Tangential Alexander polynomials and non-reduced degeneration

0.48 MB·English

by  Mutsuo Oka

TANGENTIAL ALEXANDER POLYNOMIALS AND NON-REDUCED
DEGENERATION
MUTSUO OKA
6
0 Abstract. WeintroduceanotionoftangentialAlexanderpolynomialsforplanecurvesand
0
studytherelationwithθ-Alexanderpolynomial. Asanapplication,weusethesepolynomials
2
to study a non-reduced degeneration Ct, → D0+jL. We show that there exists a certain
n
surjectivity of thefundamental groups and divisibility among their Alexanderpolynomials.
a
J

2
1
1. Introduction

]
G
Let C be a plane curve. We are interested in the geometry of plane curves. Choose a line
A
L P2 and put C2 := P2 L. As geometrical invariants, we consider
. ⊂ L −
h (a) Fundamental groups: π (P2 C) and π (C2 C)
t 1 − 1 L−
a
(b) Alexander polynomial ∆ (t;L).
m C
Zariski studied π (P2 C) systematically [36] and further developments have been made by
[ 1 −

many authors. To compute the Alexander polynomial, we need to choose a line at infinity L.
2
v HoweverforagenericL,theAlexanderpolynomialhastoomuchrestrictionsandwehaveoften
6 the trivial case ∆ (t;L) = (t 1)r−1 where r is the number of the irreducible components. In
3 C −
2 ourpreviouspaper[24],wehaveintroducedthenotionofθ-Alexanderpolynomials. Thisgives
1
more informations for certain reducible curves but it does not give any further information
0
6 for irreducible curves.
0
The purpose of this paper is to introduce the notion of the tangential Alexander polyno-
/
h
mials. Namely we consider all tangent lines T C for the line at infinity. It turns out that
t P
a
m tangential Alexander polynomials are related to θ-Alexander polynomials. We apply these
: polynomials to study non-reduced degenerations. Let Ct, t ∆ be a analytic family of re-
v ∈
t→∞
i duced curves for t = 0 such that Ct C0 = D0 +jL where L is a line. The case j 2 is
X 6 −→ ≥
a typical non-reduced degeneration. In this situation we study the geometry of D using that
r 0
a of C . One of our results is the surjectivity assertion of the natural homomorphism:
t
φ: π (C2 D ) π (C2 C )
1 L− 0 → 1 L− t
Here the point is that L is the line component of the limit curve C ( Theorem 14, 5). This
0
§
paper consists of the following sections.
2 Fundamental groups
§
3 Alexander polynomial
§
4 Dual stratification and tangential fundamental groups
§
5 Degeneration into non-reduced curves with a multiple line
§
1991 Mathematics Subject Classification. 14H30,14H45, 32S55.
Key words and phrases. tangential Alexanderpolynomial, non-reduceddegeneration.
1
2 M.OKA
2. Fundamental groups
Let L be a fixed line and put C2 := P2 L. We say L is generic with respect to C if L
L −
and C intersects transversely. The topology of C2 C does not depend on L if L is generic
L−
and we call it the generic affine complement and we often write as C2 instead of C2. The
L
following Lemma describes the relation of two fundamental groups.
Lemma 1. ([17]) Let ω be a lasso for L and N(ω) be the subgroup normally generated by ω.
(1) The following sequence is exact.
1 N(ω) π (C2 C,b ) π (P2 C,b ) 1
→ → 1 L− 0 → 1 − 0 →
(2) Assume that L is generic. Then
(i) ω is in the center of π (C2 C) and N(ω)= Z.
1 L− ∼
(ii) We have the equality D(π (C2 C)) = D(π (P2 C)) among their commutator groups.
1 1
− −
Thus π (P2 C) is abelian if and only if π (C2 C) is abelian.
1 1
− −
For non-generic line L, π (C2 C) may be non-abelian even if π (P2 C) is abelian. For
1 L− 1 −
example, let C = Y2Z X3 = 0 and take L = Z = 0 . Then π (C2 C)= B where B
{ − } { } 1 L− ∼ 3 3
is the braid group of three strings and we recall that B = a,b aba = bab ([4]).
3 ∼
h | i
2.1. First homology group H (P2 C). Assume that C is a projective curve with r ir-
1
−
reducible components C ,...,C of degree d ,...,d respectively. By Lefschetz duality, we
1 r 1 r
have the following.
Proposition 2. H (P2 C,Z) is isomorphic to Zr−1 (Z/d Z) where d = gcd(d ,...,d ).
1 0 0 1 r
− ×
In particular, H (C2 C)= Zr.
1 L− ∼
Take a lasso g for each component C of C for i = 1,...,r. Then the corresponding
i i
homology classes [g ], i = 1,...,r give free abelian generators of H (C2 C).
{ i } 1 L−
2.2. Degenerations and fundamental groups. Let C beareducedplane curve. Thetotal
Milnor numberµ(C)is definedbythesumof thelocal Milnornumbersµ(C,P) atthesingular
points P of C. Let ∆ := ζ C ζ 1 the unit disk. We consider an analytic family of
{ ∈ || | ≤ }
projective curves C = F (X,Y,Z) = 0 , t ∆ where F (X,Y,Z) are reduced homogeneous
t t t
{ } ∈
polynomial of degree d for any t. We call C ;t ∆ a reduced degeneration. We assume that
t
{ ∈ }
C , t = 0 have the same configuration of singularities sothat they aretopologically equivalent
t
6
but C may obtain more singularities, i.e., µ(C ) µ(C ).
0 t 0
≤
Theorem 3. For a given reduced degeneration C ;t ∆ , there is a canonical surjective
t
{ ∈ }
homomorphism for t = 0:
6
ϕ: π (P2 C ) ։ π (P2 C )
1 0 1 t
− −
In particular, if π (P2 C ) is abelian, so is π (P2 C ).
1 0 1 t
− −
See for example, [24] and also Theorem 14 of 5 for another simple proof.
§
TANGENTIAL ALEXANDER POLYNOMIALS 3
2.3. Product formula. AssumethatC isacurveofdegreed , i = 1,2whichareintersecting
i i
transversely at d d distinct points. We denote the transversality as C ⋔ C . Take a line L
1 2 1 2
such that L C C = . Note that L need not be generic for C or C .
1 2 1 2
∩ ∩ ∅
Theorem 4. (Oka-Sakamoto[28]) Under the above assumption, we have
π (C2 C C ) = π (C2 C ) π (C2 C )
1 L− 1∪ 2 ∼ 1 L− 1 × 1 L− 2
For further information about fundamental groups, we refer to [4, 12, 18, 20, 32].
2.4. Example.
2.4.1. Abelian cases. A curve C with small singularities has often commutative fundamental
group π (P2 C). Some examples are here:
1
−
– C is a smooth irreducible curve.
– Irreducible curves with only A -singularities (i.e., nodes) by [36, 9, 8, 10, 16, 29], or irre-
1
ducible curve of degree d with a nodes and b cusps (i.e., A ) with 6b+2a< d2 ([16]).
2
– π (P2 C) (respectively π (C2 C)) is abelian for any irreducible curve of degree d if it
1 − 1 L −
has a flex of order d 3 in C2 (resp. of order d 2) ([36]).
≥ − L −
Let f : C2 C be a polynomial mapping. Recall that α is a atypical value at infinity if the
→
topological triviality at infinity fails at t = α for the family of curves C := f−1(t) (see [34]).
t
Proposition 5. ([20]) Let f :C2 C be a polynomial mapping and assume that 0 is not an
→
atypical value at infinity and C = f−1(0) is smooth in C2. Then π (C2 C)= Z.
1 − ∼
2.4.2. Non-abelian case. Assume that p,q are positive integers greater than 1 and consider
the curve
C : f (X,Y,Z)q +f (X,Y,Z)p = 0
p,q p q
where f , f are polynomials of degree p, q respectively. C is called a curve of (p,q)-torus
p q p,q
type. Assume that two curves f = 0 and f = 0 intersect transversely and there is no
p q
{ } { }
other singularities of C . Then π (P2 C ) = G(p,q,q) and π (C2 C) = G(p,q). In
p.q 1 − p,q ∼ 1 − ∼
particular, if p, q are coprime, π (P2 C ) = Z Z . For the definition of G(p,q) and
1 − p,q ∼ p ∗ q
G(p,q,r), we refer to [18].
2.5. Class formula and flex formula. For the study of curves of low degree, it is often
important to know the existence of flex points. Let d= degree(C), dˇbe the degree of the dual
curve Cˇ, let Σ(C) be the singular points of C and let α(C) be the number of the flex points.
Then dˇand α(C) are given by the formula:
dˇ= d(d 1) (µ(C,P)+m(C,P) 1)
− − P∈Σ(C) −
α(C) = 3dP(d 2) γ(C,P)
− − P∈Σ(C)
where m(C,p) is the multiplicity of C at P andPγ(C,P) is the flex defect of the singularity
(C,P) [15, 21]. (In [21], we have denoted γ(C,P) as δ(C,P). To distinguish with δ-genus of
the singularity, we change the notation.)
4 M.OKA
3. Alexander polynomial
3.1. General definition. Let X be a finite connected CW-complex and let φ : π (X) Z
1
→
be a surjective homomorphism. We fix a generator t of the infinite cyclic group Z. Let
Λ is the group ring of Z. Then Λ is isomorphic to the Laurent polynomial ring C[t,t−1]
and Λ is a principal ideal domain. Consider an infinite cyclic covering p : X X such
→
that p (π (X)) = Kerφ. Then H (X,C) has a structure of Λ-module where t acts as the
# 1 1
e
canonical covering transformation. Thusbythe structuretheorem of modulesover aprincipal
e e
ideal domain, we have an identification:
H (X,C)= Λ/λ Λ/λ
1 ∼ 1 n
⊕···⊕
as Λ-modules. We normalize the deenominators so that λi is a polynomial in t with λi(0) = 0
6
for each i = 1,...,n. The Alexander polynomial associated to φ is defined (see [11]) by the
product ∆ (t) := n λ (t).
φ i=1 i
Q
3.2. Alexander polynomials of plane curves. In our situation, we consider a plane curve
C = C C whereC ,...,C areirreduciblecomponentsofdegreed ,...,d respectively.
1 r 1 r 1 r
∪···∪
Take a line L as the line at infinity and let φ be the composition
θ
φ : π (C2 C) ξ H (C2 C,Z) = Zr θ Z
θ 1 L− −→ 1 L− ∼ →
where θ is a surjective homomorphism. Recall that θ is determined by giving an integer n to
i
eachcomponentC suchthatgcd(n ,...,n ) = 1. Wecalln the weight for C . The Alexander
i 1 r i i
polynomial of C with respect to (L,θ) is defined by ∆ (t) and we denote it as ∆ (t;L,θ).
φθ C
(1) (Generic case) Assume that L to be generic and θ = θ where θ is defined by the
sum sum
canonical summation θ (a ,...,a )= r a (weight 1 foreach component.) Inthis case,
sum 1 r i=1 i
we simply write as ∆ (t) and call it the generic Alexander polynomial of C, as it does not
C P
depend on the choice of a generic L.
(2) If θ is the canonical summation θ but L is not generic, we denote it as ∆ (t;L),
sum C
omitting θ. In particular, when L is the tangent line of a smooth point P C, we call
∈
∆ (t;L) the tangential Alexander polynomial at P and we also use the notation ∆ (t;P).
C C
(3) If L is generic but θ is not θ , we called ∆ (t;L,θ) the θ-Alexander polynomial and we
sum C
denote it as ∆ (t;θ). In [24] we denoted it by ∆ (t), but for the consistency of the notation
C C,θ
with (2), we use the notation ∆ (t;θ).
C
Recall that (t 1)r−1 ∆ (t) ([24]). Thus this is also the case for ∆ (t;L) with any line L,
C C
− |
as ∆ (t) ∆ (t;L). We say that ∆ (t :L) is trivial if ∆ (t;L) = (t 1)r−1.
C C C C
| −
3.3. Fox calculus. Supposethat G is a group and φ: G Z is a given surjective homomor-
→
phism. Assume that G has a finite presentation as
G = x ,...,x R ,...,R
∼ 1 n 1 m
h | i
This corresponds to a surjective homomorphism ψ : F(n) G so that Kerψ is normally
→
generated by the words R ,...,R where F(n) is a free group of rank n, generated by
1 m
TANGENTIAL ALEXANDER POLYNOMIALS 5
x ,...,x . Consider the group ring C(F(n)) of F(n) with C-coefficients. The Fox differ-
1 n
entials ∂ : C(F(n)) C(F(n)) for j = 1,...,n, are additive homomorphisms which are
∂xj →
characterized by the following properties.
∂ ∂ ∂u ∂v
(1) x = δ , (2)(Leibniz rule) (uv) = +u , u,v C(F(n))
i i,j
∂x ∂x ∂x ∂x ∈
j j j j
The composition φ ψ : F(n) Z gives a ring homomorphism γ : C(F(n)) C[t,t−1]. The
◦ → →
Alexander matrix A is an m n matrix with coefficients in C[t,t−1] and its (i,j)-component
×
is given by γ(∂Ri). Then the Alexander polynomial ∆ (t) is defined by the greatest common
∂xj φ
divisor of (n 1) (n 1)-minors of A. In the case of G = π (X) for some connected
1
− × −
topological space X, this definition coincides with the previous one (Fox [6]).
3.3.1. Examples. We gives several examples.
1. Consider the trivial case: G = Zr and φ= θ , the canonical one. Then
sum
1-1. G = Z = x . Then ∆(t) = 1.
∼ ∼ 1
h i
1-2. If G = Zr = x ,...,x R = x x x −1x −1, 1 i < j r , we have ∆(t) =
∼ h 1 r| i,j i j i j ≤ ≤ i
(t 1)r−1. This follows from the Fox derivation:
−
∂ ∂
(x x x −1x −1)= 1 x x x−1, (x x x −1x −1)= x x x x−1x−1.
∂x i j i j − i j i ∂x i j i j i− i j i j
i j
2. Let C = Y2Z X3 = 0 and L = Z = Y , L = Z = 0 . Note that (0,1,0) is a flex
gen
{ − } { } { }
point of C and L is the flex tangent. Then
π (C2 C)= Z, ∆ (t;L )= 1
1 Lgen − ∼ C gen
π (C C)= x ,x x x x =x x x = B , ∆ (t;L) = t2 t+1
1 L− h 1 2| 1 2 1 2 1 2i∼ 3 C −
3. Let us consider the curve C = Y2Z3 X5 = 0 C2 and L = Z = 0 , M = Y = 0 .
{ − } ⊂ { } { }
Then π (P2 C)= Z/5Z and π (C2 C)= G(2,5) and π (C2 C) = G(3,5). In this case,
1 − ∼ 1 L− ∼ 1 M − ∼
we get
(t10 1)(t 1) (t15 1)(t 1)
∆ (t)= 1, ∆ (t;L) = − − , ∆ (t;M) = − −
C C (t5 1)(t2 1) C (t5 1)(t3 1)
− − − −
3.4. Weakness of the generic Alexander polynomial ∆ (t). The following Lemma de-
C
scribes the relation between the Alexander polynomial and local singularities.
Lemma 6. (Libgober[11]) Let P ,...,P be the singular points of C (including those at infin-
1 k
ity) and let ∆ (t) be the characteristic polynomial of the Milnor fibration of the germ (C,P ).
i i
Then the generic Alexander polynomial satisfies the divisibility: ∆ (t;L) k ∆ (t).
C | i=1 i
Lemma7. (Libgober[11])LetdbethedegreeofC. ThentheAlexanderpolyQnomial ∆ (t;L )
C ∞
divides the Alexander polynomial at infinity ∆ (t). If L is generic, ∆ (t) is given by
∞ ∞ ∞
(td 1)d−2(t 1). In particular, the roots of the generic Alexander polynomial are d-th roots
− −
of unity.
Corollary 8. ([11], See also [36]) Assume that C is an irreducible curve of degree d and
assume that the singularities are either nodes (i.e., A ) or ordinary cusp singularities (i.e.,
1
A ). If d is not divisible by 6, the generic Alexander polynomial ∆ (t) is trivial.
2 C
6 M.OKA
This implies that there does not exist any non-trivial generic Alexander polynomials of
degree n with n 0 mod 6, for example, this is the case for cubic, quartic and quintic
6≡
curves, whose singularities are copies of A or A . However even though there does exist
1 2
interesting geometry on these curves. We will show by examples that certain tangential
Alexander polynomial gives non-trivial Alexander invariants and we will give an explanation
from viewpoint of non-reduced degeneration in 5.
§
Another weakness of generic Alexander polynomials is for reducible curves. Let C and C
1 2
becurves which intersect transversely each other. We take a line L so that L does not contain
any points of C C . Note that L need not be generic for C C . Theorem 4 says that
1 2 1 2
∩ ∪
π (C2 C C ) =π (C2 C ) π (C2 C )
1 L− 1∪ 2 ∼ 1 L− 1 × 1 L− 2
However the Alexander polynomial ∆ (t;L) loses these informations. In fact, we have
C1∪C2
Theorem 9. ([24]) Assume that C and C intersect transversely and let C = C C . Let
1 2 1 2
∪
L be a line such that L C C = . Then ∆ (t;L) = (t 1)r−1 where r is the number of
1 2 C
∩ ∩ ∅ −
irreducible components.
For further information about Alexander polynomials, we refer to [7, 11, 13, 14, 31]
4. Dual stratification and tangential fundamental groups.
4.1. Dual stratification of curves. Let Σ be a finite set of topological equivalent class
of curve singularities and let (Σ,d) be the configuration space of plane curves of degree
M
d with a fixed singularity configuration Σ. Take two curves C, C′ (Σ,d) in the same
∈ M
connected component and two smooth points P C and Q C′. We consider their tangent
∈ ∈
lines L = T C, L′ = T C. Though the topology of (P2,C) and (P2,C′) are topologically
P Q
equivalent, this may not the case for (P2,C L) and (P2,C′ L′). To analyze this, we
∪ ∪
introduce the dual stratification (C) for C ( (Σ,d)) and ( (Σ,d)) of (Σ,d) as
S ∈ S M S M M
follows.
Let Pˇ2 be the dual projective space. Recall that a point α = (α : α : α ) Pˇ2 (resp. a
1 2 3
∈
point P = (p : p : p ) P2) can be considered as a line L = α X +α Y +α Z = 0
1 2 3 α 1 2 3
∈ { }
in P2 (resp. a line L = p U + p V +p W = 0 in Pˇ2). First take C (Σ,d). Let
P 1 2 3
{ } ∈ M
Σ(C) = P ,...,P be the singular points of C. Let (d) be the set of partition of the
1 k
{ } P
integer d. We consider the mappings ψ : Cˇ (d) and ψˇ : C (dˇ) defined as follows.
→ P → P
Let α Cˇ (resp. P C) and let L C = R ,...,R (resp. L Cˇ = S ,...,S ).
α 1 ν P 1 µ
∈ ∈ ∩ { } ∩ { }
We define ψ(α) = I(C,L ;R ), i = 1,...,ν where I(C,L ;R ) is the local intersection
α i α i
{ }
multiplicity. Respectively we define ψˇ(P) = I(Cˇ,L ;S );j = 1,...,µ . Note that for a
P j
{ }
generic line α Cˇ, L is a simple tangent line and therefore ψ(α) = 2,1,...,1 . For a
α
∈ { }
generic flex point P, the tangent line L = T C gives the partition ψ(L) = 3,1,...,1 . A
P
{ }
line α Cˇ is called an multi-tangent line if ψ(α) has at least two members 2. A simple
∈ ≥
bi-tangent line is a typical such line which is simply tangent at two smooth points. A smooth
point P C is called tangentially generic if it is smooth and the tangent line T C gives the
P
∈
partition 2,1,...,1 . Recall thattheGauss mapassociated with C,denotedas G :C Cˇ,
C
{ } →
is defined by G (P) = T C. Let Σntg(C)= P ,...,P besmooth points which are not
C P k+1 k+t
{ }
TANGENTIAL ALEXANDER POLYNOMIALS 7
tangentially genericandputΣ(C)= Σ(C) Σntg(C)= P ,...,P . Thedualstratification
1 k+t
∪ { }
(C) of C is by definition, := C Σ(C),Σ(C) . Thus if C is irreducible, (C) has one
S S { − } S
e
open dense stratum made of tangentially generic points and k +t starata made of isolated
e e
points.
4.2. Dual stratification of the configuration space (Σ,d). Now we consider the dual
M
stratification of (Σ,d). To distinguish a point in (Σ,d) and the corresponding curve,
M M
we denote points in (Σ,d) by α (Σ,d) and the corresponding curve by C . The
α
M ∈ M
configuration of the singularities of the dual curve Σ(Cˇ ) is not unique for α (Σ,d) but
α
∈ M
it has only finite possible types, say Σ∗,...,Σ∗ when we fix the configuration space (Σ,d).
1 ℓ M
We consider the partition of the configuration space by the following sets:
α (Σ,d); Σ(Cˇ ), (C ), (Cˇ ),ψ ,ψˇ are constant
α α α α α
{ ∈ M S S }
The dual stratification ( (Σ(cid:0),d)) is defined by the strata w(cid:1)hich are the connected compo-
S M
nents of these partitions. Thus for a stratum M ( (Σ,d)), each C and Cˇ , α M have
α α
∈ S M ∈
constant dual stratifications.
For a stratum M ( (Σ,d)), we can associate a family of plane curves C , α M such
α
∈S M ∈
that the dual family of curves Cˇ , α M is a family in (Σ∗,dˇ) for some j. Observe that
α ∈ M j
any α M, the dual stratification (C ) and (Cˇ ) are constant for α by definition. Thus
α α
∈ S S
for two α,β M, C , C are homeomorphic as a stratified sets. More precisely we have
α β
∈
Proposition 10. Take α M and take a point L Cˇ . This induces a continuous
0 ∈ α0 ∈ α0
family of lines L Cˇ such that ψ (L ) is constant. Then the topology of the affine pair
α α α α
∈
(C2 ,C) does not depend on α M. In particular the fundamental group π (C2 C) does
Lα ∈ 1 Lα −
not depend on α M.
∈
Proof. Recall that the local topology of C L at an intersection point P is determined
α α
∪
by the local Milnor number µ(C L ,P) and this is determined by µ(C ,P) and the local
α α α
∪
intersection multiplicity I(C ,L ;P). The definition of the dual stratification of ( (Σ,d))
α α
S M
guarantees the µ-constancy of the family of plane curves C L , α M of degree d+1. (cid:3)
α α
∪ ∈
4.3. Tangential fundamental group and Tangential Alexander polynomial. For a
line L Cˇ, we call π (C2 C) the tangential fundamental group and ∆ (t;L) the tangential
∈ 1 L− C
Alexander polynomial. If L = T C for some simple point P C, we also use the notation
P
∈
∆ (t;P) for ∆ (t;T C). We also define k-fold Alexander polynomial ∆ (t;P ,...,P ) by
C C P C 1 k
∆ (t;P ) with L = T C. It is easy to observe that π (C2 C) and ∆ (t;P) are
C∪L1∪···∪Lk−1 k j Pj 1 L− C
constantontheopen(denseifC isirreducible)strataofthedualstratification (C). However
S
in general it may give a different polynomial for singular lines L Cˇ (they are the images
∈
of isolated strata of (C) by the Gauss map). We will see some examples later. Thus the
S
tangential Alexander polynomials altogether contain more geometrical informations than the
generic Alexander polynomials. The main purposeof this paper is to investigate the property
of thetangential Alexander polynomials. Note that ifC is irreducible, there is only onechoice
of θ (up to ) but there are many choices for L, even for irreducible C.
±
8 M.OKA
4.4. Alexander spectrum. We also consider the set of tangential Alexander polynomials
t-AS(C):= ∆ (t;P);P C
C
{ ∈ }
and we call t-AS(C) the tangential Alexander spectrum of C. There exist at most finite
polynomials in the spectrum. In fact, it is bounded by the number of strata of (C).
S
We can also define the k-fold tangential Alexander spectrum of C by
t-AS(k)(C) := ∆ (t;P ,...,P );P C
C 1 k j
{ ∈ }
It often happens that even when the Alexander spectrum t-AS(C) is trivial, 2-fold Alexander
spectrum t-AS(2)(C) (or higher one) is not trivial.
4.5. Example. We consider (2A +A ,4) and (E ,4). By class formula, the dual curve
2 1 6
M M
Cˇ of a generic member C of (2A +A ,4) or (E ,4) is a quartic with 2A +A in both
2 1 6 2 1
M M
cases. (C has generically 2 flex points.) In both configuration spaces (2A +A ,4) and
2 1
M
(E ,4), there arestrata which correspondto degenerated members, namely curves with one
6
M
flex of order 2. Thisimplies that the dualcurve has an E singularity. For these configuration
6
spaces, there is a beautiful work by C.T.C.Wall [35]. Consider the subsets:
M := C (2A +A ,4);Σ(Cˇ)= 2A +A ,
1 2 1 2 1
{ ∈ M { }}
M := C (2A +A ,4);Σ(Cˇ) = E
2 2 1 6
{ ∈ M { }}
N := C (E ,4);Σ(Cˇ)= 2A +A ,
1 6 2 1
{ ∈M { }}
N := C (E ,4);Σ(Cˇ)= E
2 6 6
{ ∈ M { }}
We can easily see that M , M , N , N are respective dual stratifications of the configu-
1 2 1 2
{ } { }
ration spaces (2A +A ,4) and (E ,4). We observe that under the Gauss map,
2 1 6
M M
–M and N are self-dual and
1 2
–M and N are dual each other.
2 1
We observe also that M ∂M and N ∂N .
2 1 2 1
⊂ ⊂
(1-1) We consider quartic C M with Σ(C ) = 2A +A with two flexes. By the class
1 1 1 2 1
∈
formula, such a curve has a bi-tangent line. As an example, we take:
17
(1) C : y4+8y3+1/4+7/2y2 7/2y2x2+1/4x4 1/2x2 = 0
1
4 − −
C has two cusps at P = (1,0), P = ( 1,0) and one A at (0, 1). Two flexes are at
1 1 2 1
− −
Q = (10√10,9), Q = ( 10√10,9). We have a bi-tangent line y = 1 which are tangent
1 2
−
at B = (2√2,1), Q = ( 2√2,1). For the dual stratification (C ), we have to take two
1 2 1
− S
more points S = (25/7,8/7), S =( 25/7,8/7) whose tangent lines pass through P and P
1 2 2 1
−
respectively. Using Zariski-van Kampen pencil method, we can compute π (C2 C ) as
1 L− 1
π (C2 C ) = ξ ,ξ ,ξ ξ ξ ξ = ξ ξ ξ , ξ ξ ξ = ξ ξ ξ , ξ ξ = ξ ξ , ξ ξ ξ = ξ ξ ξ
1 L− 1 h 1 2 3| 1 2 1 2 1 2 2 3 2 3 2 3 1 3 3 1 1 2 3 3 2 1i
where L = y = 1 . This gives ∆ (t;L) = t2 t+1 by Fox calculus. Other tangent lines
{ } C1 −
give the trivial Alexander polynomial. We leave the proof of this assertion as an exercise.
Thus t-AS(C ) = 1,t2 t+1 .
1
{ − }
TANGENTIAL ALEXANDER POLYNOMIALS 9
(1-2) We consider the following quartic C M with Σ(C ) = 2A +A with a degenerated
2 2 2 2 1
∈
flex of order 2 at infinity L = Z = 0 :
{ }
C : 1 12y+36y2 32y3 2x2+12x2y+x4 = 0
2
− − −
π (C2 C ) = ξ ,ξ ,ξ ξ ξ ξ = ξ ξ ξ , ξ ξ ξ = ξ ξ ξ , ξ ξ = ξ ξ
1 L− 2 h 1 2 3| 1 3 1 3 1 3 3 2 3 2 3 2 2 1 1 2i
We can also see that ∆ (t;L) = t2 t+1. Note that two A singularities are at ( 1,0) and
C2 − 2 ±
one A is at (0,1/2). In the dual stratification (C ), there are two more ’singular’ points
1 2
S
R = (9,8) and R = ( 9,8) whose tangent line pass through the cusps. However these
1 2
−
tangent lines give trivial tangential Alexander polynomials.
(2-1) Consider a quartic C N , defined by y3+x4 x2y2 = 0 which has one E -singularity
3 1 6
∈ −
at O. Two flex points are at ( 6√6/5,36/5). The bi-tangent line is given by y = 4. By an
±
easy computation, we observe that L = y = 4 gives π (C2 C ) = B and ∆ (t;L) =
{ } 1 L − 3 ∼ 3 C3
t2 t+1. Other tangent Alexander spectra are trivial. So t-AS(C ) = 1,t2 t+1 . The
3
− { − }
dual stratification has 5 isolated points.
(2-2) Consider the following quartic C N , f(x,y)= y3+x4 = 0, with E and one flex of
4 2 6
∈
order 2 at P = (0,1,0). Take the flex line Z = 0 as L. Then
π (C2 C)= G(3,4) = ξ ,ξ ,ξ ξ = ωξ ω−1, ξ = ωξ ω−1
1 L− ∼ h 0 1 2| 0 1 1 2 i
and ∆ (t;L) = (t2 t+1)(t4 t2+1) where ω = ξ ξ ξ .
C4 − − 2 1 0
1 1 3
(3) Let C : f(x,y)= y4 3x2y2+y2+ x4 4x3+3x2 = 0 be a 3 cuspidal quartic.
−2 −2− 2 −
As is well-known [36, 19], the generic affine fundamental group is a finite group of order 12,
with presentation
π (C2 C)= ξ,ζ ξζξ = ζξζ, ξ2 = ζ2 .
1
− h | i
Though the fundamental group is not abelian, the generic Alexander polynomial is trivial.
For L = y = 0 (this is the tangent cone of a cusp and L corresponds to a flex of Cˇ),
{ }
π (C2 C)= ξ,ζ ξζξ = ζξζ = B .
1 L− h | i 3
By the class formula, the dual curve Cˇ is a cubic curve with a node. (C) has 3 singular
S
points from 3A and two ’singular points’ from the bi-tangent line. We can also see that
2
π (C2 C)= B for an arbitrary tangent line L except the bitangent line L . The bi-tangent
1 L− ∼ 3 b
line is given by x= 2/3 in this example. By an easy computation, we see that
π (C2 C)= ξ ,ξ ,ξ ,ζ ξ ξ ξ = ξ ξ ξ , ξ ξ ξ = ξ ξ ξ , ξ ζξ =ζξ ζ, ζ = ξ−1ξ ξ
1 Lb − ∼ h 0 1 2 | 0 1 0 1 0 1 1 2 1 2 1 2 2 2 2 1 0 1i
and we have
Proposition 11. For the bitangent line L , ∆ (t;L )= (t2 t+1)2. For any other tangent
b C b
−
line L, ∆ (t;L) = t2 t+1. In particular, this implies that t-AS(C) = t2 t+1, (t2 t+1)2 .
C
− { − − }
10 M.OKA
4.5.1. Further example. In the above examples of quartics, the geometry of C T C does not
P
∪
change for flexes of the same order. However this is not the case in general.
ConsiderafixedmarkpointP C. Wecall(C,P)a curve with a marked point P. Twocurves
∈
withmarkedpoints(C,P)and(C′,P′)arecalledamarked Zariskipairif C TPC,C′ TP′C′
{ ∪ ∪ }
is a Zariski pair. For further information about Zariski pairs, see [1, 2, 3, 24, 25]. In [26],
we have shown that for any quintic B with configuration in the next list, there exist two
5
different flex points P, P′ such that (B ,P) and (B ,P′) are marked Zariski pairs.
5 5
4A , 4A +A , A +2A , A +2A +A , E +2A
2 2 1 5 2 5 2 1 6 2
(♯)

E +A , 2A , A +A , A +A +A , A
 6 5 5 8 2 8 2 1 11
In fact their generic Alexander polynomials are given as t2 t+1, 1 respectively. This implies

−
that the among flexes of these quintics, there are two classes of different topological nature:
one class which does not contribute the tangential Alexander spectrum and the other which
contributes by (t2 t+1). We give one example. The following quintic B : f(x,y) = 0 has
5
−
A singularity at the origin and 9 flex points. Among them, the flex at P = (0,1) is different
11
from others (a flex of torus type). In fact, B T B is a sextic of torus type [26]. All other
5 P 5
∪
flex points gives trivial tangential Alexander polynomial.
33 7 129 5 15
f(x,y)= y5+ x+ y4+ x2 x 5/2 y3+
−64 8 64 −4 − 8 −
(cid:18) (cid:19) (cid:18) (cid:19)
15 13 3
x3+ x2+x+1 y2+ x4 2x3 2x2 y+x5+x4
8 4 −4 − −
(cid:18) (cid:19) (cid:18) (cid:19)
Figure 1. Quintic with A
11
4.6. θ-Alexander polynomials. To cover the weakness of Alexander polynomials for ir-
reducible curves, we have proposed θ-Alexander polynomials in [24]. First recall that the
radical q(t) of a polynomial q(t) = ν (t ξ )µi is defined by q(t) := ν (t ξ ). Here
i=1 − i i=1 − i
µ 1, i. The following theorem shows the importance of θ-Alexander polynomial.
i ≥ ∀p Q p Q