A plane sextic with finite fundamental group
aa r X i v : . [ m a t h . AG ] N ov A PLANE SEXTIC WITH FINITE FUNDAMENTAL GROUP
ALEX DEGTYAREV AND MUTSUO OKA
Abstract.
We analyze irreducible plane sextics whose fundamental groupfactors to D . We produce explicit equations for all curves and show that, inthe simplest case of the set of singularities 3 A , the group is D × Z . Introduction
Motivation and principal results.
In this paper, we use the term D n -sextic for an irreducible plane sextic B ⊂ P whose fundamental group π ( P r B )factors to the dihedral group D n , n >
3. The D n -sextics are classified in [3]and [4] (the case of non-simple singularities). The integer n can take values 3, 5,and 7. All D -sextics are of torus type ( i.e. , they are given by equations of the form p + q = 0); in particular, their fundamental groups are infinite. The D -sexticsform 13 equisingular deformation families, and their fundamental groups are known,see [5] and [7]; with one exception, they are all finite. Finally, the D -sextics formtwo equisingular families, see Proposition 2.1.1 below, and their groups are notknown. In this paper, we compute one of the two groups. Our principal result isthe following statement.1.1.1. Theorem.
The fundamental group of the complement of a D -sextic withthe set of singularities A is D × Z . Theorem 1.1.1 is proved in Section 3.3.Our result can be regarded as another attempt to substantiate a modified ver-sion [5] of Oka’s conjecture [6] on the fundamental group of an irreducible planesextic, stating that the group of an irreducible sextic with simple singularities thatis not of torus type is finite. (Note that the finiteness of the group is sufficient toconclude that the Alexander polynomial of the curve is trivial, see, e.g. , [10].)1.2.
Contents of the paper.
In Section 2 we analyze the geometric propertiesof D -sextics, whose existence was proved in [3] purely arithmetically. We usethe theory of K D -sextic admits a Z -symmetry, seeTheorem 2.1.2, and we use this symmetry to obtain explicit equations defining all D -sextics, see Theorem 2.1.3. The curves form a dimension one family, dependingon one parameter t ∈ C , t = 1. Most calculations involving polynomials were doneusing Maple .The heart of the paper is Section 3. We use a particular value t = 5 / t = 1, where the curve degenerates to a triple cubic) andanalyze the real part of the curve obtained. With respect to an appropriatelychosen real pencil of lines, it has sufficiently many real critical values, and we apply Mathematics Subject Classification.
Primary: 14H30; Secondary: 14H45.
Key words and phrases.
Plane sextic, non-torus sextic, fundamental group, dihedral covering. van Kampen’s method (ignoring all non-real critical values) to produce an ‘upperestimate’ on the fundamental group, see Theorem 3.1.1. Comparing the latter withthe known ‘lower estimate’ (the fact that the curve is known to be a D -sextic),we prove Theorem 1.1.1. 2. The construction
Statements. A Z -action on P is called regular if it lifts to a regular rep-resentation Z → GL (3 , C ). An order 3 element c ∈ PGL (3 , C ) is called regular if it generates a regular Z -action. Any regular order 3 automorphism of P hasthree isolated fixed points (and no other fixed points). Conversely, any order 3automorphism c of P with isolated fixed points only is regular (as isolated fixedpoints correspond to dimension one eigenspaces of the lift of c to C ).The following statement is proved in [3] (see also [4], where sextics with non-simple singular points are ruled out).2.1.1. Proposition.
All D -sextics form two equisingular deformation families,one family for each of the sets of singularities A and A ⊕ A . (cid:3) The principal results of this section are the following two theorems.2.1.2.
Theorem.
Any D -sextic B is invariant under a certain regular order automorphism c : P → P acting on the three type A singular points of B by acyclic permutation. Theorem.
Up to projective transformation, the D -sextics form a connectedone parameter family B ( t ); in appropriate homogeneous coordinates, they are givenby the polynomial (2.1) 2 t ( t − z z z + z z z + z z z )+ ( t − z z + z z + z z )+ t ( t − z z + z z + z z )+ 2 t ( t + 1)( z z + z z + z z )+ 4 t ( t + 2)( z z z + z z z + z z z )+ 2( t + 4 t + 1)( z z z + z z z + z z z )+ t ( t + 13 t + 10) z z z , where t ∈ C and t = 1 . The restriction of B ( t ) to the subset t = 1 , − is anequisingular deformation, all curves having the set of singularities A . The threecurves with t = − are extra singular ; their sets of singularities are A ⊕ A . Remark.
We do not assert that all curves B ( t ) are pairwise distinct. In fact,one can observe that the substitution t ǫt , ǫ = 1, results in an equivalent curve,the corresponding change of coordinates being ( z : z : z ) ( z : ǫ z : ǫz ). Inparticular, all three extra singular curves are equivalent.Theorems 2.1.2 and 2.1.3 are proved, respectively, in Sections 2.3 and 2.8 below.2.2. Discriminant forms.
An ( integral ) lattice is a finitely generated free abeliangroup L supplied with a symmetric bilinear form b : L ⊗ L → Z . We abbreviate b ( x, y ) = x · y and b ( x, x ) = x . A lattice L is even if x = 0 mod 2 for all x ∈ L . As the transition matrix between two integral bases has determinant ± PLANE SEXTIC WITH FINITE FUNDAMENTAL GROUP 3 the determinant det L ∈ Z ( i.e. , the determinant of the Gram matrix of b in anybasis of L ) is well defined. A lattice L is called nondegenerate if det L = 0; it iscalled unimodular if det L = ± L , the bilinear form extends to L ⊗ Q by linearity. If L isnondegenerate, the dual group L ∗ = Hom( L, Z ) can be identified with the subgroup (cid:8) x ∈ L ⊗ Q (cid:12)(cid:12) x · y ∈ Z for all x ∈ L (cid:9) . In particular, L ⊂ L ∗ . The quotient L ∗ /L is a finite group; it is called the discrim-inant group of L and is denoted by discr L or L . The discriminant group L inheritsfrom L ⊗ Q a symmetric bilinear form L ⊗ L → Q / Z , called the discriminant form ,and, if L is even, its quadratic extension L → Q / Z . When speaking about thediscriminant groups, their (anti-)isomorphisms, etc., we always assume that thediscriminant form (and its quadratic extension if the lattice is even) is taken intoaccount. One has L = | det L | ; in particular, L = 0 if and only if L is unimodular.An extension of a lattice L is another lattice M containing L , so that the formon L is the restriction of that on M . An isomorphism between two extensions M ⊃ L and M ⊃ L is an isometry M → M whose restriction to L is theidentity. In what follows, we are only interested in the case when both L and M are even and [ M : L ] < ∞ . Next two theorems are found in V. V. Nikulin [9].2.2.1. Theorem.
Given a nondegenerate even lattice L , there is a canonical one-to-one correspondence between the set of isomorphism classes of finite index extensions M ⊃ L ( by even lattices ) and the set of isotropic subgroups K ⊂ L . Under thiscorrespondence, M = { x ∈ L ∗ | x mod L ∈ K} and discr M = K ⊥ / K . (cid:3) The isotropic subgroup
K ⊂ L as in Theorem 2.2.1 is called the kernel of theextension M ⊃ L . It can be defined as the image of M/L under the homomorphisminduced by the natural inclusion
M ֒ → L ∗ .2.2.2. Theorem.
Let M ⊃ L be a finite index extension of a nondegenerate evenlattice L ( by an even lattice M ) , and let K ⊂ L be its kernel. Then, an auto-isometry L → L extends to M if and only if the induced automorphism of L preserves K . (cid:3) Proof of Theorem 2.1.2.
Fix a D -sextic B ⊂ P and consider the doublecovering X → P ramified at B and its minimal resolution ˜ X . Since all singularpoints of B are simple, see Proposition 2.1.1, ˜ X is a K P of B , denote by D P the set of exceptional divisors in ˜ X over P , as wellas its incidence graph, which is the Dynkin graph of the same name A – D – E as P .Let Σ P ⊂ H ( ˜ X ) be the sublattice spanned by D P . (Here, H ( ˜ X ) is regarded as alattice via the intersection index form.) Let, further, Σ ′ = L P Σ P , the summationrunning over all type A singular points P of B (see Proposition 2.1.1), and let˜Σ ′ ⊃ Σ ′ be the primitive hull of Σ ′ in H ( ˜ X ), i.e. , ˜Σ ′ = (Σ ′ ⊗ Q ) ∩ H ( ˜ X ). It is afinite index extension; denote by K ⊂ discr Σ ′ its kernel.Let P , P , P be the type A points. For each point P = P i , i = 0 , ,
2, fixan orientation of its (linear) graph D P and let e i , . . . , e i be the elements of D P numbered consecutively according to the chosen orientation. Denote by e ∗ i , . . . , e ∗ i the dual basis for Σ ∗ P . The discriminant group discr Σ P ∼ = Z is generated by e ∗ i mod Σ P , and, for each k = 1 , . . . ,
6, one has e ∗ ik = ke ∗ i mod Σ P . Let(2.2) γ = e ∗ + e ∗ + e ∗ , γ = e ∗ + e ∗ + e ∗ , γ = e ∗ + e ∗ + e ∗ . According to [3], under an appropriate numbering of the type A singular pointsof B and appropriate orientation of their Dynkin graphs, the kernel K ∼ = Z is A. DEGTYAREV AND M. OKA generated by the residue γ mod Σ ′ . (For the convenience of the further exposition,we use an indexing slightly different from that used in [3].) Observe that each ofthe residues γ = 2 γ mod Σ ′ and γ = 4 γ mod Σ ′ also generates K .Define an isometry c Σ : Σ ′ → Σ ′ via e k e k e k e k , k = 1 , . . . c = id and the induced action on discr Σ ′ is a regular representation of Z over F . Hence, discr Σ ′ splits into direct (not orthogonal) sum of 1-dimensionaleigenspaces, discr Σ ′ = V ⊕ V ⊕ V , corresponding to the three cubic roots of unity1 , , ∈ F . Since c Σ ( γ ) = γ = 2 γ mod Σ ′ , see above, one has K = V and itis immediate that K ⊥ / K can be identified with V . Hence, c Σ extends to an auto-isometry ˜ c Σ : ˜Σ ′ → ˜Σ ′ , see Theorem 2.2.2, and the induced action on discr ˜Σ = V ,see Theorem 2.2.1, is trivial. Applying Theorem 2.2.2 to the finite index extension H ( ˜ X ) ⊃ ˜Σ ′ ⊕ ( ˜Σ ′ ) ⊥ , one concludes that the direct sum ˜ c Σ ⊕ id extends to anorder 3 auto-isometry ˜ c ∗ : H ( ˜ X ) → H ( ˜ X ).By construction, ˜ c ∗ preserves the class h of the pull-back of a generic line in P and the class ω of a holomorphic 2-form on ˜ X (as both h, ω ∈ ( ˜Σ ′ ) ⊥ ). Furthermore,˜ c ∗ preserve the positive cone V + 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; in the case under consideration, itis uniquely characterized by the requirement that V + · e > e over a singular point of B and that the closure of V + should contain h .)The usual averaging argument shows that ˜ X has a K¨ahler metric with ˜ c ∗ -invariantfundamental class ρ ∈ V + . The pair ( ω mod C ∗ , ρ mod R ∗ ) represents a point inthe fine period space of marked quasipolarized K c ∗ , there is a unique automorphism ˜ c : ˜ X → ˜ X inducing ˜ c ∗ in the homology. It is of order 3 (as the only automorphism inducing˜ c ∗ = id is the identity), symplectic ( i.e. , preserving holomorphic 2-forms), andcommutes with the deck translation of the ramified covering ˜ X → P (as the map˜ X → P is defined by the linear system h ∈ Pic ˜ X preserved by ˜ c ∗ ). Thus, ˜ c descends to an order 3 automorphism c : P → P . The latter preserves B (as itlifts to ˜ X ) and has isolated fixed points only (as so does its lift ˜ c , as any symplecticautomorphism of a K c is regular. (cid:3) Geometric properties of D -sextics. The following geometric characteri-zation of D -sextics is found in [3].2.4.1. Proposition.
The three type A singular points P , P , P of a D -sextic B can be ordered so that there are three conics Q , Q , Q such that each Q i , i =0 , , , intersects B at P i − k , k = 1 , , , with multiplicity k . Remark.
Here and below, to shorten the notation, we use the cyclic indexing P i +3 s = P i and Q i +3 s = Q i for s ∈ Z . In fact, the points should be orderedas explained in Section 2.3; then Q i is the projection to P of the rational curverealizing the ( − γ i + h ∈ Pic ˜ X , where γ i is given by (2.2).2.4.3. Lemma.
The automorphism c given by Theorem 2.1.2 acts on the set ofconics { Q , Q , Q } as in Proposition 2.4.1 by a cyclic permutation.Proof. For each i = 0 , ,
2, the incidence conditions described above define at mostone conic Q i (as otherwise two conics would intersect at six points). Since c per-mutes the singular points of B , it must also permute the conics. (cid:3) PLANE SEXTIC WITH FINITE FUNDAMENTAL GROUP 5
Lemma.
Let Q , Q , Q be the conics as in Proposition 2.4.1. Then, eitherall Q i , i = 0 , , , are irreducible or else, for each i = 0 , , , one has a splitting Q i = ( P i − P i ) + ( P i P i +1 ) . In the latter case, B is tangent to ( P i − P i ) at P i .Proof. Due to Lemma 2.4.3, if one of Q i is reducible, so are the others. Assumethat Q splits into two lines, Q = L ′ + L ′′ . If the intersection point L ′ ∩ L ′′ isa singular point of B , one immediately concludes that Q = ( P P ) + ( P P ) andextends this splitting to the other conics via c .Otherwise, assume that it is L ′ that intersects B at P with multiplicity 6. Thenthe component L ′′ = c ( L ′′ ) of Q = c ( Q ) is tangent to B at P (we assume that c acts via P P P P ); hence, L ′′ = L ′ and this line cannot passthrough P . (Neither can the other component L ′ = c ( L ′ ), as it intersects B at P with the maximal multiplicity 6.) (cid:3) Theorem 2.1.3: the generic case.
Fix a D -sextic B and denote by P , P , P its three type A singular points, ordered as explained above. Let Q , Q , Q be the conics as in Proposition 2.4.1, and let c : P → P be the order 3automorphism given by Theorem 2.1.2. In this section, we assume that Q , Q , Q are irreducible, see Lemma 2.4.4.Perform the triangular transformation centered at P , P , P , i.e. , blow up thethree points and blow down the proper transforms of the lines ( P i P j ), 0 i < j B has three type A singular points ¯ P i , i = 0 , ,
2. The transforms ¯ Q i , i = 0 , , Q i passes through ¯ P i +1 and is tangent to ¯ B at ¯ P i . Besides,¯ B has three (at least) nodes ¯ S i , i = 0 , ,
2, located at the blow-up centers of theinverse triangular transformation. Note that (under appropriate indexing) the line( ¯ S i ¯ S i +1 ) contains ¯ P i , i = 0 , , c , the transformation commuteswith c and the new configuration is still Z -symmetric.Choose homogeneous coordinates ( u : u : u ) in P so that ¯ P = (1 : 0 : 0),¯ P = (0 : 1 : 0), ¯ P = (0 : 0 : 1), and (1 : 1 : 1) is one of the fixed points of c .Then c acts via a cyclic permutation of the coordinates, and its three fixed pointsare (1 : ǫ : ǫ ), ǫ = 1. The condition that ¯ P i ∈ ( ¯ S i ¯ S i +1 ) and that the triple ¯ S ,¯ S , ¯ S is c -invariant translates as follows: there is a parameter t ∈ C such that¯ S = (1 : t : t ), ¯ S = ( t : 1 : t ), and ¯ S = ( t : t : 1). In order to get three distinctpoints other than ¯ P , ¯ P , ¯ P , one must have t = 0, t = 1.In the chosen coordinates, ¯ B has three type A singular points located at thevertices of the coordinate triangle and tangent to its edges. Since ¯ B is also preservedby c , it must be given by a polynomial F ( u , u , u ) of the form(2.3) a ( u u + u u + u u ) + b ( u u u + u u u + u u u )+ c ( u u u + u u u + u u u ) + du u u for some a, b, c, d ∈ C . Conversely, any curve ¯ B given by a polynomial as above ispreserved by c and has three singular points adjacent to A and situated in theprescribed way with respect to the coordinate lines ¯ Q i , i = 0 , ,
2. Due to thesymmetry, it suffices to make sure that ¯ B is singular at ¯ S and that its singularityat ¯ P is adjacent to A . The former condition results in the linear system6 at + (3 t + 3 t ) b + (5 t + t ) c + 2 dt = 0 , (4 t + 2 t ) a + (2 t + 4 t ) b + (4 t + 2 t ) c + 2 dt = 0 , A. DEGTYAREV AND M. OKA (4 t + 2 t ) a + ( t + 5 t ) b + (3 t + 3 t ) c + 2 dt = 0 , and the latter condition is equivalent to b = 2 a or b = − a . In both cases, thesolution space of the linear system has dimension one; it is spanned by( a, b, c, d ) = ( t , t , − t ( t + 2) , ( t + 2) )and ( a, b, c, d ) = (1 , − , − t , t ( t + 8)) , respectively. The first solution, with b = 2 a , results in a reducible polynomial( tu u − u u u − u u u t + tu u + tu u ) ;hence, it should be disregarded. For the second solution, the substitution u = 1, u = x , u = y + x results in a polynomial with the principal part y − t x , i.e. ,the singularity of ¯ B at ¯ P (and, due to the symmetry, at ¯ P and ¯ P as well) is oftype A exactly. In particular, the curve is irreducible. (Indeed, the only possiblesplitting would be into an irreducible quintic and a line, but in this case all nodesof ¯ B would have to be collinear.)To obtain the original curve B , one should perform the substitution u = v + t v + tv , u = tv + v + t v , u = t v + tv + v (passing to an invariant coordinate triangle with the vertices at ¯ S i , i = 0 , , v = z z , v = z z , v = z z .The resulting polynomial is the one given by (2.1) with t = 0.Counting the genus and taking into account the symmetry, one concludes thatthe singularities of ¯ B at ¯ S i , i = 0 , ,
2, are either all nodes or all cusps, the latterpossibility corresponding to t = − t = − /
3. Note that the cusps of ¯ B merelymean that the original curve B is tangent to the lines ( P i P j ), 0 i < j
2; thesecurves are still in the same equisingular deformation family.2.5.1.
Remark. If t = 1, the polynomial (2.1) becomes reducible. For example, if t = 1, it turns into 4( z z + z z + z z ) .2.6. Theorem 2.1.3: the case of reducible conics.
Now, assume that theconics Q , Q , Q are reducible, see Lemma 2.4.4. In this case, we can start directlyfrom (2.3), placing the singular points so that P = (1 : 0 : 0), P = (0 : 0 : 1), and P = (0 : 1 : 0). Note that a = 0; hence, we can let a = 1.As above, in view of the symmetry it suffices to analyze the singularity at P .The condition that the singularity is adjacent to A is equivalent to b = ±
2. If b = 2, the substitution u = 1, u = x , u = y − x + cx / x, y ) with the principal part y + ( d − c / x . Hence, d = c /
4. However, inthis case the original polynomial F is reducible: F = 14 (2 u u + 2 u u + 2 u u + cu u u ) . Let b = −
2. Then, substituting u = 1, u = x , u = y + x , one obtains y + 3 cyx + 2 cx + dx − x + (higher order terms) . Hence, c = d = 0, and in this case the singularity at the origin is exactly A . Thecurve is irreducible (as any sextic with three type A singular points) and, afterthe coordinate change ( u : u : u ) ( z : z : z ), the resulting equation is (2.1)with t = 0. PLANE SEXTIC WITH FINITE FUNDAMENTAL GROUP 7
Extra singular D -sextics. Since the total Milnor number of a plane sexticdoes not exceed 19, a curve B = B ( t ) can have at most one extra singular point,which must be of type A . Since, in addition, B is preserved by c , this extrasingular point must be fixed by c , i.e. , it must be of the form (1 : ǫ : ǫ ), ǫ = 1.Solving the corresponding linear system shows that B is singular at (1 : ǫ : ǫ ), ǫ = 1, whenever it passes through this point, and this is the case when t = − /ǫ .In conclusion, B has an extra node (the set of singularities 3 A ⊕ A ) if and onlyif t = − Proof of Theorem 2.1.3.
As shown in Sections 2.5 and 2.6, any D -sexticbelongs to the connected family B ( t ), t = 1. According to Section 2.7, this familyrepresents two equisingular deformation classes: the restriction to the connectedsubset t = 1 , −
27 (the set of singularities 3 A ) and three equivalent isolated curvescorresponding to t = −
27 (the set of singularities 3 A ⊕ A ; the equivalence isgiven by the coordinate change ( z : z : z ) ( z : ǫz : ǫ z ), ǫ = 1, cf .Remark 2.1.4). Comparing this result with Proposition 2.1.1, one concludes thatany curve given by (2.1) with t = 1 is a D -sextic. (cid:3) The fundamental group
Calculation of the group.
For the calculation, we choose a real curve B ( t )given by Theorem 2.1.3 and close to the triple conic B (1), see Remark 2.5.1.3.1.1. Theorem.
For the curve B = B (5 / , there is an epimorphism G := h ω, ξ | ξ = e, ω = ξω ξ i ։ π ( P r B ) . Proof.
Let B = B (5 /
6) and make the following change of coordinates: z = 1 / Z − / Y + 1 / X, z = − / X + 2 / Z − / Y, z = Y. Then, in the affine space C = P r { Z = 0 } , B is defined by g ( x, y ) = 0, where g ( x, y ) = 71619683 x + 17872177147 y − xy − xy + 3225595668704 x y + 3568177147 − x y − x − xy + 5626129322674816 y − y − y + 8148537711337408 y − y − x + 2243209952 x − x + 27265797558272 x y + 10926231259712 xy + 986875711337408 x y + 117188933779136 x y + 7776841911337408 xy − yx − x y − x y − x y + 12505944784 x + 3971752834352 x y. Its three singularities are located at P = ( − , , P = (2 , , P = ( − / , / , and the graph in the real plane is given in Figure 1. A. DEGTYAREV AND M. OKA –0.6–0.4–0.200.20.4–1 –0.5 0.5 1 1.5 2
Figure 1.
The graph of B For the calculation, we apply van Kampen’s method [8] to the horizontal pencil L η = { y = η } . The singular pencils corresponds to the roots of(90617210907008 y − y − y +38781803208839 y + 8841431367018 y − y − y + 512733413664 y − y − y (2 y − = 0 . Note that we have five real singular pencil lines L η , η = η i , i = 1 , , . . . , ,η ≈ − . , η ≈ − . , η = 0 , η ≈ . , η = 1 / , where L η i , i = 1 , , B , which come from the first factor of degree9. There also are three pairs of complex conjugate singular fibers, but we donot use them; that is why we only assert that the map constructed below is anepimorphism, not an isomorphism. We take the base point at infinity b = (1 : 0 : 0),and we fix generators ρ , . . . , ρ on the regular pencil line y = − ε (where ε is asufficiently small positive real number) as in Figure 2. The bullets are lassos, whichare counterclockwise oriented loops going around a point of B .The monodromy relations at y = η , η are tangent relations, they are given as( R ) : ρ = ρ , ρ = ρ − ρ ρ Thus, hereafter we eliminate the generator ρ . PLANE SEXTIC WITH FINITE FUNDAMENTAL GROUP 9 ρ ρ ρ ρ ρ ρ O Figure 2.
Generators in the fiber y = − ε The monodromy relations at y = 0 are two A -cusp relations, they are given as( R ) : ( ω ρ = ρ ω , ω = ρ ρ ,τ ρ = ρ τ , τ = ρ ρ . To see the relations at y = η , η effectively, we take new elements ρ ′ , ρ ′ , ρ ′ , ρ ′ asin Figure 3. The new elements are defined as( R ) : ρ ′ = ω − ρ ω , ρ ′ = ω − ρ ω, ρ ′ = τ − ρ τ, ρ ′ = τ − ρ τ . Note that they satisfy the relations ρ ′ ρ ′ = ω, ρ ′ ρ ′ = τ.ρ ′ ρ ′ ρ ρ ρ ′ ρ ′ Figure 3.
Generators in the fiber y = ε Now, the monodromy relation at y = η is given as( R ) : ρ = ρ ′ or ρ = τ − ρ τ. The relation at y = 1 / A -cusp relation, which is given as( R ) : ( ρ ρ ′ ρ ′ ρ ′ − ) ρ = ( ρ ′ ρ ′ ρ ′ − )( ρ ρ ′ ρ ′ ρ ′ − ) . Finally, the vanishing relation at infinity is given as( R ∞ ) : ω τ = e. We eliminate the generator ρ using ( R ). Then ( R ) is translated into thefollowing relation:( R ′ ) : ρ − ρ ρ = τ − ρ τ. The relation ( R ∞ ) can be rewritten as( R ′∞ ) : τ = ω − . From ( R ′∞ ) and ( R ′ ), we get ρ = ( τ ρ − ) ρ ( ρ τ − ) = ω − ρ − ρ ρ ω = R ρ ρ ρ − . As ρ = τ ρ − , this implies( R ′′ ) : ρ = ρ ρ ρ − , ρ = ω − ρ − ω − . We can rewrite ρ ′ , using the above relations, as follows: ρ ′ = ω ρ − ω − = R ρ − ρ ρ . Thus, ρ ′ ρ ′ ρ ′ − = ω − ρ , and ( R ) can be rewritten in ρ , ρ as follows:( R ′ ) : ( ρ ρ ω − ) ρ = ( ρ ω − )( ρ ρ ω − ) . We have to rewrite the relations in the words of ρ , ρ . The relation τ ρ = ρ τ gives ω − ( ω − ρ − ω − ) = ( ρ ρ ρ − ) ω − , which reduces to ω ρ = ρ ω . Using the relation ω ρ = ρ ω several times, weget ρ = ωρ ω . Now, we eliminate ρ using ω = ρ ρ to obtain( R ) : ( ωρ ) = e. Replace the generator ρ by ξ = ωρ , so that the new generators are ω , ξ ; then ρ , ρ are expressed as ρ = ω − ξ and ρ = ξ − ω , and ( R ) is written as ξ = e . Therelation ω ρ = ρ ω reduces to( R ) : ω = ξω ξ. Thus, we have shown that π ( P r B ) is generated by two elements ω , ξ , which aresubject to the relations ξ = e, ω = ξω ξ. This establishes the required epimorphism. (cid:3)
The group structure of G . Below, we analyze the group G obtained inTheorem 3.1.1 and show that it is isomorphic to D × Z = h a, b, ξ | ξ = e, a = e, b = e, ξbξ = b , [ a, b ] = [ a, ξ ] = e i , where [ a, b ] = aba − b − is the commutator.3.2.1. Lemma.
The map ξ ξ , ω ab establishes an isomorphism G ∼ = D × Z . PLANE SEXTIC WITH FINITE FUNDAMENTAL GROUP 11
Proof.
Putting c = ab , we see that a = c , b = c . Thus we can use two generators ξ , c , and D × Z = h ξ, c | ξ = e, c = e, ξc ξ = c , [ c , ξ ] = e i . Now we consider our group G : G = h ω, ξ | ξ = e, ξω ξ = ω i . First we see that ξ ω ξ = ω = ( ξ ω ξ ) = ξω ξ . Thus we get ω = e . We assertthat ω is in the center of G . Indeed, ξω ξ = ( ξ ω ξ ) = ω = ω . Thus, ω isin the center, and so is ω = ( ω ) . Observe that ξω ξ = ( ω ) = ω . Thus, wehave another presentation of G , G = h ω, ξ | ξ = e, ω = e, ξω ξ = ω , [ ω , ξ ] = e i , which coincides with that of D × Z . (The original relation ξω ξ = ω is recoveredby squaring the relation ξω ξ = ω , taking into account ω = e , and cancellingthe central element ω .) (cid:3) Proof of Theorem 1.1.1.
According to Theorem 2.1.3, any curve B = B ( t ), t = 1, is a D -sextic, i.e. , its fundamental group π = π ( P r B ) factors to D .On the other hand, π/ [ π, π ] = Z . The smallest group with these properties is D × Z , i.e. , one has ord π > ord( D × Z ). In view of Theorem 3.1.1 andLemma 3.2.1, there is an epimorphism D × Z ։ π ; comparing the orders, oneconcludes that it is an isomorphism. (cid:3) References [1] A. Beauville
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On the fundamental groups of the complements of plane singular sextics ,J. Math. Soc. Japan, (2005), no. 1, 37–54.[7] C. Eyral, M. Oka, On a conjecture of Degtyarev on non-torus plane curves , this volume.[8] E. R. van Kampen,
On the fundamental group of an algebraic curve , Amer. J. Math., (1933), 255–260.[9] V. V. Nikulin, Integer quadratic forms and some of their geometrical applications , Izv. Akad.Nauk SSSR, Ser. Mat., (1979), no. 1, 111–177 (Russian); English transl. in Math. USSR–Izv., (1979), no. 1, 103–167.[10] M. Oka A survey on Alexander polynomials of plane curves , S´eminair & Congr`es, (2005),209–232. A. Degtyarev: Department of Mathematics, Bilkent University, Bilkent, Ankara06533, Turkey
E-mail address : [email protected] M. Oka: Department of Mathematics, Tokyo University of Science, 26 Wakamiya-cho, Shinjuku-ku, Tokyo 162-8601, Japan
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