→Cbe the function given by f=xp+yq with gcd(p, q) = 1. Consider π(q,p):b C2 (q,p)→C2the (p, q)-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. 16 Chapter I. Quotient Singularities and Embedded Q-Resolutions 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.2. Embedded Q-resolution of {xp+yq= 0} ⊂ C2. (ii) The point Pis of type (d;p, q).Assume X=X(d;p, q) and it is written in a normalized form, i.e. gcd(d, p) = gcd(d, q) = 1. Also assume π=πω,d :b C2 ω,d →X(d;p, q) is the weighted blow-up at the origin with respect to ω= (p, q). The new ambient space is cover as b C2 ω,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]ω. Second chart X(q;p, −d)−→ U2, (x, yd)7→ [(xyp, yq)]d,[x: 1]ω. As above, the exceptional divisor E=π−1 ω(0) is identified with P1 ωwhich is isomorphic to P1under the map [x:y]ω7−→ [xq1:yp1], p1=p gcd(p, q), q1=q gcd(p, q). The singular points of b C2 ω,d are cyclic quotient singularities and coincide with the origins of the two charts. They are written in a normalized form if gcd(p, q) = 1. Example (I.3.10). 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 π(q,p):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 at 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. §I.3. Weighted Blow-ups and Embedded Q-Resolutions 17 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.3. Embedded Q-resolution of f= (xp+yq)(xq+yp). Let us consider π(p,q2−p2),q the (p, q2−p2)-weighted blow-up at the origin of X(q;−1, p), π(p,q2−p2),q :b C2 (p,q2−p2),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(p,q2−p2),q ◦π(q,p)is an embedded Q-resolution of {f= 0} ⊂ C2 where all quotient spaces are written in a normalized form. Figure I.3 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 π=π(d;a,b),ω :\ X(d;a, b)ω−→ X(d;a, b) is the weighted blow-up at the origin of X(d;a, b) with respect to ω= (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 . 18 Chapter I. Quotient Singularities and Embedded Q-Resolutions The charts are given by First chart Xp−1q pd a pb −qa −→ b U1, (x, y)7→ ((xp, xqy),[1 : y]ω)(d;a,b). Second chart Xq p −1 qd qa −pb b −→ b U2, (x, y)7→ ((xyp, yq),[x: 1]ω)(d;a,b). The exceptional divisor E=π−1 (d;a,b),ω(0) is identified with the quotient space P1 ω(d;a, b) := P1 ω/µdwhich is isomorphic to P1under the map P1 ω(d;a, b)−→ P1 [x:y]ω7→ [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. (I.3.11). Let us give another expression for the previous charts. We follow the proof of Lemma (I.1.8) and Remark (I.1.7). Let αand βsatisfying αd +βa = gcd(d, a). One has the following isomorphisms induced by the identity map2. p−1q pd a pb −qa =pd d −qd pd a pb −qa  =pd (d, a)−q(d, a) + βpb pd 0dpb (d,a) (7) For the last equality note that α(−qd) + β(pb −qa) = −qgcd(d, a) + βpb and the determinant of the minor of the matrix representing the second quotient space is dpb. From (7), assuming gcd(d, a, b) = 1 , one also has the isomorphism Xpd (d, a)−q(d, a) + βpb (d, a) 0 b∼ = −→ Xpd (d, a); 1,−q(d, a) + βpb, [(x, y)] 7→ [(x, y(d,a))]. Analogously one can proceed with the second chart. Choose λ,µsatisfying B´ezout’s identity λd +µb = gcd(d, b). 2Recall once again the notation (i1,...,ik) = gcd(i1,...,ik) for long formulas. §I.3. Weighted Blow-ups and Embedded Q-Resolutions 19 Then the equations of the two charts in these new coordinates are given by the isomorphism First chart Xpd (d, a); 1,−q(d, a) + βpb−→ b U1, (x, y(d,a))7→ ((xp, xqy),[1 : y]ω). Second chart Xqd (d, b);−p(d, b) + µqa, 1−→ b U2, (x(d,b), y)7→ ((xyp, yq),[x: 1]ω). These spaces are written in a normalized form if and only if the following greatest common divisor equals one: dp (d, a),−q(d, a) + βpb=dp, dq, pb −qa=dq (d, b),−p(d, b) + µqa. Although elementary, the proof of the preceding equalities are not intuitive. That is why the first one is commented separately in the result below. Lemma (I.3.12). With the assumption above, dp (d, a),−q(d, a) + βpb=dp, dq, pb −qa. Moreover, dp (d,a),dq (d,b), pb −qaalso equals the previous number. Proof. Note that pd gcd(d,a)= gcd pd, dpb gcd(d,a), since gcd(d, a, b) = 1, and consequently dp (d, a),−q(d, a) + βpb=dp, dpb (d, a),−q(d, a) + βpb. The following two couples of equalities complete the first part of the proof. •a (d,a)·−q(d, a) + βpb +α·dpb (d,a)=pb −qa. •−d (d,a)·−q(d, a) + βpb +βdpb (d,a)=dq. • −q(d, a) + βpb =α(−qd) + β(pb −qa). •dpb (d,a)=d (d,a)(pb −qa) + a (d,a)(dq). The second part of the statement is again rather artificial but elementary; the details are left to the reader.  20 Chapter I. Quotient Singularities and Embedded Q-Resolutions Remark (I.3.13). Assume that gcd(d, a, b) = 1. The (p, q)-weighted blowup at the origin of X(d;a, b) is isomorphic to the ω0-weighted blow-up at the origin of Xd0;a0, b0, where the new vectors are ω0=p·gcd(d, b), q ·gcd(d, a), (d0, a0, b0) = d gcd(d, a) gcd(d, b),a gcd(d, a),b gcd(d, b). In fact, there is a commutative diagram of blowing-ups \ X(d;a, b)ω # H// π(d;a,b),ω  \ X(d0;a0, b0)ω0 π(d0;a0,b0),ω0  X(d;a, b)h//X(d0;a0, b0) where Hand hare isomorphisms of analytic spaces defined by [((x, y),[u:v])ω](d;a,b) H 7−→ [((x(d,b), y(d,a)),[u(d,b):v(d,a)])ω0](d0;a0,b0); [(x, y)](d;a,b) h 7−→ [(x(d,b), y(d,a))](d0;a0,b0), and Hgives rise to the identity map on each chart. Note also that if C={f= 0} ⊆ X(d;a, b),C0={f0= 0} ⊆ X(d0;a0, b0) such that h∗(C0) = C, then ordω(f) = ordω0(f0) = ord(f(xp, yq)). Hence the order is preserved under this construction. Remark (I.3.14). Using the notation in (I.3.11), assume gcd(p, q) = 1 and X(d;a, b) is written in a normalized form. To normalize the last cyclic quotient spaces obtained in that paragraph, let e= gcd(pd, −q+βpb) = gcd(d, pb −qa). Then one has the isomorphism X(pd; 1,−q+βpb)∼ = −→ Xpd e; 1,−q+βpb e, [(x, y)] 7→ [(xe, y)]. §I.3. Weighted Blow-ups and Embedded Q-Resolutions 21 One proceeds analogously with the second chart. Finally the equations of the two charts in these new coordinates are given by (8) First chart Xpd e; 1,−q+βpb e−→ b U1, (xe, y)7→ ((xp, xqy),[1 : y]ω)(d;a,b). Second chart Xqd e;−p+µqa e,1−→ b U2, (x, ye)7→ ((xyp, yq),[x: 1]ω)(d;a,b). Recall that βand µare the inverse of aand bmodulo d, respectively. Note that both quotient spaces are now written in their normalized form. Example (I.3.15). 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.3.10), one obtains the following picture 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.4. 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.3.10) assuming r=qand s=p. (I.3.16). (Puiseux expansion). Let us study the behavior of Puiseux pairs under weighted blow-ups. Let C={f= 0} ⊂ C2be the irreducible plane curve given by d Y j=1 −y+ (a1jx p1 q+· · · +akjx pk q)+(b1jxr1 s+· · · +bljxrl s) + · · · , where p1<· · · < pk,r1<· · · < rl,p1 q<ri sand each fraction is irreducible. 22 Chapter I. Quotient Singularities and Embedded Q-Resolutions Let π(q,p1):b C2 (q,p1)→C2be the (q, p1)-blow-up at the origin. In the first chart, that is, after performing the substitution (x, y)7−→ (xq, xp1y), one obtains the following equation for the total transform xp1d· d Y j=1 −y+ (a1j+a2jxp2−p1+· · · +akjxpk−p1)+ (b1jxr1q−p1s s+· · · +bljxrlq−p1s s) + · · · = 0. At first sight the exceptional divisor and the strict transform intersect at ddifferent smooth points. However, since aq 1jdoes not depend on j by conjugation, all of them are the same. After the following change of coordinates y7−→ y+ (a2jxp2−p1+· · · +akjxpk−p1), the local equation of the total transform π−1 (q,p1)(C) at this point is xp1d· d/q Y j=1 −y+ (b1jxr1q−p1s s+· · · +bljxrlq−p1s s) + · · · = 0. This proves that in the irreducible case, only a weighted blow-up is needed for each Puiseux pair in order to compute an embedded Q-resolution. Moreover, the embedded Q-resolution obtained is as in Figure I.5. E1 E2 E3 E4 E5 Figure I.5. Embedded Q-resolution of an irreducible plane curve. In the non-irreducible case, the situation is a bit more complicated but can still be described in terms of the Puiseux pairs of each irreducible component and their intersection multiplicities. I.3–2. Dimension 3 Let Xbe a 3-dimensional variety with abelian quotient singularities and consider π:b X→Xthe weighted blow-up at a point P∈Xwith respect to ω= (p, q, r). Two special situations are considered. II Cartier and Weil Divisors on V-Manifolds: Pull-Back of a Q-Divisor This chapter is based on [AMO11a] and its aim is to show that when X is a V-manifold there is an isomorphism of Q-vector spaces between Cartier and Weil divisors, see Theorem (II.2.6) below. It is explained in (II.2.14) how to write explicitly a Q-Weil divisor as a Q-Cartier divisor. Also, the case of the exceptional divisor of a weighted blow-up in dimension 2 (which is in general just a Weil divisor) is treated in Example (II.2.15). Following the theory of holomorphic line bundles, the pull-back of a Qdivisor can be defined using this approach, see Section II.4. This provides all the necessary ingredients to develop a rational intersection theory on variety with quotient singularities. Although Chapter III is devoted to the details, an illustrative example is shown at the end, see (II.4.5). Section §II.1 Divisors on Complex Analytic Varieties Let Xbe an irreducible complex analytic variety. As usual, consider OX the structure sheaf of Xand 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. Remark (II.1.1). By a complex analytic variety we mean a reduced complex space. A subvariety Vof Xis a reduced closed complex subspace of X, or equivalently, an analytic set in X, cf. [GR84]. An irreducible subvariety Vcorresponds to a prime ideal in the ring of sections of any local complex model space meeting V. 30 Chapter II. Cartier and Weil Divisors on V-Manifolds Definition (II.1.2). ACartier divisor on Xis a global section of the sheaf K∗ X/O∗ X, that is, an element in Γ(X, K∗ X/O∗ X) = H0(X, K∗ X/O∗ X). Any Cartier divisor 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. 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∈J, then 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)}iwith 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. That is, a Cartier divisor is principal if it is in the image of the natural map Γ(X, K∗ X)−→ Γ(X, K∗ X/O∗ X). Two Cartier divisors Dand Eare linearly equivalent, denoted by D∼E, if they differ by a principal divisor. The Picard group Pic(X) denotes the group of linear equivalence classes of Cartier divisors. The support of a Cartier divisor D, denoted by Supp(D) or |D|, is the subset of Xconsisting of all points xsuch that a local equation for Dis not in O∗ X,x. The support of Dis a closed subset of X. Definition (II.1.3). 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. Remark (II.1.4). In the algebraic category meromorphic functions are assumed to be regular functions and hence the locally finite sum of Definition (II.1.3) is automatically finite. Therefore WeDiv(X) is the free abelian group on the codimension one irreducible algebraic subvarieties of X. Similar considerations hold if Xis a compact analytic variety. §II.1. Divisors on Complex Analytic Varieties 31 Given a Cartier divisor there is a Weil divisor associated with it. To see this, the notion of order of a divisor along an irreducible subvariety of codimension one is needed. (II.1.5). (Order function). Let V⊂Xbe an irreducible subvariety of codimension one. It corresponds to a prime ideal in the ring of sections of any local complex model space meeting V. The local ring of Xalong V, denoted by OX,V , is the localization of such ring of sections at the corresponding prime ideal; it is a one-dimensional local domain. For a given f∈ OX,V define ordV(f) to be ordV(f) := lengthOX,V OX,V hfi, where lengthOX,V denotes the length as an OX,V -module. This determines a well-defined group homomorphism ordV: Γ(X, K∗ X)−→ Z that satisfies, for a given f∈Γ(X, K∗ X), the following local finiteness property: (Uxis assumed to be an open neighborhood of x) ∀x∈X, ∃Ux⊂X|#{ordV(f)6= 0 |V∩Ux6=∅} <+∞. The previous length, Xbeing a complex analytic variety of dimension n≥2, can be computed as follows. Choose x∈Vsuch that xis smooth in Xand (V, x) defines an irreducible germ. Thus, this germ is the zero set of an irreducible g∈ OX,x. Then ordV(f) = ordV,x(f), where ordV,x(f) is the classical order of a meromorphic function at a smooth point with respect to an irreducible subvariety of codimension one; it is known to be given by the equality f=gord ·h∈ OX,x, h -g. The same applies if Xis 1-dimensional and smooth. Remark (II.1.6). The order ordV,x(f) does not depend on the defining equation g, as long as we choose girreducible. In fact, two irreducible g, g0∈ OX,x with V(g) = V(g0) only differ by a unit in OX,x. Moreover, ordV,x(f) does not depend on x, since the set of regular points Vred is connected if V is irreducible. 32 Chapter II. Cartier and Weil Divisors on V-Manifolds Now if Dis a Cartier divisor on X, one writes ordV(D) = ordV(fi) where fiis a local equation of Don any open set Uiwith Ui∩V6=∅. This is well defined since fiis uniquely determined up to multiplication by units and the order function is a homomorphism. Define the associated Weil divisor of a Cartier divisor Dby setting TX: CaDiv(X)−→ WeDiv(X) D7→ X V⊂X ordV(D)·[V], where the sum is taken over all codimension one irreducible subvarieties V of X. The previous sum is locally finite, i.e. for any x∈Xthere exists an open neighborhood Usuch that the set {ordV(D)6= 0 |V∩U6=∅} is finite. By the additivity of the order function, the mapping TXis a homomorphism of abelian groups. A Weil divisor is principal if it is the image of a principal Cartier divisor under TX; they form a subgroup of WeDiv(X). If Cl(X) denotes the quotient group of their equivalence classes, then TXinduces a morphism Pic(X)−→ Cl(X). These two homomorphisms (TXand the induced one) are in general neither injective nor surjective. In this sense one has the following result. Theorem (II.1.7). (cf. [GD67, 21.6]). If Xis normal (resp. locally factorial) then the previous maps CaDiv(X)→WeDiv(X)and Pic(X)→Cl(X) are injective (resp. bijective). The image of the first map is the subgroup of locally principal1Weil divisors.  Remark (II.1.8). Locally factorial essentially means that every local ring OX,x is a unique factorization domain. In particular, every smooth analytic variety is locally factorial. In such a case, Cartier and Weil divisors are identified and denoted by Div(X) := CaDiv(X) = WeDiv(X). Their equivalence classes coincide under this identification and we often write Pic(X) = Cl(X). 1A Weil divisor Don Xis said to be locally principal if Xcan be covered by open sets Usuch that D|Uis principal for each U. §II.2. Divisors on V-Manifolds: Q-Divisor 33 Example (II.1.9). Let Xbe the surface in C3defined by the equation z2=xy. The line V={x=z= 0}defines a Weil divisor which is not a Cartier divisor. In this case Pic(X) = 0 and Cl(X) = Z/(2). Note that Xis normal but not locally factorial. However, the associated Weil divisor of {(X, x)}is TX{(X, x)}=X Z⊂X, irred codim(Z)=1 ordZ(x)·[Z] = 2[V]. Thus [V] is principal as an element in WeDiv(X)⊗ZQand corresponds to the Q-Cartier divisor 1 2{(X, x)}. Using the notation of Chapter I, this fact can be interpreted as follows. First note that identifying our surface Xwith X(2; 1,1) under [(x, y)] 7−→ (x2, y2, xy), the previous Weil divisor corresponds to D={x= 0}. Although f=x defines a zero set on X(2; 1,1), it does not induce a function on the abelian quotient space. However, x2:X(2; 1,1) →Cis a well-defined function and gives rise to the same zero set as f. Hence as Q-Cartier divisors D=1 2{(X(2; 1,1), x2)}. Section §II.2 Divisors on V-Manifolds: Q-Divisor Example (II.1.9) above illustrates the general behavior of Cartier and Weil divisors on V-manifolds, namely Weil divisors are all locally principal over Q. To prove it we need some preliminaries. (II.2.1). If Xis smooth, contractible, and Stein, then Hi(X, O∗ X) = 0, ∀i≥1. Indeed, there is a short exact sequence of sheaves of abelian groups 0−→ ZX−→ (OX,+) exp −→ (O∗ X,·)−→ 0 that gives rise to the following long exact sequence in cohomology 0−→ H0(X, ZX)−→ H0(X, OX)−→ H0(X, O∗ X)−→ H1(X, ZX)−→ H1(X, OX)−→ H1(X, O∗ X)−→ H2(X, ZX)−→ H2(X, OX)−→ H2(X, O∗ X)−→ · · · Let i≥1. Since Xis contractible, Hi(X, ZX) = 0. The cohomology Hi(X, OX) vanishes too because Xis Stein and OXis a coherent sheaf. Hence Hi(X, O∗ X) = 0 as claimed and the previous long exact sequence is nothing but 0 −→ ZX(X)−→ OX(X)−→ O∗ X(X)−→ 0. 34 Chapter II. Cartier and Weil Divisors on V-Manifolds (II.2.2). The short exact sequence of sheaves of multiplicative groups 0−→ O∗ X−→ K∗ X−→ K∗ X/O∗ X−→ 0 gives the long exact sequence in cohomology 0−→ H0(X, O∗ X)−→ H0(X, K∗ X)−→ H0(X, K∗ X/O∗ X)−→ H1(X, O∗ X)−→ H1(X, K∗ X)−→ H1(X, K∗ X/O∗ X)−→ H2(X, O∗ X)−→ H2(X, K∗ X)−→ H2(X, K∗ X/O∗ X)−→ · · · If, as above, Hi(X, O∗ X) = 0, ∀i≥1, then the previous long exact sequence gives rise to the short exact sequence 0−→ O∗ X(X)−→ K∗ X(X)−→ CaDiv(X)−→ 0 together with an isomorphism Hi(X, K∗ X)→Hi(X, K∗ X/O∗ X), ∀i≥1. In particular, every Cartier divisor on Xis principal, that is, it is of the form {(X, f)}where f∈Γ(X, K∗ X). Remark (II.2.3). As an easy consequence of (II.2.1) and (II.2.2), one has that every effective Weil divisor on an open ball B⊂Cnis given by the zero set of a holomorphic function f:B→C. The Weil divisor is irreducible on Bif and only if fdefines a prime ideal in OCn(B). In the algebraic category the corresponding holomorphic function is a polynomial. Lemma (II.2.4). Let B⊂Cnbe an open ball and let Gbe a finite group acting on B. Then one has Cl(B/G)⊗ZQ= 0. Proof. Let V⊂B/G =: Ube an irreducible subvariety of codimension one. We shall prove that there exists k≥1 such that k[V]∈WeDiv(U) is principal. Consider the natural projection π:B→U. Then W:= π−1(V) gives rise to an effective Weil divisor on the open ball B. By Remark (II.2.3), there exists f:B→Ca holomorphic function such that W={f= 0} ⊂ B. Thus, V=π(W) = {[x]|x∈B, f(x)=0}={f= 0} ⊂ U. Moreover, by construction the holomorphic function fsatisfies the following property (10) ∀P∈U, f(P) = 0 =⇒f(σ·P) = 0,∀σ∈G. Note that fdoes not necessarily defines an analytic function on U. This reflects the fact that, although Vis given by just one equation, [V]∈ WeDiv(U) is not principal, see Example (II.1.9). Now the main idea is to change fby another holomorphic function Fsuch that V={F= 0}but now with F∈Γ(U, OU). §II.2. Divisors on V-Manifolds: Q-Divisor 35 Let us consider F=Qσ∈Gfσwhere fσ(x) = f(σ·x); clearly it verifies the previous conditions. Then {(U, F)}is a principal Cartier divisor and its associated Weil divisor is TU{(U, F)}=X Z⊂U, irred codim(Z)=1 ordZ(F)·[Z] = ordV(F)·[V]. Note that ordZ(F)6= 0 implies Z=V, since Vis irreducible.  Remark (II.2.5). The proof of this result is based on an idea extracted from [Ful98, Ex. 1.7.6]. Theorem (II.2.6). Let Xbe a V-manifold. 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 X, there always exists k∈Zsuch that kD ∈CaDiv(X). Proof. By Proposition (I.1.19), the variety Xis normal and then Theorem (II.1.7) applies. Therefore the linear map TX⊗1 is injective and its image is the Q-vector space generated by the locally principal Weil divisors on X. Let V⊂Xbe an irreducible subvariety of codimension one. Consider {Ui}ian open covering of Xsuch that Uiis analytically isomorphic to Bi/Gi where Bi⊂Cnis an open ball and Giis a finite subgroup of GL(n, C). By Lemma (II.2.4), Cl(Ui)⊗Q= 0 for all i. Thus [V|Ui] is principal as an element in WeDiv(Ui)⊗ZQwhich implies that Vis locally principal over Q; hence it belongs to the image of TX⊗1.  Definition (II.2.7). 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). II.2–1. Writing a Weil divisor as a Q-Cartier divisor Following the proofs of Lemma (II.2.4) and Theorem (II.2.6), every Weil divisor on Xcan locally be written as Q-Cartier divisor like [V|U] = 1 ordV(F){(U, F)} where F=Qσ∈Gfσand V∩U={f= 0}with f:B→Cbeing holomorphic on an open ball and satisfying (10). 36 Chapter II. Cartier and Weil Divisors on V-Manifolds The rest of this section is devoted to explicitly calculating ordV(F). First, in Proposition (II.2.8), it is shown that Fis essentially a power of f, if the latter is chosen properly. Then, ordV(F) is computed in Proposition (II.2.12). Proposition (II.2.8). Let f:B→Cbe a non-zero holomorphic function on an open ball B⊂Cnsuch that the germ fx∈ OB,x is reduced for all x∈B. Let Gbe a finite subgroup of GL(n, C)acting on B. As above, consider F=Y σ∈G fσ where fσ(x) = f(σ·x)for σ∈G. The following conditions are equivalent: (1) ∀P∈B, f(P) = 0 =⇒f(σ·P)=0,∀σ∈G. (2) ∀σ∈G,∃hσ∈Γ(B, O∗ B)such that fσ=hσf. (3) ∃h∈Γ(B, O∗ B)such that F=hf|G|. (4) ∃k≥1,∃h∈Γ(B, O∗ B)such that hfk∈Γ(B/G, OB/G). Proof. For (1) ⇒(2), first note that fσ∈IV (f). Now fix x∈B. Since fxis reduced, there exists a holomorphic function hon a small enough open neighborhood of xsuch that as germs (fσ)x=hxfx. The order of the converging power series (fσ)xand fxare equal because the action is linear. Thus hxis a unit in OB,x. In particular, fσ fis holomorphic and does not vanish at x∈B. For (2) ⇒(3), consider h=Qσ∈Ghσ. Then one has F=Y σ∈G fσ=Y σ∈G (hσf) = Y σ∈G hσ·f|G|=hf|G|. For (3) ⇒(4), since F:B/G →Cis analytic, take k=|G|. Finally, note that ∀P∈B, f(P) = 0 ⇐⇒ (hfk)(P)=(hfk)(σ·P)=0⇐⇒ f(σ·P)=0. Hence (4) ⇒(1) follows and the proof is complete.  The following example shows that the reduceness condition in the statement of the previous result is necessary. Example (II.2.9). Let f= (x2+y)(x2−y)3∈C[x, y] and consider the cyclic quotient space M=X(2; 1,1). Then {f= 0} ⊂ Mdefines a zero set, i.e. condition (1) holds, but there are no k≥1 and h∈Γ(B, O∗ B) such that hfkis a well-defined function over M. §II.2. Divisors on V-Manifolds: Q-Divisor 37 Remark (II.2.10). If the holomorphic function f:B→Cin Proposition (II.2.8) is given by a polynomial, then the condition [fx∈ OB,x reduced ∀x∈B] holds if and only if fis reduced as a polynomial. In such a case, the holomorphic nowhere-vanishing function hσ(and hence h) above is a non-zero constant. Therefore f|G|itself (without multiplying by a unit) is a well-defined analytic function on B/G, cf. (IV.4.1). The situation of Remark (II.2.10) above is specific for polynomials and, in general, it does not apply in the holomorphic case, as the following example indicates. Example (II.2.11). Let f:C2→Cbe the holomorphic function given by f=exxy and consider the quotient space M=X(2; 1,1). Then f defines a zero set on Mand it verifies the four equivalent conditions of Proposition (II.2.8). However, there is no k≥1 such that fkinduces a function over M. As it is said, this happens because fis not a polynomial. Proposition (II.2.12). Let B⊂Cnbe an open ball and Ga finite subgroup of GL(n, C)acting on B. Let V⊂B/G =: Ube an irreducible subvariety of codimension one and consider F=Y σ∈G fσ where f:B→Cis a holomorphic function defining V. If Gis small and fis chosen so that fx∈ OB,x is reduced ∀x∈B, then ordV(F:U→C) = |G|. Proof. Choose [P]∈Vsuch that [P] is smooth in Uand (V, [P]) defines an irreducible germ, then ordV(F) = ordV,[P](F), see (II.1.5). By Theorem (I.1.4), since Gis small and [P]∈Uis smooth, using the covering π:B→U, one finds an isomorphism of germs (U, [P]) ∼ =(B/GP,[P]) = (B, P) induced by the identity map2. The germ (V, [P]) is converted under this isomorphism into (W, P) where Wis the zero set of fP∈ OB,P . On the other hand, by Proposition (II.2.8), there exists h∈Γ(B, O∗ B) such that F=hf|G|. Putting all together the wanted order is ordV,[P](F:U→C) = ordV(fP),P (hf|G|:B→C) = |G| as claimed.  2See also Lemma (I.1.16) where the abelian case in treated in detail. 38 Chapter II. Cartier and Weil Divisors on V-Manifolds Remark (II.2.13). Recall that π:B→B/G =: Udenotes the projection. Without any condition on Gand f(i.e. neither Gsmall nor fx∈ OB,x reduced ∀x∈Bare required), the order can still be computed as follows ordV(F) = X i deg(Wi/V )·ordWi(f), where the Wi’s are the irreducible components of π−1(V) (assumed to be a finite number) and deg(Wi/V ) is the degree of the restriction mapping π|Wi:Wi→V. Note that under the assumption of Proposition (II.2.12), ordWi(f)=1 and Pideg(Wi/V ) = |G|. (II.2.14). Here we summarize how to write a Weil divisor as an element in CaDiv(X)⊗ZQwhere Xis 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 polynomial fi,j :Uj→Csatisfying the condition [(fi,j)x∈ OBj,x reduced ∀x∈Bj] and such that Vi∩Uj={fi,j = 0}. Then, [Vi|Uj] = 1 |Gj|{(Uj, f|Gj| i,j )}. (3) Identifying {(Uj, f|Gj| i,j )}with its image under the natural inclusion 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 )}. We finish this section with an example where the exceptional divisor of a weighted blow-up (which is in general just a Weil divisor) is explicitly written as a Q-Cartier divisor. Example (II.2.15). Let Xbe a surface with abelian quotient singularities. Let π:b X→Xbe the weighted blow-up at a point of type (d;a, b) with respect to ω= (p, q). In general, the exceptional divisor E:= π−1(0) ∼ = P1 ω(d;a, b) is a Weil divisor on b Xwhich does not correspond to a Cartier divisor. Let us write Eas an element in CaDiv( b X)⊗ZQ. §II.4. Pull-Back of a Q-Divisor 45 As for the behavior with respect to the sections, if s:X→Eis a nonzero global meromorphic section of Edefined by a collection of meromorphic functions {fi:Ui→C}i∈I, then its pull-back, denoted by F∗s:Y→F∗E, (F∗s)(y) := ((s◦F)(y), y) is the global meromorphic section of F∗Eassociated with fi◦F|F−1(Ui):F−1(Ui)→Ci∈I. Moreover, F∗sis the zero section of F∗Eif and only if F(Y)⊆Supp(s) := Supp((s)). The following diagram represents the pull-back of a line bundle with a global meromorphic section. Note that locally e Fis F×1C. F∗E # e F// F∗π  E π  YF// F∗s 66 X s ii Definition (II.4.2). Let F:Y→Xbe a morphism between irreducible complex analytic varieties. Let Dbe a Cartier divisor on Xand consider [D] its equivalence class in Pic(X). Define F∗[D] to be the equivalence class in Pic(Y) of the divisor associated with any non-zero global meromorphic section of the bundle F∗OX(D), i.e. F∗[D] = [(t)] where tis a non-zero meromorphic section as above. Remark (II.4.3). The pull-back is well defined and it has the following properties: (1) In our setting, there always exist non-zero global meromorphic sections of a line bundle of the form F∗OX(D). (2) The pull-back F∗[D]∈Pic(Y) only depends on the equivalence class of D. Assume D∼D0and consider tand t0two non-zero global meromorphic sections of F∗OX(D) and F∗OX(D0), respectively. Then, using Lemma (II.3.10)(1) and the functoriality of the pull-back, one sees that [(t)] = [(t0)] ∈Pic(Y). (3) If F(Y)*Supp(D), then F∗[D] coincides with the one given at the beginning of this section. This follows from (II.4.1) and the fact that t=F∗sDis a non-zero global meromorphic section of F∗OX(D). Hence Definition (II.4.2) gives rise to a group homomorphism F∗: Pic(X)→Pic(Y) as claimed. 46 Chapter II. Cartier and Weil Divisors on V-Manifolds (4) The pull-back is a contravariant functor, that is, if Zis another irreducible complex analytic variety and G:Z→Yis a morphism, then (G◦F)∗=F∗◦G∗. Although F:Y→Xinduces a group homomorphism between the Picard groups of Xand Y, in practice it is convenient to treat the following two cases separately: (Here Dis a Cartier divisor on X) •If F(Y)*Supp(D), then F∗(D)∈CaDiv(Y). •Otherwise F∗(D) is only defined up to linear equivalence. This approach is essentially the one presented by Fulton in [Ful98, Ch. 2] where the notion of pseudo-divisor is introduced. There, if F(Y)⊆Supp(D), then the pull-back F∗[D] is defined as the equivalence class in Pic(Y) of any Cartier divisor Eon Ywhose line bundle OY(E) is isomorphic to F∗OX(D). Definition (II.4.4). Let F:Y→Xbe a morphism between two irreducible V-manifolds and consider D∈Q-Div(X). Then Dcan be written as a finite sum Pr i=1 αiDiwhere Di∈CaDiv(X) and αi∈Q. The pull-back of Dis defined as F∗(D) := r X i=1 αi·F∗(Di), where F∗(Di) is the pull-back of a Cartier divisor as in (II.4.2). Hence F∗(D) is an element in CaDiv(Y)⊗ZQif F(Y)*|Di|, for all i= 1, . . . , r, and it is only defined up to Q-linear equivalence if F(Y)⊆ |Di0| for some i0∈ {1, . . . , r}. In any case, [F∗(D)] ∈Pic(Y)⊗ZQ. Now we have all the necessary ingredients to develop a rational intersection theory on varieties with quotient singularities. Chapter III is devoted to working out all the details, but first the following illustrative example will be given. Example (II.4.5). Let X=X(2; 1,1) and consider the Weil divisors D1= {x= 0}and D2={y= 0}. Let us compute the Weil divisor associated with j∗ D1D2, where jD1:D1,→Xis the inclusion. Following (II.2.14), the divisor D2can be written as 1 2{(X, y2)}. By definition, since D1*D2, the pull-back is j∗ D1D2=1 2(D1, y2|D1). §II.4. Pull-Back of a Q-Divisor 47 Thus its associated Weil divisor is TD1(j∗ D1D2) = 1 2X P∈D1 ordP(y2|D1)·[P] =1 2ord[(0,0)](y2|D1)·[(0,0)] = 1 2·[(0,0)]. Note that there is an isomorphism D1=X(2; 1) ≃C, [y]7→ y2, and the function y2:D1→Cis converted into the identity map C→Cunder this isomorphism. Hence ord[(0,0)](y2|D1) = 1. It is natural to define the (global and local) intersection multiplicity as D1·D2= (D1·D2)[(0,0)] =1 2. III Intersection Theory on Surfaces with Quotient Singularities Previously in Chapter I we saw how useful weighted blow-ups can be to compute embedded Q-resolutions. In this chapter, to study this special kind of resolutions, we develop an intersection theory on varieties with quotient singularities. Roughly speaking, given Xa complex analytic variety, the intersection product D·Eis well understood whenever Dis a compact Weil divisor on X and Eis a Cartier divisor on X. Over varieties with quotient singularities the notion of Cartier and Weil divisor coincide after tensoring with Q, see Theorem (II.1.7), and hence a rational intersection theory can be defined on this kind of varieties. This theory was first introduced by Mumford on normal surfaces, see [Mum61]. We give an alternative equivalent definition, without involving an embedded resolution of the ambient space, that allows us to compute the self-intersection numbers of the exceptional divisors of weighted blow-ups in dimension two. Also B´ezout’s theorem for quotients of weighted projective planes is studied. See [AMO11b] for further applications including the computation of abstract resolutions of surfaces via Jung method. Also, see [AMO11c] for an overview on this chapter and [Ort10] for a more direct approach. Section §III.1 Intersection Numbers: Generalities Base on Example (II.4.5) the intersection number of two Q-divisors is defined in terms of the degree map as follows. 50 Chapter III. Intersection Theory on V-Surfaces (III.1.1). (Degree of a Q-divisor). Let Cbe an irreducible analytic curve. Given a Weil divisor on Cwith finite support, D=Pr i=1 ni·[Pi], its degree is defined as deg(D) = r X i=1 ni∈Z. It is a group homomorphism. Moreover, if Cis compact, the degree of a principal divisor is zero and thus passes to the quotient yielding the map deg : Cl(C)→Z, cf. [Ful98, Prop. 1.4]. The degree of a Cartier divisor is the degree of its associated Weil divisor, that is, by definition deg(D) := deg(TCD). Finally, extending to rational coefficients, one obtains a group homomorphism (13) deg : nD∈Q-Div(C) with finite support o−→ Q that passes to the quotient Pic(C)⊗ZQwhen the curve is compact. Definition (III.1.2). Let Xbe a V-manifold of dimension 2 and consider D1, D2∈Q-Div(X). If D1is irreducible, then the intersection number, denoted by D1·D2, is defined as D1·D2:= deg j∗ D1D2∈Q, where jD1:|D1|,→Xdenotes the inclusion and deg is the map in (13). The expression above extends linearly if D1is a finite sum of irreducible Q-divisors. Following (III.1.1) and Definition (II.4.4), this number is only well defined if either |D1|*|D2|and |D1|∩|D2|is finite, or the divisor D1has a compact support. Let us discuss these two cases separately. To simplify assume D1is an irreducible Q-divisor. •If D1has compact support, then extending the order function to rational coefficients ordP: CaDiv(|D1|)⊗ZQ→Q, one writes the intersection number D1·D2as deg(E) = deg X P∈D1 ordP(E)·[P]=X P∈D1 ordP(E), where Eis any Q-Cartier divisor on |D1|representing the rational class [j∗ D1D2]∈Pic(|D1|)⊗ZQ. §III.1. Intersection Numbers: Generalities 51 •If |D1|*|D2|, then j∗ D1D2∈CaDiv(|D1|)⊗ZQand its support is the set |D1|∩|D2|. In this situation the order at Pits-self (D1·D2)P:= ordPj∗ D1D2∈Q is well defined and it is called the local intersection number at P. In addition, if |D1|∩|D2|is finite, then by definition D1·D2=X P∈|D1|∩|D2| (D1·D2)P. If D1is not irreducible, then the local intersection number (D1·D2)Pis extended by linearity so that the previous formula still holds. In the following result the main usual properties of intersection numbers are collected. Its proofs is omitted since it is well known for the classical case (i.e. without tensoring with Q), cf. [Ful98], and our generalization is based on extending the classical definition to rational coefficients. Proposition (III.1.3). Let Xbe a V-manifold of dimension 2and consider D1, D2, D3∈Q-Div(X). Then the local and the global intersection numbers, provided the indicated operations make sense according to Definition (III.1.2), satisfy the following properties: (α∈Q,P∈X) (1) Bilinear: Global D1·(D2+D3) = D1·D2+D1·D3 (D1+D2)·D3=D1·D3+D2·D3 (αD1)·D2=D1·(αD2) = α(D1·D2) Local D1·(D2+D3)P= (D1·D2)P+ (D1·D3)P (D1+D2)·D3P= (D1·D3)P+ (D2·D3)P (αD1)·D2P= (D1·(αD2))P=α(D1·D2)P (2) Commutative: If D1·D2and D2·D1are both defined, then D1·D2=D2·D1. Analogously (D1·D2)P= (D2·D1)Pif both local numbers are defined. (3) Non-negative: Assume D1and D2are effective, irreducible, and distinct. Then D1·D2and (D1·D2)Pare greater than or equal to zero if they are defined. Moreover, (D1·D2)P= 0 if and only if P /∈ |D1|∩|D2|, and hence D1·D2= 0 if and only if |D1|∩|D2|=∅. (4) Non-rational: If D2∈CaDiv(X)and D1∈WeDiv(X), then D1·D2and (D1·D2)Pare integral numbers. By the commutative property, the same holds if D1is a Cartier divisor and D2is a Weil divisor. 52 Chapter III. Intersection Theory on V-Surfaces (5) Q-Linear equivalence: Assume D1has compact support. If D2 and D3are Q-linearly equivalent, i.e. [D2]=[D3]∈Pic(X)⊗ZQ, then D1·D2=D1·D3. Due to the commutativity, the roles of D1and D2can be exchanged. In particular, D1·D2= 0 for every principal Q-divisor D2. (6) Normalization: Let ν:g |D1| → |D1|be the normalization of the support of D1and jD1:|D1|,→Xthe inclusion. Then D1·D2= deg jD1◦ν∗D2. Observe that in this situation the normalization is a smooth complex analytic curve.  Remark (III.1.4). This rational intersection number was first introduced by Mumford for normal surfaces, see [Mum61, Pag. 17]. Our Definition (III.1.2) coincides with Mumford’s because it has good behavior with respect to the pull-back, see Theorem (III.1.5). The main advantage is that ours does not involve a resolution of the ambient space and, for instance, this allows us to easily find formulas for the self-intersection numbers of the exceptional divisors of weighted blow-ups, without computing any resolution, see Proposition (III.3.2). The following result (the pull-back formula) is essential for obtaining B´ezout’s Theorem on quotients of weighted projective planes as well as for studying the local intersection number on X(d;A). Again its proofs follows from the fact that our generalization is based on extending the classical definition to rational coefficients. Theorem (III.1.5). Let F:Y→Xbe a proper morphism between two irreducible V-manifolds of dimension 2, and D1, D2∈Q-Div(X). (1) The cardinal of F−1(P),P∈X, is finite and generically constant. This generic number is denoted by deg(F). (2) If D1·D2is defined, then so is F∗(D1)·F∗(D2). In such a case, one has F∗(D1)·F∗(D2) = deg(F) (D·E). (3) If (D1·D2)Pis defined for some P∈X, then so is the local number (F∗(D1)·F∗(D2))Q,∀Q∈F−1(P). In such a case, it is verified that PQ∈F−1(P)(F∗(D1)·F∗(D2))Q= deg(F)(D1·D2)P. The rest of this section is devoted to reviewing some classical results concerning the intersection multiplicity, namely the computation of the local intersection number at a smooth point, the self-intersection numbers of the exceptional divisors of blow-ups at a smooth point, and the classical B´ezout’s Theorem on P2. Afterward, these results are generalized in the upcoming sections. §III.1. Intersection Numbers: Generalities 53 (III.1.6). (Local intersection number at a smooth point). Let Xbe a smooth analytic surface. Consider D1,D2two effective (Cartier or Weil)1divisors on Xand P∈Xa point. From Remark (II.2.3), the divisor Diis locally given by a holomorphic function fi,i= 1,2, in a neighborhood of P. Then (D1·D2)Pequals ordP(f2|D1) = lengthOD1,P OD1,P f2|D1= dimCOX,P hf1, f2i. Moreover, Xbeing a smooth variety, OX,P is isomorphic to C{x, y}and hence the previous dimension can be computed, for instance, by means of Gr¨obner bases with respect to local orderings. (III.1.7). (Classical blow-up at a smooth point). Let Xbe an analytic surface. Let π:b X→Xbe the classical blow-up at a smooth point P. Consider Cand Dtwo (Cartier or Weil) divisors on Xwith multiplicities mCand mDat P. Denote by Ethe exceptional divisor of π, and by b C (resp. b D) the strict transform of C(resp. D). Then, (1) E·π∗(C) = 0, (2) π∗(C) = b C+mCE, (3) E·b C=mC, (4) E2=−1, (5) b C·b D=C·D−mCmD. In addition, if Dhas compact support, then b D2=D2−m2 D. Note that the exceptional divisor has multiplicity 1 at every point. This is why for the self-intersection numbers of the exceptional divisors every time we blow up a point on them, when computing an embedded resolution of a plane curve, one only has to subtract 1. Example (III.1.8). The fourth property can easily be deduced assuming the first three. Let us prove it here by using directly Definition (III.1.2). Assume X=C2and π:b C2→C2is the blow-up at the origin. By definition, E2= deg(j∗ EE) = deg(t), where t:E→j∗ EOX(E) is any non-zero global meromorphic section of j∗ EOX(E). •Let us cover b C2by U1∪U2and use coordinates ((x, y),[u:v]) for C2×P1. As a Cartier divisor, the exceptional divisor of πis E={(U1, x),(U2, y)}. 1Recall that on smooth analytic varieties, Cartier and Weil divisors are identified and their equivalence classes coincide under this identification, i.e. Pic(X) = Cl(X), see Theorem (II.1.7). 54 Chapter III. Intersection Theory on V-Surfaces •Then OX(E) is the line bundle on b C2with transition function φ12 :U1∩U2→C∗,φ12(((x, y),[u:v])) = x y. Thus j∗ EOX(E) is the line bundle on E=V1∪V2with transition function ψ12 :V1∩V2−→ C∗, ψ12([u:v]) = u v. •The family {(V1,u v),(V2,1)}gives rise to a non-zero global meromorphic section of j∗ EOX(E). Its associated Weil divisor on P1is −{v= 0} ∈ WeDiv(P1) which has degree −1. Another way to proceed is to show directly that the dual of j∗ EOX(E) is isomorphic to the line bundle on Eassociated with the Weil divisor {v= 0}. (III.1.9). (B´ezout’s Theorem on P2). Every analytic (Cartier or Weil) divisor on P2is algebraic and thus it can be written as a difference of two effective divisors. On the other hand, every effective divisor is defined by a homogeneous polynomial. The degree of an effective divisor on P2is the degree, deg(F), of the corresponding homogeneous polynomial. This degree map is extended linearly yielding a group homomorphism deg : Div(P2)→Zthat characterizes the linear equivalence classes in the following sense: ∀D1, D2∈Div(P2), (14) [D1]=[D2]∈Pic(P2) = Cl(P2)⇐⇒ deg(D1) = deg(D2). Let D1,D2be two divisors on P2, then D1·D2= deg(D1) deg(D2). In particular, the self-intersection number of a divisor Don P2is given by D2= deg(D)2. In addition, if |D1|*|D2|, then |D1|∩|D2|is a finite set of points and, by the discussion after Definition (III.1.2), one has deg(D1) deg(D2) = D1·D2=X P∈|D1|∩|D2| (D1·D2)P. The proof of this result is an easy consequence of (III.1.3), and the fact that Diis linearly equivalent to deg(Di)Li, where Liis a linear form, i= 1,2, by (14). The rest of this chapter is devoted to generalizing the classical results of (III.1.6), (III.1.7), and (III.1.9) to V-manifolds, weighted blow-ups, and quotients of weighted projective planes, respectively. Section §III.2 Computing Local Intersection Numbers Let Xbe an algebraic V-manifold of dimension 2. Consider D1and D2 two effective Q-divisors on X, and P∈Xa point. From (II.2.3), cf. proof of Lemma (II.2.4), the divisor Diis locally given in a neighborhood of Pby a reduced polynomial fi,i= 1,2. §III.3. Intersection Numbers and Weighted Blow-ups 61 (III.3.4). In the same spirit of the preceding example, let us calculate E·b C using directly Definition (III.1.2) and the fact that E·b C=X P∈E∩b C (E·b C)P. Suppose Cis locally given by a meromorphic function f(x, y) = 0 defined on a neighborhood of the origin of X(d;a, b). Consider f=fν+fν+l+· · · the decomposition of f(x, y) into (p, q)-homogeneous parts. The global equation of E∩b C={fν= 0} ⊂ P1 ω(d;a, b) can be written as fν(x, y) = xαyβ k Y i=1 (xq−εq iyp)mi. Note that ν= ord(p,q)(f) = pα +qβ +pq Pr i=1 mi. The intersection multiplicity of Eand b Cat the point [εi: 1]ωis mi, while it is αe dq (resp. βe dp ), not necessarily an integer, at the possibly singular point [0 : 1] (resp. [1 : 0]). All these statements follows from §III.2, since by (I.3.14) the local equations of Eand b Cin the second chart are Xdq e;−p+δqa e,1⊇(E:y= 0; b C:xαQk i=1(xq−εq i)mi= 0, where δis the inverse of bmodulo d. To compute the intersection multiplicity at [1 : 0] the first chart is needed, but the details are omitted. On the other hand, the isomorphism P1 ω(d;a, b)−→ P1 [x:y]ω7→ [xdq/e :ydp/e], tells us that [εi: 1]ω= [εj: 1]ω∈P1 ω(d;a, b)⇐⇒ (εq i)d e= (εq j)d e. Consequently, the cardinality of E∩b C\ {[0 : 1],[1 : 0]}is k d/e and in fact one has r X i=1 mi= r X i=1 (E·b C)[εi:1]ω=d eX P6=[0:1],[1:0] (E·b C)P. Finally, collecting all the information above, it follows that X E∩b C (E·b C)P= (E·b C)[0:1] + (E·b C)[1:0] +X P6=[0:1],[1:0] (E·b C)P= =αe dq +βe dp +e d r X i=1 mi=e dpqpα +qβ +pq r X i=1 mi=eν dpq. 62 Chapter III. Intersection Theory on V-Surfaces Another way to proceed in order to calculate E·b Cis to realize that the required intersection product is the degree of the Weil divisor on P1 ω(d;a, b) given by fν(x, y) = xαyβQr i=1(xq−εq iyp)mi. This expression is mapped to xαe dq yβe dp k Y i=1 (xe d−εq iye d)mi under the isomorphism P1 ω(d;a, b)→P1. The latter is clear to have degree eν dpq as a Weil divisor on P1. Remark (III.3.5). Although elementary, the computation of the self-intersection numbers E2and E·b Cby using directly the definition is long and tedious. That is why Proposition (III.3.2) is proven with the pull-back formula (III.1.5) so that the proof becomes simpler and clearer. Let us discuss two special cases of Prop. (III.3.2) according to I.3–1, namely the point P∈Xis smooth and the point Pis of type (d;p, q) with gcd(d, p) = gcd(d, q) = 1. Consider the ω-blow-up π:= πω:b C2 ω→C2 (resp. π:= πω,d :b C2 ω,d →X(d;p, q)). The following properties hold: (1) E·π∗(C) = 0 (in both cases). (2) π∗(C) = b C+νE (resp. π∗(C) = b C+ν dE). (3) E·b C=ν pq (in both cases). (4) E2=−1 pq (resp. E2=−d pq ). (5) b C·b D=C·D−νµ pq (resp. b C·b D=C·D−νµ dpq ). Remark (III.3.6). To state formulas when P∈Xis a point represented by a type of the form (d;A), where A∈Mat(r×2,Z), one proceeds as in the proof of Proposition (III.3.2). In particular, one has to compute deg(h), e, and ν. For instance, to calculate esuch that H∗(EX) = eEY, one needs to write EXas a Q-Cartier divisor as in Example (II.2.15), or equivalently, to find the number esuch that the map X(d;A),[(0,1)]−→ C2,(0,1) [(x, y)] 7→ (xe, y) is an isomorphism of analytic germs. In fact, one can show that e=deg(pr) deg(pr |x=0), where pr : C2→Q1is the projection on the first chart. §III.3. Intersection Numbers and Weighted Blow-ups 63 When (d;A)=(d;a, b) but the type is not necessarily normalized or gcd(p, q)6= 1, then e=gcd(dp, dq, pb −qa) gcd(d, a, b). For deg(h) a generalization of Lemma (III.4.2) is needed. The details are left to the reader. Example (III.3.7). Let us consider the following divisors on C2, C1={((x3−y2)2−x4y3)=0}, C2={x3−y2= 0}, C3={x3+y2= 0}, C4={x= 0}, C5={y= 0}. The local intersection numbers (Ci·Cj)0,i, j ∈ {1,...,5},i6=j, are encoded in the intersection matrix associated with any embedded Q-resolution of C=S5 i=1 Ci, see [AMO11b] for a proof of this result. Let π1:C2 (2,3) →C2be the (2,3)-weighted blow-up at the origin. The new space has two cyclic quotient singular points of type (2; 1,1) and (3; 1,1) located at the exceptional divisor E1. The local equation of the total transform in the first chart is given by the function x29 ((1 −y2)2−x5y3) (1 −y2) (1 + y2)y:X(2; 1,1) −→ C, where x= 0 is the equation of the exceptional divisor and the other factors correspond in the same order to the strict transform of C1,C2,C3,C5 (denoted again by the same symbol). To study the strict transform of C4 one needs the second chart, the details are left to the reader. Hence E1has multiplicity 29 and self-intersection number −1 6; it intersects transversely C3,C4, and C5at three different points, while it intersects C1and C2at the same smooth point P, different from the other three. The local equation of the divisor E1∪C2∪C1at this point Pis x29 y(x5−y2)=0, see Figure III.1 below. E1(−1 6)E1(−17 30 ) C4 (3) C3 C1 C2(2) E2(−1 10 ) (5)(2) C5 (2) (3) C2C3C4 C5 C1 P π2 ←− Figure III.1. Embedded Q-resolution of C=S5 i=1 Ci⊂C2. 64 Chapter III. Intersection Theory on V-Surfaces Let π2be the (2,5)-weighted blow-up at the point Pabove. The new ambient space has two singular points of type (2; 1,1) and (5; 1,2). The local equations of the total transform of E1∪C2∪C1are given by the following two functions: 1st chart x73 |{z} E2 ·y |{z} C2 ·(1 −y2) | {z } C1 :X(2; 1,1) −→ C 2nd chart x29 |{z} E1 ·y73 |{z} E2 ·(x5−1) | {z } C1 :X(2; 1,1) −→ C Thus the new exceptional divisor E2has multiplicity 73 and intersects transversely the strict transform of C1,C2, and E1. Hence the composition π2◦π1is an embedded Q-resolution of C=S5 i=1 Ci⊂C2. Figure III.1 above illustrates the whole process. As for the self-intersection numbers, E2 2=−1 10,E2 1=−1 6−22 1·2·5=−17 30. The intersection matrix associated with the embedded Q-resolution obtained and its opposite inverse are A=−17/30 1/5 1/5−1/10, B =−A−1=6 12 12 34. Now one observes the intersection number is encoded in Bas follows. For i= 1,...,5, set ki∈ {1,...,5}such that ∅ 6=Ci∩ Eki=: {Pi}. Denote by O(Ci) the order of the cyclic group acting on Pi. Then, (Ci·Cj)0=bki,kj O(Ci)O(Cj). Looking at the figure one sees that (k1, . . . , k5) = (2,2,1,1,1), (O(C1), . . . , O(C5)) = (1,2,1,3,2). Hence, for instance, (C1·C2)0=bk1,k2 O(C1)O(C2)=b22 1·2=34 2= 17, which is indeed the intersection multiplicity at the origin of C1and C2. Analogously for the other indices. §III.4. B´ezout’s Theorem for Weighted Projective Planes 65 Remark (III.3.8). Consider the group action of type (5; 2,3) on C2. The previous plane curve Cis invariant under this action and then it makes sense to compute an embedded Q-resolution of C:= C/µ5⊂X(5; 2,3). Similar calculations, as in the previous example, lead to a figure as the one obtained above with the following relevant differences: • E1∩ E2is a smooth point. • E1(resp. E2) has self-intersection number −17 6(resp. −1 2). •The intersection matrix is A0=−17/6 1 1−1/2and its opposite inverse is B0=−(A0)−1=6/5 12/5 12/5 34/5. Hence, for instance, (C1·C2)0=b0 22 1·2=34/5 2=17 5, which is exactly the intersection number of these two curves, since that local number can also be computed as (C1·C2)0=1 5(C1·C2)0. Analogous considerations hold for (Ci·Cj)0,i, j = 1,...,5. Section §III.4 B´ezout’s Theorem for Weighted Projective Planes For a given weight vector ω= (p, q, r)∈N3and an action on C3of type (d;a, b, c), consider the quotient weighted projective plane P2 ω(d;a, b, c) := P2 ω/µd and the corresponding morphism τ(d;a,b,c),ω :P2→P2 ω(d;a, b, c) defined by (16) τ(d;a,b,c),ω([x:y:z]) = [xp:yq:zr]ω. Recall that P2 ω(d;a, b, c) is a variety with abelian quotient singularities; its charts are described in (I.3.19). The degree of a Q-divisor on P2 ω(d;a, b, c) is the degree of its pull-back under the map τ(d;a,b,c),ω, that is, by definition, D∈Q-Div P2 ω(d;a, b, c),degω(D) := deg τ∗ (d;a,b,c),ω(D). Thus if D={F= 0}is a Q-divisor on P2 ω(d;a, b, c) given by a ωhomogeneous polynomial that indeed defines a zero set on the quotient projective space, then degω(D) is the classical degree, denoted by degω(F), of the quasi-homogeneous polynomial. 66 Chapter III. Intersection Theory on V-Surfaces (III.4.1). The degree of a Q-divisor on P2 ω(d;a, b, c) has the following behavior with respect to the normalization process of weighted projective planes. •Let ω= (p, q, r)∈N3and ω0=1 gcd(p,q,r)ω. Consider the morphism P2 ω→P2 ω0induced by the identity map. Let D0be a Q-divisor on P2 ω0and Dits pull-back under the previous map. Then, degω0(D0) = 1 gcd(p, q, r)degω(D). •Let ω= (p, q, r)∈N3and ω0=p (p,q)·(p,r),q (q,p)·(q,r),r (r,p)·(r,q). Consider the morphism P2 ω→P2 ω0defined by [x:y:z]ω7−→ [xgcd(q,r):ygcd(p,r):zgcd(p,q)]ω0. Let D0be a Q-divisor on P2 ω0and Dits pull-back under the previous map which is a Q-divisor on P2 ω. Then, degω0(D0) = degω(D) gcd(p, q)·gcd(p, r)·gcd(q, r). The following result can be stated in a more general setting. However, it is presented in this way to keep the exposition as simple as possible. Lemma (III.4.2). The degree of the projection pr : C2→Xd;a b e;r s is given by the formula d·e gcd d·gcd(e, r, s), e ·gcd(d, a, b), as −br. Proof. Assume gcd(d, a, b) = gcd(e, r, s) = 1; the general formula is obtained easily from this one, since d a b e r s = d gcd(d,a,b) a gcd(d,a,b) b gcd(d,a,b) e gcd(e,r,s) r gcd(e,r,s) s gcd(e,r,s)!. The degree of the required projection C2→Xd;a b e;r s is de `, where `is the order of the abelian group H=n(ξ, η)∈µd×µe|ξaηr= 1, ξbηs= 1oC(µd×µe). To calculate `, let us consider (ξ, η)∈µd×µeand solve the system (ξaηr= 1, ξbηs= 1. Raising both equations to the e-th power, one obtains ξae = 1 and ξbe = 1. Hence, ξ∈µd∩µae ∩µbe =µgcd(d,ae,be)=µgcd(d,e). Note that the assumption gcd(d, a, b) = 1 was used in the last equality. Analogously, it follows that η∈µgcd(d,e), provided that gcd(e, r, s) = 1. §III.4. B´ezout’s Theorem for Weighted Projective Planes 67 Thus there exist i, j ∈ {0,1,...,gcd(d, e)−1}such that ξ=ζiand η=ζj, where ζis a fixed primitive (d, e)-th root of unity. Now the claim is reduced to finding the number of solutions of the system of congruences ai +rj ≡0 bi +sj ≡0,mod gcd(d, e). This is known to be gcd(d, e, as −br) and now the proof is complete.  Proposition (III.4.3). Using the notation above, let us denote by m1,m2, m3the determinants of the three minors of order 2of the matrix p q r a b c . Assume that gcd(p, q, r)=1and write e= gcd(d, m1, m2, m3). Then, the intersection number of two Q-divisors on P2 ω(d;a, b, c)is D1·D2=e dpqr degω(D1) degω(D2)∈Q. In particular, the self-intersection number of a Q-divisor is given by D2=e dpqr degω(D)2. Moreover, if |D1|*|D2|, then |D1|∩|D2|is a finite set of points and (17) e dpqr degω(D1) degω(D2) = X P∈|D1|∩|D2| (D1·D2)P. Proof. For simplicity, let us write just τfor the map in (16), omitting the subindex. Note that τis a proper morphism between two irreducible V-manifolds of dimension 2. Thus by Theorem (III.1.5)(2) and the classical B´ezout’s theorem on P2(III.1.9), one has the following sequence of equalities, deg(τ) (D1·D2) = τ∗(D1)·τ∗(D2) = deg (τ∗(D1)) deg (τ∗(D2)) = degω(D1) degω(D2). The rest of the proof is the computation of deg(τ); the final part is a consequence of discussion after Definition (III.1.2). In the first chart τtakes the form C2→Xp;q r pd ;m1m2, (y, z)7→ [(yq, zr)], see (I.3.19) for details. By decomposing this morphism into C2→C2, (y, z)7→ (yq, zr) and the projection C2→Xp;q r pd ;m1m2, (y, z)7→ [(y, z)], one obtains deg(τ) = qr ·deg hC2pr −→ Xp;q r pd ;m1m2i. The determinant of the corresponding matrix is qm2−rm1=pm3. From Lemma (III.4.2), the latter degree is p·pd gcd p·gcd(pd, m1, m2), pd, pm3=dp gcd d, m1, m2, m3, and hence the proof is complete.  68 Chapter III. Intersection Theory on V-Surfaces Corollary (III.4.4). Let X,Y,Zbe the Weil divisors on the quotient space P2 ω(d;a, b, c)given by {x= 0},{y= 0}, and {z= 0}, respectively. Using the notation of Proposition (III.4.3), one has: (1) X2=ep dqr,Y2=eq dpr,Z=er dpq. (2) X·Y=e dr,X·Z=e dq,Y·Z=e dp. Remark (III.4.5). Some comments about the previous results. (1) The local intersection numbers (D1·D2)Pin (17) are computed in (III.2.1) in terms of the dimension of a C-vector space. This dimension can in turn be computed by means of Gr¨obner bases with respect to local orderings as usual. (2) If d= 1, then e= 1 too and the formulas above become a bit simpler. In particular, one obtains B´ezout’s theorem on weighted projective planes, (the last equality if |D1|*|D2|only) D1·D2=1 pqr degω(D1) degω(D2) = X P∈|D1|∩|D2| (D1·D2)P. (3) To state B´ezout’s theorem on P2 ω(d;A), where A∈Mat(r×3,Z), one proceeds in the same way. First consider the natural morphism τ:P2→P2 ω(d;A) defined by [x:y:z]7→ [xp:yq:zr]ω, then apply the pull-back formula, and finally compute the degree of τ. That is, ∀D1, D2∈QDiv P2 ω(d;A), one has D1·D2=1 deg(τ)degω(D1) degω(D2). The latter degree is reduced, as in the proof of Prop. (III.4.3), to the calculation of the degree of the projection C2→X(e;B), (y, z)7→ [(y, z)], where the type (e;B) is obtained after taking charts on the corresponding projective planes. In this sense a generalization of Lemma (III.4.2) is welcome. Example (III.4.6). Without assuming gcd(p, q, r) = 1 in (III.4.3), the degree of τis dpqr ewhere e= gcd d·gcd(p, q, r), m1, m2, m3). The general formula for the degree of τ:P2→P2 ω(d;A) is left to the reader. IV Monodromy Zeta Function and Lefschetz Numbers In this chapter the behavior of the Lefschetz numbers and the zeta function of the monodromy with respect to an embedded Q-resolution is investigated, cf. [Mar11c]. These two invariants have already been studied in different contexts by several authors. Hence before going into details, let us recall some of those approaches. Let f: (Cn+1,0) →(C,0) be a germ of a non-constant analytic function and let (H, 0) be the hypersurface singularity defined by f. Consider the Milnor fiber F={x∈Cn+1 :||x|| ≤ ε, f(x) = η}(0 < η << ε, where εsmall enough) and h:F→Fthe corresponding geometric monodromy. The induced automorphisms on the complex cohomology groups are often denoted by h:= Hq(h) : Hq(F, C)→Hq(F, C). In [A’C75], A’Campo gives a method for computing the Lefschetz number of the iterates hk:= h◦ · · · ◦ hof the geometric monodromy, defined by Λ(hk) := X q≥0 (−1)qtr Hq(hk), in terms of an embedded resolution of the singularity (H, 0) ⊂(Cn+1,0). These Lefschetz numbers are related to the monodromy zeta function Z(f) := Y q≥0 det(Id∗−tHq(h))(−1)q by the following well-known formula (18) Z(f) = exp −X k≥1 Λ(hk)tk k!. 70 Chapter IV. Monodromy Zeta Function and Lefschetz Numbers Using this relationship he derives a new expression for Z(f). More precisely, let π:X→(Cn+1,0) be an embedded resolution of (H, 0). Consider the total transform of Hwritten as π∗(H) = b H+ r X i=1 miEi, where b His the strict transform of H, and E1, . . . , Erare the irreducible components of the exceptional divisor π∗(0). Now, define ˇ Ei:= Ei\ Ei∩[ j6=i Ej∪b H!. Then, the Lefschetz numbers and the complex monodromy zeta function are given by Λ(hk) = r X i=1, mi|k miχ(ˇ Ei), Z(f) = r Y i=1 (1 −tmi)χ(ˇ Ei). The Euler characteristic of the Milnor fiber is therefore (19) χ(F) = Λ(h0) = r X i=1 miχ(ˇ Ei). When (H, 0) defines an isolated singularity, both the characteristic polynomial of the monodromy ∆(t) and the Milnor number µ= dim Hn(F, C) = deg ∆(t) can be obtained from the zeta function as follows, ∆(t) = "1 t−1 r Y i=1 (tmi−1)χ(ˇ Ei)#(−1)n ; µ= (−1)nh−1 + r X i=1 miχ(ˇ Ei)i, and in particular by (19), µ= (−1)n[−1 + χ(F)] holds. Another contribution in the same direction is found in [GLM97], where the authors give a generalization of A’Campo’s formula for the zeta function via partial resolutions, that is, the map π:X→(Cn+1,0) is assumed to be just a modification (i.e. the condition about normal crossing divisor in the embedded resolution is removed). Also Dimca, using the machinery of constructible sheaves, proved the same result allowing Xto be an arbitrary analytic space, see [Dim04, Th. 6.1.14.]. §IV.2. Partial Statement and Examples 77 The set Sm,d is not empty for m=e1e2e3/e2and d∈ {1,e1 e,e2 e,e3 e}. Since the intersection E∩b Hcan be identified with C, the Euler characteristics are χ(Sm,1) = −χ(C), χ(Sm, e1 e) = χ(Sm, e2 e) = χ(Sm, e3 e) = 1. From Theorem (IV.2.1), the characteristic polynomial of His ∆(t) = te1e2 e−1te1e3 e−1te2e3 e−1 t−1te1e2e3 e2−1χ(C). On the other hand, the Milnor number is known to be µ=e1e2 e−1e1e3 e−1e2e3 e−1. Using that µ= deg ∆(t), one finally obtains χ(C) = e1+e2+e3−e1e2e3 e. Example (IV.2.6). Let p, q, r be three positive integers and consider the polynomial function f:C3→Cgiven by f=xp+yq+zr. To simplify notation, we set e1= gcd(q, r), e2= gcd(p, r), e3= gcd(p, q), e= gcd(p, q, r), and k=e1e2e3. The following information will be useful later: gcd(qr, pr, pq) = e1e2e3 e=k e, d1:= gcd epr k,epq k=ep e2e3 ;a1:= lcm(d2, d3) = e2qr e1k=d2d3, d2:= gcd eqr k,epq k=eq e1e3 ;a2:= lcm(d1, d3) = e2pr e2k, d3:= gcd eqr k,epr k=er e1e2 ;a3:= lcm(d1, d2) = e2pq e3k. Take the weight vector ω=e k(qr, pr, pq) and let π:b C3 ω→C3be the weighted blow-up at the origin with respect to ω. The new space b C3 ω=U0∪U1∪U2 has three lines (each of them isomorphic to P1) of singular points located at the exceptional divisor E=π∗(0) ∼ =P2 ω. They actually coincide with the three lines L0, L1, L2at infinity of P2 ω. 78 Chapter IV. Monodromy Zeta Function and Lefschetz Numbers In the third chart U2=X(epq k;eqr k,epr k,−1), an equation of the total transform is zepqr k(xp+yq+ 1), where z= 0 is the exceptional divisor and the other equation corresponds to the strict transform. L0 L1 L2 epr k epq keqr k er e1e2ep e2e3 eq e1e3 b H∩EE∼ =P2 ω epqr k e2points e1pts e3pts Figure IV.5. Embedded Q-resolution of f=xp+yq+zr. Working in this coordinate system, one sees that the line L0(resp. L1) and b Hintersect at exactly e1(resp. e2) points. Analogously, L2∩b Hconsists of e3points. Moreover, using Lemma (I.1.16), we have that b Hand Eare smooth varieties that intersect transversely. Hence the map πis an embedded Q-resolution of {f= 0} ⊂ C3where all the cyclic quotient spaces are presented in their normalized form. The Euler characteristics as well as the fractions m/d for the non-empty sets Sm,d are calculated in the two tables below. Sepqr k,1Sepqr k,ep e2e3 Sepqr k,eq e1e3 Sepqr k,er e1e2 m d epqr k qr e1 pr e2 pq e3 χe1+e2+e3 −χ(C)−e1−e2−e3 Sepqr k,eqr kSepqr k,epr kSepqr k,epq k m/d p q r χ1 1 1 §IV.2. Partial Statement and Examples 79 Here we denote by Cthe variety in P2 ωdefined by the quasi-homogeneous polynomial xp+yq+zr. Recall that from Proposition (I.2.5), the map P2 ω→P2(e1 e,e2 e,e3 e) given by [x:y:z]ω7−→ [x ep e2e3:y eq e1e3:zer e1e2](e1 e,e2 e,e3 e) is an isomorphism and it maps the hypersurface Cto xe2e3 e+ye1e3 e+ze1e2 e= 0. By Example (IV.2.5), its Euler characteristic is χ(C) = e1+e2+e3−e1e2e3 e, and finally, from Theorem (IV.2.1), one obtains the characteristic polynomial of f, ∆(t) = t epqr e1e2e3−1e1e2e3 etp−1tq−1tr−1 t−1t qr e1−1e1t pr e2−1e2t pq e3−1e3. Note that the Euler characteristic of Ccould also be obtained using that the Milnor number is µ= (p−1)(q−1)(r−1) = deg ∆(t), as in the preceding example. Example (IV.2.7). Let f:C3→Cbe the polynomial function defined by f=zm+k+hm(x, y, z). Assume that C={hm= 0} ⊆ P2has only one singular point P= [0 : 0 : 1], which is locally isomorphic to the cusp xq+yp, gcd(p, q) = 1. Denote k1= gcd(k, p) and k2= gcd(k, q). Consider the classical blow-up at the origin π1:b C3→C3. In the third chart, the local equation of the total transform is zm(zk+xq+yp) = 0. The strict transform b Hand the exceptional divisor E0intersect transversely at every point but in P∈C≡E0∩b H. Also b H\Pis smooth. E1∩E0 m E0kp k1k2 kq k1k2 k k1k2 P2 b H∩E0 k k1k2 pq k1k2 kp k1k2 b H∩E1 q k2p k1 kq k1k2E0∩E1 x= 0 y= 0 z= 0 k1pts k2pts P2 ω pq k1k2(m+k)E1 Figure IV.6. Intersection of E0(resp. E1) with the rest of components. 80 Chapter IV. Monodromy Zeta Function and Lefschetz Numbers One is therefore interested in the blowing-up at the point Pwith respect to (kp, kq, pq). However, in order to obtain cyclic quotient spaces in their normalized form, it is more suitable to choose ω= ( kp k1k2,kq k1k2,pq k1k2) instead. Let π2be the weighted blow-up at Pwith respect to the vector ω. The local equation of the total transform in the second chart is given by the polynomial function ny pq k1k2(m+k)zm(zk+xq+ 1) = 0o⊂Xkq k1k2 ;kp k1k2 ,−1,pq k1k2, where y= 0 represents the new exceptional divisor E1. The composition π=π1◦π2is an embedded Q-resolution. The final situation is illustrated in Figure IV.6, see Chapter VII for details. The sets for which the Euler characteristic has to be computed are Sm,1, S`,1, S`, p k1 , S`, q k2 , S`, pq k1k2 ;`=pq k1k2 (m+k). Clearly χ(S`, pq/k1k2) = 1, χ(S`, p/k1) = −k2, and χ(S`, q/k2) = −k1, since they are homeomorphic to a point, P1\{k2+2 points}and P1\{k1+2 points} respectively. The set Sm,1is P2\C. Finally, we use the additivity of the Euler characteristic to compute χ(S`,1). Indeed, let D⊂P2(k1, k2,1) be the variety defined by the equation zk1k2+xk2+yk1= 0. Note that, by Proposition (I.2.5), Dis isomorphism to the surface b H∩E1={zk+xq+yp= 0} ⊂ P2 ω and, by Example (IV.2.5) (using e1=k1,e2=k2,e3= 1), its Euler characteristic is k1+k2+ 1 −k1k2. Then, χ(S`,1) = 3 −(2+2+2+χ(D)) + k1+k2+ 4 = k1k2. Every cyclic quotient singularity is written in their normalized form and thus the generalized A’Campo’s formula can be applied with d0=d, ∆(t) = tm−1χ(P2\C) t−1·tm+k−1t pq k1k2(m+k)−1k1k2 t p k1(m+k)−1k1t q k2(m+k)−1k2 =tm−1χ(P2\C) t−1·∆k P(tm+k). Let us explain the notation. The symbol ∆P(t) denotes the characteristic polynomial of Cat P= [0 : 0 : 1], where the curve is locally isomorphic to xq+yp, and if ∆(t) = Qi(tmi−1)ai, then ∆k(t) denotes ∆k(t) = Y it mi gcd(mi,k)−1gcd(mi,k)ai. §IV.3. Proof of the Theorem 81 The family of examples zm+k+hm(x, y, z), where hmdefines a reduced projective plane curve such that Sing(hm)∩ {z= 0}=∅as a subset in P2, i.e. Yomdin-Lˆe surface singularities, is studied in Chapter VII. We conclude by emphasizing that in the classical A’Campo’s formula one has to pay attention to compute the Euler characteristic while the multiplicities remain trivial. Using our formula we also have to take care of computing the multiplicities and the order of the corresponding cyclic groups, especially when the quotient singularity is not in its normalized form. Discussion (I.1.15) and Lemma (I.1.16) are very useful in this sense. Section §IV.3 Proof of the Theorem One way to proceed is to rebuild A’Campo’s paper [A’C75], thus giving a model of the Milnor fibration in our setting. This method is very natural but perhaps a bit long and tedious. In [GLM97], the authors give a generalization of A’Campo’s formula for the monodromy zeta function via partial resolution but the ambient space considered there is still smooth and the proof can not be generalized to an arbitrary analytic variety. That is why a very general result by Dimca is used instead, see Theorem (IV.3.6) below. This leads us to talk about constructible complexes of sheaves with respect to a stratification and also about the nearby cycles associated with an analytic function. Using this theorem, only the monodromy zeta function of a monomial defining a function over a quotient space of type X(d;A) is needed. IV.3–1. A result by Dimca To state the result we need some notions about sheaves and constructibility. We refer, for instance, to [Dim04] and the references listed there for further details. (IV.3.1). Consider Sh(X, VectC) the abelian category of sheaves of C-vector spaces on a topological space X. To simplify notation its derived category is often denoted by D∗(X). The constant sheaf corresponding to Cis denoted by CX; it is by definition the sheaf associated with the constant presheaf which sends every open subset of Xto C. If U⊂Xis connected open, then CX(U) = C. 82 Chapter IV. Monodromy Zeta Function and Lefschetz Numbers Let f:X→Ybe a continuous mapping between two topological spaces. The direct image functor f∗:Sh(X, VectC)→Sh(Y, VectC) is defined on objects by (f∗F)(V) = F(f−1(V)), for any sheaf Fon Xand any open set V⊂Y. This functor is additive and left exact; its derived functor is denoted by Rf∗:D∗(X)→D∗(Y). The inverse image functor f−1:Sh(Y, VectC)→Sh(X, VectC) is defined as f−1Gbeing the sheaf associated with the presheaf U7−→ lim −→ f(U)⊂V G(V). Here Gis a sheaf on Yand U⊂Xis open. This functor is exact and hence the corresponding derived functor Rf−1:D∗(Y)→D∗(X) is usually denoted again by f−1. If f(U)⊂Yis open, then (f−1G)(U) = G(f(U)). In particular, if the map iU:U ,→Xdenotes the inclusion of an open set, then i−1 UF=F|U. The restriction to an arbitrary subspace Z⊂Xis defined by F|Z:= i−1 ZF, where iZ:Z ,→Xis the inclusion. Using this notation one has CX|Z:= i−1 ZCX=CZ. (IV.3.2). Let Xbe a complex analytic space and S={Xj}j∈Ja locally finite partition of Xinto non-empty, connected, locally closed subsets called strata of S. The partition Sis called a stratification if it satisfies the following conditions: (1) The boundary condition, i.e. each boundary ∂Xj=Xj\Xjis a union of strata in S. (2) Constructibility, i.e. for all j∈Jthe spaces Xjand ∂Xjare closed complex analytic subspaces in X. (3) Stratification, i.e. all the strata are smooth constructible subvarieties of X. Definition (IV.3.3). Let S={Xj}j∈Jbe a stratification on X. (i) A sheaf complex F•∈D∗(X) is called S-constructible if the restriction of each cohomology sheaf Hq(F•)|Xjis a CXj-local system of finite rank, that is, one has the isomorphisms of CXj-vector spaces Hq(F•)|Xj≃Crj,q Xj. (ii) Given u:F•→ F•an automorphisms of CX-vector spaces, the complex F•is called equivariantly S-constructible with respect to u, if it is S-constructible and the induced automorphisms on the cohomology groups Hq(u)x:Hm(F•)x→ Hm(F•)xare all conjugate. §IV.3. Proof of the Theorem 83 (IV.3.4). Let Xbe a complex analytic variety and g:X→Ca nonconstant analytic function. Consider the diagram, g−1(0)i//XX\g−1(0) #f  ?_ j ooE ˆ f  ˆπ oo C∗e C∗ exp oo where i:g−1(0) ,→Xand j:X\g−1(0) ,→Xare inclusions, e C∗is the universal cover of C∗, and Edenotes the pull-back. Definition (IV.3.5). Let F•∈D∗(X) be a complex. The nearby cycles of F•with respect to the function g:X→Cis defined to be the sheaf complex given by ψgF•:= i−1R(j◦ˆπ)∗(j◦ˆπ)−1F•∈D∗(g−1(0)). The nearby cycles is a local operation in the sense that if U⊂Xis an open set, then (ψgF•)|W=ψg|WF•|Wholds. Also note that ψgF•only depends on gand F•|X\g−1(0). There is an associated monodromy deck transformation h:E→E coming from the action of the natural generator of π1(C∗) which satisfies ˆπ◦h= ˆπ. This homeomorphism induces an isomorphism of complexes Mg:ψgF•−→ ψgF•. For every point x∈g−1(0) there is a natural isomorphism from the stalk cohomology of ψgF•at xto the cohomology of the Milnor fiber at xwith coefficients in F•, that is, for all  > 0 small enough and all t∈C∗with |t|<< , one has Hq(ψgF•)x≃Hq(g−1(t)∩B(x),F• |)(21) ≃Hq(g−1(t)∩B(x),F• |), where the open ball B(x) is taken inside any local embedding of (X, x) in an affine space. The monodromy morphism Mg,x on the left-hand side corresponds to the morphism on the right-hand side induced by the monodromy homeomorphism of the local Milnor fibration associated with g: (X, x)→(C,0). Now we are ready to state Dimca’s theorem. To be precise, he only considered the case when the ambient space is smooth M=Cn+1, see below. Repeating exactly the same arguments, one obtains the result for any analytic variety. 84 Chapter IV. Monodromy Zeta Function and Lefschetz Numbers Theorem (IV.3.6) ([Dim04], Th. 6.1.14).Let f: (M, p)→(C,0) be the germ of a non-constant analytic function which is defined on a small neighborhood Uof p. Let Hbe the hypersurface {x∈U|f(x)=0}. Assume π:X→Uis a proper analytic map such that πinduces an isomorphism between X\π−1(H)and U\H. Let g=f◦πdenote the composition and j:X\π−1(H),→Xthe inclusion. Let Sbe a finite stratification of the exceptional divisor π−1(0) such that ψgRj∗CX\π−1(H)is equivariantly S-constructible with respect to the semisimple part of Mg. Then, Λ(h) = X S∈S χ(S)Λ(g, xS) ; Z(f) = Y S∈S Z(g, xS)χ(S), where xSis an arbitrary point in the stratum Sand Z(g, xS),Λ(g, xs)are the zeta function and the Lefschetz number of the germ gat xS. Remark (IV.3.7). Let F•=Rj∗CX\π−1(H). Using the notation of the previous theorem, the isomorphism of (21) tells us that Hq(ψgF•)x=Hq(Fx,C) where Fxis the Milnor fiber at x. This clarifies when the complex of sheaves ψgF•is equivariantly Sconstructible with respect to the semisimple part of Mg. In particular, this condition is satisfied, for instance, when the local equation of galong each stratum is the same. IV.3–2. Zeta function of a normal crossing divisor Let M=Cn/µdbe a quotient space of type X(d;A), not necessarily cyclic or written in a normalized form. Recall the multi-index notation: X(d;A) = X   d1a11 . . . a1n . . .. . ..... . . drar1. . . arn   ,d= (d1, . . . , dr), aj= (a1j, . . . , arj). In Section III, we have seen that for each j= 1, . . . , n there is an isomorphism (22) X(d;aj)−→ C [xj]7→ x`j j, where `j= lcm d1 gcd(d1, a1j),..., dr gcd(dr, arj). §IV.3. Proof of the Theorem 85 Given a homogeneous polynomial defined over M, the classical formula for the monodromy zeta function depending on the degree of the polynomial and the Euler characteristic of the Milnor fiber seems to be more complicated in this setting. Using the techniques developed in Chapter VI, one can provide formulas at least for plane curves and surfaces but the trick of applying the fixed point theorem does not work anymore. However, for our purpose, only the normal crossing case is needed. (IV.3.8). We first proceed to compute the geometric monodromy of a homogeneous polynomial f:M→Cof degree N:= deg(f). Let α: [0,1] →C∗ be a generator of the fundamental group of C∗, for example, α(t) = exp(2πit) and consider [x]∈F=f−1(1). The path eα: [0,1] −→ M\f−1(0), t7→ (e2πi Ntx1, . . . , e2πi Ntxn) defines a lifting of αwith initial point [(x1, . . . , xn)]. Thus the geometric monodromy h:F→Fcorresponds to the map eα(0) = (x1, . . . , xn)h 7−→ (e2πi Nx1, . . . , e2πi Nxn)=eα(1). As in the case M=Cn, this also works for quasi-homogeneous polynomials, replacing the exponentials for suitable numbers according to the weights. (IV.3.9). Let us study the monodromy zeta function in the simplest normal crossing case, i.e. f=xm1 1:M→C. The Milnor fiber F:= f−1(1) = {[x]∈M|xm1 1= 1} has the same homotopy type as F0:= {[(x1,0,...,0)] ∈M|xm1 1= 1}which can be identified with [x1]∈X(d;a1)|xm1 1= 1. In fact, r:F→F0: [x]7→ [x1] is a strong deformation retraction. Since h(F0)⊂F0, the geometric monodromy h:F→Fis homotopic to its restriction h0:= h|F0:F0→F0. Using the isomorphism (22), X(d;a1)≃C: [x]7→ x`1, the claim is reduced to the calculation of the zeta function of the polynomial xm1/`1 1:C→C. But this is known to be 1 −tm1/`1. 86 Chapter IV. Monodromy Zeta Function and Lefschetz Numbers (IV.3.10). Assume now that f=xm1 1· · · xmk k:M→C,k≥2. The Milnor fiber F:= f−1(1) has the same homotopic type as F0:= n(x1,...xk)∈S1×(k) · · · ×S1 µdxm1 1· · · xmk k= 1o, where µddefines an action of type (d;a1,...,ak) on the space (S1)k. As above, there is a strong deformation retraction r:F−→ F0, [x]7→ hx1 |x1|,..., xk |xk|,0,...,0i satisfying that h(F0)⊂F0. We shall see that the Lefschetz numbers Λ((h0)j) = Λ(hj) equal zero for all j≥1. This would imply Zf(t) = 1 by virtue of (18). Two cases arise: •If (h0)jdoes not have fixed points, then by the classical fixed point theorem Λ((h0)j) = 0. •Otherwise, (h0)jis the identity map and Λ((h0)j) = χ(F0) = 0. Note that there is an unramified covering (S1)k⊃f F0:= {xm1 1· · · xmk k= 1}π −→ F0 with a finite number of sheets. The first of the preceding spaces f F0has e= gcd(m1, . . . , mk) disjoint components, each of them homotopically equivalent to a real (k−1)-dimensional torus Tk−1= (S1)k−1. It follows that χ(F0) = 1 deg πe χ(Tk−1)=0. Note that the condition k≥2 has only been used at the end. In the case k= 1, one has deg π=`1, e =m1, χ(T0)=1, χ(F0) = m1/`1. We summarize the previous discussion in the following lemma. Lemma (IV.3.11). The monodromy zeta function of a normal crossing divisor given by xm1 1· · · xmk k:X(d;A)→C,k≥1, is Zxm1 1· · · xmk k:X(d;A)→C;t=(1−t m1 `1k= 1; 1k≥2, where `1= lcm d1 gcd(d1, a11),..., dr gcd(dr, ar1). §IV.4. Zeta Function of Not-Well-Defined Functions 93 (IV.4.5). In the previous example, X(d;p, q) can be normalized to Xd (d, p)(d, q);p (d, p),q (d, q). Under this isomorphism the polynomial f=xayb(x2+y3) is sent to xa (d,q)·yb (d,p)x2 (d,q)+y3 (d,p), which is not a polynomial in general. This seems to force one to work with non-normalized spaces. However, since d|(2p−3q) and gcd(d, p, q) = 1, then gcd(d, q)|2 and gcd(d, p)|3. Thus the preceding expression is a polynomial times a monomial with rational exponents. This fact is not a coincidence as the following result clarifies, see also Remark (VI.2.7). Although it can be stated in a more general setting, to simplify the ideas, we only consider polynomials in two variables over cyclic quotient singularities. Proposition (IV.4.6). Let d, p, q be three numbers, gcd(d, p, q)=1. Let f(x, y)∈C[x, y]be a polynomial such that f(ξp dx, ξq dy) = ξv df(x, y). If x-f(x, y)and y-f(x, y), then f(x1/gcd(d,q), y1/gcd(d,p))is again a polynomial. As a consequence, an arbitrary polynomial g(x, y)satisfying g(ξp dx, ξq dy) = ξv dg(x, y), is converted after normalizing X(d;p, q)into a polynomial times a monomial with rational exponents, that is, it can be written in the form, gx1 gcd(d,q), y 1 gcd(d,p)=xaybh(x, y), where h(x, y)∈C[x, y]and a, b ∈Q≥0. Proof. Since y-f(x, y), there exists k0≥0 such that xk0is a monomial of f. The action is diagonal and does not change the form of the monomials. Hence xk0has the same behavior with respect to the action as f, that is, ξk0p dxk0=ξv dxk0. This implies that d|(k0p−v). Take k≥0 such that k≡ −k0 modulo d. Now xkf(x, y) : X(d;p, q)→Cis a function with x-f(x, y). Then gcd(d, q)|kand f(x1/gcd(d,q), y) is a polynomial, see Remark (VI.2.7). Analogously the expression f(x, y1/gcd(d,p)) is a polynomial too and the proof is complete.  94 Chapter IV. Monodromy Zeta Function and Lefschetz Numbers (IV.4.7). As for weighted projective planes, let F∈C[x, y, z] be a (p, q, r)- homogeneous polynomial with gcd(p, q, r) = 1. The monodromy zeta function of F(x, y, z) at a point of the form [a:b: 1] is defined by ZF(x, y, z),[a:b: 1]; t:= Zf(x, y, 1),(a, b); t. Note that f(ξp rx, ξq r,1) = ξdeg(f) rf(x, y, 1) and thus f(x, y, 1) satisfies the conditions of Proposition (IV.4.1)(2), where the quotient space is simply M=X(r;p, q). Therefore the expression above equals Z(f(x, y, 1)r,(a, b); t1/r). Analogously, the zeta function at every point of P2(p, q, r) is defined and one sees that it is independent of the chosen chart. This can be generalized to spaces like Pn ω/µd, where µdis an abelian finite group acting diagonally as usual. Remark (IV.4.8). To define the monodromy zeta function for polynomials defining a zero set but there is no ksuch that fkis a function over the quotient space, one could use A’Campo’s formula and try to prove that the rational function obtained is independent of the chosen embedded Qresolution. We do not insist on the veracity of this fact because it is not the purpose of this work. Example (IV.4.9). We continue here with Example (IV.4.2). Blowing up the origin of X(2; 1,1) with respect to the weights (1,2), an embedded Qresolution of {f= 0} ⊂ X(2; 1,1) is computed and thus it makes sense to define the zeta function using this resolution. 8 (4; 1,1) (2; 1,1) Z(t) = (1 −t4)(1 −t2) (1 −t8) Figure IV.8. Embedded Q-resolution of {(x2+y)(x2− y)3= 0} ⊂ X(2; 1,1) and its monodromy zeta function. Section §IV.5 Why Abelian? D4as a Quotient Singularity All over the chapter, the ambient space Xis assumed to be Cn/G, where Gis an abelian finite subgroup of GL(n, C). In this final part, using D4as a quotient singularity, it is exemplified the behavior for non-abelian groups. §IV.5. Why Abelian? D4as a Quotient Singularity 95 As we shall see, double points in an embedded Q-resolution of a welldefined function f:X→Ccontributes, in general, to its monodromy zeta function. In this sense abelian groups are the largest family for which Theorem (IV.2.1) applies. Let C2with coordinate (x, y) and consider the subgroup of GL(2,C) generated by the matrices A=i0 0−i, B =0−1 1 0 . Thus A2=B2= (AB)2=−Id2. This group of order 8, often denoted by BD8, is called the binary dihedral group. The quotient singularity C2/BD8 is denoted by D4. Let us compute the zeta function of f:= (xy)m:D4→C, where mis an even positive integer so that the map is well defined. Consider π:b C2→C2 the usual blow-up at the origin. The action BD8on C2extends naturally to an action on b C2such that the induced map ¯π:b C2/BD8→C2/BD8=: D4 defines an embedded Q-resolution of {f= 0} ⊂ D4. More precisely, there are three quotient singular points all of them of type (2; 1,1) located at the exceptional divisor. They correspond to the points [0 : 1], [1 : 1], [i: 1] ∈P1/BD8. The strict transform intersects transversely the exceptional divisor at P:= ((0,0),[0 : 1]) and the equation of the total transform at this point is given by xmym:X(2; 1,1) →C, see Figure IV.9. (2; 1,1) m m (2; 1,1) (2; 1,1) P Figure IV.9. Embedded Q-resolution of {(xy)m= 0} ⊂ D4. From Theorem (IV.2.1), the monodromy zeta function of fand the Euler characteristic of the Milnor fiber are Z(t) = (1 −tm/2)2 1−tm=1−tm/2 1 + tm/2, χ(F) = deg Z(t)=0. In particular, Z(t) is not trivial although fdefines a “double point” on D4, as claimed. V Mixed Hodge Structure on the Cohomology of the Milnor Fiber Steenbrink in [Ste77] gave a mixed Hodge structure (MHS) on the cohomology of the Milnor fiber using a spectral sequence that is constructed from the divisors associated with the semistable reduction of an embedded resolution. The aim of this chapter is to describe explicitly a similar spectral sequence converging to the cohomology of the Milnor fiber starting with an embedded Q-resolution. The main idea behind this construction is that in the classical case after considering the semistable reduction the ambient space already contains quotient singularities. We prove that the same is true for embedded Qresolutions and thus the construction by Steenbrink with the spectral sequence can be adapted to provide a MHS on the cohomology Hq(F, C). Since the embedded Q-resolution can be chosen so that every exceptional divisor contributes to the complex monodromy, our spectral sequence is better in the sense that less divisors will appear in the semistable reduction and thus the combinatorial complexity of the spectral sequence will be simpler, cf. [Mar11b]. Section §V.1 The Semistable Reduction This tool was introduced by Mumford in [KKMS73, pp. 53-108] and roughly speaking the mission of the semistable reduction is to get a reduced divisor that provides a model of the Milnor fibration. The spectral sequence converging to the cohomology of the Milnor fiber will be defined in terms of this reduced divisor, see Section V.3. Here we present a more general approach than the needed for the Milnor fibration. 98 Chapter V. MHS on the Cohomology of the Milnor Fiber (V.1.1). Let Xbe a complex analytic variety and let g:X→D2 ηbe a nonconstant analytic function. Assume Xonly has abelian quotient singularities and g−1(0) is a Q-normal crossing divisor, that is, gis locally given by a function of the form xm0 0· · · xmk k:X(d;A)→C. Let ebe the least common multiple of all possible multiplicities appearing in the divisor g−1(0) and consider σ:D2 η1/e →D2 ηthe branched covering defined by σ(t) = te. Denote by (X1, g1, σ1) the pull-back of gand σ. X1 g1// σ1  D2 η1/e σ  Xg//D2 η The map σ1is a cyclic covering of esheets ramified over g−1(0). If Fdenotes the Milnor fiber of g:X→C, then σ−1 1(F) has econnected components which are projected diffeomorphically onto F. We have not yet completed the construction of the semistable reduction because X1is not normal. Indeed, given P∈g−1(0) there exist integers k≥0 and m0, . . . , mk≥1 such that g(x0, . . . , xn) = xm0 0· · · xmk k:B2n+2/µd−→ C, where B2n+2 is an open ball of Cn+1 and the group µdacts diagonally as in (d;A). Denote by P1the unique point in σ−1 1(P). Then, X1in a neighborhood of P1is of the form (24) n(x0, . . . , xn), t∈X(d;A)×Cte=xm0 0· · · xmk ko, and hence the space X1is not necessarily normal. Let ν:e X→X1be the normalization and denote by eg:= g1◦νand %:= σ1◦νthe natural maps. The normalization process has essentially two steps when the corresponding ring is a unique factorization domain (UFD). First, separate the irreducible components, and then find the normalization of each component. In the latter case, the ring in question is a domain and the following result applies. Lemma (V.1.2). Let A⊂Bbe an integral extension of commutative rings. Suppose that Bis an integrally closed domain such that Q(B)|Q(A)is a Galois extension. Then, the normalization of the ring Ais A=BGal(Q(B)|Q(A)). Proof. Since Bis normal and the extension A⊂Bis integral, then A=B∩Q(A). Now the statement follows from the Galois condition.  §V.1. The Semistable Reduction 99 Example (V.1.3). The ring of functions of X(2; 1,1) is isomorphic to C[x2, xy, y2] as an algebraic variety. In this ring the polynomial xy is irreducible but not prime. To compute the normalization of the quotient ring C[x2, xy, y2]/hxyi, one can not proceed in the same way as in a UFD. This happens because µ2does not define an action on the factors of the polynomial xy. Although the ring of functions of the previous space (24) is not a UFD, see Example (V.1.3) above, to compute the normalization of X1one can proceed in the same spirit because of the special form of the polynomial te−xm0 0· · · xmk k, see proof of Proposition (V.1.7). Before that we need to introduce some notations. Definition (V.1.4). Let Xbe a complex analytic space having only abelian quotient singularities and consider EaQ-normal crossing divisor on X. Assume P∈ |E|is a point such that the local equation of Eat Pis given by the function xm0 0· · · xmk k:X(d;A) := Cn+1/µd−→ C,(0 ≤k≤n) where x0, . . . , xnare local coordinates of Xat P,d= (d0, . . . , dr), and A= (aij)i,j ∈Mat((r+ 1) ×(n+ 1),Z). The multiplicity of Eat P, denoted by m(E, P ) or simply m(P) if the divisor is clear from de context, is defined by m(E, P) := gcd m0, . . . , mk,Pk j=0 a0jmj d0 ,...,Pk j=0 arjmj dr. If there exists T⊂ |E|such that the function P∈T7→ m(E, P) is constant, then we use the notation m(T) := m(E, P0), where P0is an arbitrary point in T. Remark (V.1.5). Using the general fact lcm(m b0,...,m br) = m gcd(b0,...,br), one easily checks that this definition coincides with the one of (IV.3.12) for k= 0, cf. (25), that is, m(E, P) := m L, L = lcm d0 gcd(d0, a00),..., dr gcd(dr, ar0), where Eis a Q-divisor on Xlocally given at the point Pby the well-defined function xm 0:X(d;A)→C. In the situation of (V.1.1), the multiplicity m(g∗(0), P) with P∈g−1(0) can be interpreted geometrically as follows. 100 Chapter V. MHS on the Cohomology of the Milnor Fiber Lemma (V.1.6). The number of prime (or irreducible) factors of the polynomial te−xm0 0· · · xmk kregarded as an element in C[x0, . . . , xn]µd⊗CC[t] is m(g∗(0), P). Hence this number also coincides with the cardinality of the fiber over Pof the covering %:e X→X. Proof. Let us denote `= gcd(m0, . . . , mk) and Ci=Pk j=0 aijmjfor i= 0, . . . , r. The polynomial te−xm0 0· · · xmk k∈C[x0, . . . , xn, t] factorizes into `different components as te−xm0 0· · · xmk k= `−1 Y i=0 te `−ζi `x m0 ` 0· · · x mk ` k, where ζ`is a primitive `-th root of unity. However, this factors are not invariant under the group µd, since they are mapped to te `−ζi `x m0 ` 0· · · x mk ` k7−→ te `−ξ C0 ` d0· · · ξ Cr ` dr·ζi `x m0 ` 0· · · x mk ` k, by the action of (ξd0, . . . , ξdr)∈µd. Recall that Cn+1/µd=X(d;A). Let Hibe the cyclic group defined by Hi:= {ξCi/` di|ξdi∈µdi}, for i= 0, . . . , r, and consider H=H0· · · Hr. Since te−xm0 0· · · xmk kdefines a function over X(d;A)×C, then dimust divide Ciand, consequently, all the previous groups are (normal) subgroups of µ`. The order of µ`/H is exactly the number of prime (or irreducible) components of the preceding polynomial regarded as an element in C[x0, . . . , xn]µd⊗CC[t]. The order of Hiis |Hi|=di gcd(di, Ci/`)=` gcd(`, Ci/di). Then, one has |H|=|H0· · · Hr|= lcm |H0|,...,|Hr| =` gcd `, C0 d0,...,Cr dr=` m(P). (25) In the expression above, a general property about greatest common divisor and least common multiple already mentioned in (V.1.5) was used.  Assume that g−1(0) = E0∪ · · · ∪ Esand let us denote Di=%−1(Ei) for i= 0, . . . , s and D=Ss i=0 Di. This commutative diagram illustrates the whole process of the semistable reduction. (26) Di// % e Xν// %  eg '' X1 g1// σ1  D2 η1/e σ  Ei//X X g//D2 η §V.1. The Semistable Reduction 101 Consider the stratification of Xassociated with the normal crossing divisor g−1(0) ⊂X. That is, given a possibly empty set I⊆ {0,1, . . . , s}, consider E◦ I:= ∩i∈IEi\∪i/∈IEi. Also, let X=Fj∈JQjbe a finite stratification of Xgiven by its quotient singularities so that the local equation of gat P∈E◦ I∩Qjis of the form xm1 1· · · xmk k:B/G −→ C, where Bis an open ball around P, and Gis an abelian group acting diagonally as in (d;A). The multiplicities mi’s and the action Gare the same along each stratum E◦ I∩Qj, i.e. they do not depend on the chosen point P∈E◦ I∩Qj. Denote mI,j := m(E◦ I∩Qj). Finally, assume that E◦ I∩Qjis connected. Proposition (V.1.7). The variety e Xonly has abelian quotient singularities located at eg−1(0) = Dwhich is a reduced divisor with normal crossings on e X. Also, %:e X→Xis a cyclic branched covering of esheets unramified over X\g−1(0). Moreover, for ∅ 6=I⊆S:= {0,1, . . . , s}and j∈J, the following properties hold. (1) The restriction %|:%−1(E◦ I∩Qj)→E◦ I∩Qjis a cyclic branched covering of mI,j sheets unramified over E◦ I∩Qj. (2) The variety %−1(E◦ I∩Qj)is a V-manifold with abelian quotient singularities with gcd({m(P)|P∈E◦ I∩Qj})connected components. (3) Let ϕ:e X→e Xbe the canonical generator of the monodromy of the covering %. Then, its restriction to %−1(E◦ I∩Qj)is a generator of the monodromy of %|:%−1(E◦ I∩Qj)→E◦ I∩Qj. (4) The Euler characteristic of each connected component of Diis X {i}⊂I⊂{0,1,...,s} j∈J mI,j ·χ(E◦ I∩Qj)gcd({m(P)|P∈Ei}). Proof. First note that the morphism %:e X→Xis a cyclic branched covering unramified over X\g−1(0), since so is σ1:X1→Xand the normalization ν:e X→X1does not change the normal points. Let P∈g−1(0) and choose coordinates x0, . . . , xnas in (V.1.1) so that X1⊂X(d;A)×Cis locally given by the polynomial te−xm0 0· · · xmk k. Let us denote for i= 0, . . . , k, m(P) = m(g∗(0), P), e0=e/m(P), m0 i=mi/m(P). 102 Chapter V. MHS on the Cohomology of the Milnor Fiber Consider the ring A=C[x0, . . . , xn, t] hte−xm0 0· · · xmk ki. The action given by X(d;A) is extended to Aso that the variable tis invariant. Then, by Lemma (V.1.6), the normalization Aµdof the ring Aµd is isomorphic to the direct sum of m(P) isomorphic copies of the normalization of C[x0, . . . , xn]µd⊗CC[t] te0−xm0 0 0· · · xm0 k k= C[x0, . . . , xn, t] te0−xm0 0 0· · · xm0 k k!µd . Therefore to compute it we only need to consider the case m(P) = 1, for which the ring Aµdis an integral domain. Now we plan to apply Lemma (V.1.2) to a ring extension Aµd⊂B, where Bis a polynomial algebra. Let ci=e/mifor i= 0, . . . , k. Denote B=C[y0, . . . , yn] and consider Aµdas subring of Bby putting        xi=yci iif 0 ≤i≤k, xi=yifor i > k, t=y0· · · yk Note that Acan not be embedded in Bbecause it is not even a domain. Since µdacts diagonally on Cn+2, there exists N0 such that yc0N 0, . . . , yckN k, yN k+1, . . . , yN n∈Aµd. This implies that the extension Aµd⊂Bis integral. Also, Bis a normal domain. It remains to prove that Q(B)|Q(Aµd) is a Galois field extension. One has C(yc0N 0, . . . , yckN k, yN k+1, . . . , yN n)⊂Q(Aµd)⊂Q(B) = C(y0, . . . , yn). Note that the largest extension is clearly Galois. Its Galois group is abelian and it is isomorphic to µc0N× · · · × µckN×µN×n−k . . . ×µN. Thus Aµd=BGal(Q(B)|Q(Aµd)). This shows that Spec(Aµd) and hence e Xare V-manifolds. Locally D is the quotient under the group Gal(Q(B)|Q(Aµd)) of the reduced divisor y0· · · yk= 0. The rest of the statement follows from the fact that the branched coverings involved are cyclic. For the last part, use the classical Riemann-Hurwitz formula.  §V.3. The Spectral Sequence by Steenbrink 109 Let us introduce some notation. •Let I= (i0, . . . , ik) with 0 ≤i0<· · · < ik≤s. DI=Di0,...,ik:= Di0∩ · · · ∩ Dik, ˇ DI=ˇ Di0,...,ik:= DI\[ j6=i0,...,ik (Dj∩DI). The first one is a projective V-manifold of dimension1n−k. The second one is a smooth complex variety of the same dimension. •Let 0 ≤i0<· · · < ik≤sand ij< i0 j< ij+1 with −1≤j≤k. Denote by κi0 j i0,...,ij,ij+1,...,ik:Di0,...,ij,i0 j,ij+1,...,ik,−→ Di0,...,ij,ij+1,...,ik, the natural inclusion. •Let D[k]:= G 0≤i0<···<ik≤s Di0,...,ik. •Let D[k] +:= G 1≤i0<···<ik≤s Di0,...,ik. Definition (V.3.7). Let k∈Zwith 0 ≤k≤nand let i, j ∈Zwith i, j ≥0. Define kEi,k−j 1as kEi,k−j 1:=    Hi(D[k] +,Q) if j= 0, Hi−2j(D[k+j],Q) if j > 0. Note that for j= 0 the divisor D+is used, while for j > 0 it is taken the divisor D. All the spaces whose cohomology is considered are compact. (V.3.8). These spaces give rise to the first term E1of our spectral sequence E={Ep,q n}: Ep,q 1:= n M k=0 kEp,q 1, where kEp,q 1= 0 if it is not defined previously. Note that the space pEi,k−j 1possesses a natural pure Hodge structure of weight i−2j, since it is defined as the cohomology of degree i−2jof a compact K¨ahler V-manifold. Performing an index shifting e Hp+j,q+j:= Hp,q, pEi,k−j 1also has a pure Hodge structure of weight i, cf. Theorem (V.3.4). 1Recall the convection on indices, e.g. for plane curve n= 1. 110 Chapter V. MHS on the Cohomology of the Milnor Fiber It still remains to define the differentials. In the first term E1the differentials are of type (0,1), i.e. upward vertical arrows. (V.3.9). Let us resume the notation of (V.3.6). Let κi0 j i0,...,ik∗:H∗Di0,...,ij,i0 j,ij+1,...,ik,Q−→ H∗Di0,...,ij,ij+1,...,ik,Q be the homomorphism induced by the inclusion on the homology groups. Using Poincar´e duality for compact V-manifolds, one has the following Gysintype maps: H∗Di0,...,ij,i0 j,ij+1,...,ik,Q# κi0 j i0,...,ik∗// DP ∼ =  H∗Di0,...,ij,ij+1,...,ik,Q DP ∼ =  H2(n−k−1)−∗Di0,...,ij,i0 j,ij+1,...,ik,Q//___ H2(n−k)−∗Di0,...,ij,ij+1,...,ik,Q These arrows are only possible if the spaces are compact. It is always the case except for k= 0 and i0= 0, where the corresponding map is defined as zero. By abuse of notation, the morphism associated with the dashed arrow that completes the previous diagram is again denoted by (κi0 j i0,...,ik)∗. Definition (V.3.10). The differentials on kE1,kδ:kEi,k−j−1 1→kEi,k−j 1 are defined by kδ|Hi−2(j+1)(Di0,...,ik+j+1 ,Q):= k+j+1 X l=0 (−1)lκil i0,...,b il,...,ik+j+1 ∗. Remark (V.3.11). The pair (kE1,kδ) is the term E1of the spectral sequence that provides the MHS of G 0≤i0<···<ik≤s ˇ Di0,...,ik, which is the complement of a divisor with normal crossings on a projective variety. To finish with the description of the differentials, the interactions between different kE1have to be taken into account. These differentials are of Mayer-Vi´etoris type. Denote by κil i0,...,ik+j∗ the corresponding homomorphism on the cohomology groups. §V.3. The Spectral Sequence by Steenbrink 111 Definition (V.3.12). The morphisms k,k+1δ:kEi,k−j 1→k+1Ei,k−k+1 1are defined as k,k+1δ|Hi−2j(Di0,...,ik+j,Q):= X `6=i0,...,ik+j (−1)e(l;i0,...,ik+j)κil i0,...,ik+j∗, where e(l;i0, . . . , ik+j) is the number of coefficients i0, . . . , ik+jless than l. Remark (V.3.13). The pair (kEi,k 1,k,k+1δ) is exactly the term E1of the spectral sequence providing the MHS of the divisor with normal crossings D+ which appears in [Del71b]. Observe that the first two columns of this spectral sequence for k= 0 coincides with the first two columns of the term E1of {Ep,q n}. Definition (V.3.14). The direct sum of the differentials kδand k,k+1δis the differential δof the term E1. n= 1 n= 2 1E1 0E1 2E1 1E1 0E1 (0,0) (1,0) (0,2) (0,1) (0,0) (1,0) (2,−1) (2,1) Figure V.2. Decomposition of E={Ep,q n}for n= 1,2. This section ends with the explicit description of the spectral sequence {Ep,q n} ⊗QCfor the cases n= 1,2. For n= 1, let us denote with a triangle the terms belonging to 1E1and with a circle the ones belonging to 0E1. NH0(D[1] +,C)(k= 1) •H0(D[0] +,C) 0,1δ OO •H1(D[0] +,C)•H2(D[0] +,C) (k= 0) •H0(D[1],C) 0δ OO Figure V.3. Steenbrink’s spectral sequence for plane curves, i.e. n= 1, with its decomposition E1=0E1⊕1E1. 112 Chapter V. MHS on the Cohomology of the Milnor Fiber For surfaces, that is n= 2, denote with a square the terms belonging to 2E1, with a triangle the ones belonging to 1E1, and finally with a circle those coming from 0E1. H0(D[2] +)(k= 2) NH0(D[1] +) 1,2δ OO NH1(D[1] +)NH2(D[1] +)(k= 1) •H0(D[0] +) 0,1δ OO •H1(D[0] +) 0,1δ OO NH0(D[2]) ⊕ •H2(D[0] +) 1δ⊕0,1δ OO •H3(D[0] +)•H4(D[0] +) (k= 0) •H0(D[1]) 0δ OO •H1(D[1]) 0δ OO •H2(D[1]) 0δ OO •H0(D[2]) 0δ OO Figure V.4. Steenbrink’s spectral sequence for surfaces, i.e. n= 2, with its decomposition E1=0E1⊕1E1⊕2E1. Section §V.4 Example of a Plane Curve 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}and C2={xr+ys= 0}. In Example (I.3.15), an embedded Q-resolution of {f= 0} ⊂ C2is computed, see Figure V.5 below. 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  P1 P2 P3 R1 R2 R3 Figure V.5. Embedded Q-resolution of f= (xp+yq)(xr+ys). §V.4. Example of a Plane Curve 113 Let us calculate here the MHS associated with the Milnor fiber of fand its complex monodromy. Before that, the notion of (weighted) dual graph in this setting is introduced. Usually one encodes a normal crossing divisor with its dual complex: one vertex for each irreducible component, one edge for the intersection of two irreducible components, one triangle for the intersection of three irreducible components, etc. This is particularly useful for normal crossing divisors on surfaces where the dual complex is converted into a (weighted) graph. (V.4.1). Let us explain in detail how to encode a Q-divisor with Q-normal crossings on a V-manifold using its weighted dual graph. We are interested in the following cases: the divisor π∗(C)=(f◦π)∗(0) ⊂b Xin an embedded Q-resolution πof a plane curve C=f∗(0), and also in its corresponding semistable reduction. Their weighted dual graph Γ is defined as follows: •The set VΓof vertices of Γ is the ordered set of irreducible components of π∗(C) (for some arbitrary order). It is decomposed in two subsets VΓ=V0 Γ`VC Γ; the first one corresponds to the exceptional components and the second one to the strict transforms, using arrow-ends. •The set EΓof edges of Γ is in bijection with the double points of π∗(C). •Each E∈V0 Γis weighted by its genus gE(omitted if gE= 0). It is also weighted by its self-intersection number eE∈Q, see Definition (III.1.2). •Each E∈VΓis weighted by mEdefined as follows: given a generic point in E, one can choose local analytic coordinates (xE, yE) centered at this point such that yE= 0 is a local equation of Eand (f◦π)(xE, yE) = ymE E. •For E∈VΓ, let Sing0(E) be the set of singular points of b Xin E which are not double points. Then, together with E, the sequence of normalized types {(dP;aP, bP)}P∈Sing0(E), where Eis the image of y= 0, is given. Note that dPdivides mE. •If the double point Pγ=E1∩E2,E1< E2, associated with γ∈EΓ is singular, we provide a normalized type (d;a, b), where E1is the image of x= 0 and E2is the image of y= 0. Note that ddivides amE1+bmE2. Remark (V.4.2). The weighted dual graph can also be considered associated with an abstract good Q-resolution. This is especially useful for describing aQ-resolution via Jung method, see [AMO11b] for details. 114 Chapter V. MHS on the Cohomology of the Milnor Fiber Example (V.4.3). The embedded Q-resolution of the preceding example, see Figure V.5 above, is computed with the (q, p)-blow-up at the origin of C2 followed by the (s, qr−ps)-blow-up at a point of type (q;−1, p). Its weighted dual graph is shown in Figure V.6. e1=−r p(rq−ps) (p;q, −1) E1E2 m1=p(s+q)m2=s(p+r) e2=−q s(rq−ps) (s;r, −1) (rq −ps;ar +bs, −1) C2 C1 Figure V.6. Dual graph of the embedded Q-resolution of {(xp+yq)(xr+ys)=0} ⊂ C2, where ap +bq = 1. The self-intersection numbers are calculated using (III.3.2). The point Q is also of type (rq−ps;ar+bs, −1) where ap+bq = 1. In fact, it is normalized since gcd(rq −ps, ar +bs) = 1. Now is the time to study the semistable reduction of the embedded Qresolution obtained in Example (V.4.3). Denote by P1the point in E1of type (p;q, −1), P2the intersection of C1with E1, and P3a generic point in E1. Analogously, denote by R1the point in E2of type (s;−1, r), R2the intersection of C2with E2, and R3a generic point in E2, cf. Figure V.5. Let E=C1∪C2∪ E1∪ E2⊂Xbe the total transform of the plane curve {f= 0} ⊂ C2for the embedded Q-resolution π:X→C2. Also, write g:= f◦πand use the notation in (V.1.1) and (26) so that E=g∗(0). Following Definition (V.1.4), the numbers m(E, P), where P∈Eis one of the previous points, are calculated below: m(E, P1) = q+s, m(E, R1) = p+r, m(E, P2)=1, m(E, R2)=1, m(E, P3) = p(q+s), m(E, R3) = s(p+r). On the other hand, by Lemma (V.1.6), the cardinality of the fiber over Q∈ E1∩ E2of the covering %:e X→X(i.e. the semistable reduction) is m(E, Q) = gcd p(q+s), s(p+r), A, B, where A=p(q+s)·s+s(p+r)·(−q) rq −ps =−s, B=p(q+s)·(−r) + s(p+r)·p rq −ps =−p. Consequently, m(E, Q) = gcd(p, s). §V.4. Example of a Plane Curve 115 From Proposition (V.1.7), one deduces the following statements. The divisors D1:= %−1(E1) and D2:= %−1(E2) have just one connected component. Their Euler characteristics are χ(D1) = q+s+ gcd(p, s)+1−p(q+s), χ(D2) = p+r+ gcd(p, s)+1−s(p+r). The preimage of the strict transforms, %−1(C1) and %−1(C2), are isomorphic to C1and C2respectively, and thus denoted again by the same letter. g1=(p−1)(q+s)−gcd(p, s)+1 2 D1D2 m2= 1 g2=(s−1)(p+r)−gcd(p, s)+1 2 C2 C1 m1= 1 . . . Q0 gcd(p, s) Figure V.7. Dual graph of the semistable reduction of f. Since the singularity defined by fis isolated, the generalized Steenbrink’s spectral sequence gives rise the exact sequences 0−→ Ker 0,1δ | {z } C −→ H0(D[0] +)0,1δ −−→ H0(D[1] +)−→ Coker 0,1δ | {z } GrW 0H1(F,C) −→ 0, and 0−→ Ker 0δ | {z } GrW 2H1(F,C) −→ H0(D[1])0δ −→ H2(D[0] +)−→ Coker 0δ | {z } 0 −→ 0. Moreover, GrW 1H1(F, C) = H1(D[0] +). The divisor D[0] +is the disjoint union of D1and D2, and D[1] (resp. D[1] +) consists of gcd(p, s) + 2 (resp. gcd(p, s)) points. Hence, H0(D[0] +)=2 H0(D[1] +) = gcd(p, s))=⇒H0,0= GrW 0H1(F, C) = Cgcd(p,s)−1. Analogously, H0(D[1]) = gcd(p, s) + 2 and H2(D[0] +) = 2, which implies that H1,1= GrW 2H1(F, C) = Cgcd(p,s). As for the (pure) Hodge structure of weight 1 associated with the cohomology H1(D[0] +) = H0,1⊕H1,0, it is known to be determined by the genus of the corresponding real surface. In this case, H0,1=Cg1⊕Cg2 and H1,0=Cg1⊕Cg2. 116 Chapter V. MHS on the Cohomology of the Milnor Fiber Remark (V.4.4). It must be satisfied that Pp,q dimCHp,q =µ. In fact, the Milnor number is the degree of the characteristic polynomial, which is by Theorem (IV.3.14) equal to ∆(t) = t−1tp(q+s)−1ts(p+r)−1 tq+s−1tp+r−1. Summarizing, the mixed Hodge structure of the cohomology of the Milnor fiber H1(F, C) obtained is H1(F, C) = H0,0 |{z} GrW 0H1(F,C) ⊕H0,1⊕H1,0 | {z } GrW 1H1(F,C) ⊕H1,1 |{z} GrW 2H1(F,C) , where H0,0=Cgcd(p,s)−1, H0,1=Cg1⊕Cg2, H1,0=Cg1⊕Cg2=H0,1, H1,1=Cgcd(p,s). The genera g1and g2are calculated in Figure V.7. The action of the monodromy on GrW 0H1(F, C) is given by the polynomial tgcd(p,s)−1 t−1. Note that this provides the eigenvalues of the monodromy with Jordan blocks of size 2. VI An Embedded Q-Resolution for Superisolated Singularities Let (V, 0) ⊂(C3,0) be a germ of surface singularity in C3. By definition, Vis the zero set of a holomorphic function f:U→C, where U⊂C3is a small neighborhood of the origin and f(0) = 0. Denote also by fthe germ at the origin of this function; it is an element of the local ring C{x, y, z}. Consider the decomposition of finto homogeneous parts, f(x, y, z) = fm(x, y, z) + fm+1(x, y, z) + · · · , where fiis homogeneous of degree iand fm6= 0. The integer mis the multiplicity of the singularity and the order of the series f. Denote by C:= V(fm)⊂P2the projective plane curve defined by the tangent cone of the singularity. The following families are considered in this work: (1) Superisolated singularity (or, shortly, SIS): the local equation f satisfies P2⊃Sing(C)∩V(fm+1) = ∅. (2) Yomdin-Lˆe singularity (YS): the decomposition of finto homogeneous polynomials is of the form f=fm+fm+k+· · · and the condition Sing(C)∩V(fm+k) = ∅holds. (3) Weighted Yomdin-Lˆe singularity (WYS): let ω:= (a, b, c)∈N3 be three positive numbers such that gcd(a, b, c) = 1. The sum f= fm+fm+k+· · · is the decomposition of finto (a, b, c)-homogeneous parts and Sing(C)∩V(fm+k) = ∅in P2 ω. Remark (VI.0.5). Recall that when fiis a quasi-homogeneous polynomial with respect to ω, it defines a curve in the weighted projective plane P2 ω. The notion of singular point in this setting is given in Chapter III. Now, the third definition above makes sense. 118 Chapter VI. An Embedded Q-Resolution for SIS These singularities have been studied by many authors. We are content to cite merely the survey [ALM06], where part of the theory of these singularities and their applications including some new and recent developments are reviewed. Although these three families can be studied simultaneously, for better exposition they are presented and treated separately. In this chapter, a detailed description of an embedded Q-resolution of superisolated surface singularities in terms of an embedded Q-resolution of its tangent cone is given. In particular, it is proven that only weighted blow-ups at points are needed. Also, we shall see that an exceptional divisor in the resolution of (V, 0) contributes to the complex monodromy if and only if so does the corresponding divisor in the tangent cone. Thus the weights can be chosen so that every exceptional divisor in the Q-resolution of (V, 0) contributes to the monodromy. The generalized A’Campo’s formula applies and the characteristic polynomial and the Milnor number are calculated as an application. Other more sophisticated invariants, including mixed Hodge structure of the cohomology of the Milnor fiber, are the subject of our study for the future. As we will see, the previous chapters are essential for describing the embedded Q-resolution. More precisely, the following sections and results will be very useful: §I.3–1, §I.3–2, (III.2.1), (III.3.2), (III.4.3). Section §VI.1 Preparations for the Q-Resolution These singularities have been introduced by Luengo and also appear in a paper by Stevens, where the µ-constant stratum is studied, see [Lue87] and [Ste89] respectively. Afterward Artal described in his PhD thesis [Art94b] an embedded resolution of such singularities using blow-ups at points and rational curves. Here an embedded Q-resolution is given and particularly it is proven that only weighted blow-ups at points are needed. By contrast, the final ambient space obtained has abelian quotient singularities. (VI.1.1). Let (V, 0) be a SIS in (C3,0) defined by a holomorphic function f:U→C. As above, denote by mthe multiplicity of V, and Cthe tangent cone. Let π0:b U→Ube the blow-up at the origin. Recall that the total transform is the divisor π∗ 0(V) = b V+mE0, where b Vis the strict transform of V, and E0is the exceptional divisor of π0. The intersection b V∩E0is identified with the tangent cone of the singularity.