On irreducible sextics with non-abelian fundamental group
aa r X i v : . [ m a t h . AG ] N ov ON IRREDUCIBLE SEXTICS WITHNON-ABELIAN FUNDAMENTAL GROUP
Alex Degtyarev
Abstract.
We calculate the fundamental groups π = π ( P r B ) for all irreducibleplane sextics B ⊂ P with simple singularities for which π is known to admit adihedral quotient D . All groups found are shown to be finite, two of them being oflarge order: 960 and 21600.
1. Introduction
Recall that a plane sextic B ⊂ P is said to be of torus type , or tame , if its equation can be represented in the form p + q = 0, where p and q aresome homogeneous polynomials of degree 2 and 3, respectively. Essentially, sexticsof torus type were introduced by O. Zariski as the ramification loci of cubic surfaces.The fundamental group π = π ( P r B ) of an irreducible sextic B of torus typeis known to be infinite; in particular, it is nonabelian. For a long time, no otherexamples of nonabelian groups were known, which lead M. Oka to a conjecture [7]that the fundamental group of an irreducible sextic that is not of torus type is alwaysabelian. The conjecture was disproved in [3], [4], and for the counterexamples forwhich the group π was computed it turned out to be finite, with the exception ofone family with non-simple singularities. Besides, it was also shown that, for eachirreducible sextic that is not of torus type, the abelinization of the commutant of π is finite. (This assertion is a restatement of the proved part of Oka’s conjecture,related to the Alexander polynomial.) Thus, the following statement seems to bea reasonable replacement for the original conjecture. Let B be an irreducible plane sextic with simple singularitiesand not of torus type. Then the fundamental group π ( P r B ) is finite. The fundamental groups of all irreducible sextics with a non-simple singularpoint are found in [3] (the case of a quadruple point) and [4] (the case of a singularpoint adjacent to J ). On the other hand, the construction of sextics with simplesingularities and nonabelian fundamental group π = π ( P r C ) suggested in [3] israther indirect; it proves that π has a dihedral quotient, but it is not suitable tocompute π exactly. In this paper, we attempt to substantiate Conjecture 1.1.1 bycomputing the groups of some of the curves discovered in [3]. Mathematics Subject Classification . Primary: 14H30 Secondary: 14H45.
Key words and phrases.
Plane sextic, non-torus sextic, fundamental group, dihedral covering.Typeset by
AMS -TEX ALEX DEGTYAREV
For a group G , denote by G ′ = [ G, G ] its commutant,or derived group, and let G ′′ = ( G ′ ) ′ etc . We use the notation Z n and D n for,respectively, the cyclic and dihedral groups of order n .To shorten the statements, we introduce the term generalized D n -sextic to standfor a plane sextic B whose fundamental group π ( P r B ) factors to D n , n >
3. A D n -sextic is an irreducible generalized D n -sextic with simple singularities. Recall,see [3], that there are D -, D -, and D -sextics; all D -sextics are of torus type,and the D -sextics form eight equisingular deformation families, one family foreach of the following sets of singularities:4 A , A ⊕ A , A ⊕ A , A ⊕ A , A ⊕ A , A ⊕ A ⊕ A , A ⊕ A ⊕ A , A . The objective of the paper is the computation of the fundamental groups of all D -sextics. The principal result is the following theorem. Let B be a D -sextic. Then the group π = π ( P r B ) is finite.Furthermore, one has π = D ⊗ Z with the following two exceptions :(1) The set of singularities of B is A ⊕ A : then ord π = 960 and one has π ′′ /π ′′′ = ( Z ) and π ′′′ = Z . (2) The set of singularities of B is A ⊕ A ⊕ A : then ord π = 21600 and π ′′ is the only perfect group of order , see [9] . ( In all cases, including the exceptional ones, π/π ′ = Z and π ′ /π ′′ = Z . )According to [3], the D -sextics form two equisingular deformation families,with the sets of singularities 3 A and 3 A ⊕ A . For the former family, the grouphas recently been shown to be D × Z , see [6]. The group of the sextics with theset of singularities 3 A ⊕ A (which are all projectively equivalent) is still unknown.The groups of the D -sextics with the sets of singularities 4 A and 4 A ⊕ A were found independently by C. Eyrol and M. Oka [8]. The starting point of the computation isTheorem 2.1.1, which provides an explicit geometric construction for D -sextics.(In [3], only the existence of D -sextics with the sets of singularities listed above isproven.) We show that each D -sextic B is a double covering of a very particularrigid trigonal curve ¯ B in the Hirzebruch surface Σ ; the curve ¯ B has two type A singular points, and various sets of singularities for B are obtained by varying theramification locus. Theorem 2.1.1 is dealt with in Section 2.The fundamental groups are computed in Section 3: we merely apply the classicalapproach due to van Kampen to the ruling of the Hirzebruch surface Σ . In a fewdifficult cases, the resulting representations are studied using GAP .Section 4 is not directly related to Theorem 1.2.1: we use the representationgiven by Theorem 2.1.1 to produce explicit equations for D -sextics. The equationdepends on three parameters a, b, c ∈ C , see (4.2.1); we analyze the parameter spaceand describe the triples ( a, b, c ) resulting in particular sets of singularities. I am thankful to the organizers of the Fourth Franco-Japanese Symposium on Singularities held in Toyama in August, 2007 for theirhospitality and for the excellent working conditions. A great deal of time consumingcomputations used in the paper was done using
GAP and
Maple , and I am taking thisopportunity to extend my gratitude to the creators of these indispensable softwarepackages.
N IRREDUCIBLE SEXTICS WITH NON-ABELIAN FUNDAMENTAL GROUP 3
2. The construction
Recall that any involutive automorphism c : P → P has afixed line L = L c and an isolated fixed point O = O c , and the quotient P ( O ) /c isthe rational geometrically ruled surface Σ . (Here, P ( O ) stands for the plane P with O blown up.) The images in Σ of O and L are, respectively, the exceptionalsection E and a generic section ¯ L , so that P ( O ) is the double covering of Σ ramified at ¯ L + E .Recall also that the semigroup of classes of effective divisors on Σ is generatedby E and the class F of a fiber of the ruling. One has E = − F = 0, F ◦ E = 1,and K Σ = − E − F .The principal result of this section is the following theorem. Let B ⊂ P be a D -sextic. Then P admits an involution c preserving B , so that O c / ∈ B , and the image of B in Σ = P ( O c ) /c is a trigonalcurve ¯ B , disjoint from E , with two type A singular points. Theorem 2.1.1 is proved at the end of this section, in 2.5 below. Theorem 2.1.1has a partial converse, see Theorem 3.2.8.
Remark.
The proof of Theorem 2.1.1 given in 2.5 uses the theory of K each D -sextic is symmetric. The fact that each of the eightequisingular deformation families of such sextics admits a symmetric representative can easily be established using the results of Section 3, where each set of singularitieslisted in the introduction is realized by a symmetric D -sextic. K -surface. Let B be a plane sextic with simple singularities.Consider the double covering X → P ramified at B and its minimal resolution ˜ X .Then, ˜ X is a K τ : ˜ X → ˜ X of the covering is aholomorphic anti-symplectic ( i.e. , reversing holomorphic 2-forms) involution.Recall that H ( ˜ X ) ∼ = 2 E ⊕ U is an even unimodular lattice of rank 22 andsignature −
16. For a singular point P of B , denote by D P ⊂ H ( ˜ X ) the set ofclasses of exceptional divisors over P ; we use the same notation D P for the incidencegraph of these divisors, which is an irreducible Dynkin diagram of the same name A – D – E as the type of P . Note that, for a type A singular point, the action of τ ∗ on D P is the only nontrivial symmetry of the graph. Let Σ P ⊂ H ( ˜ X ) be thesublattice spanned by D P ; it is an irreducible negative definite root system.Denote Σ = L Σ P , the summation running over all singular points P of B , andlet S = Σ ⊕ Z h ⊂ H ( ˜ X ), where h ∈ H ( ˜ X ) is the class realized by the pull-backof a generic line in P . One has h = 2, and the sums above are orthogonal. Let˜Σ ⊂ ˜ S ⊂ H ( ˜ X ) be the primitive hulls of Σ and S , respectively. The finite indexextension ˜ S ⊃ S is determined by its kernel K , which is an isotropic subgroup ofthe discriminant group discr S . (For the definition of the discriminant group andits relation to lattice extensions, see V. V. Nikulin [11].) As shown in [2], if B isirreducible, then K ⊂ discr Σ. According to [3], it is the kernel K that essentiallyenumerates the dihedral quotients of π ( P r B ). Let B be a plane sextic with simple singularities, anddenote by ˜ X the covering K c : ˜ X → ˜ X be a holomorphic symplectic( i.e. , preserving holomorphic 2-forms) involution. As is known, ˜ c has eight fixed ALEX DEGTYAREV points, and the quotient Y = ˜ X ′ / ˜ c is again a K X ′ is ˜ X with thefixed points of ˜ c blown up.Since the projection ˜ X → P is the map defined by the linear system h ∈ Pic ˜ X ,the two involutions ˜ c , τ commute if and only if the induced automorphism ˜ c ∗ of H ( ˜ X ) preserves h . In this case, ˜ c descends to an involution c : P → P whichpreserves B . Let O = O c and L = L c . In what follows, we always assume that B does not contain L as a component. We fix the notation ¯ L and ¯ B for the imagesof L and B , respectively, in Σ .Alternatively, if ˜ c ∗ ( h ) = h , then τ descends to an anti-symplectic involution τ Y : Y → Y , and the quotient Y/τ Y blows down to Σ . Let O , B , ¯ B , etc . be as above. If O / ∈ B , then ¯ B ∈ | E + 6 F | isa trigonal curve disjoint from E . If O ∈ B , then ¯ B ∈ | E + 6 F | is a hyperellipticcurve, ¯ B ◦ E = 2 . Proof.
The branch locus of the ramified covering Y → Σ consists of ¯ B , ¯ L , and,if O ∈ B , the exceptional section E . On the other hand, since Y is a K i.e. , it belongs to | E + 8 F | . Since¯ L ∈ | E + 2 F | , the statement follows. (cid:3) Let P be a c -invariant type A p singular point of B , and let ¯ P ∈ Σ be its image. Assume that p > and that ˜ c ∗ acts trivially on D P . Then P ∈ L and ¯ P is a type A p +1 singular point of ¯ B + ¯ L , i.e. , a point of ( p + 1) -fold intersectionof ¯ L and ¯ B at a smooth branch of ¯ B . Conversely, the double covering of a curve ¯ B as above has a type A p singular point. Proof.
Since each curve in D P is preserved by ˜ c (as a set), the intersection pointsof the curves must be fixed by ˜ c . Furthermore, each of the two outermost curvesmust contain one more fixed point of ˜ c . Blowing up the fixed points, one obtains asequence of rational curves with the following incidence graph: ❝ s ❝ s . . . s ❝ s ❝ (Here, ❝ and s stand, respectively, for ( − − p + 1.) The ( − c , and the ( − Y is a sequence of (2 p + 1) rational curves whose incidencegraph D is A p +1 . The involution τ Y induced from τ acts on D as the only nontrivialsymmetry; in particular, it is nontrivial on the middle curve. Thus, the projectionto Y/τ Y is a chain of p ( − − A p +1 singular point of the branch locus.The converse statement can be proved by analyzing a local equation of ¯ B + ¯ L ,so that ¯ L = { y = 0 } , and substituting y y . (cid:3) Let P be a c -invariant type A p singular point of B , and let ¯ P ∈ Σ be its image. Assume that either p = 1 or p > and ˜ c ∗ acts nontrivially on D P .Then p = 2 k − is odd and either (1) P ∈ L and ¯ P is a type D k +2 ( type A if p = 1) singular point of ¯ B + ¯ L , or (2) P = O and ¯ P is a type D k +1 singular point ( a pair of type A singularpoints if p = 1) of ¯ B + E . N IRREDUCIBLE SEXTICS WITH NON-ABELIAN FUNDAMENTAL GROUP 5
The type D s singular point above is formed by the section ¯ L or E intersecting ¯ B with multiplicity at a type A s − singular point of ¯ B . Conversely, the doublecovering of a curve ¯ B as above has a type A p singular point. Proof. If p were even, the two middle curves in the exceptional divisor over P wouldintersect transversally at a fixed point of ˜ c . Since ˜ c is symplectic, it cannot trans-pose two such curves. (The differential d ˜ c at each fixed point is the multiplicationby ( − p is odd.The middle curve in the exceptional divisor is fixed by ˜ c ; hence, it contains twofixed points. Blowing them up, one obtains a collection of rational curves with thefollowing incidence graph: ❝ . ❝ . . . . ❝ s . ❝ . . . . ❝❝ (Here, ❝ , . ❝ , and s stand, respectively, for ( − − − p + 2 = 2 k + 1, the action of ˜ c ∗ on the graph is the horizontalsymmetry, and ˜ c fixes the two ( − Y is a collection of ( k + 2) rational curves whose incidence graph is D k +2 (the‘short’ edges corresponding to the two ( − X ).Since τ Y is anti-symplectic, whenever two invariant curves intersect transversallyat a fixed point, exactly one of them is fixed by τ Y pointwise. Hence, the actionof τ Y on the exceptional divisor is determined by its action on the curve with threeneighbors in the incidence graph. Depending on whether this action is trivial ornot, the projection of the exceptional divisor to Y/τ Y has one of the following twoincidence graphs: ❝s ❝ s . . . ❝ or . ❝ ❝ s ❝ . . . Now, depending on the parity of k , these graphs blow down either to a singletype D k +2 singular point of the branch locus (if the last vertex corresponds to a( − − D k +1 singular point on it (if the lastvertex corresponds to a ( − − E ⊂ Σ .The converse statement is proved by analyzing a local equation. (cid:3) For completeness, we also consider the case of type D and E singular points. Let P be a c -invariant singular point of B of type D or E , andlet ¯ P ∈ Σ be its image. Then ˜ c ∗ acts nontrivially on D P ; in particular, P is oftype D q , q > , or E . Furthermore, P ∈ L and ¯ P is, respectively, a type D q − or E singular point of ¯ B + ¯ L ; in the former case, ¯ P is a simple node of ¯ B withone of the branches tangent to ¯ L . Conversely, the double covering of a curve ¯ B asabove has a corresponding type D q or E singular point. Proof.
If ˜ c ∗ acted trivially on D P , then the curve with three neighbors in thediagram would have three fixed points and thus it would be fixed by ˜ c . Hence, theaction is nontrivial. This observation rules out type E and E singular points.The further analysis is completely similar to the proof of Lemma 2.3.3, with theadditional simplification that the diagram in Y is asymmetric and, hence, the curvewith three neighbors is fixed by c Y . We omit the details. (cid:3) ALEX DEGTYAREV
All D -sextics are described in [3]; anysuch curve has ‘essential’ set of singularities 4 A , A ⊕ A , or 2 A plus, possibly,a few other singular points of type A or A .Let B be a D -sextic. Pick a point P = P i of type A (respectively, a point P = Q k of type A ), choose an orientation of its (linear) graph D P , and denote by e i , . . . , e i (respectively, f k , . . . , f k ) the vertices of D P , numbered consecutivelyaccording to the chosen orientation. Let e ∗ ij (respectively, f ∗ kj ) be the dual basisfor Σ ∗ P ⊂ Σ P ⊗ Q . Note that e ∗ ij = − e ∗ i, − j mod Σ P and f ∗ ij = − f ∗ i, − j modΣ P . According to [3], under an appropriate numbering of the singular points andappropriate orientation of their graphs D P , the kernel K of the extension ˜Σ ⊃ Σ isthe cyclic group Z generated by the residue γ = ¯ γ mod Σ, where ¯ γ is given by e ∗ + e ∗ + e ∗ + e ∗ , f ∗ + e ∗ + e ∗ , or f ∗ + f ∗ (for the set of essential singularities 4 A , A ⊕ A , or 2 A , respectively).Define an involution c S : S → S as follows:– h h ,– x x for x ∈ Σ P for P a singular point other than A or A ,– e j ↔ e , − j , e j ↔ e , − j , j = 1 , . . . , f kj ↔ f k, − j , j = 1 , . . . , c S acts identically on the p -primary part of discr S for anyprime p = 5, and the action of c S on the 5-primary part discr S ⊗ F (which canbe regarded as an F -vector space) has two dimensional ( − K as a maximal isotropic subgroup, so that K ⊥ / K can be identified withthe (+1)-eigenspace of c S . Hence, c S extends to an involution ˜ c S : ˜ S → ˜ S , the latteracts identically on discr ˜ S , and the direct sum ˜ c S ⊕ id S ⊥ extends to an involution˜ c ∗ : H ( X ) → H ( X ).By construction, ˜ c ∗ preserves h and, since it acts identically on the transcendentallattice (Pic X ) ⊥ ⊂ ˜ S ⊥ , it also preserves classes of holomorphic forms. Furthermore,˜ c ∗ preserves the positive cone of X . (Recall that the positive cone is an openfundamental polyhedron V + ⊂ (Pic X ) ⊗ R of the group generated by reflectionsdefined by vectors x ∈ Pic X with x = −
2; it is uniquely characterized by therequirement that V + · e > e ∈ S P D P and thatthe closure of V + should contain h .) Due to the description of the fine periodspace of marked K c ∗ is induced by a uniqueinvolutive automorphism ˜ c : X → X , which is symplectic and commutes with thedeck translation τ . The descent of ˜ c to P is the involution c the existence of whichis asserted by Theorem 2.1.1. The involution c : P → P is constructed in 2.4.Due to Lemmas 2.3.1–2.3.3 (and the description of the singularities of B and theaction of ˜ c ∗ on the set of exceptional divisors, see 2.4), the image ¯ B ⊂ Σ is either atrigonal curve with the set of singularities 2 A (and then O / ∈ B ) or a hyperellipticcurve with the set of singularities 2 A or A ⊕ A . The latter possibility is ruledout by the fact that the genus of a nonsingular curve in | E + 6 F | is 3. (cid:3)
3. Calculation of the groups ¯ B . The trigonal curve ¯ B ⊂ Σ with two type A singular pointsis a maximal trigonal curve in the sense of [5]. Up to automorphism of Σ , such N IRREDUCIBLE SEXTICS WITH NON-ABELIAN FUNDAMENTAL GROUP 7 a curve is unique; its skeleton Sk ⊂ P (see [5]) is shown in Figure 1. One canobserve that the skeleton is symmetric with respect to the dotted grey line (thereal structure z ¯ z on P ) and, properly drown, it is also symmetric with respectto the holomorphic involution z
7→ − /z . Hence, the curve ¯ B can be chosen realand symmetric with respect to a real holomorphic involution of Σ (see Section 4below for explicit equations). Furthermore, all singular fibers of ¯ B (two cusps andtwo vertical tangents) are also real. Figure 1 . The skeleton of ¯ B Alternatively, ¯ B can be obtained as a birational transform of a plane quartic C with the set of singularities A ⊕ A , see Figure 2. (Up to automorphism of P ,such a quartic is also unique.) In the figure, the line ( P P ) is tangent to C at P ,and the transformation consists in blowing P up twice and blowing down thetransform of ( P P ) and one of the exceptional divisors over P . Lines through P other than ( P P ) transform to fibers of Σ ; this observation gives one a fairly goodunderstanding of the geometry of ¯ B , see, e.g. , Figure 4. ¯ L Q Q ¯ L Q Q P ( A ) P ( A ) P Figure 2 . The quartic C with the set of singularities A ⊕ A To calculate the fundamental group, we fix a realcurve ¯ B as in 3.1 and choose an appropriate real section ¯ L intersecting ¯ B at realpoints. Let F , . . . , F k be the singular fibers of ¯ B + ¯ L ( i.e. , the fibers intersecting¯ B + ¯ L at less than four points). Under the assumptions, they are all real. In thefigures below, the curves ¯ B and ¯ L are shown, respectively, in black and grey, andthe singular fibers are the vertical grey dotted lines.Fix a real nonsingular fiber F ∞ intersecting ¯ B at one real point, and consider theaffine part Σ r ( E ∪ F ∞ ), cf. Figure 4. Pick a real nonsingular fiber F intersecting ¯ B at three real points and a generic real section S . We identify S with the base of theruling. Let x = S ∩ F , x ∞ = S ∩ F ∞ , and x i = S ∩ F i , i = 1 , . . . , k . Assume that S is proper in the following sense: there is a segment I ⊂ S R containing x and all x i , ALEX DEGTYAREV F R αβγδ F ∩ ¯ Lx S R σ σ σ k − σ k . . . . . . x Figure 3 . The basis α , β , γ , δ and the loops σ i i = 1 , . . . , k , and disjoint from ¯ B and ¯ L . (The usual compactness argument showsthat such a section exists.) In the figures, we assume that I lies above ¯ B and ¯ L .Let G = π ( F r ( ¯ B ∪ E ∪ ¯ L ) , x ), and let α , β , γ , δ be the basis for G shownin Figure 3, left. (All loops are oriented in the counterclockwise direction.) Wealways assume that δ is a loop around F ∩ ¯ L . (Sometimes, the generators shouldbe reordered by inserting δ between α and β or β and γ , cf. δ commutes with all subsequent generators, so thatthe reordering is irrelevant.) Let, further, σ , . . . , σ k be the basis for the group π ( S r { x , . . . , x k x ∞ } , x ) shown in Figure 3, right: each σ i is a small circle about x i connected to x by a real segment l i ⊂ S R bent to circumvent the other singularfibers in the counterclockwise direction. The braid monodromy along a loop σ i , i = 1 , . . . , k , is theautomorphism m i : G → G resulting from dragging F along σ i while keeping thebase point in S . (Since S is proper, m i is indeed a braid.) The group
Π = π (Σ r ( ¯ B ∪ E ∪ ¯ L )) is given by (3.2.3) Π = (cid:10) α, β, γ, δ (cid:12)(cid:12) m i = id , i = 1 , . . . , k , ( αβγδ ) = 1 (cid:11) , where each braid relation m i = id should be understood as a quadruple of relations m i ( α ) = α , m i ( β ) = β , m i ( γ ) = γ , m i ( δ ) = δ . Proof.
The representation (3.2.3) is the essence of van Kampen’s method, see [10],applied to the ruling of Σ . The only statement that needs proof is the last relation( αβγδ ) = 1, resulting from the patching of the fiber at infinity F ∞ . This relationis [ ∂D ] = 1, where D ⊂ S is a small disk around F ∞ ∩ S . If the base fiber F issufficiently close to F ∞ , then [ ∂D ] = ( αβγδ ) , as in this case one can take for S asmall perturbation of E +2 F . In general, due to the properness of S , the translationhomomorphism between any two nonsingular fibers is a braid; hence, it leaves theproduct αβγδ invariant and the relation has the same form for any fiber F . (cid:3) In sections 3.3–3.6 below, we attempt to calculate Π using Proposition 3.2.2. Tofind the braid monodromy m i , we represent it as the local braid monodromy alonga small circle surrounding x i , conjugated by the translation homomorphism alongthe real path l i connecting x i to x . The former is well known: it can be found byconsidering model equations. For the latter, we choose the models so that, at each N IRREDUCIBLE SEXTICS WITH NON-ABELIAN FUNDAMENTAL GROUP 9 moment, all but at most two points of the curve are real; in this case, the resultingbraids are written down directly from the pictures.The following lemma facilitates the calculation by reducing the number of fibersto be considered.
In representation (3.2.3) , ( any ) one of the braid relations m i = id can be ignored. Proof.
The product σ . . . σ k is the class of a large circle encompassing all singularfibers. Hence, m k ◦ . . . ◦ m is the so called monodromy at infinity , which is knownto be the conjugation by ( αβγδ ) . In view of the last relation in (3.2.3), each m i is a composition of the others. (cid:3) For the rest of this section, we fix the notation α , β , γ , δ for generators of thegroup Π = π (Σ r ( ¯ B ∪ E ∪ ¯ L )), chosen as explained above. The generator δ playsa special rˆole in the passage to π = π ( P r B ). The fundamental group π = π ( P r B ) is the kernel of thehomomorphism Π /δ → Z , α, β, γ , δ . Proof.
The statement is a direct consequence of the construction: one considers theappropriate double covering of Σ r ( ¯ B ∪ E ∪ ¯ L ) and patches L . (cid:3) If δ is a central element of Π /δ , then π ( P r B ) = D × Z . Proof.
Since δ is a central element, the relation ( αβγδ ) = 1 (or similar) turnsinto ( αβγ ) = 1 in Π /δ , and each braid relation becomes either trivial or a braidrelation for the group π (Σ r ( ¯ B ∪ E )) = D × Z (the latter group is calculatedin [4]). Hence, Π /δ = ( D × Z ) × Z , and Proposition 3.2.5 applies. (cid:3) Let B be a D -sextic with the set of singularities A . Thenone has π ( P r B ) = D × Z . Proof.
The curve B is the double covering of ¯ B ramified at a section ¯ L transversalto ¯ B . In this case, δ is a central element of π (Σ r ( ¯ B ∪ E ∪ ¯ L )) ( cf. Section 3.3and Figure 4 below for a much less generic situation), and the statement followsfrom Lemma 3.2.6. (cid:3)
Let ¯ B ⊂ Σ be a trigonal curve with two type A singularpoints, and let p : P → Σ /E be the double covering ramified at the vertex E/E and a section ¯ L disjoint from E . Then p − ( ¯ B ) is a generalized D -sextic. Proof.
By perturbing ¯ L to a section transversal to ¯ B , one perturbs B to a generic D -sextic as in Corollary 3.2.7. (cid:3) A . The sextic B is the double covering of ¯ B ramified at a section ¯ L passing through both cusps of ¯ B , see Figure 4; one can takefor ¯ L the transform of the secant ¯ L shown in Figure 2.We choose the generators α , δ , β , γ in a nonsingular fiber between F and F .Then, there are relations [ γ, β ] = 1, β = γ , and [ δ, α ] = 1 (from F , F , and F ,respectively); due to Lemma 3.2.6, the group is D × Z . A ⊕ A . The curve B is the double coveringof ¯ B ramified at a section ¯ L double tangent to ¯ B , see Figure 5. (The existence of adouble tangent section whose position with respect to ¯ B is as shown in the figure F F Q F F Q F F P P Figure 4 . The set of singularities 2 A F F F F F F F F P P Figure 5 . The set of singularities 4 A ⊕ A is rather obvious geometrically: one moves a sufficiently sharp parabola to achievetwo tangency points. An explicit construction of a pair ( ¯ B, ¯ L ) using equations isfound in Section 4.6 and Figure 8 below.)We choose the generators α , β , δ , γ in a nonsingular fiber between F and F .Ignoring the cusp F , see Lemma 3.2.4, the relations for Π are(3.4.1) ( αβ ) α = β ( αβ ) (from the cusp F ),(3.4.2) [ β, δ ] = [ γ, δ ] = 1 (from F and F ),(3.4.3) ( αδ ) = ( δα ) (from F ),(3.4.4) ( α − δαβ ) = ( βα − δα ) (from F ),(3.4.5) γ = ( α − δα ) − β ( α − δα ) (from the tangent F ),(3.4.6) γ − αβα − γ = ( αβ ) α ( αβ ) − (from the tangent F ),(3.4.7) ( αβδγ ) = 1 (patching F ∞ ).(The relations are simplified using (3.4.2).) The corresponding group π given byProposition 3.2.5 was analyzed using the GAP software package. According to
GAP , π is an iterated semi-direct product Z × (((( Z × Q ) ⋊ Z ) ⋊ Z ) ⋊ Z ), and theabelian factors of its derived series are as stated in Theorem 1.2.1. (Here, Q is theorder 8 subgroup {± , ± i, ± j, ± k } ⊂ H .) ⊕ A ⊕ A . The curve B is the double coveringof ¯ B ramified at a section ¯ L inflection tangent to ¯ B , see Figure 6. One can takefor ¯ L the transform of the tangent ¯ L shown in Figure 2; an explicit constructionusing equations is found in Section 4.7 and Figure 8. N IRREDUCIBLE SEXTICS WITH NON-ABELIAN FUNDAMENTAL GROUP 11 F F Q F F Q F F P P Figure 6 . The set of singularities A ⊕ A ⊕ A We choose the generators α , β , γ , δ in a nonsingular fiber between F and F .Ignoring the vertical tangent F , see Lemma 3.2.4, the relations for Π are(3.5.1) β = γ (from the tangent F ),(3.5.2) [ α, γδγ − ] = 1 (from F ),(3.5.3) ( γδ ) = ( δγ ) (from F ),(3.5.4) [ αβ, δ ] = 1 (from the cusp F ),(3.5.5) δ ( αβ ) α = β ( αβ ) δ (from the cusp F ),(3.5.6) ( β γ ) β = γ ( β γ ) (from the cusp F ),(3.5.7) ( αβγδ ) = 1 (patching F ∞ ),where δ = ( γδγ ) δ ( γδγ ) − and β = ( αγδ − γ − ) β ( αγδ − γ − ) − . The resultinggroup π , see Proposition 3.2.5, was analyzed using GAP . Its derived series is asstated in Theorem 1.2.1. ⊕ A ⊕ A and A ⊕ A . For the set ofsingularities A ⊕ A ⊕ A , we perturb the inflection tangency point Q in Figure 6to a simple tangency point and a point of transversal intersection. Then, (3.5.3)is replaced with [ γ, δ ] = 1 and one obtains [ α, δ ] = 1 (from (3.5.2) ), δ = δ , and[ β, δ ] = 1 (from (3.5.4) ); due to Lemma 3.2.6, the resulting group π is D × Z .For the set of singularities 4 A ⊕ A , the intersection point P in Figure 6 isperturbed to two points of transversal intersection. Then, (3.5.4) and (3.5.5) turninto [ α, δ ] = [ β, δ ] = 1 and ( αβ ) α = β ( αβ ) , respectively. Using GAP shows thatthe resulting group π is D × Z . ⊕ A and A ⊕ A . These sextics can beobtained by small perturbations from sextics with the sets of singularities, e.g. ,2 A (see 3.3) and A ⊕ A ⊕ A (see 3.6), respectively; the resulting groups havealready been shown to be isomorphic to D × Z .
4. The equations
The calculations in this section (substitution, factorization, discriminants, andsystem solving) are straightforward but rather tedious. Most calculations wereperformed using
Maple . ¯ B . In appropriate affine coordinates ( x, y ) in Σ the trigonalcurve ¯ B with two type A singular points is given by the Weierstraß equation f ( x, y ) = 4 y − yp ( x ) + q ( x ) = 0 , where(4.1.1) p ( x ) = x − x + 14 x + 12 x + 1 ,q ( x ) = ( x + 1)( x − x + 74 x + 18 x + 1) . The discriminant of (4.1.1) with respect to y is ∆ = (2) (3) x ( x − x − x = 0 and x = ∞ (the singular points of ¯ B ) and two simpleroots x ± = 11 / ± √ / Remark.
The point x = ∞ is a 5-fold root of ∆ as the ‘predicted’ degree of ∆ is 12.Originally, equation (4.1.1) was obtained by an appropriate birational coordinatechange from the equation y − x y + x − x y of the quartic with the set ofsingularities A ⊕ A , see Figure 2.The curve ¯ B is rational; it can be parametrized as follows(4.1.2) x ( t ) = t ( t − t + 1 , y ( t ) = ( t + 1)( t − t − t + 2 t + 1)2( t + 1) . The special points on the curve correspond to the following values of the parameter:(4.1.3) t = 0 , t ∞ = ∞ (the cusps) ,t ′ = 1 , t ′∞ = − ,t ± = − ∓ √ x = x ± ) ,t ′± = 2 ± √ x = x ± ) . Both the curve and the parametrization are real, as are all singular fibers of ¯ B .Furthermore, ¯ B is invariant under the automorphism x
7→ − /x , y y/x . In the t -line, this transformation corresponds to the automorphism t
7→ − /t . Due to Theorem 2.1.1, any D -sextic is given by an affineequation of the form(4.2.1) f ( x, y + x + bx + c ) = 0 , where f ( x, y ) is the polynomial given by 4.1.1 and y = ax + bx + c is the equationof the section ¯ L constituting the branch locus. Conversely, from Theorem 3.2.8it follows that any curve B given by (4.2.1) is a D -sextic provided that it isirreducible and all its singularities are simple. Note that B is reducible (splits intotwo cubics interchanged by the involution on P ) if and only if, at each point ofintersection of ¯ B and ¯ L , the local intersection index is even. Hence, in view ofthe classification of sections given below, B is reducible if and only if it has the(non-simple) set of singularities Y , ⊕ A , see 4.3; such a curve splits into twocubics with a common cusp.The set of singularities of a sextic B given by (4.2.1) with a generic triple ( a, b, c )(so that ¯ L is transversal to ¯ B ) is 4 A . In Sections 4.3–4.7 below, we discuss the pos-sible degenerations of the section ¯ L and express them in terms of the triple ( a, b, c ). N IRREDUCIBLE SEXTICS WITH NON-ABELIAN FUNDAMENTAL GROUP 13 (Sometimes, the condition is stated using an extra parameter t , as an attempt toeliminate t results in a multi-line Maple output.) For each degeneration, we useLemmas 2.3.2 and 2.3.3 to indicate the set of singularities of the correspondingsextic B . The results should be understood as follows: a sextic B given by (4.2.1)has a certain set of singularities Σ if and only if the triple ( a, b, c ) satisfies the con-dition corresponding to Σ but does not satisfy any condition corresponding to animmediate degeneration of Σ (see the adjacency diagram shown in Figure 7). ( W ⊕ A ) ??y ( Y , ⊕ A ) ←−− ( Y , ⊕ A ) ??y ??y A ←−− A ⊕ A ←−− A x?? x?? A ⊕ A −−→ A ⊕ A ←−− A ⊕ A ⊕ A x?? x?? A ⊕ A ←−− A ⊕ A ⊕ A Figure 7 . Immediate adjacencies of sets of singularitiesFor completeness, we also mention (parenthetically in Figure 7) the sextics B given by (4.2.1) whose singularities are not simple; this is the case if and only if thetriple ( a, b, c ) is as in (4.3.2) below. In Arnol ′ d’s notation, B may only have a non-simple singular point of one of the following two types: Y , (transversal intersectionof two cusps) or W (semiquasihomogeneous singularity of type (4 ,
5) ).
A section y = ax + bx + c passesthrough one of the cusps of ¯ B (the set of singularities A ⊕ A ) if and only if(4.3.1) c = 12 (the cusp at t = 0) or a = 12 (the cusp at t = ∞ ) . Hence, the section passes through both cusps (the set of singularities 2 A ) if andonly if a = c = 1 / B at a cusp (the set of singularities Y , ⊕ A )if and only if(4.3.2) c = 12 , b = 3 (at t = 0) or a = 12 , b = − t = ∞ ) . It passes through the other cusp (the set of singularities Y , ⊕ A ; this sexticis reducible) if and only if a = c = 1 / b = ±
3. Finally, the section passesthrough a cusp with local intersection index 5 (the set of singularities W ⊕ A )if and only if(4.3.3) ( a, b, c ) = (cid:16) − , , (cid:17) or ( a, b, c ) = (cid:16) , − , − (cid:17) . Such a section cannot pass through the other cusp.A section passing through both cusps of ¯ B or tangent to ¯ B at a cusp doesnot admit any degenerations other than described above. Indeed, if a = c = 1 / c = 1 / b = 3), then, restricting the original polynomial f ( x, y )to the section and reducing the trivial factor x (respectively, x ), one obtains apolynomial in x whose discriminant is 16( b − ( b + 3) (respectively, 12(2 a − ). Remark.
According to [3], D -sextics are characterized by the existence of twoconics in a very special position with respect to the type A and A singular pointsof the curve. These conics are the pull-backs of the two sections y = ax + bx + c with a = c = 1 / b = ±
3, each section being tangent to ¯ B at one of its cuspsand passing through the other cusp. D -sextics. From 4.3, it follows that thedouble covering construction also produces representatives of the two families ofirreducible generalized D -sextics with a quadruple singular point, see [3]. (In eachfamily, symmetric curves form a codimension one subset.) It is worth mentioningthat the remaining classes of irreducible generalized D -sextics, those with the setsof singularities J , ⊕ A , J , ⊕ A , and J , ⊕ A , see [4], are also related to thetrigonal curve ¯ B ⊂ Σ with two type A singular points: they are obtained from ¯ B by a birational transformation rather than double covering. Let(4.5.1) s ( t ) = ax ( t ) + bx ( t ) + c, where x ( t ) is given by (4.1.3). Solving s ( t ) = y ( t ) and s ′ ( t ) = y ′ ( t ), one concludesthat a section y = ax + bx + c is tangent to ¯ B at a point ( x ( t ) , y ( t )) ∈ ¯ B (the setof singularities 4 A ⊕ A ) if and only if(4.5.2) a = − b ( t + 2 t −
1) + t − t − t − t ( t − t + t − ,c = − bt ( t − t + 1) + 3 t + 5 t − t + 12( t + 1)( t + t − b ∈ C and t ∈ C r { , ± , t ± } or(4.5.3) t = 1 and ( b, c ) = ( − , −
1) or t = − a, b ) = ( − , B at t = 0 (the set of singularities A ⊕ A ⊕ A ) if and only if c = 1 /
2, see 4.3; in this case(4.5.4) a = t − t − t + 3 t + 112( t − ( t + t − , b = − t + 2 t − t − t + t − , c = 12 . The section passes through the cusp of ¯ B at t = ∞ (another implementation of theset of singularities A ⊕ A ⊕ A ) if and only if a = 1 /
2, see 4.3; in this case(4.5.5) a = 12 , b = − t + 2 t + 1)( t + 1)( t + t − , c = − t − t − t + t + 12( t + 1) ( t + t − . Relations (4.5.4) and (4.5.5) still hold for the exceptional values t = − t = 1,respectively, cf. (4.5.3).There is a somewhat unexpectedly simple relation between the two tangencypoints of a section double tangent to ¯ B . We state it below as a separate lemma.Denote by ǫ ± the roots of the polynomial t + 3 t + 1. One has ǫ ± = ( − ± √ / t ± /t ′± . Note that ǫ + ǫ − = 1. N IRREDUCIBLE SEXTICS WITH NON-ABELIAN FUNDAMENTAL GROUP 15
Let ¯ B ⊂ Σ be the trigonal curve parametrized by (4.1.2) , andlet t , t ∈ C r { , t ± } be two distinct values of the parameter. Then, there is asection tangent to ¯ B at both points ( x ( t i ) , y ( t i )) , i = 1 , , if and only if t /t = ǫ ± . Proof.
Assume that t , t = ±
1. (The case when one of t , t takes an exceptionalvalue ± a , a and c , c be the coefficients a and c in (4.5.2) evaluated at t = t , t , respectively. Then a − a = c − c = 0;hence, ( a − a ) t t ( t − t −
1) + ( c − c )( t + 1)( t + 1) = 0. The latterexpression takes the form3( t − t ) ( t + 3 t t + t )( t + t − t + t −
1) = 0 , and, taking into account the restrictions on t , t , one obtains t /t = ǫ ± . Forthe converse statement, one observes that, if t /t = ǫ ± , then the linear system a = a , c = c in one variable b has a solution (given by (4.5.7) below). (cid:3) Thus, a section y = ax + bx + c is double tangent to ¯ B (the set of singularities4 A ⊕ A ) if and only if(4.5.7) b = b ± = − t + (3 ǫ ∓ + 1) t − ǫ ∓ )( t − ǫ ± t − ǫ ∓ )( t + t ± )( t − t ± )( t − t ∓ ) ( t − t ′± ) . Here, the two tangency points are at t and ǫ ± t ; the expressions for a and c areobtained by a direct substitution to (4.5.2). This relation still holds if t = ± B at a smooth point does not admit any other degenerations.Indeed, sections passing through both singular points of ¯ B or tangent to ¯ B at asingular point are considered in 4.3, and sections inflection tangent to ¯ B are treatedin 4.7 below. A section cannot be tangent to ¯ B at three smooth points at t = t , t ,and t , as then one would have t /t = ǫ ± , t /t = ǫ ± , and t /t = ǫ ± ; this systemis incompatible unless t = t = t = 0. Finally, a double tangent cannot passthrough a singular point, say at t = 0, as substituting t = t and t = t to (4.5.4)and eliminating a and b , one obtains a system in ( t , t ) which has no solutionswith t = t . The pair ( ¯ B, ¯ L ) used to calculate thefundamental group of a sextic B with the set of singularities 2 A ⊕ A , see 3.4and Figure 5, can be obtained from (4.5.7) and (4.5.2) with the following values ofthe parameters: t = t = 1 / t = ǫ + t ≈ − . a ≈ − . b ≈ − .
93, and c ≈ . B andthe section y = ax + bx + c are at t ≈ .
281 (over x ≈ − . t ≈ . x ≈ . A section y = ax + bx + c is inflection tangent to ¯ B at a point ( x ( t ) , y ( t )) ∈ ¯ B (the set of singularities 4 A ⊕ A ) if and only if(4.7.1) a = t + 3 t − t + 12 t + 112( t + t − ,b = − t + 1)( t + 3 t − t − t + 1)( t + t − ,c = − t − t + 5 t − t + 12( t + t − , –1.5–1–0.500.5 y–0.12 –0.1 –0.08 –0.06 –0.04 –0.02 0.02 0.04 0.06x Figure 8 . Maple plot of the curve ¯ B (black), a double tangent (solidgrey), and an inflection tangent through a singular point (dotted grey) t ∈ C r { , t ± } . (To see this, one should solve for ( a, b, c ) the system s ( t ) = y ( t ), s ′ ( t ) = y ′ ( t ), s ′′ ( t ) = y ′′ ( t ), where s ( t ) is the section given by (4.5.1).) Thisinflection tangent passes through one of the cusps of ¯ B (the set of singularities A ⊕ A ⊕ A ) if and only if t = 3 / t = 0) or t = − / t = ∞ ). The corresponding values of ( a, b, c ) are(4.7.2) ( a, b, c ) = (cid:16) , , (cid:17) and ( a, b, c ) = (cid:16) , − , (cid:17) , respectively. The section corresponding to t = 3 / B cannot have any other degen-erations. Indeed, after clearing the denominators and reducing the trivial factor( u − t ) , the equation y ( u ) = ax ( u ) + bx ( u ) + c with a , b , and c given by (4.7.1)and x ( · ), y ( · ) as in (4.1.2) has solution u = t only for t = 0 or t ± (hence, noquadruple intersection points), and the discriminant of the above equation withrespect to u is, up to a constant coefficient, t (3 t + 4)(4 t − t + t − (hence,no other tangency points). References
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