# Braid monodromy and Alexander polynomials of real plane curves

Anatoly Libgober  Department of Mathematics
University of Illinois
Chicago, IL 60607<EMAIL_ADDRESS>In Memory of John W.Wood.

###### Abstract.

We describe symmetries of the braid monodromy decomposition for a class of
plane curves defined over reals including the real curves with no real points
and proving new divisibility relations for Alexander invariants of such
curves.

##  1\. Preface and statement of results

Alexander polynomial of a projective curve embedded into a smooth algebraic
surface (cf. [17]) is an invariant of the fundamental group of the complement
to the curve. It measures a degree of non-commutativity of the group and can
be expressed in terms of geometric data of the surface and the curve,
including the local types and position of singularities of the curve on the
surface. An interesting problem is understanding which polynomials can appear
as the Alexander polynomials of the fundamental groups of the complements:
this is a very special case of the fundamental problem of understanding the
quasi-projective groups.

Divisibility theorems give strong restrictions on the class of polynomials
which can occur as the Alexander polynomials of projective curves. One such
result (cf.[15],[17]) asserts that Alexander polynomial of the fundamental
group of the complement to an ample curve $C$ divides the product of the
Alexander polynomials of links of all singularities of the curve. In
particular, if such $C$ has ordinary nodes and cusps as the only singularities
than the global Alexander polynomial has a form $(t-1)^{a}(t^{2}-t+1)^{b}$.
Focusing on the curves in a complex projective plane rather than general
smooth algebraic surfaces (as we will do in this paper) one can easily see
that $a+1$ is the number of irreducible components of $C$ (cf. [17]) but the
range of multiplicities $b$ is far from clear. In the case of curves in
${\mathbb{C}}{\mathbb{P}}^{2}$, the global Alexander polynomial also divides
the Alexander polynomial of the link in the 3-sphere which is the boundary of
a small tubular neighborhood of a line with the link being defined as the
intersection of the curve with this 3-sphere. If this “line at infinity” is
transversal to the curve $C$ then the corresponding link is the Hopf link with
$d$ components where $d$ is the degree of $C$ and the Alexander polynomial of
the latter is $(t-1)(t^{d}-1)^{d-2}$. One obtains that for curves with nodes
and cusps the multiplicity of the factor $(t^{2}-t+1)$ is at most $d-2$ if
$6|d$ and is zero otherwise (for a similar divisibility relation on surfaces
more general than ${\mathbb{C}}{\mathbb{P}}^{2}$ see [17]). 111A different
divisibility relations one obtains if the line at infinity is selected to be
non-transversal to the curve or contains the singularities. This gives
divisibility by the Alexander polynomial of the complement to affine curve
which is a complement in ${\mathbb{C}}{\mathbb{P}}^{2}$ to the union of the
curve and the line. The Alexander polynomial of this affine curve may be
different than the Alexander polynomial of the projective curve. A slightly
better than $d-2$ bound on the multiplicity is given in [2] Cor.3.13.

This bound is much weaker than what so far was observed in examples. At the
moment, it is unknown if the multiplicity of a primitive root of unity of
degree $6$ in the Alexander polynomial of a curve, having as its singularities
ordinary cusps and nodes only, has a bound independent of $d$. The largest
known multiplicity is $4$ for a curve of degree 12 with 39 cusps (cf. [2]). It
is known that if the Mordell Weil ranks of isotrivial elliptic threefold (or
isotrivial elliptic surfaces) are bounded, the multiplicities of the factors
of the Alexander polynomials of curves in this class, also are bounded
independently of degree (cf. [2] for proof of both assertions).

In this note, we discuss a new type of divisibility relations for the
Alexander polynomials for the complements to curves in
${\mathbb{C}}{\mathbb{P}}^{2}$ for which the defining equations have only real
coefficients 222in the case of reducible curves, this allows for irreducible
components not to have ${\mathbb{R}}$ as the field of definition; we do not
make any assumptions about the reality of the critical values of projections
used to construct the braid monodromy as was customary in previous works, cf.
[14]. The presence of a real structure imposes restrictions on the braid
monodromy of the curve. The latter is an invariant of a curve
$C\subset{\mathbb{C}}{\mathbb{P}}^{2}$ and its projection onto a complex line
$N\subset{\mathbb{C}}{\mathbb{P}}^{2}$, given as the homomorphism
$\pi_{1}(N\setminus Cr,b)\rightarrow B_{d}$. Here $b$ is a base point, $Cr$ is
the subset of $N$ consisting of points over which the fiber of projection of
$C$ has cardinality less than the degree $d$ of $C$, and $B_{d}$ is Artin’s
braid group. If one views $B_{d}$ as the mapping class group of a disk with
boundary and $d$ marked points then the braid monodromy assigns to a loop in
$N\setminus Cr$ the class of the diffeomorphism is given by the trivialization
of the fibration of the pair $({\mathbb{C}}{\mathbb{P}}^{2}\setminus p,C)$,
over this loop (here $p$ is the center of projection onto $N$, cf. [21], [17]
for a more recent exposition or section 3 below).

The main results of the paper are the Theorem 3.1, its corollary given as
Proposition 6.4, the Theorem 5.1, and its corollary the Proposition 6.7.

In Section 3, we describe symmetry in the structure of this homomorphism
depending on the real structure of the curve. It appears that certain
operators introduced by Garside (cf. [9]) play important role in the
description of this symmetry and vice versa, the study of braid monodromy of
real curves gives geometric interpretation to some of Garside’s identities. In
particular, if the projection is defined over ${\mathbb{R}}$ and the
intersection of the finite set $Cr$ with the real locus of $N$ is empty then
the braid monodromy takes the class in $\pi_{1}(N\setminus Cr,b)$ (where $b$
is real) represented by the loop corresponding to
${\mathbb{R}}{\mathbb{P}}^{1}$ to the Garside word $\Delta$. Braids
corresponding to such loops were considered in [18] in the related context of
Hurwitz schemes. If the real part of the critical set
$Cr\cap{\mathbb{R}}{\mathbb{P}}^{1}\neq\emptyset$, then $\pi_{1}(N\setminus
Cr,b)$ contains three canonical loops: the one containing only critical points
in the real part of the critical set and two loops containing all critical
points in each of two connected components of
$N\setminus{\mathbb{R}}{\mathbb{P}}^{1}$. We describe constraints on
corresponding braids and solve the equations in the braid group to obtain an
explicit form of the braids corresponding to these canonical loops.

In section 4.1 we prove that the fundamental group of a curve over
${\mathbb{R}}$ is a quotient of the fundamental group of a link which is one
of the closed braids attached to the curve in section 3. The argument here is
purely topological and does not use an algebro-geometric structure, and can be
used in a different, for example symplectic, context.

In Section 5 we show that the global Alexander polynomial of a curve over
${\mathbb{R}}$ divides the Alexander polynomial of a link which is the closure
of a braid associated with the real structure and discussed in two previous
sections. We calculate these Alexander polynomials in some cases and make the
divisibility relations explicit. For example for real curves without real
points at all, the Alexander polynomial divides $(t^{d}-1)^{{{d-2}\over
2}}(t^{d\over 2}+1)(t-1)$. This gives ${d\over 2}-1$ as a new bound on the
multiplicity of the factor $t^{2}-t+1$ in the Alexander polynomial of a curve
with nodes and cusps over ${\mathbb{R}}$ and no real points. Moreover, we show
that this bound is sharp at least for such sextics. The last section also
contains a discussion of braid monodromy of arrangements defined over reals,
i.e. such that the equation of the union of all lines is defined over
${\mathbb{R}}$, but having only finitely many real points. Such arrangements
are perhaps of interest on their own. Note that the effect of complex
conjugation on braid monodromy was considered earlier in [3] (in connection
with a study of MacLane arrangements).

Part of this work was done while the author participated in “Braids” program
at ICERM in the spring of 2022. The author thanks B.Guerville-Balle,
A.Degtyarev, the anonymous referee for very useful comments and references in
connection with the earlier version of this note, and most importantly Juan
Gonzalez-Meneses for the answer to the author’s question (cf. [11] and Remark
3.8).

##  2\. Complex conjugation and braid groups

Let ${\mathcal{P}}_{N}=\\{P_{1},\cdots P_{N}\\}$ be an invariant under
conjugation subset in ${\mathbb{C}}$. Let $P_{0}\in{\mathbb{R}}$. Complex
conjugation induces on free group
$\pi_{1}({\mathbb{C}}\setminus{\mathcal{P}}_{N},P_{0})$ an automorphism of
order 2\. In the following system of generators, this automorphism has a
particularly simple form.

Recall that a good ordered system of generators of
$\pi_{1}({\mathbb{C}}\setminus{\mathcal{P}}_{N},P_{0})$ is given by $N$ loops
each consisting of a segment $I_{i}$ running from $P_{0}$ to the vicinity of
one of the points $P_{i}$, followed by a counterclockwise loop running along
the boundary of a small circle centered at $P_{i}$ and then returning to
$P_{0}$ along $I_{i}$. Moreover, it is assumed that these loops are non-
intersecting and ordered by the counterclockwise ordering of their
intersection points with a small circle centered at $P_{0}$.

Without loss of generality, we assume that the points in ${\mathcal{P}}_{N}$
are labeled so that $P_{1},\cdots P_{k}\in H^{+}$, $P_{N},\cdots P_{N+1-k}\in
H^{-}$, $P_{i}$ is the complex conjugate of $P_{N+1-i},i=1,\cdots,k$ 333here
$H^{+}$ (resp. $H^{-}$) denote the upper (resp. lower) half plane in
${\mathbb{C}}$, $P_{k+1},\cdots P_{N-k}$ form an increasing sequence of real
numbers i.e. $P_{k+1}<\cdots<P_{N-k}<P_{0}$. Let $\alpha_{+}:[0,1]\rightarrow
H^{+}$ be an embedding of the interval $[0,1]$ such that
$\alpha^{+}(0)=P_{1},\alpha^{+}(1)=P_{k}$ and
$P_{i}\in\alpha^{+}[0,1],i=1,\cdots,k$ and the sequence
$({\alpha^{+}})^{-1}(P_{i})\in[0,1]$ is increasing. Let $I$ be the interval in
${\mathbb{C}}$ which is the union of the interval $\alpha_{+}([0,1])$,
followed by the line segment $P_{k},P_{k+1}$, then followed by the part of the
real axis running from $P_{k+1}$ to $P_{N-k}$, then followed by the segment
$[P_{N-k},P_{N+1-k}]$ and having as the final part the interval
$\overline{\alpha_{+}([0,1])}$. Then $I\supset{\mathcal{P}}_{N}$ and the above
order in ${\mathcal{P}}_{N}$ is induced by an orientation of $I$. In the case
when $N=2k$ and ${\mathcal{P}}_{N}\cap{\mathbb{R}}=\emptyset$,
$P_{k+1}=\bar{P}_{k}$ and in the above construction of $I$, instead of the
segments with real endpoints we use the segment connecting $P_{k}$ and
$P_{k+1}$ 444the order of the points in ${\mathcal{P}}_{N}$ in $H^{+}$ can be
selected arbitrarily but, as follows from this description, is consistent with
the orientation of $\alpha_{+}([0,1])$ and the order of points in $H^{+}$
determines the order of the conjugate points in $H^{-}$. In practice, it is
convenient to order the points in $H^{+}$ according to the order of imaginary
parts of the points..

We select a good ordered system of generators $x_{1},\cdots,x_{N-k}$ of the
fundamental group
$\pi_{1}(H^{+}_{\epsilon}\setminus\bigcup_{1}^{N-k}P_{i},P_{0})$ of the
complement to the set $P_{1},\cdots P_{N-k}$ in a $\epsilon$-neighborhood
$H^{+}_{\epsilon}$ of the closure of $H^{+}$ in ${\mathbb{C}}{\mathbb{P}}^{1}$
and extend it to a good ordered system $x_{1},\cdots,x_{N}$ adding loops, as
the set each being the conjugate of a loop among the first $k$ loops in the
already selected system but using orientation and the order given by the above
definition of a good ordered system. With these notations, the involution on
$\pi_{1}({\mathbb{C}}\setminus{\mathcal{P}}_{N},P_{0})$ induced by the complex
conjugation $\gamma\rightarrow\bar{\gamma}$ of the oriented loops is given by
(writing from the right to the left):

(1) $\bar{x}_{i}=x_{N+1-i}^{-1},i=1,...k\ \ \ \bar{x}_{i}=x_{N-k}\cdots
x_{i+1}x_{i}^{-1}x^{-1}_{i+1}\cdots x_{N-k}^{-1},\ \ i=k+1,\cdots N-k$

Note that if at most one of $P_{i},\ i=1,\cdots N$ is real, then the action is
just $\bar{x}_{i}=x_{N-i}^{-1}$. In particular, one has

$\overline{x_{N}\cdots x_{1}}=(x_{N}\cdots x_{1})^{-1}$

Let $Diff(D^{2},{\mathcal{P}}_{N})$ be the group of diffeomorphisms of a
conjugation invariant disk in ${\mathbb{C}}$ containing ${\mathcal{P}}_{N}$
and taking the set ${\mathcal{P}}_{N}$ into itself. Let
$Diff^{+}(D^{2},{\mathcal{P}}_{N},\partial D^{2})$ be its subgroup consisting
of diffeomorphisms which are orientation preserving and constant on the
boundary of the disk. The latter is a normal subgroup of the former and the
same is the case for the groups of connected components of each of these
groups. The group $\pi_{0}(Diff^{+}(D^{2},{\mathcal{P}}_{N},\partial D^{2})$
is Artin’s braid group $B_{N}$. The complex conjugation is an orientation-
reversing element in $Diff(D^{2},{\mathcal{P}}_{N})$ and conjugation by this
element, acting as an inner automorphism of
$\pi_{0}(Diff(D^{2},{\mathcal{P}}_{N}))$, acts as the outer automorphism of
the normal subgroup $B_{N}=\pi_{0}(Diff^{+}(D^{2},{\mathcal{P}}_{N},\partial
D^{2})$. We denote this automorphism as $\beta\rightarrow\bar{\beta},\beta\in
B_{N}$. The group $B_{N}$ is a subgroup of the group of automorphisms of
$\pi_{1}({\mathbb{C}}\setminus{\mathcal{P}}_{N},P_{0})$ and from the above
definition, one has

(2) $\bar{\beta}(x)=\overline{\beta(\bar{x})}\ \ \
x\in\pi_{1}({\mathbb{C}}\setminus{\mathcal{P}}_{N},P_{0}),\beta\in B_{N}$

It is immediate that (2) implies the following for the action of the
conjugation on the standard generators $s_{1},\cdots s_{N-1}$ of $B_{N}$
555these generators depend on a choice of $I$ or, more specifically, the
choice of the part of $I$ containing the points in $H^{+}$ since the rest of
interval $I$ described above canonically. Recall, that generators of the braid
group, viewed as $\pi_{1}(Diff^{+}(D^{2},{\mathcal{P}}_{N},\partial D^{2}))$
are the half-twists corresponding to simple proper arcs connecting consecutive
points of ${\mathcal{P}}_{N}$ (cf. [8], 9.3.1). The choice of $I$ with
orientation provides both the ordering and the arcs between consecutive
points. The action of the braid group on the free group in these generators is
given by
$s_{i}(x_{i})=x^{-1}_{i}x_{i+1}x_{i},s_{i}(x_{i+1})=x_{i},s_{i}(x_{j})=x_{j},\forall
j\neq i,i+1$. In particular $s_{i}(x_{N}\cdots x_{1})=x_{N}\cdots x_{1}$. ,
given by the orientation of $I$ from $P_{1}$ to $P_{N}$ i.e. the
counterclockwise Dehn half-twists corresponding to the subintervals of $I$
connected consecutive points in ${\mathcal{P}}_{N}$, with the order along $I$:

(3) $\bar{s}_{i}=s_{N-i}^{-1}\ \ \ i=1,..,k-1,N-k+1,\cdots,N-1;$ (4) $\ \
\bar{s}_{i}=s_{i}^{-1}\ \ i=k+1,\cdots N-k-1$

Conjugates of generators in the remaining pair are given by 666$\bar{s}_{k}$
is the clockwise Dehn twist about the segment
$\overline{[P_{k},P_{k+1}]}=[P_{N-k+1},P_{k}]$ and $\bar{s}_{N-k}$ is the
clockwise Dehn twist about the segment $[P_{N-k},P_{k}]$

(5) $\bar{s}_{k}=s_{N-k}\cdots s_{k+1}s_{k}^{-1}s_{k+1}^{-1}\cdots
s_{N-k}^{-1};\ \ \bar{s}_{N-k}=s_{N-k-1}\cdots
s_{k+1}s_{k}^{-1}s^{-1}_{k+1}\cdots s^{-1}_{N-k-1}$

Conjugation on the braid group depends on the number of real points in the set
${\mathcal{P}}_{N}$ and has particularly form (3) if ${\rm
Card}({\mathcal{P}}_{N}\cap{\mathbb{R}})\leq 1$ or (4) if
${\mathcal{P}}_{N}\subset{\mathbb{R}}$ (the latter is the case considered in
[3]).

Following Garside (cf. [9] sec. 1.2 and 2.1) we will use the involution
${\mathfrak{R}}:s_{i}\rightarrow s_{N-i}$ and the anti-homomorphism
$rev:B_{N}\rightarrow B_{N}$ which is rewriting a word in generators $s_{i}$
of the braid group or their inverses in reversed order. One has
$rev(gh)=rev(h)rev(g),\forall g,h\in B_{N}$. We will use similar operations
${\mathfrak{R}}x_{i}=x_{N+1-i}$ and $rev$ on generators $x_{1},..,x_{N}$ of a
free group. In particular, if ${\rm Card}{\mathcal{P}}_{N}\cap{\mathbb{R}}\leq
1$ (equivalently $k={N\over 2}$ or $k={{N-1}\over 2},N\ odd$) i.e. the action
is given by (3) then

(6) $\bar{s}_{i}={\mathfrak{R}}(s_{i})^{-1}\ \ \
\bar{x}_{i}={\mathfrak{R}}(x_{i})^{-1}$

It follows from [9] that with such restriction on $k$, the action of complex
conjugation on Garside word satisfies $\bar{\Delta}=\Delta^{-1}$. Also, note
that the complex conjugation and inner automorphisms generate $AutB_{N}$ cf.
[7]: the automorphism $\epsilon_{n}$ in that paper is the product of
${\mathfrak{R}}$ and complex conjugation; in the case
${\mathcal{P}}_{N}\in{\mathbb{R}}$, the automorphism $\epsilon_{n}$ is the
complex conjugation cf. (4).

##  3\. Braid monodromy of curves over ${\mathbb{R}}$

Recall the definition of braid monodromy in the context of real curves. Let
$P\in{\mathbb{R}}{\mathbb{P}}^{2}$,
$p_{{\mathbb{R}}}:{\mathbb{R}}{\mathbb{P}}^{2}\setminus P\rightarrow
N_{{\mathbb{R}}}$ be the projection from $P$ onto a line $N_{{\mathbb{R}}}$
and $\mathcal{L}_{{\mathbb{R}}}$ be the corresponding pencil of lines in
${\mathbb{R}}{\mathbb{P}}^{2}$. For each of these objects, the corresponding
complexification will be denoted by the same letter but with ${\mathbb{R}}$
changed to ${\mathbb{C}}$. Complex conjugation acts on the set of
${\mathbb{C}}$-points of each of these sets, having the set of real points as
the fixed point set. The fiber of projection $p_{{\mathbb{C}}}$ over $c\in
N_{{\mathbb{C}}}$ will be denoted $L_{{\mathbb{C}},c}$.

Let $b\in N_{{\mathbb{R}}}$ be a point selected so that $L_{{\mathbb{C}},b}$
is transversal to the complexification $C_{{\mathbb{C}}}$ of a curve
$C_{{\mathbb{R}}}$. Let $Cr\subset N_{{\mathbb{C}}}$ be the subset of the
points $c$ such that ${\rm Card}(p^{-1}(c)\cap C_{{\mathbb{C}}})<d,d={\rm
deg}(C)$.

Let $\gamma(t)$ be a loop in $N_{{\mathbb{C}}}$ with initial and endpoints
being at $b\in N_{{\mathbb{R}}}$ and situated in the upper half plane of
$N_{{\mathbb{C}}}$. Let $\bar{\gamma}(t)=\overline{\gamma(t)}$ be its
conjugate. Consider a trivialization of projection of the pair:
$p:(p_{{\mathbb{C}}}^{-1}(\gamma),C_{{\mathbb{C}}}\cap
p_{{\mathbb{C}}}^{-1}(\gamma))\rightarrow\gamma$ i.e. a continuous map of
pairs
$\Phi:(I\times{\mathbb{C}},I\times[d])\rightarrow(p_{{\mathbb{C}}}^{-1}(\gamma),C_{{\mathbb{C}}}\cap
p_{{\mathbb{C}}}^{-1}(\gamma))$ (here $I=[0,1]$ and $[d]$ is a fixed subset of
${\mathbb{C}}$ of cardinality $d$) such that

  1. $(i)$

$\Phi$ is compatible with projections of its source and target onto $I$ and
$\gamma$ respectively and in particular $\Phi(0,z)=\Phi(1,z)\in
p_{{\mathbb{C}}}^{-1}(b)$ for any $z\in{\mathbb{C}}$.

  2. $(ii)$

Restrictions of $\Phi$ onto $[0,1)\times{\mathbb{C}}$ and
$(0,1]\times{\mathbb{C}}$ are homeomorphisms onto their targets.

  3. $(iii)$

The trivialization is constant outside of a disk in $L_{{\mathbb{C}},b}$
containing $L_{{\mathbb{C}},b}\cap C$ (in particular, for any $x\in
L_{{\mathbb{C}},b}$ outside of this disk and $z\in{\mathbb{C}}$ such that
$\Phi(0,z)=x$ one has $\Phi(1,z)=x)$.

The monodromy along the loop $\gamma(t)$ is a diffeomorphism of the pair
$(L_{{\mathbb{C}},b},C_{{\mathbb{C}}}\cap L_{{\mathbb{C}},b})$ into itself
sending $x\in L_{{\mathbb{C}},b}$ to $\Phi(1,z(x))$ where $z(x)$ is the
solution to $\Phi(0,z(x))=x$. For a trivialization satisfying
$(i),(ii),(iii)$, the braid corresponding to the isotopy class of such
diffeomorphism via identification of Artin’s braid group with the mapping
class group of a disk with marked points will be denoted as $\beta(\gamma)$.

######  Definition 3.1.

The braid monodromy of a plane curve is the homomorphism
$\pi_{1}(N_{{\mathbb{C}}}\setminus Cr,b)\rightarrow B_{d}$ which assigns to
the class of a loop the braid in Artin’s braid group corresponding to the
diffeomorphism given by the monodromy obtained from a trivialization over the
loop as described above.

Following [21] we present braid monodromy as a factorization of the word
$\Delta^{2}$ written as the product of braids representing the value of braid
monodromy on a sequence of a good ordered system of generators of the
fundamental group of the complement $\pi_{1}(N_{{\mathbb{C}}}\setminus Cr,b)$.

Complex conjugation acts on trivializations as follows. Clearly, since $C$ is
defined over ${\mathbb{R}}$, for a loop $\gamma$ with a base point
$b\in{\mathbb{R}}$, one has

$(\overline{p_{{\mathbb{C}}}^{-1}(\gamma),C_{{\mathbb{C}}}\cap
p_{{\mathbb{C}}}^{-1}(\gamma)})=(p_{{\mathbb{C}}}^{-1}(\bar{\gamma}),C_{{\mathbb{C}}}\cap
p_{{\mathbb{C}}}^{-1}(\bar{\gamma}))$

######  Definition 3.2.

Conjugate of a trivialization
$\Phi:(I\times{\mathbb{C}},I\times[d])\rightarrow(p^{-1}(\gamma),C_{{\mathbb{C}}}\cap
p_{{\mathbb{C}}}^{-1}(\gamma))$ is the trivialization of $p_{{\mathbb{C}}}$
over the loop $\bar{\gamma}$ given by

(7) $\bar{\Phi}(t,z)=\overline{\Phi(t,z)}$

In particular, the monodromy diffeomorphism of
$(p_{{\mathbb{C}}}^{-1}(b),C\cap p_{{\mathbb{C}}}^{-1}(b))$ corresponding to
trivialization $\bar{\Phi}$, in terms of trivialization $\Phi$ is given by:
$x\rightarrow\overline{\Phi(1,\tilde{z})},x\in p_{{\mathbb{C}}}^{-1}(b)$ where
$\tilde{z}$ is determined by $\Phi(0,\tilde{z})=\bar{x}\in
p_{{\mathbb{C}}}^{-1}(b)$

######  Remark 3.3.

In general, it is impossible to trivialize over a loop the pair
$(p_{{\mathbb{C}}}^{-1}(b),C\cap p_{{\mathbb{C}}}^{-1}(b))$ together with
involution given by conjugation. The type of involution (given by the number
of fixed points, i.e. the number of real points in the fiber) changes while
one moves along $\gamma$, but this procedure provides a well-defined
diffeomorphism of pairs with involution.

######  Remark 3.4.

An alternative way to define a braid monodromy is to use the so-called
coefficient homomorphism defined as the holomorphic map assigning to an affine
plane curve given by Weierstrass polynomial
$y^{d}+\sum_{i=0}^{d-1}a_{i}(x)y^{i}=0$ the map
${\mathbb{C}}\rightarrow{\mathbb{C}}^{d}$ given by
$x\rightarrow(a_{d-1}(x),\cdots,a_{0}(x))$. The restriction of this map to the
complement to the set of critical values of projection of this curve onto
$x$-plane takes it to the space of the coefficients of polynomials in one
variable without multiple roots i.e. the complement in ${\mathbb{C}}^{d}$ to
the discriminant hypersurface $Discrim$. This complement is the base of a
locally trivial fibration of the complement in ${\mathbb{C}}^{d+1}$ to the
hypersurface given by equation
$y^{d}+a_{d-1}y^{n-1}+\cdots+a_{0}\in{\mathbb{C}}^{d+1}$ onto
${\mathbb{C}}^{d}$ with coordinates $(a_{d-1},\cdots,a_{0})$. The action of
the fundamental group $\pi_{1}({\mathbb{C}}^{d}\setminus Discrim,b)$, which is
isomorphic to Artin’s braid group, is induced by its action on the fundamental
group of the fiber of this fibration over the base point $b$. If
$b\in{\mathbb{R}}^{d}$ then the complex conjugation acts on the braid group
(since discriminant hypersurface is defined over ${\mathbb{R}}$) but the
specific form of this action on generators depends on the choice of $b$. This
complex conjugation on $B_{d}$ is given by (3) and (4) with $d=N$ and depends
on the number of real roots of $y^{d}+a_{d-1}(b)y^{n-1}+\cdots+a_{0}(b)$. In
particular, if the number of real roots is at most one, the action is given by
(6) and if all roots are real then one has $s_{i}\rightarrow s_{i}^{-1}$ for
all $i$ (as in [3]).

Definition 3.2 implies the following relation between the braids that the
braid monodromy assigns to conjugate loops:

######  Proposition 3.5.

Let $\beta:\pi_{1}(N_{{\mathbb{C}}}\setminus Cr)\rightarrow B_{d}$ be the
homomorphism of braid monodromy of a plane curve over ${\mathbb{R}}$ and let
$\bar{\gamma}$ be the complex conjugate of a loop
$\gamma\in\pi_{1}(N_{{\mathbb{C}}}\setminus Cr,b)$. Then

$\beta(\bar{\gamma})=\overline{\beta(\gamma)}$

and depending on ${\rm Card}(p^{-1}(b)\cap C_{{\mathbb{R}}})$ the action of
the complex conjugation on a factorization of $\beta(\gamma)$ is given by
(3),(4),(5). In particular, if $p_{{\mathbb{C}}}^{-1}(b)$ has at most one real
point then

(8) $\beta(\bar{\gamma})=({\mathfrak{R}}rev(\beta(\gamma)))^{-1}$

If all points of $p_{{\mathbb{C}}}^{-1}(b)$ are real then one has:

(9) $\beta(\bar{\gamma})=(rev(\beta(\gamma)))^{-1}$

i.e. coincides with the outer automorphism $\epsilon_{d}$ used in [7] [3].

###### Proof.

Indeed, the braid $\beta(\bar{\gamma})$ being interpreted as an automorphism
of the free group $\pi_{1}(p_{{\mathbb{C}}}^{-1}(b)\setminus
p_{{\mathbb{C}}}^{-1}(b)\cap C,b)$ is the composition of conjugations, the
automorphism corresponding to $\beta$ and the conjugation i.e. is
$\bar{\beta}$. The equalities (8) and (9) follow from the identities (6) (or
(3)) and (4). ∎

Next, we will find a conjugation invariant form of the braid monodromy
factorization of a real curve. Singularities of such a curve
$C_{{\mathbb{C}}}$ are either the points with real coordinates or come in
complex conjugate pairs. So are the critical values of projections on the
complex locus of a ${\mathbb{R}}$\- line: they are real values at real
critical points or come as complex conjugate pairs. Recall that the image of
$c\in C$ of projection on $x$-axis $N$ from a point in
${\mathbb{R}}{\mathbb{P}}^{2}\subset{\mathbb{C}}{\mathbb{P}}^{2}$ is a
critical value if the line through the center of projection and $c$ intersects
$C$ in fewer than ${\rm deg}C$ points and a projection from a center
$p\in{\mathbb{C}}{\mathbb{P}}^{2}$ is called generic if each line in the
pencil of lines through $p$ contains at most one singular or critical point on
$C$ and each critical point in the smooth locus of $C$ is the simple tangency.
We call a point in ${\mathbb{R}}{\mathbb{P}}^{2}$ generic if it is an
arbitrary point in a Zariski dense subset of ${\mathbb{R}}{\mathbb{P}}^{2}$.

The following Proposition describes which critical points of generic
projection are unavoidable on the real part of $x$ axis (the target of the
projection map).

######  Proposition 3.6.

Let $C$ be a projective plane curve over ${\mathbb{R}}$ transversal to the
line at infinity. Let, as above, $p_{P}:{\mathbb{C}}{\mathbb{P}}^{2}\setminus
P\rightarrow N_{{\mathbb{C}}}$ be a projection from a point
$P\in{\mathbb{R}}{\mathbb{P}}^{2}$ onto a fixed line $N$. If $P$ is generic
then the only critical points of $p_{P}$ on the real locus
${\mathbb{R}}{\mathbb{P}}^{1}\subset N_{{\mathbb{C}}}$ are either singular
points of $C$ with coordinates both being real or images of critical points of
restriction of $p_{P}$ on the real locus $C_{{\mathbb{R}}}$ of $C$.

###### Proof.

We will work in a coordinate system such that the projection from $P$ is just
the projection $(x,y)\rightarrow x$ onto the $x$-axis. First, notice that if a
coordinate system in ${\mathbb{R}}{\mathbb{P}}^{2}$ is generic then each
singular point either has both coordinates real or both coordinates have non-
zero imaginary parts. We assume that $C$ is in a such coordinate system. Then
note also that the number of real lines through a point
$(z,w,1)\in{\mathbb{C}}^{2}\subset{\mathbb{C}}{\mathbb{P}}^{2}$ is either
infinite (if $(z,w)\in{\mathbb{R}}^{2})$ or is either 1 ($z,w$ are on a line
${\mathbb{C}}={\mathbb{R}}^{2}$ defined over ${\mathbb{R}}$) or zero. Consider
the incidence correspondence ${\mathcal{I}}\subset
N_{{\mathbb{C}}}\times{\mathbb{R}}{\mathbb{P}}^{2}$ consisting of pair $(a,P)$
such that $a$ is the image of a critical point of projection onto
$N_{{\mathbb{C}}}$ from $P$. The projection
${\mathcal{I}}\rightarrow{\mathbb{R}}{\mathbb{P}}^{2}$ is a finite cover and
${\mathcal{I}}$ is a real two-dimensional manifold. Each real tangent
transversal to the real locus of $C$ either is a bitangent or contains
singular points with one of the coordinates being not in ${\mathbb{R}}$. There
are no points in the latter class by genericity assumption and only a finite
set of points in the former class. Taking $P\in{\mathbb{R}}{\mathbb{P}}^{2}$
which pre-image in ${\mathcal{I}}$ has an empty intersection with this finite
set in ${\mathcal{I}}$ we get the required projection. ∎

We select coordinates in ${\mathbb{C}}{\mathbb{P}}^{2}$ so that the base point
of the pencil is at infinity and the lines of the pencils are the lines $x=c$
(i.e. the center of projection is $(0,1,0$)) and we have the projection
$p:{\mathbb{C}}^{2}\rightarrow{\mathbb{C}}_{x}$ onto the $x$ axis given by
$y=0$. Moreover, the trivialization of $p$ over ${\mathbb{C}}_{x}=\\{y=0,z\neq
0\\}$ is given by projection $(x,y)\rightarrow y$ onto $y$ axis
${\mathbb{C}}_{y}$.

Let $Cr\subset{\mathbb{C}}_{x}$ be the critical set of the projection and
$N={\rm Card}Cr$. Let us view $Cr$ as a subset ${\mathcal{P}}_{N}$ discussed
in Section 2, select an order in $Cr$, a base point $b$, and a good ordered
system of generators in $\pi_{1}({\mathbb{C}}\setminus Cr,b)$ as described
there. We call this a complete good ordered system of generators compatible
with the real structure and denote its elements777the order of the loops in
this system is from the left to the right but subscripts specify the points in
the set $Cr$ with the ordering described above.

(10)
$\gamma_{1},\cdots\gamma_{h},\gamma^{r}_{l},\cdots,\gamma^{r}_{1},\bar{\gamma}^{-1}_{h},....\bar{\gamma}^{-1}_{1}$

Here $\bar{\gamma}$ is the class of the loop containing the image of the
conjugation map $H^{+}\rightarrow H^{-}$ applied loop to $\gamma$.

Finally, we will denote by $Dc\subset{\mathbb{C}}_{x}$ be a closed subset
bounded by a loop with base point $b$ and such that $Dc\cap Cr$ is the set of
real critical values.

######  Definition 3.7.

The factorization

$\beta_{1}\cdots\cdots\beta_{N}=\Delta^{2},\ \ \ \ \ N=2h+l$

where the braids $\beta_{i}$ are the images of the braid monodromy
homomorphism $\pi_{1}({\mathbb{C}}_{x}\setminus Cr,b)\rightarrow B_{d}$
corresponding to the loops (10) will be called compatible with the real
structure.

The product of the braids corresponding to the loops
$\gamma_{1},\cdots,\gamma_{h}$ will be denoted ${\mathcal{B}}_{H^{+}}$, the
product of the braids corresponding to $\gamma^{r}_{l},\cdots,\gamma^{r}_{1}$
we denote ${\mathcal{B}}_{{\mathbb{R}}}$, and the product of the braids
corresponding to the remaining loops in this system we denote
${\mathcal{B}}_{H^{-}}$ so that

$\Delta^{2}={\mathcal{B}}_{H^{+}}{\mathcal{B}}_{{\mathbb{R}}}{\mathcal{B}}_{H^{-}}$

######  Theorem 3.1.

Let $C$ be a projective plane curve over ${\mathbb{R}}$. Let
$p:{\mathbb{C}}^{2}\rightarrow{\mathbb{C}}=N_{{\mathbb{C}}}$ be a projection
of the affine part of ${\mathbb{C}}{\mathbb{P}}^{2}$ with the line at infinity
being transversal to $C_{{\mathbb{C}}}$. Let $b$ be the base point in the real
locus of $N_{{\mathbb{C}}}$ such that ${\rm Card}p^{-1}_{{\mathbb{C}}}(b)\cap
C_{{\mathbb{R}}}\leq 1$. The braid monodromy factorization corresponding to a
good ordered system of generators of $\pi_{1}(N_{{\mathbb{C}}}\setminus Cr,b)$
compatible with real structure induces decomposition

(11)
$\Delta^{2}={\mathcal{B}}_{H^{+}}\cdot{\mathcal{B}}_{{\mathbb{R}}}\cdot{\mathfrak{R}}(rev({\mathcal{B}}_{H^{+}}))$

where the braids ${\mathcal{B}}_{H^{+}},{\mathcal{B}}_{{\mathbb{R}}}$ are as
in Definition 3.7.

###### Proof.

Since ${\mathcal{B}}_{H^{-}}=\bar{\gamma}_{h}^{-1}...\bar{\gamma}_{1}^{-1}$,
using Proposition 3.5 this braid can be written as:

$({\mathfrak{R}}(rev(\beta(\gamma_{h})))^{-1})^{-1}\cdots({\mathfrak{R}}(rev(\beta(\gamma_{1})))^{-1})^{-1}={\mathfrak{R}}(rev(\beta(\gamma_{h})))....{\mathfrak{R}}(rev(\beta(\gamma_{1})))$
$={\mathfrak{R}}(rev(\beta(\gamma_{h}))rev(\beta(\gamma_{h-1}))\cdots
rev(\gamma_{1}))={\mathfrak{R}}(rev(\beta(\gamma_{1})\cdots\beta(\gamma_{h})))=$
$Re(rev({\mathcal{B}}_{H^{+}}))$

as claimed. ∎

######  Remark 3.8.

We would like to describe the braid ${\mathcal{B}}_{H^{+}}$ (and hence
${\mathcal{B}}_{H^{-}}$) in terms of the braid ${\mathcal{B}}_{{\mathbb{R}}}$
corresponding to the real part of the critical locus i.e. to solve in the
braid group the equation (11). Unfortunately, as pointed out to the author by
the referee, the constraint of the Theorem 3.1 is not sufficient for
recovering the braids ${\mathcal{B}}_{H^{\pm}}$ and
${\mathcal{B}}_{{\mathbb{R}}}$ even in the simples cases. Consider the case
when ${\mathcal{B}}_{{\mathbb{R}}}=1$, corresponding to the case of real
curves without real points and for which the projection has no real critical
values i.e.

(12) ${\mathcal{B}}{\mathfrak{R}}rev({\mathcal{B}})=\Delta^{2}$

Since ${\mathfrak{R}}(rev(\Delta))=\Delta$ (cf. [9] Lemma 3), one expects that
in decomposition (11) ${\mathcal{B}}_{H^{+}}={\mathcal{B}}_{H^{-}}=\Delta$:

However, in $B_{4}$, there is another solution to (11) i.e.
$\Gamma{\mathfrak{R}}rev\Gamma=\Delta^{2}$.

(13) $\Delta s^{-1}_{1}s_{3}{\mathfrak{R}}(rev\Delta s_{1}^{-1}s_{3})=\Delta
s_{1}^{-1}s_{3}s_{1}s_{3}^{-1}\Delta=\Delta^{2}$

Note that

(14) $\Delta s^{-1}_{1}s_{3}=(s_{1}s_{2}s_{3})^{2}\ \ \ and\ \ \
s_{3}(s_{1}s_{2}s_{3})^{2}s_{3}^{-1}=\Delta$

Nevertheless, we show below that for the algebraic curves in question the
braids ${\mathcal{B}}_{H^{\pm}}$ are the a conjugates of ”obvious” solutions.
It would be interesting to find additional constraints on the braids
${\mathcal{B}}_{H^{\pm}},{\mathcal{B}}_{{\mathbb{R}}}$ that allow to
streamline their calculation and to understand the geometric significance of
all solutions to (11). The equation (11) has a certain similarity with the
problem of finding roots in elements in the braid group (cf. [10]) but a
direct connection is not clear.

In response to the author’s query, Juan Gonzalez-Meneses informed the author
that he obtained classification results regarding the solutions of the
equation (12) for each of the Thurston-Nielsen classes: cf. [11].

##  4\. Presentations of fundamental groups of real curves.

Recall that van Kampen’s theorem (cf. [23],[21]) gives the following
presentation in terms of the braids $\beta_{i}$ described in definition (3.7).

(15) $\pi_{1}({\mathbb{C}}^{2}\setminus C)=\\{x_{1}\cdots
x_{d}|\beta_{i}(x_{j})=x_{j}\\}\ \ \ \ j=1,\cdots d,i=1,\dots,{\rm Card}Cr$

In this section, we show that for curves over ${\mathbb{R}}$, the fundamental
groups of the complement to $C_{{\mathbb{C}}}$ are quotients of geometrically
defined and depending on real structure groups admitting van Kampen type
presentations but requiring fewer relations than in (15).

With notations as in Section 3, let $Cr_{{\mathbb{R}}}\subset Cr\subset
N_{{\mathbb{C}}}$ be the subset of real critical values of projection $p$.
Recall that $Dc$ is a disk in $N_{{\mathbb{C}}}$ bounded by a loop based at
$b$ and such that $Dc\cap Cr=Cr_{{\mathbb{R}}}$.

######  Theorem 4.1.

  1. (1)

The group $\pi_{1}(p^{-1}(H^{+}\bigcup Dc)\setminus C,b)$ has a presentation:

(16) $\pi_{1}(p^{-1}(H^{+}\bigcup Dc)\setminus
C,b)=\\{x_{1},..,x_{d}|\beta_{i}(x_{j})=x_{j}\\}\ \ \ \ \ j=1,\cdots
d,i=1,..,h+l$

  2. (2)

Inclusion $p^{-1}(H^{+}\bigcup Dc)\setminus
C\rightarrow{\mathbb{C}}^{2}\setminus C$ induces the surjection:

(17) $\pi_{1}(p^{-1}(H^{+}\bigcup Dc\setminus
C),b)\rightarrow\pi_{1}({\mathbb{C}}^{2}\setminus C,b)\rightarrow 1$

###### Proof.

Recall that each loop in a good ordered system of generators bounds a disk
containing a single critical value of projection $p$. The complement
${\mathbb{C}}^{2}\setminus C$ can be retracted onto the union of $p$-pre-
images of disks bounded by the loops in a good ordered system of generators
(cf. [16]) and the fundamental group of the preimage of a disk corresponding
to critical value $j$ has presentation $\\{x_{1},\cdots
x_{d}|\beta_{j}(x_{i})=x_{i},i=1\cdots d\\}$. Part (16) follows from the van
Kampen theorem about a union of spaces (cf. [12]).

Part (17) is a corollary of part (16) since the set of relations of
$\pi_{1}({\mathbb{C}}^{2}\setminus C,b)$ consists of the same relations as the
relations for $\pi_{1}(\pi^{-1}(H^{+}\bigcup Dc\setminus C),b)$ and additional
relations (corresponding to the critical points of projection of $C$ in the
lower half plane. ∎

######  Proposition 4.1.

Let $\beta_{1},...,\beta_{r}\in B_{d}$ be a finite set of braids and
$x_{1},\cdots,x_{d}$ be a system of generators of a free group $F_{d}$. Let
$\beta=\beta_{1}\cdots\beta_{r}$ be their product,
$G(\beta_{1},\cdots\beta_{r})$ (resp. $G(\beta)$) be the quotients of the free
group $F_{d}$ by the normal subgroup generated by elements
$\beta_{i}(x_{j})x_{j}^{-1},i=1,\cdots r,j=1,\cdots,d$ (resp. the relations
$\beta(x_{j})x_{j}^{-1},j=1,\cdots,d$). Then one has surjection:

(18) $G(\beta)\rightarrow G(\beta_{1},\cdots,\beta_{r})\rightarrow 1$

###### Proof.

We shall show this by induction over $r$. Assume that for all $1\leq j\leq d$
the element $\beta_{1}\cdots\beta_{r-1}(x_{j})x_{j}^{-1}\in F_{d}$ belongs to
the normal subgroup $N_{r-1}$ of $F_{d}$ generated by
$\beta_{i}(x_{j})x_{j}^{-1},i=1,\dots,r-1$. Let
$\beta^{\prime}=\beta_{1}\cdots\beta_{r-1}$. Then for any $x_{j},j=1,\dots,d$

(19)
$\beta(x_{j})x_{j}^{-1}=\beta^{\prime}(\beta_{r}(x_{j}))x_{j}^{-1}=\beta^{\prime}(\beta_{r}(x_{j}))\beta_{r}(x_{j})^{-1}\beta_{r}(x_{j})x_{j}^{-1}$

Let $\beta_{r}(x_{j})=y_{1}y_{2}\cdots y_{s}$ where $y_{k},k=1,\dots,s$ is one
of generators $x_{1},\dots,x_{d}$ or their inverses, with possibly several of
$y_{k}$ corresponding to the same element among $x_{j}$. In particular, we
have $\beta^{\prime}(y_{k})y_{k}^{-1}\in N_{r-1}$ by the assumption of
induction. Then the right-hand side in (19) can be written as:

(20) $\beta^{\prime}(y_{1}\cdots y_{s})y_{s}^{-1}\cdots
y_{1}^{-1}\beta_{r}(x_{j})x_{j}^{-1}=\beta^{\prime}(y_{1})\cdots\beta^{\prime}(y_{s})y_{s}^{-1}\cdots
y_{1}^{-1}\beta_{r}(x_{j})x_{j}^{-1}$

The surjection $F_{d}\rightarrow F_{d}/N_{r-1}$ takes
$\beta^{\prime}(y_{s})y_{s}^{-1}$ to $1\in F_{d}/N_{r-1}$ i.e. the last
expression in (20) goes to the same element as does
$\beta^{\prime}(y_{1})\cdots\beta^{\prime}(y_{s-1})y_{s-1}^{-1}\cdots
y_{1}^{-1}\beta_{r}(x_{j})x_{j}^{-1}$ and the latter goes to the same element
as $\beta^{\prime}(y_{1})\cdots\beta^{\prime}(y_{s-2})y_{s-2}^{-1}\cdots
y_{1}^{-1}\beta_{r}(x_{j})x_{j}^{-1}$ since
$\beta^{\prime}(y_{s-1})y^{-1}_{s-1}$ goes to $1\in F_{d}/N_{r-1}$ and so on.
In particular, the last expression in (20) is the normal subgroup
$N_{r}\subset F_{d}$ generated by $N_{r-1}$ and the element
$\beta_{r}(x_{j})x_{j}^{-1}$ i.e. the subgroup of $F_{d}$ generated by the
relations of $G(\beta_{1},\cdots,\beta_{r})$ which shows the claim. ∎

######  Remark 4.2.

It is well known that the fundamental group of the complement to a singular
curve is an invariant of equisingular isotopy of complex algebraic curves on
surfaces (cf. [17] for references therein). In the case of real curves a
natural problem is to understand the rigid equisingular isotopy classes or at
least the classes of equivariant (with respect to complex conjugation)
equisingular isotopy (cf. [22] Sect. 4 for non-singular case). The fundamental
group $\pi_{1}({\mathbb{C}}{\mathbb{P}}^{2}\setminus
C_{{\mathbb{C}}},b),b\in{\mathbb{R}}{\mathbb{P}}^{2}$ endowed with the
involution provides an invariant of classes of such restricted isotopy in the
sense that for any two real curves $C_{1},C_{2}$ isotopic via equivariant
equisingular isotopy, there exist an isomorphism of the fundamental groups
equivariant with respect to involutions induced by conjugation. For example,
if $C$ is a pair of lines in ${\mathbb{C}}^{2}$, then
$\pi_{1}({\mathbb{C}}^{2}\setminus
C,b,{\mathbb{Z}})={\mathbb{Z}}^{2},i=1,2,b\in{\mathbb{R}}^{2}$, and the
involution induced by conjugation has the matrix $\left(\begin{matrix}-1&0\\\
0&-1\\\ \end{matrix}\right)$ (resp. $\left(\begin{matrix}0&-1\\\ -1&0\\\
\end{matrix}\right)$) if both lines are defined over ${\mathbb{R}}$ (resp.
both lines are imaginary) i.e. the additional structure distinguishes the
classes of rigid isotopy.

For a curve over ${\mathbb{R}}$, it follows from Prop. 3.5 that van Kampen
presentation (15 for the braid monodromy in generators used in Proposition 3.5
the automorphism of the free group given by (3) and (4) passes to an
involution of the fundamental group of the complement. Hence we obtained a
calculation of this extra structure on the fundamental group.

This involution can be encoded into an exact sequence:

(21) $0\rightarrow\pi_{1}({\mathbb{C}}^{2}\setminus
C_{{\mathbb{C}}},b)\rightarrow\pi^{{\mathbb{R}}}_{1}({\mathbb{C}}^{2}\setminus
C_{{\mathbb{C}}},b)\rightarrow{\mathbb{Z}}_{2}\rightarrow 0$

in which the action of ${\mathbb{Z}}_{2}$ on
$\pi_{1}({\mathbb{C}}^{2}\setminus C_{{\mathbb{C}}},b)$ is given by
conjugation. The group in the middle is a topological analog of Grothendieck’s
fundamental group of a variety over ${\mathbb{R}}$.

##  5\. Alexander Polynomials

Surjections of the previous section imply the divisibility relations between
the Alexander polynomials of the groups considered there.

Recall (cf. [17] and the references therein) that given a group $G$ and a
surjection $\phi:G\rightarrow{\mathbb{Z}}$ one defines the Alexander
polynomial as the order of the torsion part of the module over the ring of
Laurent polynomials ${\mathbb{Q}}[t,t^{-1}]$ with underlying ${\mathbb{Q}}$\-
vector space being the quotient of ${\rm{Ker}}\phi$ by its commutator with
constants extended to ${\mathbb{Q}}$. The module structure is defined by
requiring that the action of $t$ be given by the automorphism induced on
${\rm{Ker}}\phi$ by the conjugation by a lift to $G$ of the positive generator
of the target ${\mathbb{Z}}$ of $\phi$ (this action when considered on
abelianization of ${\rm{Ker}}\phi$ is independent of a lift).

In terms of cyclic decomposition

(22)
${\rm{Ker}}\phi/({\rm{Ker}}\phi)^{\prime}\otimes{\mathbb{Q}}=\oplus{\mathbb{Q}}[t,t^{-1}]^{a}\oplus(\oplus_{i=1}^{b}{\mathbb{Q}}[t,t^{-1}]/(\Delta_{i}(t))\
\ \ \Delta_{i}|\Delta_{i+1},$

this order is given by

$\Delta(t)=\Delta_{1}(t)\cdots\Delta_{b}(t)$

if $a=0$ and $\Delta(t)=0$ if $a\geq 1$.

Recall also (cf. [1] Th. 2.2) that Artin showed that the fundamental group of
the complement to a link in $S^{3}$ represented by a closed braid $\beta$ has
the same presentation as the group $G(\beta)$ from Proposition 4.1 888in fact,
[1] shows the presentation of the fundamental group of link in $S^{3}$ is the
group $G(\beta)$ i.e.
$G(\beta)=\\{x_{1},\cdots,x_{d}|\beta(x_{i})x_{i}^{-1},i=1,\cdots d\\}$ with
one relation deleted, but there it is also pointed out that this relation is
the combination of the remaining $d-1$ relations.

######  Proposition 5.1.

Let $G(\beta)$ and $G(\beta_{1},\cdots,\beta_{r})$ be associated with braids
$\beta$ and $\beta_{1},\cdots\beta_{r}$ groups with generators and relations
described in Proposition 4.1. Let $\Delta_{\beta}$ and
$\Delta_{\beta_{1},\cdots,\beta_{r}}$ be the Alexander polynomials of these
groups relative to surjections $\phi_{\beta}$ and
$\phi_{\beta_{1},\cdots,\beta_{r}}$ onto ${\mathbb{Z}}$ which send each of
their generators $x_{1},\cdots,x_{d}$ to the positive generator of
${\mathbb{Z}}$. Then $\Delta_{\beta_{1},\cdots,\beta_{r}}$ divides
$\Delta_{\beta}$.

###### Proof.

The surjection (18) and the commutative diagram

$\begin{matrix}G(\beta)&&\rightarrow&&G(\beta_{1},\cdots\beta_{r})\\\
&\searrow&&\swarrow&\\\ &&{\mathbb{Z}}&&\end{matrix}$

induce the surjections of ${\mathbb{Q}}[t,t^{-1}]$-modules:

(23)
${\rm{Ker}}\phi_{\beta}/({\rm{Ker}}\phi_{\beta})^{\prime}\otimes{\mathbb{Q}}\rightarrow{\rm{Ker}}\phi_{\beta_{1},\cdots,\beta_{r}}/({\rm{Ker}}\phi_{\beta_{1},\cdots,\beta_{r}})^{\prime}\otimes{\mathbb{Q}}\rightarrow
0$

Since the left group in (18) is the fundamental group of the complement to a
link in 3-sphere for which the Alexander module is a torsion module, it
follows from (23) that the right group in (18) has as its Alexander module a
torsion module and the claim follows. ∎

######  Theorem 5.1.

The Alexander polynomial of $\pi_{1}({\mathbb{C}}^{2}\setminus C)$ divides the
Alexander polynomials of the closed braids in 3-sphere associated with each of
the braids
${\mathcal{B}}_{H^{+}},{\mathcal{B}}_{{\mathbb{R}}},{{\mathcal{B}}}_{H^{+}}\cdot{{\mathcal{B}}}_{{\mathbb{R}}}$
in the braid group $B_{d}$.

###### Proof.

The theorem 4.1 identifies the group $\pi_{1}(p^{-1}(H^{+}\bigcup Dc)\setminus
C,b)$ with the group $G(\beta_{1},\cdots\beta_{h+l})$ and Proposition 5.1
shows that the Alexander polynomial of this group divides the Alexander
polynomial of the closed braid
${\mathcal{B}}_{H^{+}}{\mathcal{B}}_{{\mathbb{R}}}$. The surjection in Theorem
4.1 implies that the Alexander polynomial of
$\pi_{1}({\mathbb{C}}^{2}\setminus C)$ divides the Alexander polynomials of
$\pi_{1}(p^{-1}(H^{+}\bigcup Dc)\setminus C,b)$ as in the proof of Proposition
5.1 which completes the proof. ∎

##  6\. Examples.

In this section we discuss the braids
${\mathcal{B}}_{H^{+}},{\mathcal{B}}_{{\mathbb{R}}}$ for the curves satisfying
assumptions on the real part of the critical set and use them to get refined
divisibility conditions for the Alexander polynomials in corresponding classes
of plane singular curves over ${\mathbb{R}}$.

The considered extreme cases are the case of the arrangements of real lines
and real arrangements of lines with zero-dimensional real locus, the curves of
even degree with empty real locus, and related curves of odd degrees. To
obtain a non-trivial divisibility relation one makes a different selection of
the braid ${\mathcal{B}}_{H^{+}}$ or ${\mathcal{B}}_{{\mathbb{R}}}$ or their
product. In the case of arrangements of real lines, the braid
${\mathcal{B}}_{H^{+}}$ is trivial, the Alexander polynomial of the closed
braid in $S^{1}\times{\mathbb{C}}$ is zero, and the divisibility relation is
empty. On the other hand, ${\mathcal{B}}_{{\mathbb{R}}}=\Delta^{2}$, and one
obtains a known divisibility relation mentioned in Section 1. Note that the
Alexander module of a link in $S^{3}$ (a closed braid) is torsion of the
linking number of any two components is non zero (cf. [4]).

On the other hand, for the real curves with no real points, we show that
${\mathcal{B}}_{H^{+}}=\Delta$ which leads to refined divisibility constraints
(cf. Prop 6.7).

######  Example 6.1.

We consider maximally componentwise unreal arrangements which are the
arrangements over ${\mathbb{R}}$ with the minimal number of real points. Note
the following:

######  Proposition 6.2.

Let ${\mathcal{A}}$ be an arrangement over ${\mathbb{R}}$. Then the set of its
real points is not empty.

###### Proof.

This is immediate since the intersection point of a pair of conjugate lines is
real. ∎

Let ${\mathcal{A}}_{k}$ be an arrangement over ${\mathbb{R}}$ with $k<\infty$
real multiple points. Such an arrangement has the form

(24)
$\prod_{j=1}^{r_{1}}(((x+ay)-n_{1})^{2}+m^{1}_{j}y^{2})\cdots\prod_{j=1}^{r_{i}}(((x+ay)-n_{i})^{2}+m_{j}^{i}y^{2})\cdots\prod_{j=1}^{r_{k}}(((x+ay-
n_{k})^{2}+m_{j}^{r_{k}}y^{2})$

where

$0\neq a\in{\mathbb{R}},n_{i}\in{\mathbb{R}},\ \ \ n_{1}>n_{2}\cdots>n_{k},\ \
\ m_{i}^{j}\in{\mathbb{R}}^{+},\pm m_{i^{\prime}}^{j^{\prime}}\neq\pm
m_{i}^{j}\ \ \forall(i,j)\neq(i^{\prime},j^{\prime})$

${\mathcal{A}}_{k}$ contains $d=2\sum_{1}^{k}r_{i}$ lines with $k$ real
multiple points $(n_{i},0),i=1,..,k$ having respective multiplicities
$2r_{i},i=1,\cdots,k$. The singular points, which are the intersection points
of lines with different $n_{i}$’s for $a\neq 0$, have $x$-coordinates with
non-zero imaginary parts and come in conjugate pairs i.e.
$Cr{\mathcal{A}}_{k}=\\{x\in{\mathbb{C}}|x=n_{i}\\}$. The only real points of
${\mathcal{A}}_{k}$ are the points $(n_{i},0),i=1,\cdots,k$.

Let us describe the braid ${\mathcal{B}}_{{\mathbb{R}}}$ in this case and more
specifically the braids corresponding to (the classes of the) loops
$\gamma_{i}^{r}\in\pi_{1}({\mathbb{C}}_{x}\setminus Cr,b)$ (cf. Definition
3.7) i.e. corresponding to paths running from $b$ along real axis to one of
the points $(n_{i},0)$ while circumventing points in $Cr$ following a small
semi-circle in the upper half plane, then upon reaching the vicinity of
$(n_{i},0)$ following the full circle around $(n_{i},0)$ and finally returning
to $b$ along the same path.

######  Proposition 6.3.

There is a collection of non-intersecting segments
$\delta_{1},\cdots\delta_{i}\cdots\delta_{k}$ in $L_{{\mathbb{C}},b}$ each
containing a set $A_{i}$ or $2r_{i}$ points belonging to the $i$-th group of
lines (24) in ${\mathcal{A}}_{k}$ such that braid
${\mathcal{B}}_{{\mathbb{R}}}$ is a product of the conjugates of the full
twists $\Delta^{2}_{A_{k}}$ about $\delta_{i}$. In particular $\Delta_{i}^{2}$
commute.

###### Proof.

Explicit form of the lines (24) shows that $y$ coordinates of the
intersections of lines in this arrangement with the fibers
$L_{{\mathbb{C}},t}$ of projection used to calculate the braid monodromy have
the form

(25) $Re(y)=\lambda^{i}_{j}Im(y),\ \ \ \
i=1,\cdots,k,j=1,\cdots,r_{i},\lambda^{i}_{j}\neq\lambda^{i^{\prime}}_{j^{\prime}},(i,j)\neq(i^{\prime},j^{\prime})$

and the braids corresponding to $\gamma_{i}^{r}$ can de described in terms of
the motion of $d$ points ${\mathcal{A}}_{k}\cap\mathcal{L}_{{\mathbb{C}},b}$
along these lines 999Explicitly the intersection points of
$L_{{\mathbb{C}},t}$ and the lines corresponding to the $i$-th factor in (24)
are $y={{(n_{i}-t)(a\pm m_{j}\sqrt{-1})}\over{a^{2}+m_{j}^{2}}}$ and
$\lambda^{i}_{j}=\pm{a\over m^{i}_{j}}$. Hence these $d$ points are split into
$k$ groups each containing $2r_{i},i=1,...,k$ points. Each group moves toward
the origin along the respective group of lines in ${\mathbb{C}}$ and arrives
at $0\in{\mathbb{C}}$ at $k$ different moments. A group of $2r_{i}$ points
while moving along their respective lines undergoing slight deviations at the
moments when other groups reach their critical points (corresponding to
$\gamma^{r}_{i}$ deviations from the real axis in $x$-plane), Just before the
time when the $i$-th group should arrive at the origin the points in this
group undergo full twist (the braid corresponding to the the singularity of
$2r_{i}$ pairwise transversal lines) and returning to the the original
position of the group along the same path.

With each of $k$ critical points is associated “vanishing segment” containing
$2r_{i}$ points merging into $(n_{i},0)$ which is located at
$L_{{\mathbb{C}},b^{\prime}}$ where $b^{\prime}$ is point in the real part of
$x$ axis and where $\gamma_{i}^{r}$ starts the circle around $x=n_{i}$.
Transporting this segment along the path of $\gamma_{i}^{r}$ back from
$b^{\prime}$ to $b$ produces the vanishing segment of this group in
$L_{{\mathbb{C}},b}$. While $t$ moves from $b^{\prime}$ toward $n_{i}$ and
then completes the move along semi-circle, the set of $2r_{i}$ points merging
in $(n_{i},0)$ move from half-plane $Re<0$ to half-plane $Re>0$. If $t$
continues to move in a negative direction instead of completing the full
circle around $(n_{i},0)$, this segment will remain in the right half plane
when $t$ moves to the next critical value. Hence vanishing segments
corresponding to the critical values $n_{i}$ can be selected inductively so
that the segment corresponding to $n_{i}$ does not intersect the segments
corresponding to $n_{j},j<i$. As a result, we obtain that the braids
corresponding to the loops $\gamma_{i}^{r}$ are full twists along the
collection of non-intersecting conjugation invariant segments in
$L_{{\mathbb{C}},b}$:

(26) $\Delta_{A_{1}}^{2},\Delta_{A_{2}}^{2}\cdots\Delta_{A^{k}}^{2}$

where $\Delta_{A_{i}}$ is a rotation by 180 degrees of the group of points
$A_{i}$. In particular, these braids commute. The braid
${\mathcal{B}}_{{\mathbb{R}}}$ is the product of those twists. ∎

Next, consider other classes of algebraic curves.

######  Proposition 6.4.

Let $C$ be a real curve having degree $2d$ without real points and admitting a
generic projection
$\pi_{{\mathbb{C}}}:C_{{\mathbb{C}}}\rightarrow{\mathbb{C}}$ having no
critical values on the real axis. Then the braid ${{\mathcal{B}}}_{H+}$ is a
conjugate of the Garside element $\Delta_{2d}$ of the braid group $B_{2d}$. In
particular, the braid monodromy of such curves satisfies the relation:

(27) $\beta_{1}\cdots\beta_{h}=\gamma^{-1}\Delta_{2d}\gamma$

where $\beta_{1},\cdots,\beta_{h}$ are the braids corresponding to the loops
in a good ordered system of generators corresponding to the critical values in
the upper half-plane and $\gamma\in B_{2d}$.

######  Remark 6.5.

Consider the following conditions on a curve and its projection:

a) empty (i.e. without real points) over ${\mathbb{R}}$ which the projection
onto the $x$-axis is generic (as described before Prop.3.6) and has no real
critical values.

b) real curves given by an equation $f(x,y)=0$ where $f(x,y)$ has a constant
sign for all $(x,y)\in{\mathbb{R}}^{2}$ and projection onto $x$ axis is
generic.

c) curves of an even degree admitting generic projection without real critical
values.

Clearly a) implies b) and c) implies a). Indeed, a curve of an odd degree does
have real points. If $c$ is a real critical value of projection of a curve as
in c), then the critical point is either real, i.e. the curve is not empty, or
a pair of conjugate critical points i.e. the projection is not generic. A
curve of an even degree, admitting generic projection without real critical
values does not have to be empty (e.g. projection from an interior point of a
real circle).

###### Proof.

(of Prop. 6.4). Let $C_{G}\subset{\mathbb{C}}{\mathbb{P}}^{2}$ be a union of
real smooth quadrics given by equation
$G(x,y)=\prod_{j=1}^{d}(j(x^{2}+N^{2})+y^{2})=0,\ (N\in{\mathbb{N}})$.
Projection of $C_{G}$ onto $x$ axis has two critical values $x=\pm N\sqrt{-1}$
each corresponding to a singular point of $C_{G}$ (for $d>1$) each being a
union $d$ smooth branches tangent to the fiber of the projection. The
intersection index of each two branches at the critical point is 2. The
alternative local model of this singularity (at the origin) is
$\prod_{1}^{d}(j^{2}x-y^{2})$. The subset of $j$-th branch of the latter germ
over the loop $\gamma(s)=e^{{2\pi is}}$ in the $x$-axis is $(e^{2\pi
is},je^{\pi is}),j=1,\cdots d,0\leq s\leq 1$. It follows that the
corresponding braid of this singularity is the half-twist $\Delta_{2d}$.
Since, the loop $\gamma(s)$ (with the base point on the real axis) is
homotopic to the loop represented by the real axis in $H^{+}\setminus
i\sqrt{N}$, the braid of $C_{G}$ corresponding to the real axis is conjugate
to $\Delta_{2d}$.

Let $F=0$ be the equation of $C$ and let $h$ be the number of critical points
of projection of $C_{{\mathbb{C}}}$ onto $x$-axis located in $H^{+}$. Consider
deformation $F_{s}=(1-s)F+sG,0\leq s\leq 1$ and the corresponding family of
curves $C_{F_{s}}$. Critical points of projection of $C_{F_{s}}$ located in
$H^{+}$ (resp. $H^{-}$) move to $N\sqrt{-1}$ (resp. $-N\sqrt{-1}$) along paths
$\gamma_{i}(s)$ starting at $h$ critical points of $F$ formed by the critical
points of $C_{F_{s,{\mathbb{C}}}},s<0<1$ and ending at the respective critical
point of $G$.

As long as the projection of the curve $C_{F_{s},{\mathbb{C}}}$ onto $x$ axis
does not have real critical values, this curve remains transversal to
${\mathbb{R}}\times{\mathbb{C}}\subset{\mathbb{C}}^{2}\subset{\mathbb{C}}{\mathbb{P}}^{2}$
where the first factor is the real locus of $x$-axis and the second one is
$y$-axis. In particular, the braid ${\mathcal{B}}_{H^{+}}$ corresponding to
the curve $C_{F_{s},{\mathbb{C}}}$ does not change. For $s$ such that
$\gamma_{i}(s)$ crosses the real axis from $H^{+}$ to $H^{-}$, the conjugate
$\bar{Q}$ of the corresponding critical point $Q$ of $C_{F_{s},{\mathbb{C}}}$
has the same local type as does $Q$. The critical value of projection of
$\bar{Q}$ crosses from the opposite half-plane of the critical value at $Q$
and moves along $\overline{\gamma(s)}$. In particular, the factor of
${\mathcal{B}}_{H^{+}}$ which the braid monodromy associates with the element
in the good ordered system of generators corresponding to the critical value
on the path $\gamma_{i}(s)$ moving toward the real axis is the braid
corresponding to the same element in $\pi_{1}(H^{+}\setminus Crit(F(s))$ and
the same singularity as before the crossing the real axis. So there will be no
change in the braid ${\mathcal{B}}_{H^{+}}$ and the braid corresponding to
$C_{F}$ coincides with a conjugate of the braid of $C_{G}$ i.e.
$\gamma\Delta_{2d}\gamma^{-1},\gamma\in B_{2d}$. ∎

Recall that an acnode of multiplicity $l$ (cf. [24]) is a germ of a curve
defined over ${\mathbb{R}}$ which set of real points consists of a single
point and the set of complex points is a union of $l$ transversal smooth
branches. Note that this implies that must be even i.e. $l=2k$ and the
branches are pairwise conjugate.

######  Proposition 6.6.

Let $C$ be a curve of degree $2d$ defined over ${\mathbb{R}}$ for which the
set of real points consists of a single acnode of multiplicity $2k$. Then up,
to conjugation in $B_{2d}$, one has
${\mathcal{B}}_{{\mathbb{R}}}=\Delta_{2k}^{2}$ and
$B_{H^{+}}=\gamma\Delta_{2d}\Delta_{2k}^{-1}\gamma^{-1}={\mathcal{B}}_{H^{-}}$
where $\Delta_{2k}$ is a 180 degrees rotation of a complex conjugation
invariant subset of $2d$ points.

###### Proof.

Let $F(x,y)$ be an equation of $C$. As in Prop. 6.4 the assumption of Prop.
6.6 implies that if there exists $(x_{0},y_{0})$ such that $F(x_{0},y_{0})>0$
then $F(x,y)\geq 0$ for all $(x,y)\in{\mathbb{R}}^{2}$ with the vanishing
taking place only at the acnode. As in Prop. 6.4 consider the family of curves
$C_{s}$ with the equation $F_{s}(x,y)=(1-s)F+sG=0$. Then $F_{s}(x,y)>0$ for
all $(x,y),0<s\leq 1$. The local braid of acnode is (a conjugate in $B_{2d}$
of) the full twist $\Delta_{2k}^{2}$ on a set of $2k$ points. It coincides
with the component $B_{{\mathbb{R}}}(C_{0})$ of the braid of the curve $C$
corresponding to the real part of the set of critical values. Moreover, when
$s$ moves away from zero, the critical value $\kappa$ of the projection at the
acnode splits for $0<s<<\epsilon$ into a finite set ${\mathcal{S}}(s)$ of $K$
conjugate pairs of points. We select the base point $b$ (cf. beginning of
Section 3) in proximity to $\kappa$ so that Garside operations on elements of
$B_{2k}\subset B_{2d}$ coincide with those induced from $B_{2d}$ i.e.
${\mathfrak{R}}_{B_{2d}}(rev_{B_{2d}}\Delta_{2k})=\Delta_{2k}$. A small loop
$\gamma_{\epsilon}$ around $\kappa$, also containing the set ${\mathcal{S}}$
inside it, is taken by the braid monodromy to
${\mathcal{B}}_{{\mathbb{R}}}(C_{0})$ but the loop $\gamma_{\epsilon}$ is also
split into a product of two conjugate loops
$\gamma^{+}_{\epsilon},\gamma_{\epsilon}^{-}$ each containing inside the
subsets ${\mathcal{S}}\cap H^{+}$ and ${\mathcal{S}}\cap H^{-}$ respectively.
The local version of the argument in Prop.6.4 deforming the germ of the acnode
to the special germ having only two conjugate critical values shows that the
braid monodromy takes each of $\gamma_{\epsilon}^{\pm}$ to the rotation by 180
degrees $\Delta_{k}$. This implies that the loop
${\mathcal{B}}_{H^{+}}(C_{0})\beta(\gamma_{\epsilon}^{+})$ is a conjugate of
$\Delta_{2d}$. Hence ${\mathcal{B}}_{H^{+}}$ is a conjugate of
$\Delta_{2d}\Delta_{2k}^{-1}$. ∎

Now we turn to the Alexander polynomials and explicit divisibility relations.

######  Proposition 6.7.

Let $C$ be a real curve of even degree $d$ admitting a projection with no real
critical values. Then

(28) $\Delta_{C}(t)|(t^{d}-1)^{{d\over 2}-1}(t^{d\over 2}+1)(t-1)$

In particular, the multiplicity of the root $exp{{2\pi i}\over 6}$ of the
Alexander polynomial of such a curve, having nodes and cusps as the only
singularities, is at most ${d\over 2}-1$.

###### Proof.

We need to find the Alexander polynomial of the link which is represented by
the closed braid on $d$ strings given by $\Delta$ i.e. the rotation by 180
degrees. As was already mentioned, such a link also can be described as the
link of plane curve singularity given by local equation $\prod_{j=1}^{d\over
2}(j^{2}x-y^{2}),$ (when preimage of $x=1$ are integers $\pm j,j\leq{d\over
2}$) or equivalently $x^{d\over 2}-y^{d}=0$. The Alexander polynomial of such
a link is the characteristic polynomial of its monodromy. This is a weighted
homogeneous singularity with weights of $x$ and $y$ being ${d\over 2}$ and
$d$. Now the claim follows from Brieskorn-Pham-Milnor classical calculations
(cf. [19] Sec. 9). ∎

For the curves with a single acnode we obtain as an immediate consequence of
Proposition 6.6 since ${\mathcal{B}}_{{\mathbb{R}}}=\Delta_{k}^{2}$:

######  Corollary 6.8.

The Alexander polynomial of a curve over ${\mathbb{R}}$ with a single real
point which is an acnode divides the Alexander polynomial of the link which is
the closure of the braid $\Delta_{d}\Delta^{-1}_{k}$.

######  Example 6.9.

The above Proposition gives a sharp “estimate of the degree of Alexander
polynomials” of real curves of degree 6 without real points.

Recall that the Alexander polynomial of a plane curve of degree $6$ with $k>6$
cusps and at most nodes as other singularities is given by
$(t^{2}-t+1)^{k-6}$. More specifically, for a curve of degree $d$ with cusps
and nodes as the only singularities, denoting the zero-dimensional subset of
${\mathbb{P}}^{2}$ formed by cusps as $\Xi$, the Alexander polynomial is equal
to $1$ if $d$ is not divisible by $6$ and otherwise is equal to
$(t^{2}-t+1)^{s}$ where $s={\rm
dim}H^{1}({\mathbb{P}}^{2},{\mathcal{I}}_{\Xi}(d-3-{d\over 6}))$ with
${\mathcal{I}}_{\Xi}$ denoting the ideal sheaf of functions vanishing at
points of $\Xi$. For $d=6$ the latter is equal to
$H^{0}({\mathbb{P}}^{2},{\mathcal{I}}_{\it
cusps}(2))-\chi({\mathbb{P}}^{2},{\mathcal{I}}_{cusps}(2))=k-6$ since ${\rm
dim}H^{0}({\mathbb{P}}^{2},{\mathcal{I}}_{\Xi}(2))=0$ if the number of cusps
is greater than $6$ (if the cardinality of ${\Xi}$ is $6$ this dimension can
be either $1$ or $0$ depending on 6 cusps being positioned on a conic or not).

The divisibility theorem of [15] for a complex curve of degree $d$ transversal
to the line at infinity tells that the Alexander polynomial of the curve
divides $(1-t)(1-t^{d})^{d-2}$. Indeed, the latter is the Alexander polynomial
of the link at infinity which is the link of the closure of the braid
$\Delta^{2}$ (the Hopf link). This bounds the exponent of the Alexander
polynomial of sextic by $4$. On the other hand by 6.7 the Alexander polynomial
of a sextic with no real points should divide
$(t^{3}-1)(t+1)^{2}(t^{2}-t+1)^{2}(t-1)$. There is no sextic with 9 cusps with
no real points since the number of cusps that are not real is even. On the
other hand, there exist a real sextic with 8 cusps and no real points (cf.
[6], [13]) for which the multiplicity of the factor $(t^{2}-t+1)$ is 2.

######  Remark 6.10.

It is not hard to calculate the Alexander polynomial of the link which is the
closed braid corresponding to a curve of an odd degree having no real critical
values. In this case the projection without critical points yields again
rotation of a set with an odd number of points and the corresponding braid is
again $\Delta$. Such link appears as the link of singularity $yx^{{d-1}\over
2}-y^{d+1}$ and calculation of the characteristic polynomial of this weighted
homogeneous singularity (with non-integer weights, in which case one can use
[20]) one obtains

(29) $(t^{d}-1)^{{d-1}\over 2}(t-1)$

This may lead for example to a restriction on the degrees of the Alexander
polynomials with only ordinary triple points (in which case $d$ must be
divisible by $3$). It is unlikely however that this can give a sharp bound on
the degree.

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