aa r X i v : . [ m a t h . AG ] A p r ZARISKI k -PLETS VIA DESSINS D’ENFANTS ALEX DEGTYAREV
Abstract.
We construct exponentially large collections of pairwise distinctequisingular deformation families of irreducible plane curves sharing the samesets of singularities. The fundamental groups of all curves constructed areabelian. Introduction
Motivation and principal results.
Throughout this paper, the type of asingular point is its embedded piecewise linear type, and equisingular deformationsof curves in surfaces are understood in the piecewise linear sense, i.e. , the PL -typeof each singular point should be preserved during the deformation. This conventionis essential as some of the curves considered have non-simple singularities.Recall that a Zariski k -plet is a collection C , . . . , C k of plane curves, all of thesame degree m , such that(1) all curves have the same combinatorial data (see [5] for the definition; forirreducible curves, this means the set of types of singular points), and(2) the curves are pairwise not equisingular deformation equivalent.Note that Condition (2) in the definition differs from paper to paper, the mostcommon being the requirement that the pairs ( P , C i ) (or complements P r C i )should not be homeomorphic. In this paper, we choose equisingular deformationequivalence, i.e. , being in the same component of the moduli space, as it is thestrongest topologically meaningful ‘global’ equivalence relation. In any case, theconstruction of topologically distinguishable Zariski k -plets usually starts withfinding curves satisfying (2) above.Historically, the first example of Zariski pairs was found by O. Zariski [33],[34], who constructed a pair of irreducible sextics C , C , with six cusps each,which differ by the fundamental groups π ( P r C i ). Since then, a great numberof other examples has been found. Citing recent results only, one can mentiona large series of papers by E. Artal Bartolo, J. Carmona Ruber, J. I. Cogol-ludo Agust´ın, and H. Tokunaga (see [5], [6] and more recent papers [2]–[4] forfurther references), A. Degtyarev [11], [13], [14] (paper [11] deals with a directgeneralization of Zariski’s example: pairs of sextics distinguished by their Alexan-der polynomial), C. Eyral and M. Oka [16], [17], [25], G.-M. Greuel, C. Lossen, Mathematics Subject Classification.
Primary: 14H50; Secondary: 14H30, 14D05 .
Key words and phrases.
Zariski pair, trigonal curve, dessin d’enfants , braid monodromy. and E. Shustin [19] (Zariski pairs with abelian fundamental groups), Vik. S. Ku-likov [23], A. ¨Ozg¨uner [28] (a complete list of Zariski pairs of irreducible sextics thatare distinguished by their Alexander polynomial), I. Shimada [29]–[31] (a completelist of Zariski pairs of sextics with the maximal total Milnor number µ = 19, aswell as a list of arithmetic Zariski pairs of sextics), and A. M. Uluda˘g [32]. Theamount of literature on the subject definitely calls for a comprehensive survey!With very few exceptions, the examples found in the literature are those ofZariski pairs or triples . To my knowledge, the largest known Zariski k -plets arethose constructed in Artal Bartolo, Tokunaga [6]: for each integer m >
6, thereis a collection of ([ m/ −
1) reducible curves of degree m sharing the same com-binatorial data. The principal result of this paper is the following Theorem 1.1.1,which states that the size of Zariski k -plets can grow exponentially with the degree.(Theorem 1.1.3 below gives a slightly better count for reducible curves.)1.1.1. Theorem.
For each integer m > , there is a set of singularities shared by Z ( m ) = 1 k (cid:18) k − k − (cid:19)(cid:18) k [ k/ (cid:19)(cid:18) [ k/ ǫ (cid:19) pairwise distinct equisingular deformation families of irreducible plane curves C i of degree m , where k = [( m − / and ǫ = m − k − ∈ { , } . The fundamentalgroups of all curves C i are abelian : one has π ( P r C i ) = Z m . Recall that a real structure on a complex surface X is an anti-holomorphicinvolution conj : X → X . A curve C ⊂ X is called real (with respect to conj) ifconj( C ) = C , and a deformation C t , | t |
1, is called real if C ¯ t = conj C t . Upto projective equivalence, there is a unique real structure on P ; in appropriatecoordinates it is given by ( z : z : z ) (¯ z : ¯ z : ¯ z ).For completeness, we enumerate the families containing real curves.1.1.2. Theorem. If m = 8 t + 2 for some t ∈ Z , then Z (4 t + 2) of the families givenby Theorem 1.1.1 contain real curves ( with respect to some real structure in P ) .All other curves ( and all curves for other values of m ) split into pairs of disjointcomplex conjugate equisingular deformation families. Theorem.
For each integer m > , there is a set of combinatorial datashared by R ( m ) = 1 m − (cid:18) m − m − (cid:19) pairwise distinct equisingular deformation families of plane curves C i of degree m ( each curve splitting into an irreducible component of degree ( m − and a line ) .The fundamental groups of all curves C i are abelian : one has π ( P r C i ) = Z .If m = 2 t +1 is odd, then R ( t +3) of the families above contain real curves ( withrespect to some real structure in P ) . All other curves ( and all curves for m even ) split into pairs of disjoint complex conjugate equisingular deformation families. Theorems 1.1.1–1.1.3 are proved in Sections 7.2–7.4, respectively.
ARISKI k -PLETS VIA DESSINS D’ENFANTS 3 It is easy to see that the counts Z ( m ) and R ( m ) given by the theorems growfaster than a m/ and a m , respectively, for any a <
2. A few values of Z and R are listed in the table below. m . . .
20 40 80 Z ( m ) 6 6 30 60 140 280 840 . . . · · · R ( m ) 2 5 14 42 132 429 1430 . . . · · · Note that we are not trying to set a record here; probably, there are much largercollections of curves constituting Zariski k -plets. The principal emphasis of thispaper is the fact that Zariski k -plets can be exponentially large.1.2. Other results and tools.
The curves given by Theorems 1.1.1 and 1.1.3are plane curves of degree m with a singular point of multiplicity ( m − m −
2) or ( m −
1) do not produce Zariski pairs, see [10].) When the singular pointis blown up, the proper transform of the curve becomes a (generalized) trigonalcurve in a rational ruled surface. We explain this relation in Section 2, and thebulk of the paper deals with trigonal curves, whose theory is rather parallel toKodaira’s theory of Jacobian elliptic fibrations.A trigonal curve can be characterized by its functional j -invariant, which is arational function j : P → P , so that the singular fibers of the curve are encodedin terms of the pull-back j − { , , ∞} (see Table 1). To study the j -invariants, wefollow S. Orevkov’s approach [26], [27] (see also [15]) and use a modified versionof Grothendieck’s dessins d’enfants , see Section 4, reducing the classification oftrigonal curves with prescribed combinatorial type of singular fibers to a graphtheoretical problem. The resulting problem is rather difficult, as the graphs areallowed to undergo a number of modifications (see 4.4) caused by the fact that j may have critical values other than 0, 1, or ∞ . To avoid this difficulty, weconcentrate on a special case of the so called maximal curves, see 4.4.4, whichcan be characterized as trigonal curves not admitting any further degeneration(Proposition 4.4.8); the classification of maximal curves reduces to the enumera-tion of connected planar maps with vertices of valency
3, see Theorem 4.5.1.We exploit this relation and use oriented rooted binary trees to produce largeZariski k -plets of trigonal curves, see Proposition 7.0.4 and a slight modificationin Proposition 8.0.1.It is worth mentioning that the curves given by Propositions 7.0.4 and 8.0.1 aredefined over algebraic number fields (like all maximal curves), and in Theorem 8.0.2we use this fact to construct a slightly smaller, but still exponentially large, Zariski k -plet of plain curves with discrete moduli space. All these examples seem to begood candidates for exponentially large arithmetic Zariski k -plets (in rational ruledsurfaces and in the plane) in the sense of Shimada [29], [30].An important question that remains open is whether the curves constitutingvarious Zariski k -plets constructed in the paper can be distinguished topologically. ALEX DEGTYAREV
As a first step in this direction, we calculate the braid monodromy of the trigonalcurves, see 7.1. (For the relation between the braid monodromy and the topologyof the curve, see Orevkov [27], Vik. S. Kulikov and M. Teicher [24], or CarmonaRuber [9].) In 6.5, we give a general description of the braid monodromy of atrigonal curve in terms of its dessin; it covers all maximal curves with the exceptionof four explicitly described series. As a simple application, we obtain a criterion ofreducibility of a maximal trigonal curve in terms of its skeleton, see Corollary 6.6.1.As another direct application of the construction, we produce exponentiallylarge Zariski k -plets of Jacobian elliptic surfaces, see 8.1. (Here, by a Zariski k -pletwe mean a collection of not fiberwise deformation equivalent surfaces sharing thesame combinatorial type of singular fibers.) The series given by Theorem 8.1.2 arerelated to positive definite lattices of large rank; this gives one hope to distinguishthe surfaces, and hence their branch loci, topologically.1.3. Contents of the paper.
In Section 2, we introduce trigonal curves in ratio-nal ruled surfaces and discuss their relation to plane curves with a singular pointof multiplicity degree −
3. Section 3 reminds the basic properties of the j -invariantof a trigonal curve, and Section 4 introduces the dessin of a trigonal curve and theskeleton of a maximal curve. In Section 5, we prove a few technical statements onthe fundamental group of a generalized trigonal curve. Section 6 deals with thebraid monodromy. The principal results of the paper, Theorems 1.1.1–1.1.3, areproved in Section 7. Finally, in Section 8, we discuss a few modifications of theconstruction and state a few open problems.2. Trigonal models
Hirzebruch surfaces.
Recall that the
Hirzebruch surface Σ k , k >
0, is arational geometrically ruled surface with a section E of self-intersection − k . If k >
0, the ruling is unique and there is a unique section E of self-intersection − k ;it is called the exceptional section . In the exceptional case k = 0, the surfaceΣ = P × P admits two rulings, and we choose and fix one of them; any fiber ofthe other ruling can be chosen for the exceptional section. The fibers of the rulingare referred to as the fibers of Σ k . The semigroup of classes of effective divisorson Σ k is generated by the classes of the exceptional section E and a fiber F ; onehas E = − k , F = 0, and E · F = 1.An elementary transformation of a Hirzebruch surface Σ k is the birational trans-formation consisting in blowing up a point O ∈ Σ k and blowing down the propertransform of the fiber through O . If the blow-up center O does (respectively,does not) belong to the exceptional section E ⊂ Σ k , the result of the elementarytransformation is the Hirzebruch surface Σ k +1 (respectively, Σ k − ).2.2. Trigonal curves. A generalized trigonal curve on a Hirzebruch surface Σ k is a reduced curve not containing the exceptional section E and intersecting eachgeneric fiber at three points. Note that a generalized trigonal curve B ⊂ Σ k maycontain fibers of Σ k as components; we will call them the linear components of B . ARISKI k -PLETS VIA DESSINS D’ENFANTS 5 A singular fiber of a generalized trigonal curve B ⊂ Σ k is a fiber F of Σ k thatis not transversal to the union B ∪ E . Thus, F is either a linear component of B ,or the fiber through a point of intersection of B and E , or the fiber over a criticalvalue of the restriction to B of the projection Σ k → P .A trigonal curve is a generalized trigonal curve disjoint from the exceptionalsection. (In particular, trigonal curves have no linear components.) For a trigonalcurve B ⊂ Σ k , one has | B | = | E + 3 kF | ; conversely, any curve B ∈ | E + 3 kF | not containing E as a component is a trigonal curve.Let F be a singular fiber of a trigonal curve B . If B has at most simple singularpoints on F and F is not a component of B , then locally B ∪ E is the branch locusof a Jacobian elliptic surface X , and the pull-back of F is a singular fiber of X . Inthis case, we use the standard notation for singular elliptic fibers (referring to theextended Dynkin diagrams) to describe the type of F . Otherwise, B has a singularpoint of type J k,p or E k + ǫ , see [1] for the notation, and we use the notation ˜ J k,p and ˜ E k + ǫ , respectively, to describe the type of F .2.2.1. Remark.
We will not attempt to give a formal definition of the type of asingular fiber F of a trigonal curve B . One can understand it as the topologicaltype of the boundary singularity ( B, F ), see [1] for details. As a result of theclassification, one can conclude that this type is determined by whether F is acomponent of B and (the conjugacy class of) the braid monodromy about F , seeSection 6 below for the definition. Alternatively, if F is not a component of B and B has at worst simple singularities on F (which is always the case in this paper),then the type of F is determined by Kodaira’s type of the singular fiber of theJacobian elliptic surface ramified at B ∪ F , see above.Any generalized trigonal curve B without linear components can be convertedto a trigonal curve by a sequence of elementary transformations, at each stepblowing up a point of intersection of B and the exceptional section and blowingdown the corresponding fiber.2.3. Simplified models.
Let Σ ′ be a Hirzebruch surface, and let Σ ′′ be obtainedfrom Σ ′ by an elementary transformation. Denote by O ′ ∈ Σ ′ and O ′′ ∈ Σ ′′ theblow-up centers of the transformation and its inverse, respectively, and let F ′ ⊂ Σ ′ and F ′′ ⊂ Σ ′′ be the fibers through O ′ and O ′′ , respectively. The transform B ′′ ⊂ Σ ′′ of a generalized trigonal curve B ′ ⊂ Σ ′ is defined as follows: if B ′ doesnot (respectively, does) contain F ′ as a linear component, then B ′′ is the propertransform of B ′ (respectively, the union of the proper transform and fiber F ′′ ). Inthe above notation, there is an obvious diffeomorphism(2.1) Σ ′ r ( B ′ ∪ E ′ ∪ F ′ ) ∼ = Σ ′′ r ( B ′′ ∪ E ′′ ∪ F ′′ ) , where E ′ ⊂ Σ ′ and E ′′ ⊂ Σ ′′ are the exceptional sections.A trigonal curve B ⊂ Σ k is called simplified if all its singular points are double, i.e. , those of type A p . Clearly, each trigonal curve has a unique simplified model ¯ B ⊂ Σ l , which is obtained from B by a series of elementary transformations: ALEX DEGTYAREV one blows up a triple point of the curve and blows down the corresponding fiber,repeating this process until there are no triple points left.2.4.
Deformations.
Let B ⊂ Σ k be a generalized trigonal curve and E ⊂ Σ k the exceptional section. We define a fiberwise deformation of B as an equisin-gular deformation (path in the space of curves) preserving the topological typesof all singular fibers. Alternatively, a fiberwise deformation can be defined as anequisingular deformation of the curve B ∪ E ∪ (all singular fibers of B ).A degeneration of a generalized trigonal curve B is a family B t , | t |
1, ofgeneralized trigonal curves such that B = B and the restriction of B t to theannulus 0 < | t | nontrivial if B is not fiberwise deformation equivalent to B .Let B k ⊂ Σ k and B k +1 ⊂ Σ k +1 be two generalized trigonal curves related byan elementary transformation, and let E i ⊂ Σ i , i = k, k + 1, be the respectiveexceptional sections. In general, it is not true that an equisingular deformationof B k or B k ∪ E k is necessarily followed by an equisingular deformation of B k +1 (respectively, B k +1 ∪ E k +1 ) or vice versa : it may happen that a singular fibersplits into two and this operation affects the topology of one of the curves with-out affecting the topology of the other. However, it obviously is true that thefiberwise deformations of B k are in a natural one-to-one correspondence with thefiberwise deformations of B k +1 . A precise statement relating deformations of B k and B k +1 would require simple but tedious analysis of a number of types of sin-gular fibers. Instead of attempting to study this problem in full generality (whichbecomes even more involved if the two curves are related by a series of elementarytransformations), we just make sure that, in the examples considered in this paper(see 7.2, 7.4, and 8.0.2), generic equisingular deformations of each curve B ∪ E arefiberwise. (In 7.4, a linear component is added to the curve for this purpose.) Inmore details this issue is addressed in 7.5.2.5. The trigonal model of a plane curve.
Let C ⊂ P be a reduced curve,deg C = m , and let O be a distinguished singular point of C of multiplicity ( m − m > m > C is irreducible.) By a linear component of C we mean a component of degree 1 passing through O .Blow O up and denote the result by Y ; it is a Hirzebruch surface Σ , and theproper transform ˜ C = B ⊂ Y of C is a generalized trigonal curve. Clearly, thecombinatorial type of C determines and is determined by that of B ∪ E , thetype of O itself being recovered from the singularities of B ∪ E located in theexceptional section E . Furthermore, equisingular deformations of the pair ( C, O )are in a one-to-one correspondence with equisingular deformations of B ∪ E .Let B ′ be the curve obtained from B by removing its linear components. Asin 2.2, one can apply a sequence of elementary transformations to get a sequenceof curves B i , B ′ i ⊂ Y i ∼ = Σ i , i = 1 , . . . , k , so that B ′ k is a true trigonal curve. (Here, B i +1 is the transform of B i , and B ′ i +1 is obtained from B i +1 by removing its linearcomponents. In other words, we pass to the trigonal model of B ′ while keepingtrack of the linear components of C .) The curve B ′ k is called the trigonal model ARISKI k -PLETS VIA DESSINS D’ENFANTS 7 of C . Finally, passing from B ′ k to its simplified model B ′ ⊂ Y ∼ = Σ l , one obtainsthe simplified trigonal model B ′ of C .3. The j -invariant The contents of this section is a translation to the language of trigonal curvesof certain well known notions and facts about elliptic surfaces; for more details werefer to the excellent founding paper by K. Kodaira [22] or to more recent mono-graphs [18] and [7]. In the theory of elliptic surfaces, trigonal curves (in the senseof this paper) arise as the branch loci of the Weierstraß models of Jacobian ellipticsurfaces over a rational base. These curves have at most simple singularities andbelong to even Hirzebruch surfaces Σ s . However, most notions and statementsextend, more or less directly, to trigonal curves in odd Hirzebruch surfaces Σ s +1 .3.1. Weierstraß equation.
Let Σ k → P be a Hirzebruch surface. Any trigonalcurve B ⊂ Σ k can be given by a Weierstraß equation ; in appropriate affine chartsit has the form x + g x + g = 0 , where g and g are certain sections of O P (2 k ) and O P (3 k ), respectively, and x is a coordinate such that x = 0 is the zero section and x = ∞ is the exceptionalsection E ⊂ Σ k . The sections g , g are determined by the curve uniquely up tothe transformation(3.1) ( g , g ) ( t g , t g ) , t ∈ C ∗ . The following statement is straightforward.3.1.1.
Proposition.
A trigonal curve B as in 3.1 is simplified if and only if thereis no point z ∈ P which is a root of g of multiplicity > and a root of g ofmultiplicity > . (cid:3) The ( functional ) j -invariant of a trigonal curve B ⊂ Σ k is the meromorphicfunction j = j B : P → P given by j = 4 g ∆ , ∆ = 4 g + 27 g , where g and g are the coefficients of the Weierstraß equation of B , see 3.1. Here,the domain of j is the base of the ruling Σ k → P , whereas its range is the standardprojective line P = C ∪ {∞} . If the fiber F z over z ∈ P is nonsingular, then thevalue j ( z ) is the usual j -invariant (divided by the magic number 1728 = 12 ) ofthe quadruple of points cut on F z by the union B ∪ E (or, in more conventionalterms, the j -invariant of the elliptic curve that is the double of F z ∼ = P ramifiedat the four points above). The values of j at the finitely many remaining pointscorresponding to the singular fibers of B are obtained by analytic continuation.Since j B is defined via affine charts and analytic continuation, it is obviously in-variant under elementary transformations. In particular, the notion of j -invariantcan be extended to generalized trigonal curves (by ignoring the linear components ALEX DEGTYAREV and passing to a trigonal model), and the j -invariant of a trigonal curve B is thesame as that of the simplified model of B . The j -invariant j B : P → P has three ‘special’ values: 0, 1, and ∞ . Thecorrespondence between the type of a fiber F z , see remark in Section 2.2, and thevalue j ( z ) (and the ramification index ind z j of j at z ) is shown in Table 1. (Weconfine ourselves to the curves with at worst simple singular points. In fact, in viewof the invariance of the j -invariant under elementary transformations, it wouldsuffice to consider type ˜ A singular fibers only. For the reader’s convenience, we alsocite Kodaira’s notation for the types of singular elliptic fibers, cf. Section 2.2.) If B is a curve in Σ k , the maximal degree of j B is 6 k . However, deg j B drops if B hastriple singular points or type ˜ A ∗∗ , ˜ A ∗ , or ˜ A ∗ singular fibers, see ∆ deg j in Table 1.It is worth mentioning that the j -invariant of a generic trigonal curve is highly non -generic, as it takes values 0 and 1 with multiplicities 3 and 2 respectively(see Comments to Table 1); conversely, a generic function j : P → P would ariseas the j -invariant of a trigonal curve with a large number of type ˜ A ∗∗ and ˜ A ∗ singular fibers. Table 1.
The values j ( z ) at singular fibers F z Type of F z j ( z ) ind z j ∆ deg j mult F z ˜ A p ( ˜ D p +5 ), p > p +1 (I ∗ p +1 ) ∞ p + 1 0 ( − p + 1 ( p + 7)˜ A ∗ ( ˜ D ) I (I ∗ ) ∞ −
6) 1 (7)˜ A ∗∗ ( ˜ E ) II (II ∗ ) 0 1 mod 3 − −
8) 2 (8)˜ A ∗ ( ˜ E ) III (III ∗ ) 1 1 mod 2 − −
9) 3 (9)˜ A ∗ ( ˜ E ) IV (IV ∗ ) 0 2 mod 3 − −
10) 4 (10)
Comments.
Fibers of type ˜ A (Kodaira’s I ) are not singular. For a nonsingularfiber F z with complex multiplication of order 2 (respectively, 3) one has j ( z ) = 1and ind z j = 0 mod 2 (respectively, j ( z ) = 0 and ind z j = 0 mod 3). Singular fibersof type ˜ D (Kodaira’s I ∗ ) are not detected by the j -invariant, except that eachsuch fiber decreases the degree of j by 6. The multiplicity mult F z is the number ofsimplest ( i.e. , type ˜ A ∗ ) singular fibers resulting from a generic perturbation of F z . A trigonal curve B ⊂ Σ k is called isotrivial if j B = const.All simplified isotrivial curves can easily be classified.(1) If j B ≡
0, then g ≡ g is a section of O P (3 k ) whose all roots aresimple or double, see Proposition 3.1.1. The singular fibers of B are oftype ˜ A ∗∗ (over the simple roots of g ) or ˜ A ∗ (over the double roots of g ).(2) If j B ≡
1, then g ≡ g is a section of O P (2 k ) with simple rootsonly, see Proposition 3.1.1. All singular fibers of B are of type ˜ A ∗ (overthe roots of g ). ARISKI k -PLETS VIA DESSINS D’ENFANTS 9 (3) If j B = const = 0 ,
1, then g ≡ λg for some λ ∈ C ∗ ; in view of Proposi-tion 3.1.1, this implies that k = 0 and g , g = const, i.e. , B is a union ofdisjoint sections of Σ . (In particular, B has no singular fibers.)Note that an isotrivial trigonal curve cannot be fiberwise deformation equivalentto a non-isotrivial one, as a non-constant j -invariant j B would take value ∞ andhence the curve would have a singular fiber of type ˜ A ∗ or ˜ A p , p >
0, see Table 1.
Any non-constant meromorphic function j : P → P is the j -invariant of a certain simplified trigonal curve B ⊂ Σ k ; the latter is unique upto the change of coordinates given by (3.1) .Proof. For simplicity, restrict all functions/sections to an affine portion C ⊂ P ,which we assume to contain all pull-backs j − (0) and j − (1). Represent the func-tion l = j/ (1 − j ) by an irreducible fraction p/q . Since l ( ∞ ) = 0 ,
1, one hasdeg p = deg q . For each root a of p of multiplicity 1 mod 3 (respectively, 2 mod 3),multiply both p and q by ( z − a ) (respectively, ( z − a ) ), and for each root b of q ofmultiplicity 1 mod 2, multiply both p and q by ( z − b ) . In the resulting represen-tation l = ¯ p/ ¯ q , the multiplicity of each root of ¯ p (respectively, ¯ q ) is divisible by 3(respectively, 2), and ¯ p and ¯ q have no common roots of multiplicity >
6. Hence,one has ¯ p = 4 g and ¯ q = 27 g for some polynomials g , g satisfying the conditionin Proposition 3.1.1, and the function j = l/ ( l + 1) = ¯ p/ (¯ p + ¯ q ) is the j -invariantof the simplified trigonal curve B ⊂ Σ k given by the Weierstraß equation withcoefficients g , g , where k = deg ¯ p = deg ¯ q . Clearly, the polynomials g , g asabove are defined by l uniquely up to the transformation given by (3.1). (cid:3) A fiberwise deformation of a non-isotrivial trigonal curve B results in a deformation of its j -invariant j = j B : P → P with the followingproperties :(1) the degree of the map j : P → P remains constant ;(2) distinct poles of j remain distinct, and their multiplicities remain constant ;(3) the multiplicity of each root of j remains constant mod 3;(4) the multiplicity of each root of j − remains constant mod 2 .Conversely, any deformation of nonconstant meromorphic functions j : P → P satisfying conditions (1) – (4) above results in a fiberwise deformation of the corre-sponding (via Proposition 3.4.1 ) simplified trigonal curves. Remark.
Condition 3.4.2(3) means that a root of j of multiplicity divisi-ble by 3 may join another root and, conversely, a root of large multiplicity maybreak into several roots, all but one having multiplicities divisible by 3. Condi-tion 3.4.2(4) should be interpreted similarly. Remark.
Note that just an equisingular (not necessarily fiberwise) deforma-tion of trigonal curves does not always result in a deformation of their j -invariants.In the case of simplified curves, the degree of j B drops whenever a type ˜ A ∗ singu-lar fiber of B joins another singular fiber, of type ˜ A ∗ , ˜ A , or ˜ A , to form a fiberof type ˜ A ∗∗ , ˜ A ∗ , or ˜ A ∗ , respectively, see Table 1. Proof.
The direct statement follows essentially from Table 1. Indeed, the multiplic-ities of the poles of j B , (mod 3)-multiplicities of its roots, and (mod 2)-multiplicitiesof the roots of j B − B , and the degree deg j B can be found as the sum of the multiplicities of all poles of j B . Since the expressionfor j B depends ‘continuously’ on the coefficients of the Weierstrass equation anddeg j B remains constant, there is no extra cancellation during the deformation andthe map j B : P → P changes continuously.The converse statement follows from the construction of the simplified trigonalcurve B from a given j -invariant j , see the proof of Proposition 3.4.1. Since thedegree deg l = deg j remains constant, the polynomials p and q in the irreduciblerepresentation l = p/q change continuously during the deformation. Crucial isthe fact that the passage from p/q to ¯ p/ ¯ q depends only on the roots of p and q whose multiplicity is not divisible by 3 and 2, respectively. Hence, due to Con-ditions 3.4.2(3) and (4), the degree deg ¯ p = deg ¯ q will remain constant, the poly-nomials ¯ p and ¯ q will change continuously, and so will the coefficients g , g ofthe Weierstrass equation. The fact that the resulting deformation of the trigonalcurves is fiberwise follows again from Table 1. (cid:3) Dessins d’enfants and skeletons
According to Propositions 3.4.1 and 3.4.2, the study of simplified trigonalcurves in Hirzebruch surfaces is reduced to the study of meromorphic functions j : P → P with three ‘essential’ critical values 0, 1, and ∞ and, possibly, a fewother critical values. Following S. Orevkov [26], [27], we employ a modified versionof Grothendieck’s dessins d’enfants . Below, we outline briefly the basic conceptsand principal results; for more details and proofs we refer to [15], Sections 5.1and 5.2. Note that [15] deals with a real version of the theory, where functions(graphs) are supplied with an anti-holomorphic (respectively, orientation revers-ing) involution; however, all proofs apply to the settings of this paper literally,with the real structure ignored.Since, in this paper, we deal with rational ruled surfaces only, we restrict thefurther exhibition to the case of graphs in the sphere S ∼ = P . Given a graph Γ ⊂ S , we denote by S the closedcut of S along Γ. The connected components of S are called the regions of Γ.(Unless specified otherwise, in the topological part of this section we are workingin the PL -category.)A trichotomic graph is an embedded oriented graph Γ ⊂ S decorated with thefollowing additional structures (referred to as colorings of the edges and verticesof Γ, respectively):(1) ”–” each edge of Γ is of one of the three kinds: solid, bold, or dotted;(2) ”–” each vertex of Γ is of one of the four kinds: • , ◦ , × , or monochrome(the vertices of the first three kinds being called essential )and satisfying the following conditions: ARISKI k -PLETS VIA DESSINS D’ENFANTS 11 (1) the valency of each essential vertex of Γ is at least 2, and the valency ofeach monochrome vertex of Γ is at least 3;(2) the orientations of the edges of Γ form an orientation of the boundary ∂S ;this orientation extends to an orientation of S ;(3) all edges incident to a monochrome vertex are of the same kind;(4) × -vertices are incident to incoming dotted edges and outgoing solid edges;(5) • -vertices are incident to incoming solid edges and outgoing bold edges;(6) ◦ -vertices are incident to incoming bold edges and outgoing dotted edges.In (4)–(6) the lists are complete, i.e. , vertices cannot be incident to edges of otherkinds or with different orientation.Condition (2) implies that the orientations of the edges incident to a vertexalternate. In particular, all vertices of Γ have even valencies. In view of 4.1(3), the monochrome vertices of a trichotomic graphΓ can further be subdivided into solid, bold, and dotted, according to their incidentedges. A path in Γ is called monochrome if all its vertices are monochrome. (Then,all vertices of the path are of the same kind, and all its edges are of the same kindas its vertices.) Given two monochrome vertices u, v ∈ Γ, we say that u ≺ v if thereis an oriented monochrome path from u to v . (Clearly, only vertices of the samekind can be compatible.) The graph is called admissible if ≺ is a partial order.Since ≺ is obviously transitive, this condition is equivalent to the requirement thatΓ should have no oriented monochrome cycles.In this paper, an admissible trichotomic graph is called a dessin . Remark.
Note that the orientation of Γ is almost superfluous. Indeed, Γmay have at most two orientations satisfying 4.1(2), and if Γ has at least oneessential vertex, its orientation is uniquely determined by 4.1(4)–(6). Note alsothat each connected component of an admissible graph does have essential vertices(of all three kinds), as otherwise any component of ∂S would be an orientedmonochrome cycle. Remark.
In fact, all three decorations of a dessin Γ (orientation and thetwo colorings) can be recovered from any of the colorings. However, for clarity weretain both colorings in the diagrams.
Any orientation preserving ramified cov-ering j : S → P defines a trichotomic graph Γ( j ) ⊂ S . As a set, Γ( j ) is thepull-back j − ( P R ). (Here, P R ⊂ P is the fixed point set of the standard real struc-ture z ¯ z .) The trichotomic graph structure on Γ( j ) is introduced as follows: the • -, ◦ -, and × -vertices are the pull-backs of 0, 1, and ∞ , respectively (monochromevertices being the ramification points with other real critical values), the edgesare solid, bold, or dotted provided that their images belong to [ ∞ , , , ∞ ], respectively, and the orientation of Γ( j ) is that induced from the positiveorientation of P R ( i.e. , order of R ).As shown in [15], a trichotomic graph Γ ⊂ S is a dessin if and only if it has theform Γ( j ) for some orientation preserving ramified covering j : S → P ; the latter is determined by Γ uniquely up to homotopy in the class of ramified coveringshaving a fixed trichotomic graph.We define the dessin Γ( B ) of a trigonal curve B as the dessin Γ( j ) of its j -invariant j : P → P . The correspondence between the singular fibers of a simpli-fied trigonal curve B and the vertices of its dessin Γ( B ) is given by Table 1 (see j ( z ) ), the valency of a vertex z being twice the ramification index ind z j . The • - (respectively, ◦ -) vertices of Γ( B ) of valency 0 mod 6 (respectively, 0 mod 4)correspond to the nonsingular fibers of B with complex multiplication of order 3(respectively, 2); such vertices are called nonsingular , whereas all other essentialvertices of Γ are called singular . Let Γ ⊂ S be a trichotomic graph, and let v be avertex of Γ. Pick a regular neighborhood U ⊂ S of v and replace the intersectionΓ ∩ U with another decorated graph, so that the result Γ ′ is again a trichotomicgraph. If Γ ′ ∩ U contains essential vertices of at most one kind, then Γ ′ is calleda perturbation of Γ (at v ), and the original graph Γ is called a degeneration of Γ ′ .A perturbation Γ ′ of a dessin is also a dessin if and only if the intersection Γ ′ ∩ U contains no oriented monochrome cycles. There are no simple local criteria for theadmissibility of a degeneration. Remark.
Assume that the perturbation Γ ′ is a dessin. Since the intersectionΓ ′ ∩ ∂U is fixed, the assumption on Γ ′ ∩ U implies that Γ ′ ∩ U either is monochrome(if v is monochrome) or consists of monochrome vertices, essential vertices of thesame kind as v , and edges of the two kinds incident to v .A perturbation Γ ′ of a dessin Γ at a vertex v (and the inverse degenerationof Γ ′ to Γ) is called equisingular if v is not a × -vertex and the intersection Γ ′ ∩ U contains at most one singular • - or ◦ -vertex. Two dessins Γ ′ , Γ ′′ ⊂ S aresaid to be equivalent if they can be connected by a chain Γ ′ = Γ , Γ , . . . , Γ n =Γ ′′ of dessins, where each Γ i , 1 i n , either is isotopic to Γ i − or is anequisingular perturbation or degeneration of Γ i − . Clearly, equivalence of dessinsis an equivalence relation. Remark.
By an isotopy between two dessins Γ ′ and Γ ′′ we mean a PL -family φ t of PL -autohomeomorphisms of S such that φ = id and φ (Γ ′ ) = Γ ′′ , the lattermap taking vertices to vertices and edges to edges and preserving both colorings ofthe dessins. Note that, since the mapping class group of S is trivial, one can justrequire that Γ ′ is taken to Γ ′′ by an orientation preserving PL -autohomeomorphismof S respecting the graph structure and the colorings.The following statement, essentially based on the Riemann existence theorem,is an immediate consequence of Propositions 3.4.1 and 3.4.2 and the results of [15](particularly, Corollaries 5.1.8 and 5.2.3, with the real structure ignored). The map B Γ( B ) sending a trigonal curve B to its dessinestablishes a one-to-one correspondence between the set of fiberwise deformation ARISKI k -PLETS VIA DESSINS D’ENFANTS 13 classes of simplified trigonal curves in Hirzebruch surfaces and the set of equiva-lence classes of dessins. (cid:3) (Maximal curves and dessins) . A dessin Γ ⊂ S is called max-imal if it satisfies the following conditions:(1) all vertices of Γ are essential;(2) all • - (respectively, ◦ -) vertices of Γ have valency B is called maximal if its dessin Γ( B ) is maximal. Remark.
Conditions 4.4.4(1) and (3) in the definition of a maximal dessincan be restated as the requirement that the function j : S → P constructedfrom Γ, see 4.3, should have no critical values other than 0, 1, and ∞ . Remark.
Any maximal trigonal curve is defined over an algebraic numberfield. Indeed, as any function with three critical values, the rational function j : P → P has finitely many Galois conjugates and hence is defined over analgebraic number field. Then, the construction in the proof of Proposition 3.4.1shows that the coefficients g , g of the Weierstraß equation are defined over thesplitting field of j . (One may need to add to the field some roots and poles of j .) A trigonal curve B ′ is fiberwise deformation equivalent to amaximal trigonal curve B if and only if the dessins Γ( B ′ ) and Γ( B ) are isotopic.Furthermore, a permutation of the singular fibers of a maximal trigonal curve B isrealized by a fiberwise self-deformation if and only if the corresponding permutationof the vertices of Γ( B ) is induced by an isotopy of Γ( B ) .Proof. A maximal dessin does not admit nontrivial equisingular perturbations (dueto Conditions (1) and (2), as an equisingular perturbation requires a vertex ofhigh valency) or degenerations (due to Condition (3), as a perturbation producesmore than triangle regions). Hence, any equivalence to a maximal dessin is anisotopy. (cid:3)
A trigonal curve B is maximal if and only if it does notadmit a nontrivial degeneration, see 2.4, to a non-isotrivial curve.Proof. Let Γ = Γ( B ). After a small deformation, we can assume that Γ satisfiesthe general position assumptions 4.4.4(1) and (2). Then, if B is not maximal, Γhas a region R whose boundary contains at least two × -vertices, and these twovertices can be brought together within R . This degeneration of Γ results in anontrivial degeneration of the curve.Conversely, assume that B has a nontrivial degeneration to a curve B , whichis necessarily trigonal. Then, up to isotopy, Γ is obtained from Γ( B ) by removingdisjoint regular neighborhoods of some of its vertices and replacing them with newdecorated graphs. (Since deg j may change, it is no longer required that each ofthe new graphs should contain essential vertices of at most one kind. Note that wedo not discuss the realizability of any such modification by an actual degeneration of curves.) If this procedure is nontrivial, it results in a graph Γ with at least onenon-triangular region. (cid:3) An abstract skeleton is a connected planar map Sk ⊂ S whosevertices have valencies at most three; we allow the possibility of hanging edges , i.e. ,edges with only one end attached to a vertex. An isomorphism between two ab-stract skeletons Sk ′ and Sk ′′ is an orientation preserving PL -autohomeomorphismof S taking Sk ′ to Sk ′′ .The skeleton of a maximal trigonal curve B is the skeleton Sk( B ) ⊂ S obtainedfrom the dessin Γ( B ) by removing all × -vertices and incident edges ( i.e. , all solidand dotted edges) and disregarding the ◦ -vertices. (Note that the resulting graphis indeed connected due to Condition 4.4.4(3).) Clearly, a maximal dessin Γ isuniquely (up to homotopy) recovered from its skeleton Sk: one should place a ◦ -vertex at the middle of each edge (at the free end of each hanging edge), placea × -vertex v R at the center of each region R of Sk, and connect this vertex v R tothe • - and ◦ -vertices in the boundary ∂R by appropriate (respectively, solid anddotted) edges. The last operation is unambiguous as, due to the connectednessof Sk, each open region R is a topological disk. The map B Sk( B ) sending a maximal trigonal curve B toits skeleton establishes a one-to-one correspondence between the set of fiberwisedeformation classes of maximal trigonal curves in Hirzebruch surfaces and the setof isomorphism classes of abstract skeletons in S .Proof. The statement follows from the correspondence between maximal dessinsand skeletons described above, Theorem 4.4.3, and Proposition 4.4.7. (cid:3)
Remark.
Removing from a dessin Γ( j ) all × -vertices and incident edgesresults in a classical dessin d’enfants in the sense of Grothendieck, i.e. , the bipar-tite graph obtained as the pull-back j − ([0 , ◦ -vertices have valency at most two. Remark.
Theorem 4.5.1 suggests that, in general, the classification of maxi-mal trigonal curves with a prescribed combinatorial type of singular fibers is a wildproblem: one would have to enumerate all planar maps with prescribed valencies ofvertices and numbers of edges of regions. The only general result in this directionthat I am aware of is the Hurwitz formula [20] (see also [8]), which establishes arelation between a certain weighed count of planar maps (more precisely, ramifiedcoverings of P , not necessarily connected) and characters of symmetric groups. We conclude this section with a few simple counts. For adessin Γ, denote by ∗ = ∗ (Γ) the total number of ∗ -vertices (where ∗ is either • ,or ◦ , or × ), and by ∗ ( i ), i ∈ N , the number of ∗ -vertices of valency 2 i . (Recall thatvalencies of all vertices of a dessin are even.) Consider a trigonal curve B ⊂ Σ k , its j -invariant j : P → P , and its dessin Γ = Γ( B ). Counting the number of points ARISKI k -PLETS VIA DESSINS D’ENFANTS 15 in one of the three special fibers of j , one obtains(4.1) deg j = X i> i • ( i ) = X i> i ◦ ( i ) = X i> i × ( i ) . Since B can be perturbed to a generic trigonal curve in the same surface Σ k , anda generic curve has deg ∆ = 6 k simplest singular fibers, Table 1 yields(4.2) 6 k = X i> i × ( i ) + 2 X i =1(3) • ( i ) + 3 X i =1(2) ◦ ( i ) + 4 X i =2(3) • ( i ) . (Alternatively, one can notice that the first term in (4.2) equals deg j , see (4.1),and the remaining part of the sum is 6 k − deg j , see ∆ deg j in Table 1.) Finally,the Riemann–Hurwitz formula applied to j results in the inequality(4.3) • + ◦ + × > deg j + 2 , which turns into an equality if and only if j has no critical values other than 0, 1,and ∞ , i.e. , Conditions 4.4.4(1) and (3) are satisfied. The fundamental group
Recall that the braid group B can be defined as thegroup of automorphisms of the free group G = h α , α , α i sending each generatorto a conjugate of another generator and leaving the product α α α fixed. Weassume that the action of B on G is from the left . One has B = h σ , σ | σ σ σ = σ σ σ i , where σ : ( α , α , α ) ( α α α − , α , α ) , σ : ( α , α , α ) ( α , α α α − , α ) . We will also consider the elements σ = σ − σ σ and τ = σ σ = σ σ = σ σ .The center of B is the infinite cyclic group generated by τ .Note that the maps ( σ , σ ) ( σ , σ ) ( σ , σ ) define automorphisms of B ;in particular, the pairs ( σ , σ ) and ( σ , σ ) are subject to all relations that holdfor ( σ , σ ). In what follows, we use the convention σ l + i = σ i , i = 1 , , l ∈ Z .The degree deg β of a braid β ∈ B is defined as its image under the abelinizationhomomorphism B → Z , σ , σ
1. A braid is uniquely recovered from its degreeand its image in the quotient B /τ . Let B ⊂ Σ = Σ k be a generalized trigonal curve,and let E ⊂ Σ be the exceptional section. The fundamental group π (Σ r ( B ∪ E ))can be found using an analogue of van Kampen’s method [21] applied to the rulingof Σ. Pick a fiber F ∞ (singular or not) over a point ∞ ∈ P and trivialize theruling over P r ∞ . Let F , . . . , F r be the singular fibers of B other than F ∞ . Picka nonsingular fiber F distinct from F ∞ and a generic section S disjoint from E and intersecting all fibers F, F , . . . , F r , F ∞ outside of B .Clearly, F r ( B ∪ E ) is the plane C = F r E with three punctures. Considerthe group G = π ( F r ( B ∪ E ) , F ∩ S ), and let α , α , α be a standard setof generators of G . Let, further, γ , . . . γ r be a standard set of generators of thefundamental group π ( S r ( F ∞ ∪ S rj =1 F r ) , S ∩ F ), so that γ j is a loop around F j , j = 1 , . . . , r . For each j = 1 , . . . , r , dragging the fiber F along γ j and keeping thebase point in S results in a certain automorphism m j : G → G , called the braidmonodromy along γ j . Strictly speaking, m j is not necessarily a braid (unless B isdisjoint from E ); however, it still has the property that the image m j ( α i ) of eachstandard generator α i , i = 1 , ,
3, is a conjugate of another generator α i ′ .According to van Kampen, the group π (Σ r ( B ∪ E ∪ F ∞ ∪ S rj =1 F r ) , S ∩ F )is given by the representation (cid:10) α , α , α , γ , . . . , γ r (cid:12)(cid:12) γ − j α i γ j = m j ( α i ), i = 1 , , j = 1 , . . . , r (cid:11) , and patching back a fiber F j , j = 1 , . . . , r , results in an additional relation γ j = 1.Thus, if B has no linear components, the resulting representation for the group π (Σ r ( B ∪ E ∪ F ∞ ) , S ∩ F ) is (cid:10) α , α , α (cid:12)(cid:12) α i = m j ( α i ), i = 1 , , j = 1 , . . . , r (cid:11) . Patching back the remaining fiber F ∞ gives one more relation γ = 1, where γ isthe class of a small loop in S around S ∩ F ∞ ; an expression of γ in terms of α , α , α in the special case of trigonal curves is found below, see Remark in 6.2. Remark.
Van Kampen’s approach applies as well in the case when the curvehas linear components: for each such component, one should keep the correspond-ing generator γ j and keep the relation γ − j α i γ j = m j ( α i ) instead of α i = m j ( α i ). Let B k ⊂ Σ k and B k +1 ⊂ Σ k +1 be two generalized trigonalcurves, so that B k is obtained from B k +1 by an elementary transformation whoseblow-up center O does not belong to B k +1 . Then there is a natural isomorphism π (Σ k r ( B k ∪ E k )) = π (Σ k +1 r ( B k +1 ∪ E k +1 )) , where E i ⊂ Σ i , i = k, k + 1 , are the exceptional sections.Proof. Let F k +1 ⊂ Σ k +1 be the fiber through O , and let F k ⊂ Σ k be the fibercontracted by the inverse elementary transformation. The diffeomorphism (2.1)induces an isomorphism π (Σ k r ( B k ∪ E k ∪ F k )) = π (Σ k +1 r ( B k +1 ∪ E k +1 ∪ F k +1 )) . The group π (Σ k r ( B k ∪ E k )) is obtained from π (Σ k r ( B k ∪ E k ∪ F k )) by addingthe relation [ ∂ Γ k ] = 1, where Γ k ⊂ Σ k is a small analytic disk transversal to F k anddisjoint from all other curves involved. Similarly, patching the fiber F k +1 resultsin an additional relation [ ∂ Γ k +1 ] = 1, where Γ k +1 ⊂ Σ k +1 is a small analyticdisk transversal to F k +1 and disjoint from the other curves in Σ k +1 . Under theassumptions, one can choose Γ k +1 passing through the blow-up center O ; then itsproper transform can be taken for Γ k . Hence, one has [ ∂ Γ k ] = [ ∂ Γ k +1 ], and thetwo quotient groups are isomorphic. (cid:3) Let C ⊂ P be an algebraic curve of degree m with a distin-guished singular point O of multiplicity ( m − and without linear components.Assume that C has a branch b at O of type E . Then C is irreducible and thefundamental group π ( P r C ) = Z m is abelian. ARISKI k -PLETS VIA DESSINS D’ENFANTS 17 Proof.
Blow O up and consider the proper transform B ⊂ Σ of C , see 2.5. Thetransform of b is a type E singular point of B , and the elementary transformationcentered at this point converts B to a generalized trigonal curve B ⊂ Σ with atype ˜ A ∗∗ singular fiber. In particular, the curve is irreducible.The inverse transformation is as in Proposition 5.2.2, i.e. , its blow-up centerdoes not belong to the curve B or the exceptional section E . Hence, one has π ( P r C ) = π (Σ r ( B ∪ E )) = π (Σ r ( B ∪ E )) . (The first isomorphism is obvious; the second one is given by Proposition 5.2.2.)The last group can be found using van Kampen’s method, see 5.2. Under anappropriate choice of the generators α , α , α , the braid monodromy m about atype ˜ A ∗∗ singular fiber is τ ∈ B , and the relations m ( α i ) = α i , i = 1 , ,
3, yield α = α = α . Hence, the group is abelian. (cid:3) Let C be the union of an irreducible curve as in Proposi-tion 5.2.3 and r > linear components none of which is tangent to the branch b of type E . Then one has π ( P r C ) = Z × h γ , . . . γ r − i . In particular, if r ,the group is still abelian.Proof. As in the proof of Proposition 5.2.3, there is a relation α = α = α ,and due to the properties of the braid monodromy (each generator is taken to aconjugate of a generator) the relations γ − j α i γ j = m j ( α i ) turn into [ γ j , α i ] = 1. (cid:3) The braid monodromy
In this section, we describe the braid monodromy of a simplified trigonal curve.We fix such a curve B ⊂ Σ = Σ k and let Γ = Γ( B ). Further, we denote by F z thefiber over a point z ∈ P , and let B z = B ∩ F z and E z = E ∩ F z , where E ⊂ Σ isthe exceptional section. Note that F z r E z is an affine space over C ; in particular,one can speak about its orientation, lines, circles, angles, and length ratios. Weuse the notation F ◦ z for the punctured plane F z r ( B z ∪ E z ). The definition of the j -invariant gives an easyway to recover the topology of B from its dessin Γ. The set B z consists of a singletriple point if z is a singular • - or ◦ -vertex. If z is a × -vertex, B z consists of twopoints, one simple and one double. In all other cases, B z consists of three simplepoints, whose position in F z r E z can be characterized as follows.(1) If z is an inner point of a region of Γ, the three points of B z form a trianglewith all three edges distinct. Hence, the restriction of the projection B → P to the interior of each region of Γ is a trivial covering.(2) If z belongs to a dotted edge of Γ, the three points of B z are collinear.The ratio (smallest distance) / (largest distance) is in (0 , ); it tends to 0(respectively, ) when z approaches a × - (respectively, ◦ -) vertex.(3) If z belongs to a solid (bold) edge of Γ, the three points of B z forman isosceles triangle with the angle at the vertex less than (respectively, greater than) π/
3. The angle tends to 0, π/
3, or π when z approaches,respectively, a × -, • -, or ◦ -vertex.Furthermore, a simple model example proves the following statement.4 For a point z as in (1), arrange the vertices of B z in ascending order basedon the length of the opposite edge. The resulting orientation of B z iscounterclockwise if and only if ℑ j B ( z ) > To define the braid monodromy, we need to fix a ‘fiber atinfinity’ F ∞ , see 5.2, and a generic section S that would provide the base points S z = S ∩ F z ∈ F ◦ z . We take for F ∞ the fiber over a fixed point ∞ / ∈ Γ, andconstruct S as a small perturbation of E + kF ′ , where F ′ is the fiber over apoint z ′ in the same open region of Γ as ∞ . If the perturbation is sufficientlysmall, the section S has the following property: there is a closed neighborhood K ∋ ∞ disjoint from Γ and such that, for each point z ∈ P r K , the base point S z ∈ F z is outside a disk U z ⊂ F z containing B z and centered at its barycenter (cf. Figure 1, right, below ) . In what follows, a section S satisfying this propertyis called proper and, when speaking about the fundamental group π ( F ◦ z , S z ), wealways assume that the point z is outside the above closed neighborhood K .Note that, together with the exceptional section E and the zero section given by z (the barycenter of B z ), a proper section S gives a trivialization of the rulingover P r K , which is necessary to define the braid monodromy. Remark.
From the construction of a proper section S it follows that theclass γ of a small loop in S surrounding F ∞ ∩ S (see 5.2) is, up to conjugation,given by γ = ( α α α ) k γ . . . γ r . Hence, in this case, the final relation in vanKampen’s method is ( α α α ) k = 1. Let z ∈ Γ be a nonsingular • -vertex.According to 6.1(3), the three points of the set B z form an equilateral triangle.There is a natural one-to-one correspondence between the bold edges incident to z and the points of B z : an edge e corresponds to the point p ∈ B z that turns into thevertex of the isosceles triangle when z slides from its original position along e . Infact, the same point p turns into the vertex of the isosceles triangle when z slidesalong the solid edge e ′ opposite to e , so that the two other points are broughttogether over the × -vertex ending e ′ .In what follows, we always assume that the three bold edges e , e , e incidentto z are oriented in the counterclockwise direction, as in Figure 1, left. Such anordering is called a marking at z , and an edge e i incident to z is said to have index i at z . A marking at z is uniquely determined by assigning an index to oneof the three bold edges incident to z . Alternatively, a marking is determined byassigning an index to one of the three points constituting B z .A marking of a dessin Γ is defined as a collection of markings at each nonsingular • -vertex of Γ. The notion of marking and index of edges extends to skeletons inthe obvious way. ARISKI k -PLETS VIA DESSINS D’ENFANTS 19 Using 6.1(1)–(3), from 6.1(6 .
1) it follows that if e , e , e is a marking at anonsingular • -vertex z , the corresponding points p , p , p ∈ B z form the clockwiseorientation of the triangle B z ( Figure 1, right ) . e e e e ′ e ′ e ′ S z ∂U z p p p Figure 1.
A canonical basis for G z Pick a proper section S , see 6.2, and consider the group G z = π ( F ◦ z , S z ).A canonical basis for G z is a basis α , α , α shown in Figure 1, right, wherethe space F ◦ z is regarded as the affine line F z r E z punctured at p , p , p ∈ B z . More precisely, each element α i is the class of the loop formed by a smallcounterclockwise circle about p i , i = 1 , ,
3, which is connected to S z by a radialsegment, an arc of a circle ∂U z separating S z from B z ( cf. π/ π . As a result, a canonicalbasis α , α , α is determined by a marking at z uniquely up to conjugation by α α α , i.e. , up to the central element τ ∈ B . A canonical basis defines an isomorphism ρ z : G z → G to the ‘standard’ freegroup G = h α , α , α i . This isomorphism is determined by a marking at z upto τ . Below, all braids involving ρ z are considered up to a power of τ . Theisomorphism ρ ′ z defined by the cyclic permutation e , e , e of the bold edges isgiven by ρ ′ z = τ ◦ ρ z . Remark.
In a similar way, one can define a canonical basis and isomorphism ρ z : G z → G for a nonsingular ◦ -vertex z . The basis and the isomorphism aredetermined up to a power of τ by an ordering of the two bold edges incident to z .Our choice of • -vertices is motivated by the fact that we will apply the results toskeletons. For the rest of this section, we make the fol-lowing assumptions about Γ:(1) Γ has no monochrome vertices, all its • -vertices have valency
6, and allits ◦ -vertices have valency • - and ◦ -vertices of Γ and its bold edges is connected;(3) Γ has at least one nonsingular • -vertex. Note that Condition (1) means that the curve is generic within its fiberwise defor-mation class, and (2) can be satisfied after a sequence of equisingular perturbationsand degenerations, cf. [15]. Thus, the only true restriction is (3). In particular,any maximal dessin satisfies (1) and (2), and the remaining Condition (3) rules outfour series of maximal curves: those whose skeleton is a simple cycle (one curvein Σ k for each k >
1) or a linear tree (two curves in Σ and three curves in Σ k for k >
2; a curve is determined by the number of hanging edges in the skeleton). Allthese curves are irreducible.Chose and fix the ‘fiber at infinity’ F ∞ over a point ∞ / ∈ Γ and a propersection S , see 6.2. Denote by S ◦ ⊂ P ∼ = S the affine plane P r ∞ puncturedat the singular fibers of B . (Since S is a section, S ◦ can as well be regarded as asubset of S .)As above, let G = h α , α , α i be the free group on three generators. Fix amarking of Γ and consider the corresponding isomorphisms ρ z : G z → G , see 6.3.Given a path γ in S ◦ connecting two nonsingular • -vertices z ′ and z ′′ , consider themonodromy ˜ m γ : G z ′ → G z ′′ and define the automorphism m γ = ρ z ′′ ◦ ˜ m γ ◦ ρ − z ′ of G . It is a braid (due to the fact that S is proper). We consider m γ as an elementof the reduced group B /τ , thus removing the ambiguity in the definition of ρ . Inthe special case z ′ = z ′′ , i.e. , when γ is a loop, m γ is a well defined element of B .It can be recovered from its image in B /τ using the following obvious statement. The degree of the monodromy ˜ m γ : G z → G z defined by asimple loop γ in S ◦ is equal to the total multiplicity P mult F i ( see Table 1 ) of thesingular fibers of B encompassed by γ , i.e. , separated by γ from ∞ . (cid:3) To uniformize the formulas below, we use the convention e t + i = e i , i = 1 , , l ∈ Z , for the ordered edges incident to a given nonsingular • -vertex ( cf. similar convention for the braid group in 5.1).Let z ′ , z ′′ be two nonsingular • -vertices, connected by the path γ in Γ formedby two bold edges incident to the same ◦ -vertex. Denote m γ = m i,j ∈ B /τ ,where i , j are the indices of the edges constituting γ at z ′ and z ′′ , respectively.Then(6.1) m i,i +1 = σ i , m i +1 ,i = σ − i , and m i,i = σ i σ i − σ i . More generally, let s > γ be a simple path from z ′ to z ′′ composed of 2 s bold edges, ( s + 1) ◦ -vertices, and s • -vertices of valency 4.Perturb γ so that each singular • -vertex is circumvented in the counterclockwisedirection, and denote by m i,j ( s ) ∈ B /τ the resulting monodromy. Then, for allintegers s, t >
0, there is a reciprocity relation(6.2) m j +1 ,i ( t ) · m i +1 ,j ( s ) · σ s + t +2 i = 1 , which can be used to find m ∗ , ∗ ( s ) in terms of m ∗ , ∗ (0) = m ∗ , ∗ . One has m i,i +1 ( s ) = σ − si +1 σ i , m i +1 ,i ( s ) = σ − s − i , and m i,i ( s ) = σ − s − i σ − i +1 . Now, let γ be the loop composed of a small counterclockwise circle around a × -vertex of valency 2 d connected along a solid edge e ′ i (see Figure 1, left) to a ARISKI k -PLETS VIA DESSINS D’ENFANTS 21 nonsingular • -vertex z . The resulting monodromy c i ( d ) = m γ ∈ B is given by(6.3) c i ( d ) = σ di +1 . Finally, consider a chain of distinct bold edges starting from an edge e i at anonsingular • -vertex z and ending at a singular vertex. Let γ be a simple loopat z encompassing all vertices of the chain (except z itself) and oriented in thecounterclockwise direction, and let l i ( d ) = m γ ∈ B be the monodromy, where d = deg l i ( d ). (If the chain contains s • -vertices of valency 4, then d can take thevalues 4 s , 4 s + 2, or 4 s + 3, depending on whether the chain ends at a • -vertex ofvalency 4, ◦ -vertex, or • -vertex of valency 2.) One has(6.4) l i (4 s ) = σ − si σ − si − τ s and l i (4 s + ǫ ) = σ − s − ǫi σ − i +1 τ s +3 , where ǫ = 2 or 3. But for the choice of the trivialization of the ruling, which is alsoaccountable for the τ -ambiguity, the monodromy m γ is local with respect to γ ,and it can be found using the description of the geometry of the fibers given in 6.1.We do use this straightforward approach to establish relations (6.1) and (6.3). Theexpression for l i (4 s ) in (6.4) follows from Proposition 6.4.1 and the obvious relation l i (4 s ) = m j,i ( s ) · m i,j ( s ) , j ∈ Z , in B /τ , which is due to our convention that the paths are perturbed so as tocircumvent all singular vertices in the counterclockwise direction.For the rest, we observe that the monodromy related to a fragment of Γ can befound in any other dessin containing this fragment. The reciprocity relation (6.2)is obtained assuming that the two paths resulting in the two m ∗ , ∗ monodromiesform the boundary (oriented in the clockwise direction) of a single region R of theskeleton of the dessin, so that R contains a single × -vertex. (The factor σ s + t +2 i inthe relation is, in fact, c i − ( s + t + 2).) The expressions for l i are obtained in asimilar way: we close the unused bold edges e i − , e i +1 at z ‘around’ the chain ofedges in question and place a single × -vertex at the center of the resulting region R .Computing the monodromy around ∂R gives the relations c i − (2 s + 5 − ǫ ) · l i (4 s + ǫ ) = l i (4 s + ǫ ) · c i +1 (2 s + 5 − ǫ ) = m i − ,i +1 in B /τ (where ǫ = 2 or 3), which can be used to find l i . A maximal trigonal curve B is reducible if and only if all ver-tices of its skeleton Sk are nonsingular (i.e. , have valency and Sk admits amarking with the following properties :(1) each hanging edge has index at the ( only ) vertex incident to it ;(2) any other edge has indices (1 , , (2 , , or (3 , at its two endpoints. Remark.
Clearly, a marking at any vertex of Sk extends to at most onemarking satisfying Condition 6.6.1(2). If Sk has a hanging edge, it admits at mostone marking satisfying 6.6.1(1) and (2).
Remark.
Corollary 6.6.1 still makes sense for a trigonal curve B , not nec-essarily maximal, whose dessin Γ satisfies Conditions 6.4(1) and (2). In this case,the existence of a marking as in Corollary 6.6.1 is necessary for B to be reducible;in general, it is not sufficient. Proof.
Let B ◦ be the portion of the curve over S ◦ , and let pr : B ◦ → S ◦ be therestriction of the projection Σ k → P . It is a triple covering whose monodromyis obtained by downgrading the braid monodromy to the symmetric group S .From (6.4) it follows that the monodromy about a singular • -vertex acts transi-tively on the decks of pr, and hence any curve with such a vertex is irreducible. (Asthis argument is local, it applies as well to the four exceptional series mentionedin 6.4, proving that they are all irreducible.)Assume that B is reducible. Then it contains as a component a section of theruling. Any such section B ⊂ B defines a marking of Sk: one assigns index 1to the point B ∩ F z ∈ B z , see 6.3, and Conditions 6.6.1(1) and (2) merely listall monodromies l i (3) and m i,j preserving p . Conversely, for any marking as inthe statement, the points p ∈ B z over all • -vertices z ∈ Γ( B ) belong to a deckof pr which is preserved by the monodromy. (One needs to take into account theobvious fact that, for a maximal curve without singular • -vertices, the inclusionhomomorphism π (Sk) → π ( S ◦ ) is an isomorphism.) Hence, the curve containsa section of the ruling as a component. (cid:3) The construction
Proofs of Theorems 1.1.1 and 1.1.3 are based on the existence of large Zariski k -plets of maximal trigonal curves in Hirzebruch surfaces. For each integer k > , there exists a collection of C ( k −
1) = 1 k (cid:18) k − k − (cid:19) pairwise distinct fiberwise deformation families of irreducible maximal trigonalcurves B ⊂ Σ k with the following properties :(1) each curve has one fiber of type ˜ A ∗∗ , one fiber of type ˜ A k − , and k fibersof type ˜ A ∗ ( and no other singular fibers );(2) none of the curves admits a fiberwise self-deformation inducing a non-trivial permutation of the singular fibers of the curve.Proof. Denote by T s , s >
1, the set of all binary rooted trees on s vertices. Recallthat the cardinality of T s is given by the Catalan number C ( s ), T s = C ( s ) = 1 s + 1 (cid:18) ss (cid:19) . Each tree T ∈ T s admits a standard ‘monotonous’ geometric realization | T | ⊂ R ,see Figure 2, left. For example, one can map the level l , l >
0, vertices of T to thepoints v l,i = ( − i + 1) / l , l ), i = 0 , . . . , l −
1, so that the left (respectively,right) edge originating at v l,i connects v l,i to v l +1 , i (respectively, v l +1 , i +1 ). ARISKI k -PLETS VIA DESSINS D’ENFANTS 23 Figure 2.
Extending a binary tree T to a skeleton Sk( T )Pick a tree T ∈ T k − and extend its geometric realization | T | ⊂ R ⊂ P toa skeleton Sk( T ) as follows: mark the root of T by adding a monovalent vertexat (0 , −
1) and connecting it to v , by an edge, and complete the valency of eachvertex of | T | to three by replacing the missing branches with ‘leaves’, each leafconsisting of a vertex (at an appropriate point v l,i , l > v l − , [ i/ . (See Figure 2, right, wherethe trunk and the k leaves added to | T | are shown in grey.)The resulting skeleton Sk( T ) has one monovalent and (2 k −
1) trivalent vertices;its faces are k monogons (the interiors of the leaves) and one (5 k − T ) has a nontrivialautomorphism and that two skeletons Sk( T ), Sk( T ) are isomorphic if and onlyif T = T in T k − . Here, the key observation is the fact that the root of theoriginal tree T is ‘marked’ by the only monovalent vertex of the skeleton Sk( T ).Hence, any isomorphism of the skeletons would induce an isomorphism of oriented rooted trees (as it also preserves the orientation of S ). In particular, essentially byits very definition, an oriented rooted tree never admits an orientation preservingautomorphism.Applying Theorem 4.5.1, one obtains T k − = C ( k −
1) deformation familiesof maximal trigonal curves with the desired properties. (Each curve is irreduciblesince it has a type ˜ A ∗∗ singular fiber.) (cid:3) If k = 2 s is even, then C ( s − of the trigonal curves givenby Proposition 7.0.4 are real ( with respect to some real structure on Σ k ) . All othercurves ( and all curves for k odd ) split into pairs of complex conjugate curves.Proof. A maximal trigonal curve is real if and only if its skeleton is symmetricwith respect to some orientation reversing involution of the base S ( cf. [15], § j -invariant, andthe further passage from the j -invariant to a trigonal curve is equivariant, cf. theproof of Proposition 3.4.1). A binary rooted tree can be symmetric only if itsnumber of vertices is odd, and all symmetric trees in T s − can be parametrizedby their ‘left halves’, i.e. , by T s − . (cid:3) In this section, we apply the results of 6.5 todescribe the braid monodromy of the curves given by Proposition 7.0.4.Fix a curve B corresponding to a tree T ∈ T k − and let Γ = Γ( B ), Sk = Sk( B ).Let v , be the root of the original tree T . Denote by Γ × the set of × -verticesof Γ of valency 2 (equivalently, the set of type ˜ A ∗ singular fibers of B ). Eachvertex u ∈ Γ × can be encoded by a word w u in the alphabet { r, l } as follows: let¯ u be the • -vertex in the leaf encompassing u , and let ξ u be the simple path in Skfrom v , to ¯ u ; starting from v , and the empty word, walk along ξ u and, at eachvertex, add to the word r or l if the right (respectively, left) branch is chosen atthis vertex. (For example, in Figure 2, the × -vertices encompassed by the fiveleaves are encoded, from right to left, by the words rr , rl , lr , llr , and lll .) OrderΓ × lexicographically, with r < l . (This is the right to left order in the standardgeometric realization of the graph, cf. Figure 2.)As in 6.4, pick a point ∞ ∈ P r Γ and denote by S ◦ the plane P r ∞ puncturedat the singular vertices of Γ. Take v , for the base point, and consider the basis γ u , u ∈ Γ × , δ × , δ • for π ( S ◦ , v , ) defined as follows:(1) γ u , u ∈ Γ × , is the loop in Sk formed by the circumference of the leafsurrounding u connected to v , by the simple path ξ u ;(2) δ × is a small circle surrounding the × -vertex of valency 10 k −
4, connectedto v , by the left solid edge at v , ;(3) δ • is a small circle surrounding the singular • -vertex, connected to v , bythe bold edge.(All loops are oriented in the counterclockwise direction.) Then, the braid mon-odromy π ( S ◦ , v , ) → B is given by the following relations: γ u ¯ w u σ ¯ w − u , δ × σ k − , δ • σ σ , where ¯ w u is the braid obtained from w u by replacing each instance of r and l with σ and σ − , respectively. Proof.
We choose a marking at each nonsingular • -vertex of Sk so that e is theedge pointing downwards (and hence e and e are, respectively, the left and rightbranches of the tree). Then δ × c (5 k − δ • l (2), see (6.4), andthe image of each element γ u is found by composing appropriate monodromies m i,j , see (6.1). (cid:3) Let k and ǫ be as in the statement. Note that k > ǫ [ k/ B ⊂ Σ k given by Proposition 7.0.4.In order to convert B to a plane curve, we need to perform ( k −
1) elementarytransformations. We choose the transformations so as to contract the type ˜ A ∗∗ fiber of B , [ k/
2] of its k type ˜ A ∗ fibers, and [( k − /
2] nonsingular fibers. Inthe type ˜ A ∗∗ fiber and ǫ type ˜ A ∗ fibers the blow-up centers are chosen outsideof the curve and the exceptional section; in each other fiber the blow-up center istaken on a branch of B transversal to the fiber. The total number of deformation ARISKI k -PLETS VIA DESSINS D’ENFANTS 25 families thus obtained is Z ( m ) = C ( k − · (cid:18) k [ k/ (cid:19) · (cid:18) [ k/ ǫ (cid:19) , the three factors standing, respectively, for the choice of B , the choice of [ k/
2] ofits k type ˜ A ∗ fibers to be contracted, and the choice of ǫ of the [ k/
2] fibers wherethe blow-up center is not on B . In each case, the transform is an irreducible curve˜ C = B ⊂ Σ with the set of singularities(7.1) A k − + E + ǫ D + (cid:16)h k i − ǫ (cid:17) A + h k − i A , so that all points except the first A k − are in the exceptional section E ⊂ Σ andthe local intersection index of ˜ C and E at each singular point is minimal possible( i.e. , 2 at a double point and 3 at a triple point). Blowing E down, one obtainsan irreducible plane curve C of degree 2 k + 2 + ǫ = m . Since the combinatorialdata of C are determined by the those of ˜ C + E , all curves thus obtained sharethe same set of singularities.The fundamental groups π ( P r C ) are all abelian due to Proposition 5.2.3. (cid:3) A real curve C is obtained from a real trigonalcurve B ⊂ Σ k ; hence, k = 2 s is even and the number of real trigonal curves isgiven by Proposition 7.0.5. Next, one should choose a real ( i.e. , invariant underthe complex conjugation) collection of blow-up centers for the elementary trans-formations converting B to ˜ C , see 7.2. Since the k type ˜ A ∗ singular fibers of B split into k/ s conjugate pairs and k/ s blow-up centers should be chosenin these fibers, s must also be even, s = 2 t , and the number of choices is (cid:0) st (cid:1) : onechooses t of the s conjugate pairs. Finally, ǫ = 0 as one cannot choose only onespecial fiber with the blow-up center not on the curve: for the transformation tobe real, the conjugate fiber would have to have the same property. (cid:3) Remark.
It is worth mentioning that, in the settings of Theorem 1.1.2,each deformation class containing a real curve splits into at least t equisingularreal deformation classes. Indeed, let P R ⊂ P be the real part of the base of theruling. It contains the singular • -vertex of Γ, the root of the original tree, and the × -vertex of Γ of valency 10 k −
4. Thus, the singular fibers of B divide P R intotwo distinguishable intervals, and each of the 2 t − a, b ) ∈ Z × Z suchthat a, b > a + b t −
2, and a + b is even. It is t . The proof is similar to that of Theorem 1.1.1. Let k = m −
5, and pick one of the trigonal curves B ⊂ Σ k given by Proposition 7.0.4.Blow up the only singular point of B and blow down the corresponding fiber.Repeat this procedure ( k −
2) times. The result is an irreducible curve B ⊂ Σ which intersects the exceptional section at a nonsingular point P with multiplicity( k −
2) and has a type A k +1 singular point Q in the fiber F P through P . Now, add the fiber F P as a component, perform an elementary transformation to contractthe type ˜ A ∗∗ fiber of B to a type E singular point in the exceptional section,and blow down the exceptional section. (The fiber F P is added to the curve tomake sure that, during the deformations, the intersection point P and the singularpoint Q remain in the same fiber.) The result is a plane curve C of degree m .Clearly, all C ( k −
1) curves obtained in this way share the same combinatorialdata. The fundamental groups π ( P r C ) are all abelian due to Proposition 5.2.4.The count for the number of real curves is based on Proposition 7.0.5: fromthe construction it follows that a family contains a real curve if and only if theoriginal trigonal curve B is real. (cid:3) From the construction (creating a branch oftype E and, in 7.4, adding a linear component) it follows that any equisingulardeformation of the plane curve C must preserve the type ˜ A ∗∗ and type ˜ A k − singular fibers of the original trigonal curve B . Since all other singular fibers of B are of type ˜ A ∗ and B is maximal, the resulting deformation of B is fiberwise, seeProposition 4.4.8. The blow-up centers chosen in the branches of B transversalto its type ˜ A ∗ singular fibers (see 7.2) should stay fixed, as otherwise the type ofsingularity of C at O would change. (This observation is also crucial in the proofof Theorem 8.0.2 below.) In 7.2, a blow-up center in a nonsingular fiber of B may move to a type ˜ A ∗ singular fiber (not containing another blow-up center),to the branch of B tangent to the fiber. This degeneration corresponds to one ofthe branches of one of the type A points of ˜ C , see (7.1), becoming tangent tothe fiber; it is equisingular for C . Clearly, these modifications do not affect thenumber of deformation families. Further applications
In this section, we present a slight modification of the construction used inProposition 7.0.4 and discuss a few further applications.
For each integer k > , there exists a collection of C ( k − pairwise distinct fiberwise deformation families of pairs ( B, F ) , where B ⊂ Σ k isan irreducible maximal trigonal curve with one fiber of type ˜ A k − and ( k + 1) fibers of type ˜ A ∗ ( and no other singular fibers ) and F is a distinguished type ˜ A ∗ fiber of B . None of the curves admits a fiberwise self-deformation inducing anon-trivial permutation of its singular fibers preserving F .Proof. Modify the construction of Proposition 7.0.4 by replacing the monovalentvertex with an extra leaf attached to the root of the original tree T and selectingthe corresponding type ˜ A ∗ fiber for F . All curves obtained are irreducible dueto Corollary 6.6.1: to show that a marking as in the corollary does not exist, itsuffices to consider the two leaves attached to any maximal (in the partial orderdefined by level) vertex of T . (cid:3) ARISKI k -PLETS VIA DESSINS D’ENFANTS 27 (Rigid plane curves) . For each odd integer m = 2 k + 1 > ,there is a set of singularities shared by Z ar ( m ) > (cid:18) k − k − (cid:19) pairwise distinct equisingular deformation families of irreducible plane curves C i of degree m . Within each family, all curves are projectively equivalent and definedover an algebraic number field. Remark.
The set of singularities constructed in the proof has a point oftype A k − and a point of transversal intersection of ( k −
1) branches of type A .One has Z ar (5) = 1, and the only curve of degree 5 given by the theorem is the wellknown quintic with the set of singularities A + A , see [10]; it is defined over Q .(Note that in this case the fundamental group π ( P r C ) is abelian, see [12].) Forlarge values of m , the count Z ar ( m ) grows faster than a m for any a < Remark.
The curves given by Theorem 8.0.2 seem to be good candidates forexamples of exponentially large arithmetic Zariski k -plets in the sense of Shimada,see [29], [30]. At present, I do not know whether all/some of the curves C i areindeed Galois conjugate over an algebraic number field (except the trivial case ofpairs of complex conjugate curves). Whether the pairs ( P , C i ) or complements P r C i are homeomorphic is also an open question. Proof.
Similar to 7.2, we start with a trigonal curve B ⊂ Σ k as in Proposition 8.0.1,perform ( k −
1) elementary transformations to convert Σ k to Σ , and blow downthe exceptional section of Σ to get a plane curve. The ( k −
1) blow-up centersare taken in type ˜ A ∗ singular fibers of B , on the branch of B transversal to thefiber. (This choice makes the construction rigid, so that the resulting plane curveshave 0-dimensional moduli spaces and are defined over algebraic number fields.Indeed, since B itself is defined over a certain algebraic number field k , see remarkafter 4.4.4, all its singular fibers F j are defined over a finite extension of k , andso are the intersection points B ∩ F j . Hence, each curve C i is also defined over afinite extension of k .) The total number of choices is C ( k −
1) (for the pair (
B, F ) )times k ( k + 1) / k −
1) singular fibers containing the blow-upcenters). Since, in each skeleton, the distinguished leaf can be chosen in ( k + 1)ways, we divide the resulting count by ( k + 1). (cid:3) Below, an elliptic surface is a compact complex surface X with a distinguished rational pencil of elliptic curves, i.e. , elliptic fibration over arational base. We assume that the pencil has no multiple fibers; then it is uniqueunless the topological Euler characteristic of X is 24, i.e. , X is a K fiberwise deformation of elliptic surfaces we mean a deformation preservingthe elliptic pencil and the types of its singular fibers. All surfaces mentioned inTheorems 8.1.1 and 8.1.2 are defined over algebraic number fields. For each integer s > , there are C (2 s − distinct fiberwisedeformation families of Jacobian relatively minimal elliptic surfaces of topological Euler characteristic χ = 12 s and having one fiber of type ˜ A ∗∗ , one fiber of type ˜ A s − , and s fibers of type ˜ A ∗ ( and no other singular fibers ) . For each integer s > , there are at least C (2 s − / (2 s + 1) dis-tinct fiberwise deformation families of Jacobian relatively minimal elliptic surfacesof topological Euler characteristic χ = 12 s and having one fiber of type ˜ A s − and (2 s + 1) fibers of type ˜ A ∗ ( and no other singular fibers ) .Proof of Theorems 8.1.1 and 8.1.2. The statements follow from Propositions 7.0.4and 8.0.1 applied to k = 2 s . Each surface is obtained as the minimal resolution ofsingularities of the double covering of Σ k branched over the exceptional section E and a trigonal curve B given by the appropriate proposition. (cid:3) Remark.
Let X be one of the surfaces given by Theorem 8.1.1 or 8.1.2, andlet L = H ( X ) be its intersection lattice. Consider the sublattice S ⊂ L spannedby the components of the pull-back of B ∪ E . Over Q , it is spanned by the sectionof X , its generic fiber, and the exceptional divisors over the only singular pointof B . Hence, S is nondegenerate. The advantage of Theorem 8.1.2 is the fact that,in this case, the orthogonal complement S ⊥ is an even positive definite lattice ofrank 2 s −
2. Given that positive definite lattices tend to have many isomorphismclasses within the same genus, one can hope to use Shimada’s invariant [29] todistinguish the surfaces topologically.
References [1] V. I. Arnol ′ d, A. N. Varchenko, S. M. Guse˘ın-Zade, Singularities of differentiable maps ,vol I. The classification of critical points, caustics and wave fronts, Nauka, Moscow, 1982(Russian) English translation:
Monographs in Mathematics , vol. 82, Birkh¨auser Boston,Inc., Boston, MA, 1985.[2] E. Artal Bartolo, J. Carmona Ruber, J. I. Cogolludo Agust´ın,
On sextic curves with bigMilnor number , Trends in Singularities (A. Libgober and M. Tib˘ar, eds.), Trends in Math-ematics, Birkh¨auser Verlag, Basel/Switzerland, 2002, pp. 1–29.[3] E. Artal Bartolo, J. Carmona Ruber, J. I. Cogolludo Agust´ın,
Effective invariants of braidmonodromy , Trans. Amer. Math. Soc., (2007), no. 1, 165–183.[4] E. Artal Bartolo, J. Carmona Ruber, J. I. Cogolludo Agust´ın, H. Tokunaga,
Sextics withsingular points in special position , J. Knot Theory Ramifications, (2001), no. 4, 547–578.[5] E. Artal Bartolo, H. Tokunaga, Zariski pairs of index and Mordell-Weil groups of K surfaces , Proc. London Math. Soc. (3), (2000), no. 1, 127–144.[6] E. Artal Bartolo, H. Tokunaga, Zariski k -plets of rational curve arrangements and dihedralcovers , Topology Appl., (2004), 227–233.[7] W. Barth, C. Peters, A. van de Ven, Compact complex surfaces , Ergebnisse der Mathematikund ihrer Grenzgebiete (3), Springer-Verlag, Berlin-New York, 1984.[8] M. Bauer, C. Itzykson,
Triangulations , The Grothendieck theory of dessins d’enfants (Lu-miny, 1993) London Math. Soc. Lecture Note Ser., vol. 200, Cambridge Univ. Press, Cam-bridge, 1994, pp. 179–236. (French)[9] J. Carmona,
Monodrom´ıa de trenzas de curvas algebraicas planas , Ph.D. thesis, Universidadde Zaragoza, 2003.[10] A. Degtyarev,
Isotopy classification of complex plane projective curves of degree
5, Algebrai Analis, (1989), no. 4, 78–101 (Russian); English transl. in Leningrad Math. J., (1990),no. 4, 881–904. ARISKI k -PLETS VIA DESSINS D’ENFANTS 29 [11] A. Degtyarev, Alexander polynomial of a curve of degree six , J. Knot Theory Ramifications, (1994), 439–454.[12] A. Degtyarev, Quintics in C p with nonabelian fundamental group , Algebra i Analis, (1999), no. 5, 130–151 (Russian); English transl. in Leningrad Math. J., (2000), no. 5,809–826.[13] A. Degtyarev, On deformations of singular plane sextics , J. Algeb. Geom., (2008), 101–135.[14] A. Degtyarev, Oka’s conjecture on irreducible plane sextics , arXiv:math.AG/0701671 [15] A. Degtyarev, I. Itenberg, V. Kharlamov, On deformation types of real elliptic surfaces ,Amer. J. Math. (to appear), arXiv:math.AG/0610063 [16] C. Eyral, M. Oka, π -equivalent weak Zariski pairs , Tokyo J. Math., (2005), no. 2,499–526.[17] C. Eyral, M. Oka, Fundamental groups of the complements of certain plane non-tame torussextics , Topology Appl., (2006), no. 11, 1705–1721.[18] R. Friedman, J. W. Morgan,
Smooth four-manifolds and complex surfaces , Ergebnisse derMathematik und ihrer Grenzgebiete (3), Springer-Verlag, Berlin-New York, 1994.[19] G.-M. Greuel, C. Lossen, E. Shustin,
The variety of plane curves with ordinary singularitiesis not irreducible , Internat. Math. Res. Notices (2001), no. 11, 543–550.[20] A. Hurwitz,
Uber die Anzahl der Riemannischen Fl¨achen mit gegenbener Verzweigungs-punkten , Math. Ann., (1902), 53.[21] E. R. van Kampen, On the fundamental group of an algebraic curve , Amer. J. Math., (1933), 255–260.[22] K. Kodaira, On compact analytic surfaces, II–III , Annals of Math., On Zariski’s pairs of m -th canonical discriminant curves , arXiv:math.AG/9807154 [24] Vik. S. Kulikov, M. Teicher, Braid monodromy factorizations and diffeomorphism types ,Izv. Ross. Akad. Nauk Ser. Mat., (2000), no. 2, 89–120 (Russian); English transl. in Izv.Math., (2000), no. 2, 311–341.[25] M. Oka, Zariski pairs on sextics.
I, Vietnam J. Math., (2005), Special Issue, 81–92.[26] S. Orevkov, Riemann existence theorem and construction of real algebraic curves , Annalesde la Facult´e des Sciences de Toulouse. Math´ematiques, (6) (2003), no. 4, 517-531.[27] S. Orevkov, Private communications.[28] A. ¨Ozg¨uner, Classical Zariski pairs with nodes , M.Sc. thesis, Bilkent University, 2007.[29] I. Shimada,
Non-homeomorphic conjugate complex varieties , arXiv:math/0701115 [30] I. Shimada, On arithmetic Zariski pairs in degree arXiv:math/0611596 [31] I. Shimada, Private communications[32] A. M. Uluda˘g, More Zariski pairs and finite fundamental groups of curve complements ,Manuscripta Math., (2001), no. 3, 271–277.[33] O. Zariski,
On the problem of existence of algebraic functions of two variables possessing agiven branch curve , Amer. J. Math., (1929), 305–328.[34] O. Zariski, On the irregularity of cyclic multiple planes , Ann. Math., (1931), 485–511. Department of Mathematics, Bilkent University, 06800 Ankara, Turkey
E-mail address ::