Vanishing of Higher Order Alexander-type Invariants of Plane Curves
aa r X i v : . [ m a t h . A T ] O c t VANISHING OF HIGHER ORDER ALEXANDER-TYPE INVARIANTS OFPLANE CURVES
JOS´E I. COGOLLUDO-AGUST´IN AND EVA ELDUQUE
Abstract.
The higher order degrees are Alexander-type invariants of complements to an affineplane curve. In this paper we characterize the vanishing of such invariants for transversal unionsof plane curves C ′ and C ′′ in terms of the finiteness, the vanishing properties of the invariantsof C ′ and C ′′ , and whether they are irreducible or not. As a consequence, we characterizewhich of these types of curves have trivial multivariable Alexander polynomial in terms of theirdefining equations. Our results impose obstructions on the class of groups that can be realizedas fundamental groups of complements of a transversal union of curves. Introduction
The study of curve complements goes back to the work of Zariski ([20]), who observed thatthe position of the singularities of a plane curve influenced the topology of the curve, and thatthe fundamental group of the complement of the curve detected this phenomenon. Alexander-type invariants, which appeared first in classical knot theory and were first imported to studysingularities of plane curves by Libgober ([9, 11]), are easier to handle than the fundamentalgroup, and they are also sensitive to the type and position of singularities.In knot theory, a strategy to address problems that the Alexander polynomial is not strongenough to solve is to consider non-abelian Alexander-type invariants, such as the higher orderdegrees (e.g., see Cochran, [2]), which have been shown to give better bounds for the knot genusthan the Alexander polynomial (Horn, [7]). These invariants also have striking applications inthe world of 3-manifolds (Harvey, [6]).Leidy and Maxim initiated in [13, 14] the study of higher order Alexander-type invariantsfor complex affine plane curve complements, and Maxim and the second author continued thiswork in [5]. To any affine plane curve C ⊂ C (given by the zeros of a reduced non-constantpolynomial f ∈ C [ x, y ]), in [13] one associates a sequence { δ n ( C ) } n of (possibly infinite) integers,called the higher order degrees of C . Roughly speaking, these integers measure the “sizes” ofquotients of successive terms in the rational derived series { G ( n ) r } n ≥ of the fundamental group G = π ( C \ C ) of the curve complement. It was also noted in [13] that the higher orderdegrees of plane curves (at any level n ) are sensitive to the position of singular points. Theseintegers can also be interpreted as L2-Betti numbers associated to the tower of coverings of C \ C corresponding to the subgroups G ( i ) r (the first of which is the universal abelian cover), soin principle there is no reason to expect that such invariants have any good vanishing or finitenessproperties. Some finiteness results obtained in [5, 13] are summarized in this theorem. Mathematics Subject Classification.
Theorem 1.1.
Let C ⊂ C be a plane curve of degree m . If one of the following conditions hold,then δ n ( C ) is finite: (1) C is irreducible [13, Remark 3.3] . (2) C is in general position at infinity [13, Corollary 4.8] . (3) C is an essential line arrangement [5, Theorem 2] . (4) C has only nodes or simple tangents at infinity [5, Theorem 4] .Moreover, in the cases (2) , (3) , and (4) , we have that δ n ( C ) ≤ m ( m − for all n ≥ . Thatis, there is a uniform bound for all the higher order degrees that depends only on the degree of acurve. In relation to an old question of Serre ([1, 17]), finiteness results impose restrictions on whichgroups can be realized as fundamental groups of curve complements, but vanishing results, ontop of being stronger, shed more light on what type of problems these invariants are well suitedfor. For example, if we know that δ n = 0 for a class of curves, then δ n will not distinguishcurves within that class, but it can potentially distinguish a curve in that class from a curve ina different class.An example of a “vanishing” (or “triviality”) result is the following theorem of Oka. In generalterms, it tells us that the univariable Alexander polynomial (see Definition 2.1) of a transversalunion of curves C = C ′ ∪ C ′′ does not remember information about the topology of C ′ or C ′′ ,even though the fundamental group does (See Theorem 3.4). Theorem 1.2 ([15], Theorem 34) . Let C be a plane curve of the form C = C ′ ∪ C ′′ , where C ′ and C ′′ are curves in C of degrees m ′ and m ′′ respectively. Assume that C is in general positionat infinity, and assume that C ′ ∩ C ′′ consists on m ′ m ′′ distinct points. Then, ∆ uni C ( t ) = ( t − s − where s is the number of irreducible components of C and ∆ uni C ( t ) is the (univariable) Alexanderpolynomial of C . It is natural to ask whether more involved Alexander invariants also exhibit this behavior. Inthis paper, we completely characterize the vanishing of the higher order degrees of a union C of two curves C ′ and C ′′ that intersect transversally (even when C is not in general position atinfinity). This characterization is done in terms of the finiteness and vanishing properties of thehigher order degrees of C ′ and C ′′ , obtaining vanishing results in most cases (and finiteness inall cases). More concretely, we obtain the following. Theorem 1.3.
Let C = C ′ ∪ C ′′ ⊂ C be the union of two affine plane curves, with deg C ′ = m ′ and deg C ′′ = m ′′ . Suppose that C ′ ∩ C ′′ consists on m ′ m ′′ distinct points in C . Then, (1) δ n ( C ) is finite for all n ≥ . (2) If C ′ and C ′′ are both irreducible or both not irreducible, then δ n ( C ) = 0 for all n ≥ . (3) If C ′ is irreducible and C ′′ is not irreducible, and (a) δ ( C ′ ) = 0 , then δ ( C ) = 0 ⇔ δ ( C ′′ ) < ∞ , and δ n ( C ) = 0 for all n ≥ . (b) δ ( C ′ ) = 0 , then, for all n ≥ , δ n ( C ) = 0 ⇔ δ n ( C ′′ ) < ∞ . This provides a broad generalization of the vanishing results of [5], where the fundamentalgroup of one of the curve complements was assumed to be isomorphic to Z , and δ n of the othercurve was assumed to be finite. ANISHING OF HIGHER ORDER ALEXANDER-TYPE INVARIANTS OF PLANE CURVES 3
The paper is structured as follows. In Section 2 we recall the relevant definitions of theAlexander-type invariants that are used throughout the paper and the relationships betweenthem. In Section 3 we prove the main result (Theorem 1.3). In Section 4, we characterize whichcurves have δ ( C ) = ∞ in terms of their defining equations (curves of affine pencil type, asdefined in Lemma 4.4) and arrive at Corollary 1.4 below about the triviality of the multivariableAlexander polynomial of a transversal union of curves (see Definition 2.2). This corollary providesconcrete restrictions as to which groups can be realized as fundamental groups of a complementof a transversal union of curves (see Remark 4.5). Corollary 1.4.
Under the same hypotheses as in Theorem 1.3, (1) ∆ multi C = 0 . (2) If C ′ and C ′′ are both irreducible or both not irreducible, then ∆ multi C is a non-zero con-stant. (3) If C ′ is irreducible and C ′′ is not irreducible, then ∆ multi C is a non-zero constant if andonly if C ′′ is not of affine pencil type.Moreover, if ∆ multi C ( t, t , . . . , t s ) is not a non-zero constant, then it is of the form ( t − k forsome ≤ k ≤ m ′′ − , where t is the variable corresponding to a positively oriented meridianaround C ′ , and t i is the variable corresponding to a positively oriented meridian around the i -thirreducible component of C ′′ . Acknowledgements.
The authors would like to thank Mois´es Herrad´on Cueto and Laurent¸iuMaxim for useful discussions.2.
Definitions of classical and higher order Alexander invariants
In this section we recall the basic definitions of the notions that will be used throughout thisnote. For a more detailed explanation of the different Alexander invariants used in this paper, werefer the reader to [9] (for univariable Alexander polynomials), [19] (for multivariable Alexanderpolynomials), and [13] (for higher order degrees of plane curve complements), for example.2.1.
Alexander Polynomials.
Let C = { f ( x, y ) = 0 } ⊂ C be a plane curve given by thezeros of a reduced polynomial f , with complement U := C \ C , and denote by G := π ( U ) thefundamental group of its complement. If C has s irreducible components, then(2.1) H ( G ; Z ) = H ( U ; Z ) = G/G ′ = Z s , is generated by meridian loops about the smooth parts of the irreducible components of C .Let ψ be the linking number homomorphism G ψ → Z , given by α lk( α, C ) . Since f isa reduced polynomial, ψ is the map induced in fundamental groups by f : U → C ∗ . LetAb : G → Z s be the abelianization homomorphism, which sends a positively oriented meridianabout the i -th component of C to the i -th element of the canonical basis of Z s . Let L ψ and L Ab be the local systems of Q [ t ± ]-modules and Z [ t ± , . . . , t ± s ]-modules induced by ψ and Abrespectively. More explicitely, L ψ and L Ab are given by G → Aut( Q [ t ± ]) γ (1 t ψ ( γ ) )and G → Aut( Z [ t ± , . . . , t ± s ]) γ (1 t Ab( γ ) )where t ( a ,...,a s ) := t a · . . . · t a s s for all ( a , . . . , a s ) ∈ Z s . JOS´E I. COGOLLUDO-AGUST´IN AND EVA ELDUQUE
For the following two definitions, let F i ( M ) be the i -th Fitting ideal of a module M over acommutative ring. Definition 2.1.
The univariable Alexander polynomial of U , denoted by ∆ uni C ( t ) is defined as∆ uni C ( t ) := a generator of F (cid:16) H ( U ; L ψ ) (cid:17) ∈ Q [ t ± ] , which is well defined up to multiplication by a unit of Q [ t ± ]. Definition 2.2.
The multivariable Alexander polynomial of U , denoted by ∆ multi C ( t ) is definedas ∆ multi C ( t , . . . , t s ) := gcd (cid:16) F (cid:16) H ( U, u ; L Ab ) (cid:17)(cid:17) ∈ Z [ t ± , . . . , t ± s ] , where u is a base point. It is well defined up to multiplication by a unit of Z [ t ± , . . . , t ± s ]. Remark . Note that H ( U ; L ψ ) ∼ = H ( U ψ ; Q ) ([8, Theorem 2.1]) as modules over Q [ t ± ], where U ψ is the infinite cyclic cover of U induced by ker ψ , whose deck group is isomorphic to Z . Alsonote that the definition and computations are easier in the univariable case because Q [ t ± ] is aPID. In the definition of the higher order degrees, a (noncommutative) PID is constructed tohelp generalize this construction of the univariable Alexander polynomial to other covers of U that lie above U ψ . Remark . If C is irreducible, both definitions coincide. Indeed, from [5, Remark 10] we knowthat ∆ multi C ( t ) divides ∆ uni C ( t ) in Q [ t ± ], and the same argument of the proof of [5, Theorem 11](for m = 1) shows that both polynomials are the same up to multiplication by a unit in Q [ t ± ].2.2. Higher Order Degrees.Definition 2.5.
The rational derived series of the group G is defined inductively by: G (0) r = G ,and for n ≥ G ( n ) r = { g ∈ G ( n − r | g k ∈ [ G ( n − r , G ( n − r ] , for some k ∈ Z \ { }} . It is easy to see that G ( i ) r ⊳ G ( j ) r ⊳ G , if i ≥ j ≥
0. The successive quotients of the rationalderived series are torsion-free abelian groups. In fact (cf. [6, Lemma 3.5]), G ( n ) r /G ( n +1) r ∼ = (cid:16) G ( n ) r / [ G ( n ) r , G ( n ) r ] (cid:17) / { Z − torsion } . Therefore, for G = π ( C \ C ) we get from (2.1) that G ′ = G (1) r .The use of the rational derived series instead of the usual derived series is needed in order toavoid zero-divisors in the group ring Z Γ n , whereΓ n := G/G ( n +1) r . Γ n is a poly-torsion-free-abelian group (PTFA), that is, it admits a normal series of subgroupssuch that each of the successive quotients of the series is torsion-free abelian ([6, Corollary 3.6]).Thus, Z Γ n is a right and left Ore domain, so it embeds in its classical right ring of quotients K n , which is a skew-field. Every module over K n is a free module, and such modules have awell-defined rank rk K n which is additive on short exact sequences.In [13], one associates to any plane curve C a sequence of non-negative integers δ n ( C ) asfollows. Since G ′ is in the kernel of ψ (the linking number homomorphism), we have a well-defined induced epimorphism ¯ ψ : Γ n → Z . Let ¯Γ n = ker ¯ ψ . Then ¯Γ n is a PTFA group, so Z ¯Γ n has a right ring of quotients K n = ( Z ¯Γ n ) S − n , where S n = Z ¯Γ n \ { } . Let R n := ( Z Γ n ) S − n . R n and K n are flat left Z Γ n -modules. ANISHING OF HIGHER ORDER ALEXANDER-TYPE INVARIANTS OF PLANE CURVES 5
A very important role in what follows is played by the fact that R n is a PID; in fact, R n isomorphic to the ring of skew-Laurent polynomials K n [ t ± ]. This can be seen as follows: bychoosing a t ∈ Γ n such that ¯ ψ ( t ) = 1, one obtains a splitting φ of ¯ ψ , and the embedding Z ¯Γ n ⊂ K n extends to an isomorphism R n ∼ = K n [ t ± ]. However this isomorphism depends in general on thechoice of splitting of ¯ ψ . Definition 2.6. (1) The n -th order localized Alexander module of the plane curve C is defined to be A n ( C ) = H ( U ; R n ) := H ( U ; Z Γ n ) ⊗ Z Γ n R n , viewed as a right R n -module. The coefficients in the rightmost expression are the rank 1local system induced by the projection G ։ Γ n [6, section 5]. If we choose a splitting φ to identify R n with K n [ t ± ], we define A φn ( C ) = H ( U ; K n [ t ± ]).(2) The n -th order degree of C is defined to be: δ n ( C ) = rk K n A n ( C ) = rk K n A φn ( C ) . The higher order degrees δ n ( C ) are integral invariants of the fundamental group G of thecomplement (endowed with the linking number homomorphism). Indeed, by [6], one has: δ n ( C ) = rk K n (cid:16) G ( n +1) r / [ G ( n +1) r , G ( n +1) r ] ⊗ Z ¯Γ n K n (cid:17) . Note that since the isomorphism between R n and K n [ t ± ] depends on the choice of splitting, onecannot define a higher order version of the (univariable) Alexander polynomial in a canonicalway. However, for any choice of splitting, the degree of the associated higher order Alexanderpolynomial is the same, hence this yields a well-defined invariant of G , which is exactly the higherorder degree δ n defined above.2.3. An effective method to compute δ n ( C ) . The higher order degrees of C may be com-puted by means of Fox free calculus from a presentation of G = π ( U ), where U := C \ C see [6, Section 6] for details, although the computations can be quite tedious in practice. Suchtechniques will be used freely in this paper, as summarized in this section.Consider the matrix of Fox derivatives for a presentation of π ( U ) given by G = π ( U ) = h a , . . . , a m | r j , j = 1 , . . . , l i , that, is, the matrix (cid:18) ∂r j ∂a i (cid:19) i,j , ≤ i ≤ m, ≤ j ≤ l, which has entries in Z G , and we take its involution (the Z -linear map that takes elements of G to their inverses) A = (cid:18) ∂r j ∂a i (cid:19) i,j . Let q n : G −→ Γ n be the projection, and let q ′ n : Z G −→ Z Γ n be the induced map on grouprings. Let B ( n ) = A q ′ n , that is, the matrix formed by the images of the entries of A by q ′ n .With this notation, B ( n ) is a presentation matrix for the right Z Γ n -module H ( U, u ; Z Γ n ),where u is some base point. JOS´E I. COGOLLUDO-AGUST´IN AND EVA ELDUQUE
Moreover, since R n and K n are flat over Z Γ n , we have that B ( n ) is a presentation matrix forthe right R n -module (resp. K n -module) H ( U, u ; R n ) (resp. H ( U, u ; K n )). By [6, Proposition5.6], one obtains the following property for B ( n ). Lemma 2.7.
The rank of the left K n -module generated by the rows of B ( n ) is ≤ m − , and therank of the left K n -module generated by the rows of B ( n ) is m − ⇔ δ n ( C ) is finite. By doing allowable row and column operations to B ( n ) in R n ∼ = K n [ t ± ] ([6, Lemma 9.2]), wecan turn B ( n ) into a different presentation matrix of H ( U, u ; R n ) of the form (cid:18) D ... ... (cid:19) where D is a diagonal matrix with entries in K n [ t ± ] and the last row is a row of zeroes. Thefollowing result is immediate and allows one to obtain the invariant δ n ( C ). In it, the degree of anon-zero element of K n [ t ± ] is defined as the difference between the highest and lowest exponentsof t appearing in the polynomial. Proposition 2.8.
Under the conditions above, the higher order degree δ n ( C ) is the degree ofthe product of the diagonal elements of D if all of those elements are non-zero, and δ n ( C ) = ∞ otherwise. Vanishing of higher order degrees of transversal intersections.
The goal of this section is to prove Theorem 1.3, which characterizes the vanishing of thehigher order degrees of a curve that is the union of two curves that intersect transversally anddo not intersect at infinity.
Remark . The right hand side of the “ ⇔ ” equivalences in Theorem 1.3 is always satisfied inthe cases described in Theorem 1.1.The proof of Theorem 1.3 is going to be broken down into 3 lemmas (3.6, 3.7, and 3.8). Beforewe prove those, let us write down some facts that will be used throughout the section. Proposition 3.2 ([13], Remark 3.3, Remark 3.9, Proposition 5.1) . If C is an irreducible curve,then δ ( C ) = 0 ⇐⇒ G (1) r = G (2) r ⇐⇒ δ n ( C ) = 0 for all n ≥ . Remark . There are three curves in the statement of Theorem 1.3, namely C , C ′ and C ′′ .We will use ′ or ′′ to refer to the objects corresponding to C ′ and C ′′ respectively. For example, U ′ := C \ C ′ , G ′′ := π ( U ′′ ), etc. Theorem 3.4 (The Oka-Sakamoto theorem, [16]) . Let C = C ′ ∪ C ′′ ⊂ C be the union of twoaffine plane curves, with deg C ′ = m ′ and deg C ′′ = m ′′ . Suppose that C ′ ∩ C ′′ consists on m ′ m ′′ distinct points in C . Then, G ∼ = G ′ × G ′′ . Remark . In the conditions of the Oka-Sakamoto theorem, we can consider a presentationfor G with generators a , . . . , a m ′ , b , . . . , b m ′′ , where the a i ’s are a choice of positively orientedmeridians around C ′ generating G ′ , and the b j ’s are a choice of positively oriented meridiansaround irreducible components of C ′′ generating G ′′ [10]. Let R ′ and R ′′ be a set of relations ofa presentation of G ′ and G ′′ where the generators are the a ’s and b ’s respectively. Then, we havethe following presentation for G : G = h a , . . . , a m ′ , b , . . . , b m ′′ | [ a i , b j ] for all i = 1 , . . . , m ′ and j = 1 , . . . , m ′′ ; R ′ ; R ′′ i . ANISHING OF HIGHER ORDER ALEXANDER-TYPE INVARIANTS OF PLANE CURVES 7
The first of our three key lemmas deals with the 0-th order degree of a union of two transversalirreducible curves.
Lemma 3.6.
Let C = C ′ ∪ C ′′ ⊂ C be the union of two irreducible affine plane curves, with deg C ′ = m ′ and deg C ′′ = m ′′ . Suppose that C ′ ∩ C ′′ consists on m ′ m ′′ distinct points in C .Then, δ ( C ) = 0 . Proof.
By Theorem 3.4, G ∼ = G ′ × G ′′ . We have that G (1) r /G (2) r ∼ = ( G ′ ) (1) r / ( G ′ ) (2) r × ( G ′′ ) (1) r / ( G ′′ ) (2) r , By [6], one has: δ n ( C ) = rk K n (cid:16) G ( n +1) r / [ G ( n +1) r , G ( n +1) r ] ⊗ Z ¯Γ n K n (cid:17) . Notice that the tensor product kills the Z -torsion, so this is equivalent to(3.1) δ n ( C ) = rk K n (cid:16) G ( n +1) r /G ( n +2) r ⊗ Z ¯Γ n K n (cid:17) . Note that Z ¯Γ ∼ = Z [ t ± ] in this case. Since both C ′ and C ′′ are irreducible, we have thatΓ ′ ∼ = Γ ′′ ∼ = Z , ¯Γ ′ ∼ = ¯Γ ′ are the trivial group, and K ′ ∼ = K ′′ ∼ = Q . By Proposition 3.2, δ n of anyirreducible curve is finite for all n ≥
0, so δ ( C ′ ) = rk Q (cid:16) ( G ′ ) (1) r / ( G ′ ) (2) r ⊗ Z Q (cid:17) < ∞ and the same statement holds for C ′′ , which means that ( G ′ ) (1) r / ( G ′ ) (2) r and( G ′′ ) (1) r / ( G ′′ ) (2) r are both finite rank free abelian groups. Let us call them A and B for sim-plicity.Now, δ ( C ) = rk Q ( Z [ t ± ]) (cid:0) ( A ⊕ B ) ⊗ Z [ t ± ] Q ( Z [ t ± ]) (cid:1) , where Q ( · ) denotes taking the field of quotients.Let a ∈ A , and let k be an integer bigger than the rank of A . We have that a, at, . . . , at k are linearly dependent, so a is annihilated by some polynomial in Z [ t ± ]. The same holds for all b ∈ B . Hence, δ ( C ) = 0 . (cid:3) The proofs of lemmas 3.7 and 3.8 consist on applying the techniques of Section 2.3 to conve-niently chosen presentations of the fundamental group.
Lemma 3.7.
Let n be a fixed integer, with n ≥ . Let C = C ′ ∪ C ′′ ⊂ C be the union of twoaffine plane curves, with deg C ′ = m ′ and deg C ′′ = m ′′ . Suppose that C ′ ∩ C ′′ consists on m ′ m ′′ distinct points in C . Suppose that C ′ is irreducible, with δ ( C ′ ) = 0 . Then, δ n ( C ) is finite, and δ n ( C ) = 0 ⇔ δ n ( C ′′ ) < ∞ . Proof.
By Theorem 3.4, G ∼ = G ′ × G ′′ . We first consider the case where C ′′ is also an irreduciblecurve such that δ ( C ′′ ) = 0. In this situation, we know that δ n ( C ′′ ) = 0 for all n ≥
0, and,in fact, the stronger statement ( G ′′ ) (1) r = ( G ′′ ) (2) r holds (Proposition 3.2). Since G is the directproduct of G ′ and G ′′ , we have that G ( n +1) r /G ( n +2) r ∼ = ( G ′ ) ( n +1) r / ( G ′ ) ( n +2) r × ( G ′′ ) ( n +1) r / ( G ′′ ) ( n +2) r , which is the trivial group for all n ≥
0. By equation (3.1), one gets that δ n ( C ) = 0 for all n ≥ C ′′ is either not irreducible, or if it is irreducible, then δ ( C ′′ ) =0. JOS´E I. COGOLLUDO-AGUST´IN AND EVA ELDUQUE
We consider the presentation of G described in Remark 3.5. Since ( G ′ ) (1) r = ( G ′ ) (2) r (Proposi-tion 3.2), a i a − k = 1 in Z Γ n for all i, k ∈ { , . . . , m ′ } , n ≥ n is some integer such that n ≥ C ′′ is an irreducible curve with δ ( C ′′ ) = 0,and n ≥ C ′′ is not irreducible. Note that, if C ′′ is irreducible, the result for n = 0 is alreadyproved in Lemma 3.6.Let x = a , and x i = a i a − for all i = 2 , . . . , m ′ . Let y j = b j a − for all j = 1 , . . . , m ′′ . Weobtain the presentation G = (cid:28) x , . . . , x m ′ , y , . . . , y m ′′ | [ x , y j ] , x i x y j x − i x − y − j , e R ′ , e R ′′ i = 2 , . . . , m ′ , j = 1 , . . . , m ′′ (cid:29) where e R ′ are some relations in x , . . . , x m ′ , and e R ′′ are the same relations as R ′′ if we switchthe letter b j for y j , for all j = 1 , . . . , m ′′ . Indeed, if we plug in y j x for b j in the relations R ′′ ,the x ’s cancel out because they commute with all the y j ’s and because the linking numberhomomorphism takes any word in the b letters to the sum of the exponents appearing on thatword, so the sum of the exponents of words in R ′′ must be zero.We may assume by reordering that y = y in Γ n , where n ≥ C ′′ is irreducible, and n ≥ C ′′ is not irreducible, this amounts to b and b being positively oriented meridians around different irreducible components of C ′′ , which wecan assume after reordering. If C ′′ is irreducible but δ ( C ′′ ) = 0, Proposition 3.2 says that( G ′′ ) (2) r $ ( G ′′ ) (1) r , which implies that there exist j = l in { , . . . , m ′′ } such that b j b − l = 1 in Γ ′′ .Reordering, we may assume that j = 1 and l = 2, and we get that y = y in Γ n .Consider the involution of the matrix of Fox derivatives for this presentation of G with coef-ficients in Z Γ n ( B ( n ) in the notation of Section 2.3), — (1 − y − j ) — — ( x − − y − j ) — · · · — ( x − m ′ − y − j ) —— 0 — — (1 − x − y − j ) — · · · — 0 —— 0 — — 0 — · · · — 0 — A ′ · · · — (1 − x − y − j ) —( x − − I m ′′ ( x − x − − I m ′′ · · · ( x − x − m ′ − I m ′′ A ′′ , where “— z j —” denotes a row of m ′′ elements whose j -th entry is z j for j = 1 , . . . , m ′′ , and I m ′′ is the identity matrix of dimension m ′′ × m ′′ . A ′ is the matrix corresponding to the relations˜ R ′ , and A ′′ is the matrix that computes δ n ( C ′′ ) with coefficients in Z Γ ′′ n , which is identified with Z ¯Γ n by the isomorphism of groups h : Γ ′′ n → ¯Γ n , given by h ( b j ) = y j . First, note that x i = 1 in Γ n , for all i = 2 , . . . , m ′ . In addition, the left part of this matrixconsists on m ′ blocks of dimensions ( m ′ + m ′′ ) × m ′′ . We subtract the i -th column to the i -thcolumn of the j -th block, for all i = 1 , . . . , m ′′ , j = 2 , . . . , m ′ , to get (3.2) — (1 − y − j ) — — 0 — · · · — 0 —— 0 — — (1 − x − y − j ) — · · · — 0 —— 0 — — 0 — · · · — 0 — A ′ · · · — (1 − x − y − j ) —( x − − I m ′′ · · · A ′′ . ANISHING OF HIGHER ORDER ALEXANDER-TYPE INVARIANTS OF PLANE CURVES 9
Note that 1 − y − j = 0 in Z ¯Γ n for any j = 1 , . . . , m ′′ , since 1 − y − j = 0 in Z ¯Γ . We multiplyrow m ′ + 1 by 1 − y − on the left, and add to it the first row times 1 − x − , and the ( m ′ + j )-throw times 1 − y − j for all j = 2 , . . . , m ′′ , to get — (1 − y − j ) — — 0 — · · · — 0 —— 0 — — (1 − x − y − j ) — · · · — 0 —— 0 — — 0 — · · · — 0 — A ′ · · · — (1 − x − y − j ) —— 0 — — 0 — · · · — 0 — (1 − x − ) a ′ , ∗ | x − − I m ′′ · · · A ′′ | , where a ′ , ∗ is the first row of A ′ , so its entries are polynomials in Z [ x ], which commute withelements of K n , which is identified by h with K ′′ n .We now focus on the second to m ′ -th blocks of size m ′ × m ′′ at the top of the matrix. Wecan multiply the j -th column (on the right) of each of these blocks by y j for all j = 1 , . . . , m ′′ ,and subtract the second from the first column of each of these blocks to get y − y as the firstentry and y j − x − as the j -th entry of the k -th row of the k -th block, where k = 2 , . . . , m ′ , j = 2 , . . . , m ′ . Note that y = y in Z ¯Γ n , so y − y has an inverse in R n . Now, we multiply thefirst column (on the right) by the inverse of 1 − y − , and the first column of the j -th block ofsize m ′ × m ′′ by the inverse of y − y for all j = 2 , . . . , m ′′ . Reordering the columns, putting theones corresponding to the first column of every m ′ × m ′′ block first, we get(3.3) I m ′ ∗ A ′ ∗ (1 − x − ) a ′ , ∗ A ′′ B , where B is the matrix — 0 —( x − − I m ′′ − . Hence, performing column operations we can turn matrix (3.3) into(3.4) I m ′ − x − ) a ′ , ∗ A ′′ B , Let k be the rank of the left K ′′ n -module spanned by the rows of A ′′ . By Proposition 2.8, k isat most m ′′ −
1, and k is equal to m ′′ − δ n ( C ′′ ) < ∞ . Identifying K ′′ n with K n by h , we get that the rank of the left K n -module spanned by the rows of A ′′ is k as well. Hence,doing row and column operations in K ′′ n , and noting that x commutes with K ′′ n in R n , we canturn the matrix (3.4) into I m ′ I k x − − e B ( x − − E , where e B is an m ′′ × m ′′ matrix in K ′′ n such that the rank of the left K ′′ n -module spanned by itsrows is m ′′ −
1, and E is a matrix with entries in R n . In particular, the rank of the left K ′′ n -module spanned by the last m ′′ − k rows of e B is greater or equal than m ′′ − k −
1, and at most m ′′ − k .Let us denote by D and F the matrices formed by the last m ′′ − k rows of e B and E respectively.Doing column operations, we get(3.5) I m ′ I k x − − D ( x − − F . By Lemma 2.7, the rank of the left K n -module generated by the rows of this matrix should beless or equal than m ′ + m ′′ −
1, which rules out the possibility of the rank of the left K ′′ n -modulespanned by the rows of D being m ′′ − k . Hence, the rank of the left K ′′ n -module spanned by therows of D is m ′′ − k −
1. If we keep doing row and column operations to D in K ′′ n , and perhapspermuting some of the last m ′′ − k rows of matrix (3.5) at the end, one obtains I m ′ I k x − − I m ′′ − k − — 0 — — 0 — — 0 — (1 − x − ) ∗ , where ∗ is a matrix in R n . But again, by rank considerations, the last row of this matrix mustbe identically 0. Performing column operations, we can turn (1 − x − ) ∗ into the zero matrix.Hence, δ n ( C ) = m ′′ − k −
1, which is a finite number. This means that δ n ( C ) = 0 if and only if δ n ( C ′′ ) is finite. (cid:3) Lemma 3.8.
Let C = C ′ ∪ C ′′ ⊂ C be the union of two affine plane curves, with deg C ′ = m ′ and deg C ′′ = m ′′ . Suppose that C ′ ∩ C ′′ consists on m ′ m ′′ distinct points in C . Suppose thatneither C ′ nor C ′′ are irreducible with δ = 0 . Then δ n ( C ) = 0 for all n ≥ . Moreover, if either both C ′ and C ′′ are not irreducible, or both irreducible, the equality holds forall n ≥ . However, if one of the curves, say C ′ , is irreducible and the other one ( C ′′ ) is not,then δ ( C ) is finite, and δ ( C ) = 0 ⇔ δ ( C ′′ ) < ∞ . Proof.
If both C ′ and C ′′ are irreducible, the result for n = 0 follows from Lemma 3.6.We consider the presentation of G described in Remark 3.5. If C ′ is not irreducible, we canassume that a = a in Γ by reordering, so a a − = 1 in Z ¯Γ n for any n ≥
0. Similarly, if C ′′ is not irreducible, we can assume that b b − = 1 in Z ¯Γ n for any n ≥
0. After reordering, wemay assume the same condition if C ′′ is irreducible with δ ( C ′′ ) = 0 (resp. C ′ ), but this time for n ≥
1, as justified in the proof of Lemma 3.7.We deal with the case when n is some integer greater or equal than 1 if either C ′ or C ′′ areirreducible, and ≥ x = a , and x i = a i a − for all i = 2 , . . . , m ′ . Let y = b , and y j = b j b − for all j = 2 , . . . , m ′′ . We obtain the presentation G = h x , . . . , x m ′ , y , . . . , y m ′′ | [ x i , y j ] for all i = 1 , . . . , m ′ and j = 1 , . . . , m ′′ ; e R ′ ; e R ′′ i where e R ′ (resp. e R ′′ ) are the defining relations for π ( C \ C ′ ) (resp. π ( C \ C ′′ )) in x , . . . , x m ′ (resp. y , . . . , y m ′′ ). ANISHING OF HIGHER ORDER ALEXANDER-TYPE INVARIANTS OF PLANE CURVES 11
Consider the matrix B ( n ) described in Section 2.3, that is, (3.6) — (1 − y − j ) — — 0 — · · · — 0 —— 0 — — (1 − y − j ) — · · · — 0 —... ... · · · ... ∗ — 0 — — 0 — · · · — (1 − y − j ) —( x − − I m ′′ ( x − − I m ′′ · · · ( x − m ′ − I m ′′ ∗ , where the rightmost columns correspond to e R ′ and e R ′′ .Note that x − − Z ¯Γ n if n ≥
1, and, if C ′ is not irreducible, also for n = 0.We begin by multiplying the last m ′′ rows by the inverse of x − − — (1 − y − j ) — — 0 — · · · — 0 —— ∗ — — (1 − y − j ) — · · · — ∗ —... ... · · · ... ∗ — 0 — — 0 — · · · — (1 − y − j ) —0 I m ′′ · · · . Hence, we may compute δ n ( C ) using the matrix formed by the first m ′ rows of the matrix abovewithout the columns of the second block. This new matrix consists on m ′ − m ′ × m ′′ matrices, plus another matrix at the end, represented by the rightmost submatrix after the lastvertical line. One can permute the first and second rows in this new matrix of m ′ rows, and thenpermute columns so that the first m ′ columns of the resulting matrix are the second columns ofeach of the first ( m ′ −
1) blocks of size m ′ × m ′′ . This way one obtains a matrix of the form, (cid:18) ∗ ∗ (1 − y − ) I m ′ − ∗ (cid:19) . Note that 1 − y − is non-zero in Z ¯Γ n for n ≥
1, and, if C ′′ is not irreducible, also in Z ¯Γ . Finally,multiplying each row on the left by the inverse of 1 − y − , and performing column and rowoperations, we see that δ n ( C ) = 0.Lastly, we consider the case when n = 0, C ′ is irreducible, and C ′′ is not irreducible. Theproof of this is done by considering the same presentation for G as the one explained in the proofof Lemma 3.7, and following the same computations done there, the only difference being that n = 0 in this case and that, since C ′′ is not irreducible, we know that y = y in Z ¯Γ . Note that x i = 1 in Z Γ for i = 2 , . . . , m ′ because C ′ is irreducible. Using the same notation as in the proofof Lemma 3.7, it follows that δ ( C ) = m ′′ − k − k is the rank of theleft K ′′ n -module spanned by the rows of A ′′ , and δ ( C ′′ ) is finite if and only if k = m ′′ −
1. Thismeans that δ ( C ) = 0 if and only if δ ( C ′′ ) is finite. (cid:3) Example 3.9.
Let C ′ be an irreducible curve such that δ ( C ′ ) = 0. For example, we can take C ′ to be the cuspidal cubic, which has δ ( C ′ ) = 2, and δ n ( C ′ ) = 1 for n ≥ C ′′ be a collection of m ′′ parallel lines, each of which intersects C ′ in three distinct points.Let C = C ′ ∪ C ′′ . Then, following the proof and notations of Theorem 3.8, it follows that (cid:26) δ ( C ) = m ′′ − δ n ( C ) = 0 for all n ≥ . Indeed, in this case the fundamental group of the complement to m ′′ parallel lines is the freegroup on m ′′ generators, and thus it has a presentation with no relations. Hence A ′′ is the emptymatrix, so the k that appears at the end of the proof of Theorem 3.8 is 0. This example shows that δ and δ n can differ by arbitrarily large numbers, for n ≥
1. Thiscannot happen in the case of knots ([2]), where δ ≤ δ + 1 ≤ δ + 1 ≤ . . . .4. Restatement of the main theorem in terms of multivariable Alexanderpolynomials
We start by recalling the relationship between the Alexander polynomials of a plane curve C and δ ( C ). If C is irreducible, the result below appears in [13, Remark 3.9], and the non-irreducible case was done in [5, Theorem 11]. Theorem 4.1.
Let C ⊂ C be a plane curve with s irreducible components. Then, δ ( C ) = deg ∆ multi C ( t , . . . , t s ) Remark . We are using the convention deg 0 = ∞ . The proof of [5, Theorem 11] assumes δ ( C ) is finite, but the result is also true for δ ( C ) = ∞ because δ ( C ) = ∞ ⇔ F ( H ( U, u ; R )) = 0 ⇔ F ( H ( U, u ; Z Γ )) = 0 ⇔ ∆ multi C = 0 . In this list of equivalences, we have used that the projection G ։ Γ is the abelianizationmorphism and that R is flat as a Z Γ -module. Remark . Theorem 4.1 tells us that δ ( C ) = 0 if and only if ∆ multi C has a representative whichis a non-zero homogeneous polynomial, which, under the hypotheses of Theorem 1.3, impliesthat ∆ multi C is a non-zero constant. Indeed, ∆ multi C is the gcd of the codimension-one minors of B (0), and the matrix B (0) in equation (3.6) has a codimension-one minor which does not havenon-constant homogeneous factors.Now, we characterize the plane curves with infinite δ ( C ). Lemma 4.4.
Let C be a plane curve, with s irreducible components C , . . . , C s . Then, thefollowing are equivalent. (1) δ ( C ) = ∞ . (2) s ≥ and C i is the zero set of a polynomial of the form f ( x, y ) + λ i for all ≤ i ≤ s ,where f ( x, y ) ∈ C [ x, y ] is a polynomial of degree d ≥ , and λ i ∈ C for all ≤ i ≤ s . (3) There exists an epimorphism G ։ F s onto the free group of rank s ≥ . We will refer to condition (2) as C being of affine pencil type . Proof.
Note that by Theorem 1.1, δ ( C ) = ∞ ⇒ s ≥ B (0) is a presentation matrix for H ( U, u ; Z Γ ). By [5,Proposition 3, Remark 12], we have the following result about the first homology jump loci[5, Definition 9], which we’ll use as an intermediate equivalent condition in our proof: V ( U ) = ( C ∗ ) s ⇔ all the codimension 1 minors of B (0) are 0 . This last condition is equivalent to the rank of the left K -module generated by the rows of B (0)being strictly smaller than m −
1, which by Remark 2.3 is equivalent to δ ( C ) = ∞ .Let C be the projective completion of C , and let D be the curve in P defined by D = ¯ C ∪ L ∞ ,where L ∞ is the line at infinity. By [3, Theorem 4.1], the condition V ( U ) = ( C ∗ ) s (which can bereformulated in terms of cohomology jump loci by [4, p.50, (2.1)]) is equivalent to the existenceof a primitive pencil C [ α : α ] = α P ( x, y, z ) + α P ( x, y, z ) of plane curves on P having s + 1fibers (corresponding to s + 1 different [ α : α ] ∈ P ) whose reduced support form a partitionof the set of s + 1 irreducible components of D . Hence, the reduced support of those s + 1 fibersmust be in one to one correspondence with the irreducible components of D , so we may write ANISHING OF HIGHER ORDER ALEXANDER-TYPE INVARIANTS OF PLANE CURVES 13 the pencil in the form β F ( x, y, z ) + β z d , where F ( x, y, z ) is a degree d irreducible polynomialin C [ x, y, z ] and [ β : β ] ∈ P . Restricting to the affine part (making z = 1) yields (1) ⇔ (2).For (2) ⇒ (3), we see that f ( x, y ) induces the desired epimorphism in fundamental groups.By [12, Lemma 1.2.1] (3) implies V ( U ) = ( C ∗ ) s and hence the argument above implies (2). (cid:3) As a corollary of the lemma above, Theorem 4.1, Remark 4.3 and Theorem 1.3, one obtainsCorollary 1.4, whose proof is below.
Proof of Corollary 1.4.
The only thing left to show is the last statement. ∆ multi C can be computedwith the matrix from equation (3.2), as the operations we did to get from B (0) to that matrixwere all allowed in Z Γ ∼ = Z [ t ± , t ± , . . . , t ± s ]. The abelianization morphism identifies x with t and y j with t j t in equation (3.2), and with those identifications and up to multiplication by a unitin Z [ t ± , t ± , . . . , t ± s ], ( t j − m ′ − (1 − t ) m ′′ and ( t j − t )( t j − − t ) m ′′ − are ( m ′ + m ′′ − j = 1 , . . . , m ′′ . Hence, ∆ multi C divides the greatest common divisor of all theseminors, so ∆ multi C divides ( t − m ′′ − . The equality can be achieved, [19, Theorem 9.15, case (ii)]gives an example where ∆ multi C ( t, t , . . . , t s ) = ( t − m ′′ − . (cid:3) Remark . The higher order degrees depend on the linking number homomorphism, which, ifthe curve is not irreducible, is not an invariant of the fundamental group of the curve complement.However, the multivariable Alexander polynomial only depends on the fundamental group (upto a change of basis in the variables). Thus, Corollary 1.4 gives us direct restrictions for whichgroups can be realized as fundamental groups of the complement of a union of transversal planecurves, and those restrictions can be computed from a presentation of the group.
Example 4.6.
As an example of a curve of affine pencil type one can consider the cuspidal cubic f ( x, y ) = y − x and the curve C ′′ = { ( x, y ) ∈ C | f ( f −
1) = 0 } of degree m ′′ = 6. One cancheck that π ( C \ C ′′ ) = h α , α , γ : [ γ, α ] = [ γ α , α ] = 1 , γ = γ α γ α − i . Here, α and γα are positively oriented meridians about f = 0, and α is a positively orientedmeridian about f = 1.Let r n be the rank of the left K ′′ n -module generated by the rows of the 3 × B ( n ) ofSection 2.3 computed from the presentation of π ( C \ C ′′ ) above. One has γ − − ∈ Z ¯Γ ′′ n is 0if n = 0 and a unit otherwise. Using this, it is straightforward to check that r = 1, and r n = 2for all n ≥
1. By Lemma 2.7, δ ( C ′′ ) = ∞ and δ n ( C ′′ ) < ∞ for all n ≥
1. In fact, using themethods of Section 2.3, one can check that δ n ( C ′′ ) = 0 for all n ≥
1. Indeed, B ( n ) = − uα v uα + α − uα − u − u α − uα v − α v α − γ − α − α ∼ = α v − w − α − u uw vw α − γ − α − w ∼ = (cid:20) α v − w − α wα − w − wα − + α − γ − α (cid:21) ∼ = (cid:20) α v − w − α α − (cid:21) where u = ( γ − − v = ( α − − w = u − α − uα = [ u − , α − ]. The first transformationis a result of multiplying the first row by − u − (on the left), and then adding row 1 and vu − times row 2 to row 3. The second transformation results from eliminating column 1 and row 2(since u is a unit) and substracting wα − times row 1 from row 2, using that v and w commute.The resulting last row is identically 0 because α v = 0 and r n is at most 2 by Lemma 2.7. One obtains the last matrix after substracting the first column from the second column. Finally,1 − w − α α − ∈ Z ¯Γ ′′ n is a unit, as it is non-zero in Z ¯Γ ′′ , so δ n ( C ′′ ) = 0 for all n ≥ C ′ be an irreducible curve of degree m ′ such that C ′ and C ′′ intersect in 6 m ′ distinct points,and let C = C ′ ∪ C ′′ . Using the statement above Lemma 2.7 and the proofs of Lemmas 3.7and 3.8, we get that δ ( C ) = (row corank of a presentation matrix of H ( U ′′ , u ′′ ; K ′′ )) − − r ) − δ n ( C ) = 0 for all n ≥
1. By Corollary 1.4 andTheorem 4.1, ∆ multi C ( t, t , t ) = t − References [1] D. Arapura,
Fundamental groups of smooth projective varieties , Current topics in complex algebraic geometry(Berkeley, CA, 1992/93), Math. Sci. Res. Inst. Publ., vol. 28, Cambridge Univ. Press, Cambridge, 1995,pp. 1–16.[2] T. D. Cochran,
Noncommutative knot theory , Algebr. Geom. Topol. (2004), 347–398.[3] A. Dimca, Pencils of plane curves and characteristic varieties , Progr. Math., vol. 283, Birkh¨auser Verlag,Basel, 2010, pp. 59–82.[4] ,
Sheaves in topology , Universitext, Springer-Verlag, Berlin, 2004.[5] E. Elduque and L. Maxim,
Higher Order Degrees of Affine Plane Curve Complements , Indiana Journal ofMathematics ((to appear)).[6] S. L. Harvey,
Higher-order polynomial invariants of 3-manifolds giving lower bounds for the Thurston norm ,Topology (2005), no. 5, 895–945.[7] P. D. Horn, On computing the first higher-order Alexander modules of knots , Exp. Math. (2014), no. 2,153–169.[8] P. Kirk and C. Livingston, Twisted Alexander invariants, Reidemeister torsion, and Casson-Gordon invari-ants , Topology (1999), no. 3, 635–661.[9] A. Libgober, Alexander polynomial of plane algebraic curves and cyclic multiple planes , Duke Math. J. (1982), no. 4, 833–851.[10] , On the homotopy type of the complement to plane algebraic curves , J. Reine Angew. Math. (1986), 103–114.[11] ,
The topology of complements to hypersurfaces and nonvanishing of a twisted de Rham cohomology ,AMS/IP Stud. Adv. Math., vol. 5, Amer. Math. Soc., Providence, RI, 1997, pp. 116–130.[12] ,
Characteristic varieties of algebraic curves , Applications of algebraic geometry to coding theory,physics and computation (Eilat, 2001), Kluwer Acad. Publ., Dordrecht, 2001, pp. 215–254.[13] C. Leidy and L. Maxim,
Higher-order Alexander invariants of plane algebraic curves , Int. Math. Res. Not.(2006), Art. ID 12976, 23.[14] ,
Obstructions on fundamental groups of plane curve complements , Real and complex singularities,Contemp. Math., vol. 459, Amer. Math. Soc., Providence, RI, 2008, pp. 117–130.[15] M. Oka,
A survey on Alexander polynomials of plane curves , Singularit´es Franco-Japonaises, S´emin. Congr.,vol. 10, Soc. Math. France, Paris, 2005, pp. 209–232.[16] M. Oka and K. Sakamoto,
Product theorem of the fundamental group of a reducible curve , J. Math. Soc. Japan (1978), no. 5, 599–602.[17] Jean-Pierre Serre, Sur la topologie des vari´et´es alg´ebriques en caract´eristique p , Symposium internacionalde topolog´ıa algebraica International symposium on algebraic topology, Universidad Nacional Aut´onoma deM´exico and UNESCO, Mexico City, 1958, pp. 24–53 (French).[18] Y. Su, Higher-order Alexander invariants of hypersurface complements , arXiv:1510.03467.[19] A. I. Suciu,
Fundamental groups, Alexander invariants, and cohomology jumping loci , Topology of algebraicvarieties and singularities, Contemp. Math., vol. 538, Amer. Math. Soc., Providence, RI, 2011, pp. 179–223.[20] O. Zariski,
On the irregularity of cyclic multiple planes , Ann. of Math. (2) (1931), no. 3, 485–511. ANISHING OF HIGHER ORDER ALEXANDER-TYPE INVARIANTS OF PLANE CURVES 15
Departamento de Matem´aticas, IUMA, Universidad de Zaragoza, C. Pedro Cerbuna 12, 50009Zaragoza, Spain
Email address : [email protected] Department of Mathematics, University of Michigan-Ann Arbor, 530 Church Street, Ann Arbor,MI 48109-1043, USA
Email address ::