HIGHER-ORDER ALEXANDER INVARIANTS OF PLANE ALGEBRAIC CURVES

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1 HIGHER-ORDER ALEXANDER INVARIANTS OF PLANE ALGEBRAIC CURVES CONSTANCE LEIDY AND LAURENTIU MAXIM Abstact. We define new highe-ode Alexande modules A n (C) and highe-ode degees δ n (C) which ae invaiants of the algebaic plana cuve C. These come fom analyzing the module stuctue of the homology of cetain solvable coves of the complement of the cuve C. These invaiants ae in the spiit of those developed by T. Cochan in [2] and S. Havey in [8] and [9], which wee used to study knots, 3-manifolds, and finitely pesented goups, espectively. We show that fo cuves in geneal position at infinity, the higheode degees ae finite. This povides new obstuctions on the type of goups that can aise as fundamental goups of complements to affine cuves in geneal position at infinity. 1. Intoduction The study of singula plane cuves is a subject going back to the wok of Zaiski, who obseved that the position of singulaities has an influence on the topology of the cuve, and that this phenomena can be detected by the fundamental goup of the complement. Howeve, the fundamental goup of a plane cuve complement is in geneal highly noncommutative, and thus difficult to handle. It is theefoe natual to look fo invaiants of the fundamental goup that captue infomation about the topology of the cuve, such as Alexande-type invaiants associated to vaious coveing spaces of the cuve complement. By analogy with the classical theoy of knots and links in a 3-sphee, Libgobe developed invaiants of the total linking numbe infinite cyclic cove in [10, 13, 14] and those of the univesal abelian cove in [17, 16]. In this pape, we conside cetain solvable coves of the cuve complement, and thei associated Alexande invaiants. Using techniques developed by T. Cochan, K. O, and P. Teichne in [3], S. Havey (in [8]) and T. Cochan (in [2]) defined highe-ode Alexande modules and highe-ode degees associated to 3-manifolds and knots, espectively. They used these invaiants to show that cetain goups cannot be ealized as the fundamental goup of the complement of a knot, o as the fundamental goup of a 3-manifold. In the pesent pape, we use the same type of invaiants to study the complement of complex plane algebaic cuves. Ou main esult shows that unde cetain estictions on the cuve, these invaiants ae unifomly bounded. This povides a new obstuction on the goups being ealizable as the fundamental goup of the complement to a plane cuve Suvey of esults. Let C be a educed cuve in C 2, and conside U, the complement of C in C 2, with G = π 1 (U). The multivaiable Alexande invaiant, studied in [16, 17] (but see also [6]), is defined by consideing the univesal abelian coveing space of U coesponding 2000 Mathematics Subject Classification. Pimay 32S20, 32S05; Seconday 14J70, 14F17, 57M25, 57M27. Key wods and phases. cuve complement, singulaities, Alexande invaiants, highe-ode degee. 1

2 2 CONSTANCE LEIDY AND LAURENTIU MAXIM to the map G G/[G, G] = Z s, whee s is the numbe of ieducible components of the cuve. We continue this constuction by taking iteated univesal tosion-fee abelian coves of U coesponding to the maps G G/G (n+1) Γ n, whee G (n) is the n th -tem in the ational deived seies of G (defined in 2 below). We define the highe-ode Alexande modules of the plane cuve complement to be A Z n(c) = H 1 (U; ZΓ n ), and note that A Z 0 (C) is just the (integal) univesal abelian Alexande module of C. The following is a coollay to Theoem 4.1, and povides an analogue to simila esults fom the infinite cyclic and univesal abelian cases: Coollay 4.2. If C is a educed cuve in C 2, that is in geneal position at infinity, A Z n(c) is a tosion ZΓ n -module. Futhemoe, we conside some skew Lauent polynomial ings K n [t ±1 ], which ae obtained fom ZΓ n by inveting the non-zeo elements of a paticula subing. The advantage of using K n [t ±1 ] coefficients instead of ZΓ n coefficients is that the fome is a pincipal ideal domain. We define the highe-ode degee of C to be δ n (C) = k Kn H 1 (U; K n [t ±1 ]). Even though, in pinciple, the highe-ode degees may be computed by means of Fox fee calculus (cf. [8], 6), the calculations ae tedious as they depend on a pesentation of the fundamental goup of the cuve complement. Howeve, in the case that the cuve is in geneal position at infinity, we find a unifom uppe bound on the highe-ode degees. In paticula, we pove the following esult: Theoem 4.1. Suppose C is a degee d cuve in C 2, such that its pojective completion C is tansvese to the line at infinity. If C has singulaities c k, 1 k l, then (1.1) δ n (C) Σ l k=1 (µ(c, c k ) + 2n k ) + 2g + d l, whee µ(c, c k ) is the Milno numbe associated to the singulaity gem at c k, n k is the numbe of banches though the singulaity c k, and g is the genus of the nomalized cuve. As a diect coollay of the poof of Theoem 4.1, we also find a bound on the highe-ode degees of the cuve in tems of local degees, δ k n, fo each singulaity c k of C. The latte wee defined and studied by Havey [8]. Theoem 4.5. If C satisfies the assumptions of the pevious theoem, then δ n (C) Σ l k=1( δ k n + 2n k ) + 2g + d, whee δ k n = δ n (X k ) is Havey s invaiant of the link complement X k associated to the singulaity c k. We view Theoem 4.5 as an analogue of the divisibility popeties fo the infinite cyclic Alexande polynomial of the complement as shown in [13]. Fo ieducible cuves, egadless of the position of the line at infinity, the highe-ode degees ae finite and thus the highe-ode Alexande modules ae tosion. Howeve, if the line at infinity is not tansvese to the ieducible cuve C, then the uppe bounds mentioned above will also include the contibution of the singula points at infinity (simila to [12], Theoem 4.3). To complete the analogy with the case of Alexande polynomials of the infinite cyclic cove of the complement, we also povide an uppe bound on δ n (C) by the coesponding highe-ode Alexande invaiant of the link at infinity (see Theoem 4.7). Fo a cuve of degee d, in geneal position at infinity, this is an unifom bound equal to d(d 2).

3 HIGHER-ORDER ALEXANDER INVARIANTS OF PLANE ALGEBRAIC CURVES 3 We conclude the pape with sample computations of highe-ode degees of plane cuves, also indicating some connections with the infinite cyclic and univesal abelian invaiants of the cuve. In passing we note that the highe-ode degees δ n, at any level n, ae sensitive to the position of singula points (see Example 5.6). This fact alone gives an incentive to look fo examples of Zaiski pais that ae distinguished by some δ k, but not by any othe Alexande-type invaiants. This will make the subject of futue wok by these authos Connections with othe wok. We end the intoduction with pesenting futhe motivation fo this wok and connections with pevious studies on invaiants of the fundamental goup of a plane cuve complement. Although in geometic poblems the fundamental goup of complements to pojective cuves plays a cental ole, by switching to the affine setting (i.e. by emoving also a geneic line) no essential infomation is lost. Indeed, if C is the pojective completion of C and H is the line at infinity, the two goups ae elated by the cental extension 0 Z π 1 (CP 2 ( C H)) π 1 (CP 2 C) 0. Moeove, as Oka poved (e.g., see [23], Lemma 2), the commutatos of the two fundamental goups (in the affine and esp. pojective setting) coincide, theefoe eithe of the two goups can be used fo computing the ational deived seies of the fundamental goup of the affine complement. Ou finiteness esult on the highe-ode degees povides also new obstuctions on the type of goups that can aise as fundamental goups of complements to affine hypesufaces in geneal position at infinity. Indeed, by a Zaiski-Lefschetz theoem, possible fundamental goups of complements to hypesufaces in C n ae pecisely the fundamental goups of affine plane cuve complements. Note that fo a geneal goup, one does not expect the higheode degees δ n to be finite. Fo instance, fo a fee goup with at least 2 geneatos the fee anks n ae positive (cf. [8], Example 8.2) theefoe δ n is infinite. Simila obstuctions on fundamental goups of complements to affine plane cuves wee peviously obtained by Libgobe and othes (fo a nice discussion on this topic in elation with a question of See, the eade is advised to consult [15]). Fo example, fom the study of the total linking numbe infinite cyclic cove of the complement [10, 13], it follows that the Alexande polynomial of the (affine) cuve is cyclotomic. Moe pecisely, fo a cuve in geneal position at infinity this polynomial divides the poduct of the local Alexande polynomials at the singula points, and its zeos ae oots of unity of ode d = deg( C). This esult aleady obstucts many knot goups fom being ealizable as fundamental goups of complements to affine plane cuves. Moe obstuctions wee deived by Libgobe [16, 17] and Aapua [1] fom the study of the univesal abelian cove of the affine complement. We only mention hee the poweful esult of Aapua which states that the suppot (hence all chaacteistic vaieties) of the fundamental goup of a plane cuve complement is a union of subtoi of the chaacte tous, possibly tanslated by unitay chaactes. Ou obstuctions come fom analyzing the solvable coveings associated to the ational deived seies of the fundamental goup of the affine complement. It would be inteesting at this point to undestand how the highe-ode degees ae elated to (o influenced by) the invaiants of the infinite cyclic o univesal abelian coves of the complement. Poposition 5.1 aleady povides such a elation. In connection with the univesal abelian cove, Libgobe poved that if the codimension (in the chaacte tous) of suppot of the univesal abelian

4 4 CONSTANCE LEIDY AND LAURENTIU MAXIM Alexande module is geate than 1, then δ 0 (C) = 0. Of couse, this assumption can only be satisfied if the cuve has at least 2 ieducible components, and it emains to undestand fo what type of cuves such a condition holds. Acknowledgements. The authos would like to thank Tim Cochan, Shelly Havey, Anatoly Libgobe and Julius Shaneson fo many helpful convesations about this poject. 2. Rational deived seies of a goup; PTFA goups In this section, we eview the definitions and basic constuctions that we will need fom [8] and [2]. Moe details can be found in these souces. We begin by ecalling the definition of the ational deived seies {G (i) } associated to any goup G. Definition 2.1. Let G (0) of G by: G (n) = {g G (n 1) = G. Fo n 1, define the n th tem of the ational deived seies g k [G (n 1), G (n 1) ], fo some k Z {0}}. We denote by Γ n the quotient G/G (n+1). Since G (n) is a nomal subgoup of G (i) fo 0 i n ([8], Lemma 3.2), it follows that Γ n is a goup. The use of ational deived seies, as opposed to the usual deived seies {G (n) }, is necessay to avoid zeo divisos in the goup ing ZΓ n. Howeve, if G is a knot goup o a fee goup, the ational deived seies and the deived seies coincide ([8], p. 902). If G is a finite goup then G (n) = G, hence Γ n = {1} fo all n 0. The ational deived seies is defined in such a way that the successive quotients G (n) /G (n+1) ae Z-tosion-fee and abelian. In fact ([8], Lemma 3.5): ( (2.1) G (n) /G (n+1) = G (n) /[G (n), G (n) ] ) /{Z tosion}. If G = π 1 (X) this says that G (n) /G (n+1) = H1 (X Γn 1 )/{Z tosion}, whee X Γn 1 is the egula Γ n 1 -cove of X. In paticula, G/G (1) = G (0) /G (1) = H1 (X)/{Z tosion} = Z β1(x). Remak 2.2. If G is the fundamental goup of a link complement in S 3 o that of a plane cuve complement, then G (1) = G (1) (since thee is no tosion in the fist homology of the complement). Definition 2.3. A goup Γ is poly-tosion-fee-abelian (PTFA) if it admits a nomal seies of subgoups {1} = G 0 G 1 G n = Γ such that each of the successive quotients G i+1 /G i is tosion-fee abelian. Remak 2.4. We collect hee the following facts: (1) Any subgoup of a PTFA goup is a PTFA goup. (2) If Γ is PTFA, then ZΓ is a ight (and left) Oe domain (i.e., has no zeo divisos and ZΓ {0} is a ight diviso set). Thus it embeds in its classical ight ing of quotients K, a skew field ([2], Poposition 3.2). (3) If R is an Oe domain and S is a ight diviso set, then RS 1 is flat as a left R-module. In paticula, K is a flat ZΓ-module ([24], Poposition II.3.5). (4) Evey module ove K is a fee module ([24], Poposition I.2.3). Such modules have a well-defined ank k K which is additive on shot exact sequences.

5 HIGHER-ORDER ALEXANDER INVARIANTS OF PLANE ALGEBRAIC CURVES 5 If A is a module ove an Oe domain R, then the ank of A is defined as k(a) = k K (A R K), whee K is the quotient field of R. In paticula, A is a tosion R-module if and only if A R K = 0. Poposition 2.5. ([8], Coollay 3.6) Fo any goup G, Γ n = G/G (n+1) ZΓ n embeds in its classical ight ing of quotients, K n. is PTFA. Thus Suppose X is a topological space that has the homotopy type of a connected CW-complex. Let Γ be any goup and φ : π 1 (X) Γ be a homomophism. We denote by X Γ the egula Γ-cove of X associated to φ. If φ is sujective, this is the coveing space associated to ke φ. (Fo details about the case whee φ is not sujective, we efe the eade to 3 of [2].) Let C(X Γ ; Z) be the ZΓ-fee cellula chain complex fo X Γ obtained by lifting the cell stuctue of X. If M is a ZΓ-bimodule, then define: as a ight ZΓ-module. H (X; M) = H (C(X Γ ; Z) ZΓ M) Poposition 2.6. ([3], Poposition 2.9) Let X be a connected CW-complex and Γ a PTFA goup. If φ : π 1 (X) Γ is a non-tivial coefficient system, then H 0 (X; ZΓ) is a tosion ZΓ-module. Poposition 2.7. ([2], Poposition 3.10) Let X be a connected CW-complex and Γ be a PTFA goup. Suppose π 1 (X) is finitely geneated and φ : π 1 (X) Γ is a non-tivial coefficient system. Then k (H 1 (X; ZΓ)) β 1 (X) 1, whee β 1 (X) is the fist Betti numbe of X. In paticula, if β 1 (X) = 1 then H 1 (X; ZΓ) is a tosion ZΓ-module. 3. Definitions of new invaiants Let C be a educed cuve in C 2, defined by the equation: f = f 1 f s = 0, whee f i ae the ieducible factos of f, and let C i = {f i = 0} denote the ieducible components of C. Embed C 2 in CP 2 by adding the plane at infinity, H, and let C be the pojective cuve in CP 2 defined by the homogenization f h of f. We let C i = {fi h = 0}, i = 1,, s, be the coesponding ieducible components of C. Let U be the complement CP 2 ( C H). (Altenatively, U may be egaded as the complement of the cuve C in the affine space C 2.) Then H 1 (U) is fee abelian geneated by the meidian loops γ i about the non-singula pat of each ieducible component C i, fo i = 1,, s (cf. [4], (4.1.3), (4.1.4)). If γ denotes the meidian about the line at infinity, then the equation γ + s i=1 d iγ i = 0 with d i = deg(fi h ), holds in H 1 (U) Highe-ode Alexande modules. We let G = π 1 (U), Γ n = G/G (n+1), and K n be the classical ight ing of quotients of ZΓ n. Definition 3.1. We define the highe-ode Alexande modules of the plane cuve to be: whee U Γn G (n+1) /[G (n+1) A Z n(c) = H 1 (U; ZΓ n ) = H 1 (U Γn ; Z) is the coveing of U coesponding to the subgoup G (n+1). That is, A Z n(c) =, G (n+1) ] as a ight ZΓ n -module. Definition 3.2. The n th ode ank of (the complement of) C is: n (C) = ka Z n(c).

6 6 CONSTANCE LEIDY AND LAURENTIU MAXIM Remak 3.3. (1) Note that A Z 0 (C) = G (1) /[G (1), G (1) just the univesal abelian invaiant of the complement. (2) A Z n(c)/{z tosion} = G (n+1) /G (n+2). ] = G /G, by Remak 2.2. This is (3) If C is ieducible, then β 1 (U) = 1. By Poposition 2.7, it follows that A Z n(c) is a tosion module. In Coollay 4.2, we show that unde the assumption of tansvesality at infinity, the module A Z n(c) is a tosion ZΓ n -module. Theefoe, n (C) = 0. Since U is a 2-dimensional affine vaiety, it has the homotopy type of a 2-dimensional CWcomplex. Thus the modules H k (U; ZΓ n ) ae tivial fo k > 2 and H 2 (U; ZΓ n ) is a tosion-fee ZΓ n -module. Moeove, we will show that in ou setting, the ank of H 2 (U; ZΓ n ) is equal to the Eule chaacteistic of the complement, U. Remak 3.4. Assume that the univesal abelian Alexande module of the complement is tivial, i.e. A Z 0 (C) = 0. (Note that this is the case if G is abelian o finite.) Then all highe-ode Alexande modules A Z n(c) = 0, fo n 1, ae also tivial. Indeed, by Remak 2.2, G = G (1) and A Z 0 (C) = G /G. It follows that G (n) = G = G (1), fo all n 1. Fom the definition of the ational deived seies, it is now easy to see that G (n) = G fo all n 1. Theefoe A Z n(c) = G (n+1) /[G (n+1), G (n+1) ] = G /G = 0, fo all n 0. Example 3.5. (1) If C is a non-singula cuve in geneal position at infinity, then π 1 (U) = Z (cf. [12]), hence abelian. By the above emak, it follows that A Z n(c) = 0, fo all n 0. (2) Suppose U is the complement in C 2 of a union of two lines. Then π 1 (U) is Z 2. Hence A Z n(c) = 0 fo all n 0. (3) If C is a educed cuve having only nodes as its singulaities (i.e., locally at each singula point, C looks like x 2 y 2 = 0), then it is known that π 1 (CP 2 C) is abelian (e.g., see [23]), thus has tivial commutato subgoup. Unde the assumption that the line at infinity is geneic, this implies that the commutato subgoup of π 1 (U) is also tivial ([23], Lemma 2), so π 1 (U) is abelian. Now fom Remak 3.4 it follows that A Z n(c) = 0, fo all n Localized highe-ode Alexande modules. In this section we define some skew Lauent polynomial ings K n [t ±1 ], which ae obtained fom ZΓ n by inveting the non-zeo elements of a paticula subing descibed below. This constuction is used in [8] and [2] and is descibed in algebaic geneality in [9]. We efe to those souces fo the backgound definitions. Recall ou notations: G = π 1 (U), Γ n = G/G (n+1) and K n is the classical ight ing of quotients of ZΓ n. Let ψ H 1 (G; Z) = Hom Z (G, Z) be the pimitive class epesenting the linking numbe homomophism G ψ Z, α lk(α, C). Since the commutato subgoup of G is in the kenel of ψ, it follows that ψ induces a well-defined epimophism ψ : Γ n Z. Let Γ n be the kenel of ψ. Since Γn is a subgoup of Γ n, by Remak 2.4, Γ n is a PTFA goup. Thus Z Γ n is an Oe domain and S n = Z Γ n {0} is a ight diviso set of Z Γ n. Let K n = (Z Γ n )Sn 1 be the ight ing of quotients of Z Γ n, and set R n = (ZΓ n )Sn 1. If we choose a t Γ n such that ψ(t) = 1, this yields a splitting φ : Z Γ n of ψ. As in = Poposition 4.5 of [8], the embedding Z Γ n K n extends to an isomophism R n K n [t ±1 ]. (Howeve this isomophism depends on the choice of splitting!) It follows that R n is a noncommutative pincipal left and ight ideal domain, since this is known to be tue fo any

7 HIGHER-ORDER ALEXANDER INVARIANTS OF PLANE ALGEBRAIC CURVES 7 skew Lauent polynomial ings with coefficients in a skew field ([2], Poposition 4.5). Also note that by Remak 2.4, R n is a flat left ZΓ n -module. Definition 3.6. The n th -ode localized Alexande module of the cuve C is defined to be A n (C) = H 1 (U; R n ), viewed as a ight R n -module. If we choose a splitting φ to identify R n with K n [t ±1 ], we define A φ n(c) = H 1 (U; K n [t ±1 ]). Definition 3.7. The n th -ode degee of C is defined to be: δ n (C) = k Kn A n (C) Remak 3.8. Fo any choice of φ, k Kn A n (C) = k Kn A φ n(c). So although the module A φ n(c) depends on the splitting, the ank of the module does not. The degees δ n (C) ae integal invaiants of the fundamental goup G of the complement. Indeed, we have (cf. [9], 1): ( ) (3.1) δ n (C) = k Kn G (n+1) /[G (n+1), G (n+1) ] Z Γn K n Futhemoe, fo any choice of splitting φ, since K n [t ±1 ] is a pincipal ideal domain, thee exist some nonzeo p i (t) K n [t ±1 ], i = 1,, m, such that: ( ) A φ n(c) = m K n [t ±1 ] i=1 K p i (t)k n [t ±1 n [t ±1 ] n(c) ] Theefoe, δ n (C) is finite if and only if one of the equivalent statements is tue: (1) n (C) = k Kn H 1 (U; K n ) = 0. (2) A n (C) is a tosion R n -module. (3) Fo any φ, A φ n(c) is a tosion K n [t ±1 ]-module. (4) A Z n(c) is a tosion ZΓ n -module. If this is the case, then δ n (C) is the sum of the degees of the polynomials p i (t). Remak 3.9. It is inteesting to note that if C is ieducible, then δ 0 (C) is the degee of the Alexande polynomial of C. (The latte was defined and studied by Libgobe in a sequence of papes [10, 11, 12, 13, 14].) Indeed, this follows diectly fom the above definition o fom (3.1), since in the ieducible case we have that Γ 0 = 0, theefoe K0 = Q. The invaiant δ n (C) is difficult to calculate, in geneal. weighted homogeneous affine cuves is well undestood: Howeve, the special case of Poposition Suppose C is defined by a weighted homogeneous polynomial f(x, y) = 0 in C 2, and assume that eithe n > 0 o β 1 (U) > 1. Then we have: (3.2) δ n (C) = µ(c, 0) 1, whee µ(c, 0) is the Milno numbe associated to the singulaity gem at the oigin. β 1 (U) = 1, then δ 0 (C) = µ(c, 0). Poof. The key obsevation hee is the existence of a global Milno fibation (see fo example [4], (3.1.12)): F = {f = 1} U = C 2 C f C, and the fact that F is homotopy equivalent to the infinite cyclic cove of U coesponding to the kenel of the total linking numbe homomophism ψ. The Γ n -cove of U factos though If

8 8 CONSTANCE LEIDY AND LAURENTIU MAXIM the infinite cyclic cove coesponding to ψ, which is homotopy equivalent to F. It follows that thee is an isomophism of K n -modules: H (U; R n ) = H (F ; K n ). In paticula, δ n (C) = k Kn H 1 (F ; K n ). Since F has the homotopy type of a 1-dimensional CW complex, H 2 (F ; K n ) = 0. Moeove, if eithe n > 0 o β 1 (U) > 1, the coefficient system π 1 (F ) Γ n is non-tivial. Hence, by Poposition 2.6, H 0 (F ; K n ) = 0. It follows, in this case, that δ n (C) = χ(f ) = µ(c, 0) 1. On the othe hand, if β 1 (U) = 1, then k K0 H 0 (F ; K 0 ) = k Q H 0 (F ; Q) = 1. β 1 (U) = 1, then δ 0 (C) = 1 χ(f ) = µ(c, 0). Hence, if Example Since f(x) = x 3 y 2 is a weighted homogeneous polynomial, if C is the cuve defined by f = 0, it follows fom Poposition 3.10, that δ 0 (C) = 2 and δ n (C) = 1 fo n > 0. Remak Due to the existence of Milno fibations, we note that fomula (3.2) holds fo the case of any algebaic link, by eplacing U by the link complement and δ n (C) by Havey s invaiant of the algebaic link. Fo a moe geneal discussion on fibeed 3-manifolds, see [8], Poposition 8.4, 8.5. Remak As noted in [8], 6 and 8, the highe-ode Alexande invaiants n (C) and δ n (C) can be computed fom a pesentation of the fundamental goup of the cuve complement, by means of Fox fee calculus. 4. Uppe bounds on the highe ode degee of a cuve complement In this section, we find uppe bounds fo δ n (C). In Theoem 4.1, we find an uppe bound in tems of the Milno numbe of each singulaity. In Theoem 4.5, we expess this bound in tems of the Havey s invaiants, δ n, associated to each of the singula points of C. This esult is analogous to the divisibility popeties fo the infinite cyclic Alexande polynomial of the complement (e.g., see [11, 12, 13, 19]). As a coollay to these theoems, we have that, if C is a cuve in geneal position at infinity, then δ n (C) is finite, and theefoe A n (C) is a tosion ZΓ n -module. We also give an uppe bound fo δ n (C) in tems of the highe-ode degees of the link at infinity. Theoem 4.1. Suppose C is a degee d cuve in C 2 such that its pojective completion C is tansvese to the line at infinity H. If C has singulaities c k, 1 k l, then (4.1) δ n (C) Σ l k=1 (µ(c, c k ) + 2n k ) + 2g + d l, whee µ(c, c k ) is the Milno numbe associated to the singulaity gem at c k, n k is the numbe of banches though the singulaity c k, and g is the genus of the nomalized cuve. Befoe poving Theoem 4.1, we state an immediate coollay. Coollay 4.2. If C is a plane cuve in geneal position at infinity, then δ n (C) <, i.e., A Z n(c) is a tosion ZΓ n -module.

9 HIGHER-ORDER ALEXANDER INVARIANTS OF PLANE ALGEBRAIC CURVES 9 Remak 4.3. Note that the uppe bound in (4.1) is independent of n. Remak 4.4. If C is an ieducible cuve, then independently of the position of the line at infinity, we have that β 1 (U) = 1. By Poposition 2.7, it follows that A Z n(c) is a tosion module. Howeve, if the cuve C is not in geneal position at infinity, then the uppe bound on δ n (C) also includes the contibution of the singulaities at infinity (simila to Theoem 4.3 of [12]). Poof. We fist educe the poblem to the study of the bounday, X, of a egula neighbohood of C in C 2. In ode to do this, let N( C) be a egula neighbohood of C inside CP 2 and note that, due to the tansvesality assumption, the complement N( C) ( C H) can be identified with N(C) C, whee N(C) is a egula neighbohood of C in C 2. But N(C) C etacts by defomation on X = N(C). Now by the Lefschetz hypeplane section theoem ([4], page 25), it follows that the inclusion map induces a goup epimophism π 1 (X) π 1 (U) (the agument used hee is simila to the one used in the poof of Theoem 4.3 of [12]). It follows that π 1 (X Γn ) π 1 (U Γn ). Hence, H 1 (X; ZΓ n ) H 1 (U; ZΓ n ). Since R n is a flat ZΓ n -module, thee is an R n -module epimophism H 1 (X; R n ) H 1 (U; R n ). Theefoe, we have: δ n (C) = k Kn H 1 (U; R n ) k Kn H 1 (X; R n ). Hence, it is sufficient to bound above the ight-hand side of the above inequality. This will follow by a Maye-Vietois sequence agument. Let F be the (abstact) suface obtained fom C by emoving disks D 1 D nk aound each singula point c k of C. Let N = F S 1. The bounday of N is a union of disjoint toi T1 k Tn k k fo k = 1,, l, whee l is the numbe of singula points of C. Fo each singula point c k of C we let (Sk 3, L k) be the link pai of c k, and denote by X k the link exteio, Sk 3 L k. Then X is obtained fom N by gluing the link exteios X k along the toi Ti k fo i = 1,, n k : X = N i T ( l i k=1x k k ). The gluing map sends each longitude of L k to the estiction of a section in N, and each meidian to a fibe of N. We conside the Maye-Vietois sequence in homology associated to the above cove of X and with coefficients in R n : k,i H 1 (T k i ; R n ) Ψ H 1 (N; R n ) ( l k=1h 1 (X k ; R n ) ) H 1 (X; R n ) k,i H 0 (T k i ; R n ) H 0 (N; R n ) ( l k=1h 0 (X k ; R n ) ) H 0 (X; R n ) 0 Fom Remak 2.4, we have: (4.2) k Kn H 1 (X; R n ) = k Kn H 1 (N; R n ) + Σ l k=1k Kn H 1 (X k ; R n ) Σ k,i k Kn H 1 (T k i ; R n ) +k Kn ke(ψ) + Σ k,i k Kn H 0 (T k i ; R n ) k Kn H 0 (N; R n ) Σ l k=1k Kn H 0 (X k ; R n ) + k Kn H 0 (X; R n ). Recall that, fo each singula point c k of C, the coefficient system R n on X k is induced by the following composition of maps: Zπ 1 (X k ) Zπ 1 (X) Zπ 1 (U) ZΓ n R n.

10 10 CONSTANCE LEIDY AND LAURENTIU MAXIM Since each X k fibes ove S 1 with Milno fibe F k, the Γ n -cove of X k factos though the infinite cyclic cove of X k which is homeomophic to F k R. Theefoe we have the following isomophisms of K n -modules: H (X k ; R n ) = H (F k ; K n ). Since F k has the homotopy type of a wedge of cicles, H 2 (F k ; K n ) = 0. Theefoe, χ(f k ) = k Kn H 1 (F k ; K n ) + k Kn H 0 (F k ; K n ) = k Kn H 1 (X k ; R n ) + k Kn H 0 (X k ; R n ). Simila, since N = F S 1, the Γ n -cove of N factos though the infinite cyclic cove of N which is homeomophic to F R. So if F n denotes the coesponding Γ n -cove of F, then F n is a non-compact suface and we have H 2 (F ; K n ) = 0. Theefoe: χ(f ) = k Kn H 1 (F ; K n ) + k Kn H 0 (F ; K n ) = k Kn H 1 (N; R n ) + k Kn H 0 (N; R n ). Finally, fo each k and i, we have: 0 = χ(s 1 ) = k Kn H 1 (S 1 ; K n ) + k Kn H 0 (S 1 ; K n ) = k Kn H 1 (T k i ; R n ) + k Kn H 0 (T k i ; R n ). Now we can ewite equation (4.2) as follows: (4.3) k Kn H 1 (X; R n ) = Σ l k=1χ(f k ) χ(f ) + k Kn ke(ψ) + k Kn H 0 (X; R n ). Since π 1 (X) π(u) Γ n is an epimophism, it follows that the Γ n -cove of X is connected, thus yielding that k Kn H 0 (X; R n ) = 1. Since Ψ : k,i H 1 (Ti k ; R n ) H 1 (N; R n ), it follows that k Kn ke(ψ) Σ k,i k Kn H 1 (Ti k ; R n ). Fo each k and i, we have that: k Kn H 1 (T k i ; R n ) = k Kn H 0 (T k i ; R n ) = k Kn H 0 (S 1 ; K n ) 1, since S 1 is connected. Theefoe, k Kn ke(ψ) is less than o equal to the numbe of toi, which is Σ l k=1 n k whee n k is the numbe of banches though the singulaity c k. Fom equation (4.3) we have the following: (4.4) k Kn H 1 (X; R n ) Σ l k=1( χ(f k ) + n k ) χ(f ) + 1. Futhemoe, χ(f k ) = µ(c, c k ) 1 and χ(f ) 2g + k n k + d 1, whee g is the genus of the nomalized cuve and d is the degee of the cuve, i.e. the numbe of punctues at infinity. It follows that: δ n (C) k Kn H 1 (X; R n ) Σ l k=1 (µ(c, c k ) + 2n k ) + 2g + d l. As a coollay of the poof of Theoem 4.1, we obtain the following elation between the highe-ode degees of C and the local degees at singula points: Theoem 4.5. Suppose C is a degee d cuve in C 2, such that its pojective completion C is tansvese to the line at infinity, H. If C has singulaities c k, 1 k l, then δ n (C) Σ l k=1( δ k n + 2n k ) + 2g + d, whee δ k n = δ n (X k ) is Havey s invaiant of the link complement X k associated to the singulaity c k, n k is the numbe of banches though the singulaity c k, and g is the genus of the nomalized cuve.

11 HIGHER-ORDER ALEXANDER INVARIANTS OF PLANE ALGEBRAIC CURVES 11 Poof. We have equation (4.4) in the above poof: k Kn H 1 (X; R n ) Σ l k=1( χ(f k ) + n k ) χ(f ) + 1. Futhemoe, χ(f ) 2g + k n k + d 1. Fom Poposition 8.4 of [8], we have { δ n k = δ χ(f k ) if n 0 o β 1 (X k ) 1, n (X k ) = 1 χ(f k ) if n = 0 and β 1 (X k ) = 1. In paticula, χ(f k ) δ k n, which poves the theoem. We can also give a topological estimate fo the ank of the tosion-fee ZΓ n -module H 2 (U; ZΓ n ): Coollay 4.6. If C is a plane cuve in geneal position at infinity, the ank of the tosionfee ZΓ n -module H 2 (U; ZΓ n ) is equal to the Eule chaacteistic χ(u) of the cuve complement. Poof. Let C be the equivaiant complex 0 C 2 C 1 C 1 0 of fee ZΓ n -modules, obtained by lifting the cell stuctue of U to U Γn, the Γ n -coveing of U. Then χ(c) = χ(u). On the othe hand, 2 2 χ(c) = ( 1) i k Kn H i (C ZΓn K n ) = ( 1) i kh i (C). i=0 Theefoe, by Poposition 2.6 and Coollay 4.2, it follows that χ(c) = kh 2 (U; ZΓ n ) and the claim follows. We end this section by elating the highe-ode degees of a cuve C to the highe-ode degees of its link at infinity. We pove the following theoem, simila in flavo to esults on the infinite cyclic and univesal abelian Alexande invaiants (see [11, 12, 13, 6, 19]): Theoem 4.7. Let C be an affine plane cuve, and let S 3 be a sphee in C 2 of a sufficiently lage adius (that is, the bounday of a small tubula neighbohood in CP 2 of the hypeplane H at infinity). Denote by C = S 3 C the link of C at infinity, and let X be its complement S 3 C, with G := π 1 (X ). We define δn to be the K n -ank of H 1 (X ; R n ), whee the coefficient system is induced by the map ZG ZG ZΓ n R n. Then: (4.5) δ n (C) δ n. Poof. We note that thee is a goup epimophism G G. Indeed, X is homotopy equivalent to N(H) ( C H), whee N(H) is a tubula neighbohood of H in CP 2 whose bounday is S 3. If L is a geneic line in CP 2, which can be assumed to be contained in N(H), then by the Lefschetz theoem, it follows that the composition π 1 (L L ( C H)) π 1 (N(H) ( C H)) π 1 (CP 2 ( C H)) is sujective, thus poving ou claim (this is the same agument as the one used in [12], Theoem 4.5). i=0

12 12 CONSTANCE LEIDY AND LAURENTIU MAXIM It follows that thee is a ZΓ n -module epimophism H 1 (X ; ZΓ n ) H 1 (U; ZΓ n ). Since R n is a flat ZΓ n -module, we also get a R n -module epimophism: This poves the inequality (4.5). H 1 (X ; R n ) H 1 (U; R n ). Fo a cuve in geneal position at infinity, this yields a unifom uppe bound on the highe-ode degees of the cuve, which is independent of the local type of singulaities and the numbe of singula points of the cuve: Coollay 4.8. If C is a cuve of degee d, in geneal position at infinity, then: (4.6) δ n (C) d(d 2), fo all n. Poof. The claim follows by noting that if C is tansvese to the line at infinity, then C is the Hopf link on d components (i.e., the union of d fibes of the Hopf fibation), thus an algebaic link. By the agument used in the poof of Poposition 3.10, it follows that δn = µ 1, whee µ is the Milno numbe associated to the link at infinity. On the othe hand, µ is the degee of the Alexande polynomial of the link at infinity, so it is equal to d(d 2) + 1 (cf. [13]). The inequality (4.6) follows now fom Theoem Examples In this section, we calculate the highe-ode degees fo some of the classical examples of ieducible cuves, including geneal cuspidal cuves, Zaiski s sextics with 6 cusps, Oka s cuves, and banched loci of geneic pojections. We begin with the following: Poposition 5.1. Let C C 2 be an ieducible affine cuve. Let G = π 1 (C 2 C), and denote by C (t) the Alexande polynomial of the cuve complement. If C (t) = 1, then δ n (C) = 0 fo all n. Moeove, in this case, A Z n(c) = A Z 0 (C) as Z[G/G ]-modules, fo all n. Poof. As C is an ieducible affine cuve, we have G/G = Z. Hence G = G. The Alexande polynomial C (t) is the ode of the infinite cyclic (and univesal abelian) Alexande module of the complement, that is G /G Q, egaded as a Q[Z]-module unde the action of the coveing tansfomations goup G/G (cf. [10, 13, 14]). Since C is ieducible, the infinite cyclic Alexande module is a tosion Q[t, t 1 ]-module, egadless of the position of the line at infinity (cf. [10]). In this setting, C (t) can be nomalized so that C (1) = 1. The tiviality of the Alexande polynomial means that the univesal abelian module G /G Q is tivial, i.e. G /G is a tosion abelian goup. By (2.1), we obtain: G /G = (G /[G, G ])/{Z tosion} = (G /G )/{Z tosion} = 0. Hence G = G = G. It follows by induction that G (n) = G, fo all n > 0. Theefoe, fo any n, A Z n(c) = G (n+1) /[G (n+1), G (n+1) ] = G /G = A Z 0 (C).

13 HIGHER-ORDER ALEXANDER INVARIANTS OF PLANE ALGEBRAIC CURVES 13 Now ecall that the highe-ode degees of C may be defined by (3.1): ( ) δ n (C) = dim Kn G (n+1) /[G (n+1), G (n+1) ] Z Γn K n, whee Γ n is the kenel of Γ n ψ Z = G/G. The map ψ is induced by the total linking numbe homomophism, which in ou setting is just the abelianization map G G/G = Z. It follows that fo all n we have: Γ n = G/G (n+1) = G/G, Γ n = G /G (n+1) = 0, and Kn = Q. Theefoe, fo all n, δ n (C) = dim Q (G /G Z Q) = 0. Note that if the commutato subgoup G is eithe pefect (i.e. G = G ) o a tosion goup, then the Alexande polynomial of the ieducible cuve C is tivial. In paticula, this is the case if G is abelian. The following examples deal with each of these cases. Example 5.2. Let C CP 2 be an ieducible cuve of degee d which has a cusps (these ae locally defined by the equation x 2 = y 3 ) and b nodes as the only singulaities. If d > 6a+2b, then by a esult of Noi (cf. [21], but see also [14]), it follows that π 1 (CP 2 C) is abelian. If we choose a geneic line H at infinity and set C = C H, then as in 3.5 it follows that π 1 (C 2 C) is also abelian. Hence all highe-ode degees of C vanish. Poposition 5.3. Let C CP 2 be a degee d ieducible cuspidal cuve, i.e. it admits as singulaities only nodes and cusps. Choose a geneic line H CP 2, and set C := C H and G = π 1 (C 2 C). If d 0 (mod 6), then all highe-ode degees of C vanish. Poof. This follows fom Poposition 5.1 combined with Libgobe s divisibility esults fo the Alexande polynomial of a cuve complement (see fo instance [13], Theoem 4.1). Indeed, the Alexande polynomial of a cusp is t 2 t + 1, that of a node is t 1, and we use the fact that the Alexande polynomial of an ieducible cuve C can be nomalized so that C (1) = 1. Moeove, by ou assumption of tansvesality at infinity, all zeos of C (t) ae oots of unity of ode d. Hee is a moe concete example: Example 5.4. Zaiski s thee-cuspidal quatic. Let C CP 2 be a quatic cuve with thee cusps as its only singulaities. Choose as above a geneic line H, and set C = C H. Then the fundamental goup of the affine complement is given by: G = π 1 (C 2 C) = a, b aba = bab, a 2 = b 2. It is easy to see (using fo example a Redemeiste-Sheie pocess, see [18]) that G = Z/3Z. It follows by Poposition 5.1 that δ n (C) = 0, fo all n. Moeove, the integal highe Alexande modules ae given by: A Z n(c) = Z/3Z, fo all n. Remak 5.5. If C CP 2 is an ieducible quatic cuve, but not a thee-cuspidal quatic, then the fundamental goup π 1 (CP 2 C) is abelian (cf. [4], Poposition 4.3). If H is a geneic line, and C = C H, then by [23], Lemma 2, it follows that π 1 (C 2 C) is also abelian. Thus all highe-ode degees of such a cuve vanish. Based on this obsevation and the pevious example, it follows that the highe-ode degees of any ieducible quatic cuve ae all zeo.

14 14 CONSTANCE LEIDY AND LAURENTIU MAXIM In what follows, we give examples of cuves having (some) non-tivial highe-ode degees. The key obsevation in these examples is the fact that the highe-ode degees of an affine cuve ae invaiants of the fundamental goup of the complement, see (3.1). Example 5.6. Sextics with six cusps. (a) Let C CP 2 be a cuve of degee 6 with 6 cusps on a conic. Fix a geneic line, H, and set C = C H. Then π 1 (C 2 C) = π 1 (CP 2 C H) is isomophic to the fundamental goup of the tefoil knot, and has Alexande polynomial t 2 t + 1 (see [10], 7). By Remak 3.12, the highe-ode degees of C ae the same as Cochan-Havey highe-ode degees fo the tefoil knot, i.e. δ 0 (C) = 2, and δ n (C) = 1 fo all n > 0. (b) Let C CP 2 be a cuve of degee 6 with 6 cusps as its only singula points, but this time we assume that the six cusps ae not on a conic. Then π 1 (CP 2 C) is abelian, (isomophic to Z 2 Z 3 ). Assuming the line H as above is geneic and setting C = C H, this implies that π 1 (C 2 C) is abelian as well. Theefoe, δ n (C) = 0 fo all n 0. Fom (a) and (b) we see that the highe-ode degees of a cuve, at any level n, ae also sensitive to the position of singula points. An inteesting open poblem is to find Zaiski pais that ae distinguished by some δ k, but not distinguished by any δ n fo n < k. Example 5.7. Oka s cuves. M. Oka [22] has constucted the cuves C p,q CP 2 (p, q - elatively pime), with pq singula points locally defined by x p + y q = 0, such that π 1 (CP 2 C p,q ) = Z p Z q. In fact, the cuve C p,q is defined by the equation: (x p + y p ) q + (y q + z q ) p = 0. Fix a geneic line H CP 2, and set C p,q = C p,q H. Then π 1 (C 2 C p,q ) = π 1 (CP 2 C p,q H) is isomophic to the fundamental goup of the tous knot of type (p, q). The associated Alexande polynomial is (see fo instance [10], 7): (t) = (tpq 1)(t 1) (t p 1)(t q 1). By Remak 3.12 and Poposition 3.10, we obtain: δ 0 (C p,q ) = deg (t) = (p 1)(q 1), and δ n (C p,q ) = pq p q fo all n > 0. Example 5.8. Banching cuves of geneic pojections. Baid goups. Let V k be a degee k non-singula suface in CP 3 and α : V k CP 2 be a geneic pojection. If Ck CP 2 denotes the banching locus of α, then C k is an ieducible cuve of degee k(k 1) with k(k 1)(k 2)(k 3)/2 nodes and k(k 1)(k 2) cusps. In the case k = 3, one obtains as banching locus the six-cuspidal sextic with all cusps on a conic. If C k is the affine cuve obtained fom C k by emoving the intesection with a geneic line, then Moishezon [20] showed that π 1 (C 2 C k ) is Atin s baid goup on k stands, B k. The Reidemeiste-Scheie pocess [18] leads to the explicit computation of B k /B k. Fo k 5, B k /B k = 0, hence C k has a tivial Alexande polynomial. By Poposition 5.1 we obtain that δ n (C k ) = 0, fo all n 0. Fo k = 3, B 3 is the fundamental goup of the tefoil knot, so by Example 5.6(a) we obtain: δ 0 (C 3 ) = 2 and δ n (C 3 ) = 1 fo all n > 0. The case k = 4 equies moe wok. Hee we will only calculate δ 0 and δ 1 of the coesponding cuve C 4. The Alexande polynomial of C 4 is t 2 t + 1 (see fo example [13]), thus

15 HIGHER-ORDER ALEXANDER INVARIANTS OF PLANE ALGEBRAIC CURVES 15 δ 0 (C 4 ) = 2. A pesentation fo the baid goup on fou stands is: B 4 = σ 1, σ 2, σ 3 σ 1 σ 3 = σ 3 σ 1, σ 1 σ 2 σ 1 = σ 2 σ 1 σ 2, σ 2 σ 3 σ 2 = σ 3 σ 2 σ 3. By using Reidemeiste-Scheie techniques, we can obtain a pesentation fo B 4. (This was calculated in [7].) B 4 = p, q, a, b, c pap 1 = b, pbp 1 = b 2 c, qaq 1 = c, qbq 1 = c 3 a 1 c, c = a 1 b, whee p = σ 2 σ1 1, q = σ 1 σ 2 σ1 2, a = σ 3 σ1 1, b = σ 2 σ1 1 σ 3 σ2 1, and c = σ 1 σ 2 σ1 2 σ 3 σ 1 σ2 1 σ1 1. Then, B 4/B 4 = Z Z, geneated by p and q. Notice that since B 4/B 4 is tosion-fee, (B 4 ) = B 4. Hence by (3.1), we have: δ 1 = k K1 (B 4/B 4 Z Γ1 K 1 ), whee Γ 1 = ke( ψ : B 4 /B 4 B 4 /B 4) = B 4/B 4. Theefoe, we must undestand B 4/B 4 as a Z[p ±1, q ±1 ]-module, and then detemine the ank as a Q(p, q)-vecto space. Again using Reidemeiste-Scheie techniques, we calculate a goup pesentation fo B 4: B 4 = ρ i,j, α i,j ρ i,0 = 1, ρ i,j α i+1,j ρ 1 i,j = α i,jα i,j+1, ρ i,j α i+1,j+1 ρ 1 i,j = α i,j (α i,j+1 ) 2, α i,j+2 = (α i,j+1 ) 2 α 1 i,j α i,j+1, whee ρ i,j = p i q j pq j p (i+1) and α i,j = p i q j aq j p i. Notice that p and q act on B 4 by conjugation. Futhemoe, p (q γ) = ρ 1 0,1(q (p γ))ρ 0,1, fo all γ B 4. Hence although the actions of p and q do not commute in B 4, they do commute in B 4/B 4. In paticula, B 4/B 4 is indeed a Z[p ±1, q ±1 ]-module. We have the following pesentation fo B 4 as an abelian goup: 4/B B 4/B 4 = ρ i,j, α i,j ρ i,0 = 0, α i+1,j = α i,j +α i,j+1, α i+1,j+1 = α i,j +2α i,j+1, α i,j+2 = 3α i,j+1 α i,j. To get a pesentation as a Z[p ±1, q ±1 ]-module, we note that: ρ i,j = ρ i,j = j (p i q j k ρ 0,1 ), fo j 1, k=1 j k=1 ρ i,0 = 0, (p i q j 1+k ρ 1 0,1), fo j 1, α i,j = p i q j α 0,0, fo all i, j Z. Theefoe, as a Z[p ±1, q ±1 ]-module, B 4/B 4 is geneated by ρ 0,1 and α 0,0. Futhemoe, ρ 0,1 geneates a fee submodule, while α 0,0 geneates a tosion submodule. Hence the ank as a Q(p, q)-vecto space is 1. Theefoe, δ 1 (C 4 ) = 1. Note. The backgound mateial on the constuctions mentioned in this example ae beautifully explained in Libgobe s papes [13] and [14]. In paticula, the latte contains a summay of Moishezon s esults [20].

16 16 CONSTANCE LEIDY AND LAURENTIU MAXIM Refeences [1] Aapua, D., Geomety of cohomology suppot loci fo local systems. I. J. Algebaic Geom.6(1997), [2] Cochan, T., Noncommutative knot theoy, Algebaic & Geometic Topology, Volume 4 (2004), [3] Cochan, T., O, K., Teichne, P. Knot concodance, Whitney towes and L 2 -signatues, Annals of Mathematics, 157 (2003), [4] Dimca, A., Singulaities and Topology of Hypesufaces, Univesitext, Spinge-Velag, [5] Dimca, A., Sheaves in Topology, Univesitext, Spinge-Velag, [6] Dimca, A., Maxim, L., Multivaiable Alexande invaiants of hypesuface complements, axiv: math.at/ [7] Goin, E., Lin, V., Algebaic equations with continuous coefficients, and cetain questions of the algebaic theoy of baids (Russian), Math. Sb. (N.S.) 78 (120), 1969, ; English tanslation in Math. USSR-Sb. 7 (1969), [8] Havey, S., Highe-ode polynomial invaiants of 3-manifolds giving lowe bounds fo the Thuston nom, Topology, Volume 44 (2005), Issue 5, [9] Havey, S., Monotonicity of degees of genealized Alexande polynomials of goups and 3-manifolds axiv:math.gt/ [10] Libgobe, A., Alexande polynomials of plane algebaic cuves and cyclic multiple planes, Duke Math. J., 49(1982), [11] Libgobe, A., Homotopy goups of the complements to singula hypesufaces, Bulletin of the AMS, 13(1), [12] Libgobe, A., Homotopy goups of the complements to singula hypesufaces, II, Annals of Mathematics, 139(1994), [13] Libgobe, A., Alexande invaiants of plane algebaic cuves, Singulaities, Poc. Symp. Pue Math., Vol. 40(2), 1983, [14] Libgobe, A., Fundamental goups of the complements to plane singula cuves. Algebaic geomety, Bowdoin, 1985 (Bunswick, Maine, 1985), 29-45, Poc. Sympos. Pue Math., 46, Pat 2, Ame. Math. Soc., Povidence, RI, [15] Libgobe, A., Goups which cannot be ealized as fundamental goups of the complements to hypesufaces in C N. Algebaic geomety and its applications (West Lafayette, IN, 1990), , Spinge, New Yok, [16] Libgobe, A., On the homology of finite abelian coves, Topology and its applications, 43 (1992) [17] Libgobe, A., Chaacteistic vaieties of algebaic cuves axiv: math.ag/ , in: C.Cilibeto et al.(eds), Applications of Algebaic Geomety to Coding Theoy, Physics and Computation, , Kluwe, [18] Magnus, W., Kaass, A., Solita, D., Combinatoial goup theoy: Pesentations of goups in tems of geneatos and elations, Dove Publications, Inc., New Yok, xii+444 pp. [19] Maxim, L., Intesection homology and Alexande modules of hypesuface complements, Comm. Math. Helv. (to appea). [20] Moishezon, B., Stable banch cuves and baid monodomies, Lectue Notes in Mathematics, vol. 862, [21] Noi, M., Zaiskis conjectue and elated poblems. Ann. Sci. École Nom. Sup. (4) 16 (1983), no. 2, [22] Oka, M., Some plane cuves whose complements have nonabelian fundamental goups, Math. Ann. 218 (1978), [23] Oka, M., A suvey on Alexande polynomials of plane cuves, Singulaités Fanco-Japonaises, , Sémin. Cong., 10, Soc. Math. Fance, Pais, 2005 [24] Stenstöm, B., Rings of Quotients, Spinge-Velag, New Yok, 1975.

17 HIGHER-ORDER ALEXANDER INVARIANTS OF PLANE ALGEBRAIC CURVES 17 C. Leidy: Depatment of Mathematics, Univesity of Pennsylvania, 209 S 33d St., Philadelphia, PA, , USA. addess: cleidy@math.upenn.edu L. Maxim : Depatment of Mathematics, Univesity of Illinois at Chicago, 851 S Mogan Steet, Chicago, IL 60607, USA. addess: lmaxim@math.uic.edu