Albanese varieties of cyclic covers of the projective plane and orbifold pencils
aa r X i v : . [ m a t h . AG ] F e b ALBANESE VARIETIES OF CYCLIC COVERS OF THE PROJECTIVEPLANE AND ORBIFOLD PENCILS
E. ARTAL BARTOLO, J.I. COGOLLUDO-AGUST´IN, AND A. LIBGOBER
Abstract.
The paper studies a relation between fundamental group of the complement toa plane singular curve and the orbifold pencils containing it. The main tool is the use ofAlbanese varieties of cyclic covers ramified along such curves. Our results give sufficientconditions for a plane singular curve to belong to an orbifold pencil, i.e. a pencil of planecurves with multiple fibers inducing a map onto an orbifold curve whose orbifold fundamentalgroup is non trivial. We construct an example of a cyclic cover of the projective plane whichis an abelian surface isomorphic to the Jacobian of a curve of genus 2 illustrating the extentto which these conditions are necessary.
Introduction
There is an interesting correspondence between the fundamental groups of the complementto plane algebraic curves and the structure of the pencils, possibly with multiple fibers whichone can associate with such curves. For example, if a plane curve C is composed of a pencil, i.e. C = S si =0 C i where C i are zeros of sections t i in a 2-dimensional subspace L of H ( P , O ( d )),then for each P ∈ X C := P \ C there is a well defined element t P ∈ P ( L ) such that t P ( P ) = 0and the correspondence P → t P gives a holomorphic map X C → P ( L ) \ { T i } si =0 , where T i are the points of P ( L ) corresponding to the sections t i . This map induces a surjection π ( X C ) → π ( P ( L ) \ { T i } si =0 ) and hence π ( X C ) has a free group on s generators as itsquotient.In a similar vein, the existence of pencils with multiple fibers containing C (see section 1.3)may have implications for the fundamental group even if C is irreducible . For example,suppose that an irreducible curve C ⊂ P belongs in a pencil having two multiple fibers ofmultiplicities 2 and 3, i.e., the equation F of C can be presented as F = f + g where f, g arehomogeneous polynomials. Then the rational map π : P P given by π ([ x : y : z ]) = [ f : g ] induces a regular map of X C := P \ C onto P \ { (1 , − } . This map can also be viewedas an orbifold map whose source is X C with a trivial orbifold structure and whose target isthe orbifold C , which is an affine line with two orbifold points with stabilizers of orders 2and 3. Such a dominant map yields a surjection of the fundamental group π ( X ) onto theorbifold fundamental group (cf. [4], [12, Prop.2.7]) for which one has π orb1 ( C , ) = Z ∗ Z The first two authors are partially supported. The first and second authors are partially supported bythe Spanish Government MTM2016-76868-C2-2-P and by the Departamento de Industria e Innovacin delGobierno de Aragn and Fondo Social Europeo
E15 Grupo Consolidado Geometr´ıa . The third author is alsosupported by a grant from the Simons Foundation. (isomorphic to PSL ( Z )). In the rest of the paper we call a map between orbifolds having aone-dimensional target an orbifold pencil . The classically studied pencils (whether rationalor irrational) are a special case of orbifold pencils.Previous work [12, 5, 6] has shown that sometimes the relation between the fundamentalgroup of a curve complement X C and its orbifold pencils can be reversed, namely, the structureof the fundamental group provides information on the existence of (rational) orbifold pencilson X C but the relation between fundamental groups and orbifold pencils has several aspectsnot appearing in the context of ordinary pencils. If a curve has only nodes and ordinarycusps as its singularities (or more generally, singularities called in [12] δ -essential) then thepositivity of the rank of the abelianization of the commutator π ( X C ) ′ /π ( X C ) ′′ implies theexistence of orbifold maps on X C (see Section 1 for more precise statements).In the present paper we consider the correspondence between orbifold pencils and funda-mental groups of possibly reducible curves C which may have singularities much more generalthan ordinary cusps and nodes. Our main result (see Theorem 4) describes a sufficient con-dition for the existence of orbifold pencils on P containing C in terms of the fundamentalgroup π ( X C ) of its complement. Let us describe the results of the paper in more detail.As in the case of curves with nodes and cusps only, it is convenient to state our resultsin terms of the Alexander invariants and the characters of the fundamental group. Thestatements also use the local Albanese varieties of singularities (cf. Section 1). Recall (seemore details in Section 1.1) that there is a notion of Alexander polynomial ∆ C,π ∈ Z [ t, t − ]associated with a given surjection π : π ( X C ) → Γ onto a cyclic group. Such a polynomialdepends only on the quotient of π ( X C ) by the commutator of Ker π and it contains informa-tion about the cohomology of rank one local systems on X C , namely, for χ ∈ Hom(Γ , C ∗ ) onehas H ( X C , χ ) = 0 if and only if, for a generator γ of Γ, ξ = χ ( γ ) is a root of ∆ C,π . A root ξ of the Alexander polynomial ∆ C,π can also be described as an eigenvalue of the coveringtransformation τ C acting on H ( V C , C ) where V C is a smooth model of the cyclic cover of P ofdegree deg C branched over C (cf. [17]). Note that since H ( V C , C ) is a birational invariant,the eigenvalues of τ C are independent of a choice of the smooth model V C . An alternativedescription of the multiplicity of the root ξ can be given as the superabundance of the linearsystem of plane curves described in terms of the degree and the local type of the singularitiesof C . We refer to [18] for details.The Alexander polynomial is affected by the local types of the singularities of C as wasshown in [17]. For the statement of our main results we will need a more precise than statedin [17] version of this relation and it will be shown below in Section 2. Theorem 1.
Let C be a plane curve with arbitrary singularities and let χ be a character offinite order N > of the fundamental group π ( X C ) . Assume that χ is ramified along eachirreducible component of C . Assume also that H ( X C , χ ) = 0 . Then there exists a singularity P ∈ C with local equation f P ( x, y ) = 0 for which the following property holds.Denote by B P a Milnor ball about P and let χ P be the character of π ( B P \ C ) which is thecomposition π ( B P \ C ) → π ( X C ) χ → C ∗ LBANESE VARIETIES AND ORBIFOLD PENCILS 3 where the left map is induced by the inclusion B P \ C ֒ → X C . Then the corresponding map: (T1) H ( X C , χ ) → H ( B P \ C, χ P ) has a non-trivial image (in particular H ( B P \ C, χ P ) = 0 ). The orbifold pencils on P which we attach to the curve C are obtained from irrationalpencils on V C and are constructed using the Albanese map V C → Alb( V C ). Albanese varietiesof cyclic covers of P were considered classically for covers of small degree (cf. [10, 7, 9] fora modern exposition). The work of Comessatti [10, 9] studies the irregular 3-cyclic coveringsof the plane, and he finds examples both for Albanese dimensions 1 and 2. In the latter case,he constructs an example (also found in [7] and thoroughly explained in [9]) such that thecyclic cover is the product of two copies of a special elliptic curve. Bagnera and deFranchistake another viewpoint: they study rational cyclic quotients of abelian surfaces. However, aspresented in Theorem 1, our focus is on C and its algebraic/topological properties such ascohomology conditions on its complement.Note that we obtain an explicit model of such quotients in Theorem 3.3; the ramificationcurve is described and we derived geometric properties of this curve from this fact. Ourconstruction depends on the relation between Alb( V C ) and the invariants of singularities of C described in [16]. There are also simple cases where such irrational orbifold pencils come upin a straightforward way. This is the case when Alb( V C ) is an elliptic curve, or analogously,for curves whose local Alexander polynomial equals t − t + 1. More generally we have thefollowing, Corollary 2 (cf. Theorem 4) . Let C , χ be as in Theorem and let V χC be a smooth projectivemodel of the cyclic cover associated with the kernel of χ . Assume that the Albanese dimensionof V χC is equal to one (see Theorem for explicit examples). Then C is an element of a globalquotient orbifold pencil such that χ is the pullback of a character of the orbifold fundamentalgroup of the target of this orbifold pencil. We want to relax the assumption on Albanese dimension in the Corollary 2 and assume onlythat one has a one-dimensional image in one of the isogeny χ -equivariant factor of Alb( V C ).In what follows, we will describe how, under some restriction on the analytic type of thesingularities of C , we may identify the abelian varieties which are the isogeny χ -equivariantfactors of Alb( V C ) projection onto which may lead to construction of an orbifold pencil.This restriction on the analytic type of singularities is given in terms of introduced in[12] the local Albanese varieties associated with a plane curve singularities (cf. section 1for definition.) A local Albanese variety is equipped with an automorphism i.e. a Z -actioncoming from the action of the semi-simple part of the local monodromy on the homology ofthe Milnor fiber. The relation between local Albanese varieties of singularities and globalinformation about C comes from canonical maps of each local Albanese variety into Alb( V C ).The sum of these maps over all singularities of C surjects onto Alb( V C ) (cf. [16]). These mapsfrom the local Albanese varieties of the singularities of C are Z -equivariant with respect tothe just mentioned monodromy action and the action of the (cyclic) covering group of V C . E. ARTAL, J.I. COGOLLUDO, AND A. LIBGOBER
Before stating the main result of this paper (Theorem 4) we shall state sufficient conditionsfor existence of orbifold pencil in the case when singularties of C have type A p − and for whichfewer technical assumptions can be made. Theorem 3.
Let C , χ , P , and V χC be as in Theorem and Corollary above. Assumethat C has at P an A p − -singularity, p an odd prime, and in particular, the local Albanesevariety Alb P is the Jacobian of the curve D of genus g := p − . Let alb χ,D : V χC → Jac( D ) be the composition of the Albanese map V χC → Alb( V χC ) with the projection on its isogenycomponent Jac( D ) .If the image of alb χ,D has dimension one, then there is a pencil V χC → D inducing anorbifold pencil (T3) X C D/ Im χ onto the global quotient of D by the canonical action of Im χ on D .Moreover the character χ is the pullback on π ( X C ) of a character of π orb1 ( D/ Im χ ) via thepencil (T3) . Now we are ready to state the main result of the paper with milder than in Theorem 3restriction on singularities of C but similar conclusion that global orbifold pencils exist. Theorem 4.
Let C , χ , N and P be as in Theorem . Let V χC be a smooth projective model ofthe cyclic branched cover of P associated with the kernel of χ and let τ χC be the map inducedby the deck transformation on H ( V χC , C ) . (1) Assume that the local Albanese variety
Alb P of the singularity P has an isogeny com-ponent J χ satisfying the following: (a) The action of Im χ on Alb P induces an action on J χ and the map J χ → Alb( V χC ) induced by the (Im χ ) -equivariant map Alb P → Alb( V χC ) has a finite kernel. (b) J χ is the Jacobian of a curve D such that D is a quotient of an exceptionalcurve D of positive genus in a resolution of the singularity z N = f P ( x, y ) i.e. D = D / ∆( D , χ ) where ∆( D , χ ) ⊆ Im χ is a (possibly trivial) subgroup of thecovering group Im χ , the latter being considered as an automorphism group of D .Let alb χ,D be the composition of the Albanese map V χC → Alb( V χC ) with the projectionon the factor J χ = Jac( D ) . If the dimension of the image of alb χ,D is one, then thereexists a pencil V χC → D inducing an orbifold pencil (T4) X C D orbIm χ where D orbIm χ = D/ (Im χ/ ∆( D, χ )) is the global quotient orbifold obtained via the in-duced action of (Im χ/ ∆( D, χ )) on D . For such an orbifold pencil (T4) the character χ is the pull-back on π ( X C ) of a character of π orb1 ( D orbIm χ ) via (T4) . (2) If Alb P is simple (i.e. is not isogenous to a product of abelian varieties of positivedimension) then the assumptions (a) and (b) in (1) are automatically satisfied. LBANESE VARIETIES AND ORBIFOLD PENCILS 5
Note that assumption (1)(a) means that J χ is an (Im χ )-equivariant isogeny component ofAlb( V χC ). In particular it implies that the tangent space to J χ at the identity is contained inthe χ -eigenspace of τ χC acting on the tangent space of Alb( V χC ) at the identity,The conditions for existence of orbifold pencils given by this theorem have the followingconverse showing that the existence of an orbifold pencil having the curve C as a member,implies that the Albanese variety of the corresponding cyclic cover splits up to isogeny. Somefactors of this splitting are the Jacobians of the curves with the orbifold associated with thepencils being the global quotients of these curves.More precisely (see section 1.3 for definitions related to orbifold pencils) one has: Theorem 5.
Suppose that C belongs to a global quotient orbifold pencil π (cf. Definition )of target P with orbifold points of multiplicities ¯ m = ( m , . . . , m s ) so that π induces a ho-momorphism π ( X C ) → π orb1 ( P m ) . Assume also that there is ρ ∈ Char π orb1 ( P m ) such that χ = π ∗ ( ρ ) and also that the orbifold P m is a global quotient of a curve Σ . Then Alb( V χC ) admits an (Im χ ) -equivariant surjection onto Jac(Σ) and hence one has an (Im χ ) -equivariantisogeny Alb( V χC ) ∼ Jac(Σ) × A for an abelian (Im χ ) -variety A .More generally, if there is a finite number φ , . . . , φ n of global quotient orbifold pencilsas above with targets ( P m , ρ ) ( ρ ∈ Char π orb1 ( P m ) which are Q -strongly independent , then Alb( V χC ) admits an (Im χ ) -surjection onto Jac(Σ) n , that is there is an equivariant isogeny, Alb( V χC ) ∼ Jac(Σ) n × A for an abelian (Im χ ) -variety A . The proofs of Theorems 1, 3, 4 and Theorem 5 are presented in Section 2. In Section 3 weconsider applications of Theorem 4. Firstly we discuss an example of a curve C with A g -singularities, i.e. whose singularities are locally isomorphic to u + v g +1 = 0, which belongsto an orbifold pencil. For the curves described in Theorem 3.1, all roots of the Alexanderpolynomial correspond to orbifold pencils on the complement. The Albanese variety of thecanonical cyclic cover V C is the Jacobian of a certain curve of genus g (described as a Belyicover). In Theorem 3.3 we give an example of a curve for which the Albanese variety is thesame as one of those in Theorem 3.1 (for the particular case of g = 2), but whose characterscorresponding to the roots of the Alexander polynomial cannot be obtained as pull-back viaorbifold maps. The difference between the curves in Theorems 3.1 and 3.3 comes from thedifference in the Albanese maps of the corresponding cyclic covers, namely, the images ofthe Albanese maps have different dimensions. The curve given in explicit way describedin Theorem 3.3 is particularly interesting, since its canonical cyclic cover has as a minimalmodel an abelian surface (specifically the Jacobian of a curve of genus 2 cf. also [9]). Thisconstruction of an abelian surface via cyclic coverings branched over curves given by explicitequation can be of independent interest. Finally, in Theorem 3.5 we present a family of curvescontained in more than one orbifold pencil and for which the Albanese dimension is maximal,that is, two. 1. Preliminaries
Alexander polynomials. (cf. [17])
E. ARTAL, J.I. COGOLLUDO, AND A. LIBGOBER
Let C be a plane curve with irreducible components C , C , . . . , C r where F i ( x, y, z ) = 0is a reduced equation of C i of degree d i . Then H ( X C , Z ) is an abelian group of rank r isomorphic to Z r +1 / ( d , . . . , d r ) Z . This isomorphism is given by γ (cid:18) π √− Z γ dF i F i (cid:19) ri =0 . Fix a surjection π : π ( X C ) → Γ onto a cyclic group Γ. Note that π can be factored through H ( X C , Z ) and hence induces a homomorphism Z r +1 / ( d , . . . , d r ) Z → Γ. Let K = ker π .Consider the exact sequence 0 → K/K ′ → π ( X C ) /K ′ → Γ → K/K ′ ⊗ C . The Alexander polynomial ∆ C,π ( t ) of C (relative to the surjection π ) isthe characteristic polynomial associated with the action of Γ on the vector space K/K ′ ⊗ C .Note that dim K/K ′ ⊗ C < ∞ (cf. [17]), ∆ C,π has integer coefficients and in the case ofirreducible C is independent, for all the previous choices, as an element in C [ t, t − ] modulounits. In the latter case, if Γ = Z /d Z , then K/K ′ is the abelianization of the commutatorof π ( X C ).Zeroes of the Alexander polynomial can be described in terms of the cohomology of localsystems as follows. Note that, since π ( X C ) /K ′ is abelian, π factors through a character, say χ . Let ξ ∈ C ∗ , (1 , ..., ∈ Z r +1 / ( d , . . . , d r ) Z = H ( X C , Z ) be a generator of Im χ ∈ C ∗ ; onehas:(1.1) ∆ C,π ( ξ ) = 0 ⇐⇒ dim H ( X C , χ ) > C as follows. Each singularity P ∈ C , has associated its local Alexander polynomial ∆ PC , orequivalently the characteristic polynomial of the local monodromy acting on the Milnor fiberof the singularity (cf. [21]). Then one has the divisibility relation (cf. [17])(1.2) ∆ C ( t ) | Π P ∆ PC ( t ) . Moreover the roots of the Alexander polynomial are roots of unity of the degree deg C . Example 1.1.
Let C be a curve whose singularities are topologically equivalent to the A g -singularity with local equation u = v g +1 . Since the characteristic polynomial of themonodromy for such singularity is t g +1 +1 t +1 the Alexander polynomial of C is trivial unless2(2 g + 1) | deg C and moreover it is equal to (cid:16) t g +1 +1 t +1 (cid:17) s for some s ≥ Local Albanese Varieties and singularities of CM -type. Let f = 0 be a germ of an isolated (i.e. reduced) plane curve singularity at the origin.Let M f be the Milnor fiber of f , i.e. the intersection of a sufficiently small ball B ǫ aboutthe origin and the hypersurface f = t, < | t | ≪ ǫ . The cohomology of M f (more generally,the cohomology of the Milnor fiber of an isolated hypersurface singularity) supports the limit mixed Hodge structure. It was constructed by Steenbrink and we refer to [24] for its study.Here we only note that it depends on the family of germs f = t , rather than its specificmember and record the following properties of this mixed Hodge structure used below: LBANESE VARIETIES AND ORBIFOLD PENCILS 7 (1) It has weight 2 and the weight filtration is associated with the unipotent part of themonodromy T u in the decomposition into unipotent and semisimple parts of T = T s T u ,the monodromy operator acting on H ( M f , C ).(2) The size of Jordan blocks of the monodromy operator is at most 2 and equals rk W .Moreover, rk Gr W − rk W = r − , where r is the number of branches of f = 0.(3) The Hodge filtration is invariant under the action of the semisimple part of the mon-odromy. Note that by the Monodromy Theorem the order of T s is finite (cf. [24]).(4) Let L f be the link of the singularity z n = f where n is the order of the automorphism T s . Then(1.3) Gr W H ( M f ) = Gr W H ( L f )(1)(where H (1) is the Tate twist of a Hodge structure H , cf. [16, Proposition 3.1]). Definition 1.2.
The local Albanese variety of the germ f is defined as the abelian va-riety (Gr F H ( M f )) ∗ /H (( M f , Z ), with polarization induced by the intersection form on H ( M f , Z ). Equivalently, the local Albanese is the abelian part of the semiabelian vari-ety associated by Deligne (cf. [13]) to the 1-motif in the case of the mixed Hodge structuredual to the limit mixed Hodge structure discussed above.The above definition is rather technical but it admits a simpler description in terms ofthe resolution of the singularity z n = f discussed above. Let ˜ B → B be an embedded log-resolution of the germ f = 0, V n → B be the projection of the germ of the singularity z n = f onto B . The singularities of the normalization of the fiber product S = V n × B ˜ B are cyclicquotient singularities and their (minimal) resolution e S provides a resolution of the singularityof the germ z n = f (cf. [20]). In this resolution the boundary of the tubular neighborhoodof the exceptional locus can be identified with the link L f in (1.3). Moreover H ( L f ) (andby duality H ( L f )) can be identified in an appropriate way with L H ( E i ) where E i runsthrough the set of exceptional curves in e S having a positive genus . More precisely, we have: Theorem 1.3. (cf. [16, Theorem 3.11])
Let f ( x, y ) = 0 be a singularity with a semi-simplemonodromy and let N be the order of the monodromy operator. The Albanese variety of thegerm f ( x, y ) = 0 is isogenous to a product of Jacobians of the exceptional curves of positivegenus for a resolution of (1.4) z N = f ( x, y ) . The latter description suggests an approach to defining the local Albanese for the non-reduced case, i.e. as the product of the Jacobians of curves of positive genus in the resolutionof the singularities of the germ z n = f .Finally recall the following: E. ARTAL, J.I. COGOLLUDO, AND A. LIBGOBER
Definition 1.4. (cf. [16, Definition 3.4]) A plane curve singularity has a CM -type if its localAlbanese variety is an abelian variety of CM-type. A plane curve singularity has a CM -type if its local Albanese variety is isogenous to a product of simple abelian varieties of CM-type.We refer to [23] for basic information regarding abelian varieties of CM-type. Unibranchedsingularities and singularities for which the characteristic polynomial of the monodromy op-erator has no multiple roots provide many examples of singularities of CM-type (cf. [16]). Example 1.5.
Let (
C, P ) be a simple curve singularity of type A g , with local equation y − x g +1 = 0. The local Albanese variety is associated to the surface singularity y − x g +1 = z g +1) . For any resolution of this surface singularity, there is only one non-rational irreduciblecomponent D A g of its exceptional divisor, which is a Belyi cover of the unique branchingcomponent of the minimal resolution of ( C, P ), ramified at the three intersection points withthe other components, with ramification indices 2 , g + 1 , g + 1), whose genus is g .1.3. Orbifold Pencils.Definition 1.6.
Let X be a quasi-projective manifold and S be an orbicurve (one-dimensionalorbifold). A holomorphic map φ between X and the underlying S complex curve we shallcall an orbifold pencil if the index of each orbifold point p divides the multiplicity of eachconnected component of the fiber φ ∗ ( p ) over p .We will concentrate our attention on orbifold pencils of curve complements. Let C ⊂ P be a plane curve (not necessarily irreducible) and let X C denote its complement. Consider anorbifold pencil φ : X C → S , where S is a rational orbifold curve (that is, its compactificationis P ) given by a finite number of orbifold points, say P i , i = 1 , . . . , s , with orbifold structureof order m i ∈ Z > ∪ {∞} , i = 1 , . . . , s (i.e. near which the orbifold chart is the chart given bya disk with the standard action of the cyclic group of order m i ). For convenience, m i = ∞ means that P i has been removed from S , namely, S = P \ { P i | m i = ∞} . In the future wewill denote S simply by P m , where ¯ m := ( m , . . . , m s ). Definition 1.7.
In the situation as above, we say that C belongs to an orbifold pencil oftype ¯ m . Moreover, the orbifold pencil φ will be called a global quotient orbifold pencil if thereexists a morphism Φ : X G → Σ, where X G is a quasi-projective manifold endowed with anaction of a finite group G and Σ a curve which makes the diagram(1.5) X G Σ X C P m Φ φ commutative, for which the vertical arrows are the models for the quotients by the action of G .If in addition, there is a character χ ∈ Char( X C ) and a character ρ ∈ Char orb ( P m ) suchthat χ = ρ ◦ π , and X G (resp. Σ) is the covering of X C (resp. P m ) associated with the LBANESE VARIETIES AND ORBIFOLD PENCILS 9 character χ (resp. ρ ), then we say ( C, χ ) belongs to a global quotient orbifold pencil withtarget ( P m , ρ ).Oftentimes, the set of global quotient orbifold pencils –up to the obvious equivalence byautomorphisms of the target– is infinite (see [12, 5]). A very useful (cf. Theorem 3.5 below)property to determine the different nature of such orbifold pencils is given by the following. Definition 1.8.
Global quotient orbifold pencils φ i : ( X C , χ ) → ( P m , ρ ), i = 1 , . . . , n arecalled independent if the induced maps Φ i : X G → Σ define Z [ G ]-independent morphisms ofmodules(1.6) Φ i ∗ : H ( X G , Z ) → H (Σ , Z ) , that is, independent elements of the Z [ G ]-module Hom Z [ G ] ( H ( X G , Z ) , H (Σ , Z )).In addition, if(1.7) M Φ i ∗ : H ( X G , Z ) → H (Σ , Z ) n is surjective we say that the pencils φ i are strongly independent . If the previous morphism(1.7) is considered with coefficients over Q , then we will use the term Q -strongly independent.2. Proof of theorems 1, 3, 4 and 5
Proof of Theorem . We shall use notations set up in the Introduction and in Section 1 andconsider the Alexander polynomial ∆
C,χ ( t ) of C relative to the homomorphism χ : π ( X C ) → Γ ⊂ C ∗ where Γ = Im χ is the group of N -th roots of unity by hypothesis. Let ξ ∈ C ∗ bea primitive N -th root of unity. Since H ( X C , χ ) = 0 one has ∆ C,χ ( ξ ) = 0 (cf. (1.1)). Let S χ := { P ∈ Sing( C ) | H ( B P \ C, χ P ) = 0 } ; because of (1.2), this set is non-empty.For each P ∈ S χ , consider the unbranched covering of E P := B P \ C corresponding to thesurjection π ( E P ) → Γ and denote it by ( e E P ) Γ . Then the restriction of the cyclic cover of B P given by the equation (1.4) on E P is equivalent to ( e E P ) Γ → E P . The proof of the DivisibilityTheorem (cf. [17, 16]) also shows that that there is a surjection L P ∈ S χ H (( e E P ) Γ , Q ) ξ → H ( V χC , Q ) ξ , where the subindex ξ stands for the ξ -eigenspace of the corresponding decktransformations. Hence one can take as P in (1) any singular point in S χ for which the map H (( e E P ) Γ , Q ) ξ → H ( V χC , Q ) ξ has a non trivial image. (cid:3) Remark . In fact H ( B P \ C, χ P ) = 0 is not enough to ensure that the map (T1) inTheorem 1 has a non-trivial image. For instance, consider C a sextic curve with seven ordinarycusps. It is well known (already to O.Zariski, cf. [3, 17, 12] for more recent discussions) thatthere is a conic passing through six out of the seven cusps. The Alexander polynomial of C is t − t + 1, which coincides with the local Alexander polynomials of its singularities. However,if χ is a character of order 6, the map H ( X C , χ ) → H ( B P \ C, χ P )is not trivial if and only if P is one of the six cusps on the conic. Proof of Theorem . Now let us assume that P is a singularity satisfying Theorem 4 andconsider a resolution of the associated surface singularity V P = { z N = f P ( x, y ) } , where N is the order of the character χ and f P is a local equation of C near P . Recall that sucha resolution can be obtained (Jung’s method cf. [20]) by normalizing a pull-back of anembedded resolution of the singularity at P .It follows from the A’Campo formula (cf. [1]), or from discussion in Section 1.2, that ξ is the root of the characteristic polynomial of the transformation induced on homologyby the action z ξz on a resolution of singularities of the surface V P and restricted toone of the curves of positive genus in the resolution of the singularity V P . Denote such acurve by D . Jung’s procedure implies that , D is an irreducible component of a Γ-cover ofa rational curve (namely an exceptional divisor of the resolution of P ). By Theorem 1.3(i.e. [16, Theorem 3.11]) there is an isogeny component of the Jacobian of D (possibly adirect sum of several simple components) which is also an isogeny component of Alb( V χC ).If this component is an (Im χ )-invariant Jacobian of a curve D , i.e. if the assumption (b)in Theorem 4 is fulfilled, then by Torelli’s Theorem Im χ acts on D as well (unfaithfully if D 6 = D ). Note that Theorem 4 allows non-reduced curves f = 0, which are excluded in thestatement of Theorem 1.3.As a consequence of Jung’s method, the resolutions of z n = f and z n = f red , where f red is the product of irreducible factors of f , are both obtained by pull-back and normalizationof the same embedded resolution of the curve f red = 0. In particular the conclusions ofTheorem 4(1) still hold in the non-reduced case, whereas D depends on the ramification dataof the cyclic cover V P .Returning to the proof of the existence of an orbifold pencil satisfying (1), suppose thatthe composition of the Albanese map and the projection onto Jac( D ) has a 1-dimensionalimage W . Let σ : D → D be the quotient map. Consider the diagram(2.1) D D Jac( D ) Jac( D ) = J χ V χC Alb( V χC ) W := Im alb χ,D .σ Jac( σ ) = 0This diagram shows that the image of D in Jac( D ) coincides with the image of D and henceit is contained in W . The assumption that dim Im alb χ,D = 1 hence yields that Im alb χ = D (up to a translation). Moreover the map V χC → D is Γ-equivariant and hence it induces theorbifold pencil as described in Theorem 4(1).If Jac( D ) is a simple abelian variety, then J χ = Jac( D ) as it follows from the discussionabove. This yields (2) which concludes the proof of Theorem 4. (cid:3) LBANESE VARIETIES AND ORBIFOLD PENCILS 11
Proof of Theorem . To derive this proof from Theorem 4 we have to verify that its hypothe-ses are satisfied. For A g -singularities, one has Alb P = Jac( D ), where D is a covering of thebranching component of the minimal resolution of the singularity. Note that under the hy-pothesis p = 2 g + 1 is prime, Jac( D ) is simple (cf. [23, Example 4.8(1)]). Using Theorem 4(2),the result follows. (cid:3) Proof of Theorem . Recall that π orb1 ( P m ) = π ( P \ { P i } si =1 ) / h γ m i i i si =1 where γ i are meridiansabout the points P i in π ( P \ { P i } si =1 ). Consider the composition λ ρ π ( P \ { P i } si =1 ) λ −→ π orb1 ( P m ) ρ −→ C ∗ . Following the notation introduced in Definition 1.7, consider the natural surjection morphismΛ : π ( P \ ( C ∪ S si =1 D i )) → π ( X C ). Note that the meridians about the components D i generate the normal subgroup ker Λ. Since they are taken by π onto m i -th powers of (even-tually powers of) meridians about P i , the surjection π is induced by π ( P \ ( C ∪ S si =1 D i )) → π ( P \ { P i } si =1 ). Hence we have the following commutative diagram:(2.2) π ( P \ ( C ∪ S si =1 D i )) π ( P \ { P i } si =1 ) π ( X C ) π orb1 ( P m )Π π Λ λ Since χ = π ∗ ( ρ ), the character χ is the composition π ( X C ) π −→ π orb1 ( P m ) ρ −→ C ∗ , one hasΠ(ker(Λ ◦ χ )) ⊆ ker( λ ◦ ρ ). Hence diagram (2.2) shows that π induces the map of coveringspaces(2.3) P \ ( C ∪ s [ i =1 D i ) ! Λ ◦ χ −→ (cid:0) P \ { P i } si =1 (cid:1) λ ρ corresponding to the subgroups K := ker(Λ ◦ χ ) and K ρ := ker( λ ρ ) respectively. The extensionof the map (2.3) to a smooth compactification of ( P \ ( C ∪ S si =1 D i )) Λ ◦ χ and then to aresolution of its base points yields a map of a birational model of V χC to Σ; recall that theorbifold P m is a global quotient of a Riemann surface Σ. And hence we have also a mapAlb( V χC ) → Jac(Σ). The Poincar´e Reducibility Theorem yields an isogeny between Alb( V χC )and Jac(Σ) × A .In the case of n > φ , . . . , φ n , we obtain a corresponding map for each φ i and hencea map Alb( V χC ) → Jac(Σ) n . By Definition 1.8, the corresponding map of H is surjective andhence, as above, the Poincar´e Reducibility Theorem yields the claimed isogeny. (cid:3) Curves with A g -singularities The purpose of this section is to justify the lengthy statements of the main theorems byhighlighting both their power and their subtleties through a series of examples. Simplifying the statements would only cause a more coarse description of the actual connection betweencharacteristic varieties and orbifold pencils.In what follows we present three essentially different types of situations: the pivotal exampleis shown in Theorem 3.3, where dim Im alb = 2, Alb( V χ C ) is a simple abelian variety, whichis the Jacobian of a curve, and hence the image alb projected onto the isogeny factors ofAlb( V χ C ) is never a curve. Therefore the conditions of Theorem 4 are not satisfied. Moreover,( C , χ ) does not contain a global orbifold pencil (see [4]). Another remarkable fact is that V χ C is birational to an abelian surface of CM-type corresponding to the cyclotomic field Q ( ζ ).The cyclic quotients of these abelian surfaces have been studied by Bagnera and deFranchis [7];this curve is the ramification divisor of one of such quotients.On the other hand in Theorem 3.1 a curve C is exhibited (for k = 1 and g = 2)whose Alb( V χ C ) coincides with Alb( V χ C ), however dim Im alb = 1, which implies, by Theo-rem 3, the existence of a global orbifold pencil containing ( C , χ ). Finally, in Theorem 3.5,dim Im alb = 2, as for C . However Alb( V χ C ) decomposes (up to isogeny) as a product ofthree simple Jacobians of curves and the image alb projected onto these factors are always1-dimensional. By Theorem 3 this implies the existence of three independent global orbifoldpencils containing ( C , χ ). Theorem 3.1.
Let C be an irreducible curve in P given by the equation (3.1) f g +12 k + f g +1) k = 0 , where f i is a generic homogeneous polynomial of degree i . Let χ be the character of π ( P \ C ) sending the generator of H ( P \ C ) = Z k (2 g +1) to a primitive root of unity of degree g +1) .Consider V χ C the cyclic covering of order g + 1) of P ramified along C . Let D A g be thecurve of genus g which is the cyclic Belyi cover of P , g +1 , g +1)) of degree g + 1) . Then Alb( V χ C ) ∼ Jac( D A g ) and the Albanese dimension of V χ C is 1.Remark . These curves were studied by M. Oka in [22] and the pencil provided by The-orem 3 is the one generated by f g +12 k and f g +1) k . Also note that Jac( D A g ) is the localAlbanese variety of any singularity of C , see Example 1.5. Proof.
The curve (3.1) has 2 k (2 g + 1) singularities each locally equivalent to u = v g +1 forming scheme theoretical (for generic f k , f (2 g +1) k ) complete intersection B given by f k = f (2 g +1) k = 0. The Example 1.1 provides a general form of its Alexander polynomial and acalculation using [18] shows that s = 1 i.e. it is t g +1 +1 t +1 .Consider the pencil of curves of degree 2 k (2 g + 1) given by:(3.2) π C : [ x : x : x ] [ f k ( x , x , x ) g +1 : f g +1 ( x , x , x ) k ]yielding a regular map P \ B → P . We shall view this as an orbifold pencil with target P , g +1 . Since π C ( C ) = p ∈ P , this map induces another orbifold map P \ C → P , g +1 \{ p } by restriction. Note that the inclusion P , g +1 \ { p } ֒ → P , g +1 , g +1) is a dominant map. Thelatter orbifold is a global orbifold quotient by the action of cyclic group Z g +1) of a cyclic LBANESE VARIETIES AND ORBIFOLD PENCILS 13
Belyi cover Σ having genus g (the value of the genus follows for example from the Riemann-Hurwitz formula). Moreover the pencil (3.2) lifts to the regular map ˜ π C : V χ C → Σ. Itfollows from [17] that dim H ( V χ C ) = 2 g . Hence the induced map Π C : Alb( V χ C ) → Jac(Σ)is an isogeny and one has the commutative diagram:(3.3) V C ˜ π C −→ Σ ↓ ↓ Alb( V χ C ) ˜Π C → Jac(Σ)where the vertical arrows are the Albanese map and the canonical embedding of Σ into itsJacobian. This implies that the Albanese image of V χ C is one dimensional. (cid:3) Theorem 3.3.
Let C be the union of a self-dual quintic C with 3 A -singularities and theline L which is tangent to C at one of its singularities, say P . Consider χ any characterof order 10 that ramifies along C + 5 L (the coefficients represent the ramification indices).Then (1) The canonical class of the minimal model of V χ C is trivial. (2) dim H ( V χ C , C ) = 4 . In particular this minimal model is an abelian surface. (3) This abelian surface is isomorphic to the
Jac( D A ) which is a simple abelian varietyand hence its Albanese dimension is 2.Proof. In order to prove part (1), we will construct the 10th-cyclic cover of P associated with χ . Note that K P = − H = − C . Denote by ˆ P the resulting surface (see Figure 2) afterblowing up the singular points of C to obtain a normal crossing divisor and then blowingdown the preimage of L . E E E E C L Figure 1.
Local resolution at P To understand this, we will briefly describe the local resolution of the singularity at P shown in Figure 1. The subindices of E i indicate the order of appearance of the exceptionaldivisors. Since the first two blow-ups occur on infinitely near smooth points of L , its self-intersection drops by 2. However, these first two infinitely near points are not smooth on C ,but of multiplicity 2. Since two more blow-ups on infinitely near smooth points of C arerequired to resolve the singularity, the self-intersection of C drops by 2 · (2) + 2 · (1) = 10. We denote by P ± the other singular points of type A . Note that Figure 1 (excludingthe germ of L ) also describes a resolution of P ± in C . For the corresponding exceptionaldivisors we replace the superscript 0 by ± accordingly. Analogously as mentioned above, theself-intersection of C drops by 10 at each point.By B´ezout’s Theorem, L intersects C at another point. Since its self-intersection after theblow-ups is − C and E , we can blow it down keeping the normalcrossing property. The self-intersection of both E and C increases by 1. The resultingsurface is ˆ P and the involved divisors are shown in Figure 2. By the Projection Formula weobtain K ˆ P = − C − (cid:0) E +1 + E − + E + 2 E +2 + 2 E − + 2 E (cid:1) .E ( − E E E E +2 ( − E +4 ( − E +3 ( − E +1 ( − E − E − E − E − C ( − Figure 2.
Surface ˆ P The self-intersections of the divisors are shown in parenthesis unless ( E • i ) = ( E + i ) . Sincewe have blown-up 12 points and blown-down one exceptional divisor, one can compute theEuler characteristic as follows: χ (ˆ P ) = χ ( P ) + 12 − . An alternative way to obtain a surface birationally equivalent to V χ C is to consider the 10thcyclic cover of ˆ P ramified along the total transform of C + 5 L , that is, R := C + 7 E + 14 E + 15 E + 30 E + 2 E ± + 4 E ± + 5 E ± + 10 E ± ≡ C + 7 E + 4 E + 5 E + 2 E ± + 4 E ± + 5 E ± mod 10 Pic(ˆ P ) , LBANESE VARIETIES AND ORBIFOLD PENCILS 15 where E ± i = E + i + E − i . It is easier to factor such covering as the composition of a doublecover π and a 5th-fold cover π .The double cover of ˆ P is ramified along R := C + E ± + E + E ≡ R mod 2 Pic(ˆ P )and will be denoted by X . The dual graph of the total transform π ∗ ( R ) in X is shownin Figure 3. e ( − e ( − e ( − e ( − c ( − e ++1 ( − e ++2 ( − e +4 ( − e +3 ( − e + − e + − e −− e −− e − e − +2 e − +1 e − Figure 3.
Surface X In order to compute the self-intersection of each divisor one has to apply the intersectiontheory formulas for covers (cf. [8, Chapter II. Section 10]).Note that K X = − c + 35 e − e + e − e ±± − e ±± + e ± , where e ±± i denotes the sum e ++ i + e + − i + e − + i + e −− i .By Riemann-Hurwitz, the Euler characteristic of X can be obtained as χ ( X ) = 2( χ (ˆ P ) − χ ( R )) + χ ( π ∗ ( R )) = 2(14 −
10) + 10 = 18 . After blowing down the divisors e , e , e +3 , and e − one obtains the surface Y , where K Y = − (cid:0) c + 2 e + e ±± + 2 e ±± (cid:1) and χ ( Y ) = 14 . Finally one needs to perform the 5:1 cover of Y ramified along R := c + 2 e + e ±± + 2 e ±± ,which incidentally is the support of K Y . Note that this divisor has 5 connected components,namely, e ++1 + 2 e ++2 , e + − + 2 e + − , e − +1 + 2 e − +2 , e −− + 2 e −− , and c + 2 e , each with the samecombinatorial structure as shown at the bottom of Figure 4. The appropriate ramified coveron e ++1 + 2 e ++2 is shown in Figure 4. Next to each irreducible component a list of numbersis shown: the first one being the self-intersection of the component, the second one being itsmultiplicity in the corresponding canonical Q -divisor ( K Y or K Z ), and the third one (whereapplicable) being the ramification index. The components ε ++ i are the strict transforms of e ++ i by the 5:1 cover, while the remaining components a ++ and b ++ project onto the double point. Note that the support of K Z is in the preimage of R . After blowing down all components, oneobtains a smooth surface ˆ Z with trivial canonical divisor, which is in particular the minimalmodel of V χ C . Using Riemann-Hurwitz once again, one obtains χ ( ˆ Z ) = 5( χ ( Y ) − χ ( R )) + 5 = 5(14 − ·
3) + 5 = 0 .ε ++1 ( − , a ++ ( − , b ++ ( − , ε ++2 ( − , e ++1 ( − , − , e ++2 ( − , − , Figure 4.
Surface Y From the Kodaira classification (see [8, Table 10]) the minimal model is a torus and henceit is an abelian surface.For part 2, note that the degree of the Alexander polynomial of C associated with χ (see [5,section 2.2]) is t − t + t − t +1 ([4, Theorem 4.5]). Since dim Alb( V χ C ) = deg ∆ C ,χ ( t ) = 2,the result follows. (cid:3) Remark . Note that Jac( A ) is a simple abelian variety. This follows from discussion in[16] yielding that CM-field in this case is Q ( ζ ) and explicit description of the CM-type there.More generally, for the singularity type x p + y q , where p, q are different prime numbers, recallthat Arnold-Steenbrink’s spectrum provides the CM-type for the local Albanese variety (cf.[16]), whose explicit description is well known. One can apply Shimura-Taniyama conditionsfor primitivity of a CM-type (cf. [23]) to verify that the local Albanese variety is simple inthis case. In particular Theorem 4(2) can be applied to those plane curve singularities.In general, however, local Albanese variety has several isogeny components. In the caseof uni-branched curves they all are Jacobians of Belyi cyclic covers (cf. [16]) and hence arethe components of Jacobians of Fermat curves. We refer for additional information regardingthese Jacobians to [15] and [2]. Theorem 3.5.
Let C be an irreducible curve in P given by the equation (3.4) x m + x m + x m − x m x m + x m x m + x m x m ) = 0 , where m is an odd number, say m = 2 g + 1 . Consider V χ C the cyclic covering of order m of P ramified along C . Let D A g be as above. Then Alb( V χ C ) is isogenous to Jac( D A g ) andthe Albanese dimension of V χ C is 2. LBANESE VARIETIES AND ORBIFOLD PENCILS 17
Proof.
The pencils of curves Λ i = { F i, [ α : β ] | [ α : β ] ∈ P } , (where F i, [ α : β ] = { α ( x j x k ) m + β ( x mj + x mk − x mi ) = 0 } and { i, j, k } = { , , } ) induce orbifoldmorphisms from P onto the compact orbifold P , , ([0:1] ,m ) . Since C = F i, [ − they alsodefine (by restriction) orbifold morphisms φ i : P \ C → P ,m, m defined as[ x : x : x ] φ i [ x mj x mk : ( x mj + x mk − x mi ) ] . If one shows that these morphisms are strongly independent, then by Theorem 5, theydefine a surjective morphism Alb( V χ C ) → Jac( D A g ) . Note that D A g is a curve of genus g . Moreover, the Alexander polynomial of C associated with χ is the classical Alexanderpolynomial since C is irreducible, which is ∆ C ( t ) = (cid:16) t g +1 t +1 (cid:17) (see [11]). Thusdim Alb( V χ C ) = 12 deg ∆ C ( t ) = 3 g and then Alb( V χ C ) ∼ Jac( D A g ) by dimension reasons.For the last part, consider ( φ × φ ) : P \ C → ( P ,m, m ) . Note that the preimage of ageneric point is the intersection of two generic members of the pencils Λ and Λ and hencethe morphism is finite. The same applies to (Φ × Φ ) : V χ C → Σ . By the standard propertiesof the Albanese map, alb(Φ × Φ ) : Alb( V χ C ) → Jac( D A g ) is surjective. Since the Albanesemap of V χ C factors through alb(Φ × Φ ), the result follows.It remains to show that the global quotient orbifold pencils φ , φ , and φ are stronglyindependent, in other words, that the morphisms Φ i, ∗ : H ( V χ C ) → H (Σ), i = 0 , , V χ C Σ P \ C P ,m, m Φ i φ i are Z [ µ m ]-independent ( µ m ⊂ C ∗ the cyclic group of 2 m -roots of unity) and that ⊕ i =0 Φ i, ∗ : H ( V χ C ) → H (Σ) is surjective (see Definition 1.8)Note that the base points of the pencils can be described as follows: let { i, j, k } = { , , } and consider ∆ i := { x i = 0 } ∩ Q j = { x i = 0 } ∩ Q k ,Q i := { x mj + x mk − x mi = 0 } . The 2 m base points of Λ i are ∆ j ∪ ∆ k .In order to understand V χ C we will first consider a resolution of the base points of the pencilΛ i . This is shown in Figure 5, where ˜ ℓ P (resp. ˜ C , and ˜ Q i ) represents the strict preimage of ℓ P , the axis containing P (resp. C , and the Fermat curve Q i ). The notation [ k ] next to an irreducible component E indicates the image by χ of a meridian γ around the irreduciblecomponent E as follows: χ ( γ ) = e km π √− . Unbranched components, i.e. [ k ] = [0], are shown in dashed lines.[2 g ] , E g,P E g +2 ,P , [0]˜ C , [1] E g +1 ,P , [ m ]˜ Q i , [0][2 g − , E g − ,P ...[4] , E ,P E ,P , [2][0] , ˜ ℓ P Figure 5.
In other words V χ C is the cyclic covering of order 2 m ramified along the locus C + X P ∈ ∆ (2 E ,P + 4 E ,P + · · · + (2 g − E g − ,P + 2 gE g,P + mE g +1 ,P ) , where ∆ = S i =0 ∆ i . To resolve each Λ i it would be enough to blow-up over ∆ j ∪ ∆ k , but thisway the same surface works for the three pencils.In particular, note that V χ C will contain curves Σ P which are the cyclic covering of E g +2 ,P ramified at 3 points of ramification indices 1, m −
1, and m . It is easy to check that theorders of χ at the meridians of these points are 2 m , m , and 2 respectively. Hence Σ P = Σis the curve of genus g which is the Belyi cover D A g of P ,m, m .Moreover, if P ∈ ∆ k , then Φ i | Σ P : Σ P → Σ and Φ j | Σ P : Σ P → Σ are isomorphisms since E g +2 ,P in Figure 5 is a dicritical section of Λ i and Λ j , whereas Φ k | Σ P : Σ P → Σ is a constantmap. This immediately implies the result as follows. Consider three indeterminacy pointsdistributed among the axes, for instance P := [0 : 1 : 1], P := [1 : 0 : 1], and P := [1 : 1 : 0].By the previous considerations non-trivial meridians γ i ∈ H (Σ P i ) ∼ = H (Σ) exist consideredas cycles in H ( V χ C ) via the inclusion and such thatΦ j ( γ i ) = ( γ if i = j j = i, LBANESE VARIETIES AND ORBIFOLD PENCILS 19 where γ ∈ H (Σ) is a non-trivial cycle. If Φ i, ∗ were dependent morphisms, then there shouldexist coefficients α , α , α ∈ Z [ µ m ] such that α Φ , ∗ + α Φ , ∗ + α Φ , ∗ ≡ , but using the cycle γ one obtains that α = − α , analogously, using γ (resp. γ ) one obtains α = − α (resp. α = − α ). Therefore α = α = α = α and 2 α = 0 in Z [ µ m ], whichimplies α = 0. The fact that the map ⊕ i =0 Φ i, ∗ : H ( V χ C ) → H (Σ) is surjective follows fromthe existence of the dicritical sections E g +2 ,P i and the induced isomorphisms Φ j | Σ Pi : Σ P i → Σfor j = i described above. (cid:3) References [1] N. A’Campo,
La fonction zeta d’une monodromie , Comment. Math. Helv. (1975), 233–248.[2] N. Aoki, Simple factors of the Jacobian of a Fermat curve and the Picard number of a product of Fermatcurves , Amer. J. Math. (1991), no. 5, 779–833.[3] E. Artal,
Sur les couples de Zariski , J. Algebraic Geom. (1994), 223–247.[4] E. Artal and J.I. Cogolludo-Agust´ın, On the connection between fundamental groups and pencils withmultiple fibers , J. Singul. (2010), 1–18.[5] E. Artal, J.I. Cogolludo-Agust´ın, and A. Libgober, Depth of cohomology support loci for quasi-projectivevarieties via orbifold pencils , Rev. Mat. Iberoam. (2014), no. 2, 373–404.[6] E. Artal, J.I. Cogolludo, and H.O Tokunaga, Pencils and infinite dihedral covers of P , Proc. Amer.Math. Soc. (2008), no. 1, 21–29 (electronic).[7] G. Bagnera, M. DeFranchis, Le superficie algebriche le quali ammettono una rappresentazione parametricamediante funzioni iperellittiche di due argomenti , Mem. Accad dei XL, 15 (1908), 251-343.[8] W.P. Barth, K. Hulek, C.A.M. Peters, and A. Van de Ven,
Compact complex surfaces , second ed.,Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics,vol. 4, Springer-Verlag, Berlin, 2004.[9] F. Catanese, C. Ciliberto,
On the irregularity of cyclic coverings of algebraic surfaces , Geometry ofcomplex projective varieties (Cetraro, 1990), 89115, Sem. Conf., 9, Mediterranean, Rende, 1993.[10] A.Comessatti,
Sui piani tripli ciclici irregolari , Rend. Circ.Mat.Palermo, 31 (1911), 369-386.[11] J.I. Cogolludo-Agust´ın,
Fundamental group for some cuspidal curves , Bull. London Math. Soc. (1999),no. 2, 136–142.[12] J.I. Cogolludo-Agust´ın and A. Libgober, Mordell-Weil groups of elliptic threefolds and the Alexandermodule of plane curves , J. Reine Angew. Math. (2014), 15–55.[13] P. Deligne,
Th´eorie de Hodge. III , Inst. Hautes ´Etudes Sci. Publ. Math. (1974), no. 44, 5–77.[14] E. Hironaka,
Alexander stratifications of character varieties , Ann. Inst. Fourier (Grenoble) (1997),no. 2, 555–583.[15] N. Koblitz and D. Rohrlich, Simple factors in the Jacobian of a Fermat curve , Canad. J. Math. (1978), no. 6, 1183–1205.[16] , On Mordell–Weil groups of isotrivial abelian varieties over function fields , Math. Ann. (2013), no. 2, 605–629.[17] ,
Alexander polynomial of plane algebraic curves and cyclic multiple planes , Duke Math. J. (1982), no. 4, 833–851.[18] , Alexander invariants of plane algebraic curves , Singularities, Part 2 (Arcata, Calif., 1981), Proc.Sympos. Pure Math., vol. 40, Amer. Math. Soc., Providence, RI, 1983, pp. 135–143.[19] ,
Characteristic varieties of algebraic curves , Applications of algebraic geometry to coding theory,physics and computation (Eilat, 2001), Kluwer Acad. Publ., Dordrecht, 2001, pp. 215–254. [20] J. Lipman,
Introduction to resolution of singularities , Algebraic geometry (Proc. Sympos. Pure Math.,Vol. 29, Humboldt State Univ., Arcata, Calif., 1974), Amer. Math. Soc., Providence, R.I., 1975, pp. 187–230.[21] J.W. Milnor,
Singular points of complex hypersurfaces , Annals of Mathematics Studies, vol. 61, PrincetonUniversity Press, Princeton, N.J., 1968.[22] M. Oka,
Some plane curves whose complements have non-abelian fundamental groups , Math. Ann. (1975), no. 1, 55–65.[23] G. Shimura,
Abelian varieties with complex multiplication and modular functions , Princeton Mathemat-ical Series, vol. 46, Princeton University Press, Princeton, NJ, 1998.[24] J.H.M. Steenbrink,
Mixed Hodge structure on the vanishing cohomology , Real and complex singularities(Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Oslo, 1976), Sijthoff and Noordhoff, Alphenaan den Rijn, 1977, pp. 525–563.
Departamento de Matem´aticas, IUMA, Universidad de Zaragoza, C. Pedro Cerbuna 12,50009 Zaragoza, Spain
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