数理解析研究所講究録
第 1922 巻 2014 年 21-27

21

On Alexander polynomials of some reducedcurves

東京理科大学理学部数学科川島正行

Masayuki Kawashima
Department of Mathematics
Tokyo University of science

1Introduction

Let be a plane curve in
We are interested in several topologicalinvariants. In this report, we consider Alexander polynomials of somebe a reduced
reduced curves which is defined as the following. Let,...,
besingularities. Let
quartic. Suppose that has at mostsmooth conics such that:$C$$\mathbb{P}^{2}$

$Q$

$Q$

1. Eachis tangent topoints for any$C_{i}$$Q$$C_{n}$$C_{1}$$A_{1}$

with intersection multiplicity 2 at 4 smooth$i.$

2. For all pairs $(i,j)(i\neq j)$ ,intersection points.

3.

$C_{i}\cap C_{j}\cap C_{k}=\emptyset$

and$C_{i}$

intersects transversely with

$C_{j}$at all$C_{i}\cap C_{j}\cap \mathcal{Q}=\emptyset.$

Let $B$ $:=Q+C_{1}+\cdots+C_{n}$ be the reduced curve which consists of theabove quartic and smooth conics and let $Q\cap C_{i}=\{P_{i1}, . . . , P_{i4}\}$ be thetangent points ofand for $i=1$ , . . . , . Note that the configurationsof singularities of $B$ is$C_{i}$$Q$$n$$\Sigma(B)=\Sigma(Q)+\{2n(n-1)A_{1}, 4nA_{3}\}.$

22
For example, in [7], Namba and Tsuchihashi considered the case $n=2$
and is a union of two smooth conics which are intersects transversely
each other.
In this report, we consider the case $n=3$ and then we determine their
Alexander polynomials.
$Q$2

Alexander polynomials

2.1

Definition of Alexander Polynomials

Let be an aﬃne curve of degree . Suppose that the line at infinity
be the compositionintersects transversely with . Let :of Hurewicz homomorphism and the summation homomorphism. Let bea generator of and we put the Laurent polynomial ring
We consider an infinite cyclic covering $p:\tilde{X}arrow X$ such that$ker\phi$ . Then
has a structure of -module. Thus we have
$d$

$C$

$C$

$L_{\infty}$$\phi$$\pi_{1}(X)arrow \mathbb{Z}$$t$$\Lambda$$\mathbb{Z}$$:=\mathbb{C}[t, t^{-1}].$$p_{*}(\pi_{1}(\tilde{X}))=$$\Lambda$$H_{1}(\tilde{X}, \mathbb{C})$$H_{1}(\tilde{X}, \mathbb{C})=\Lambda/\lambda_{1}(t)\oplus\cdots\oplus\Lambda/\lambda_{m}(t)$foris a polynomial in such that
where we can takeis defined by the product$i=1$ , . . . , . The Alexander polynomial.
In this report, we use the Loeser- Vaqui\’e formula ([10, 9]) for calculating
Alexander polynomials. Hereafter we follow the notations and terminologies of [4, 9] for the Loeser-Vaqui\’e formula.
$\lambda_{i}(0)\neq 0$$t$$\lambda_{i}(t)\in\Lambda$$\triangle_{C}(t)$$m$$\prod_{i=1}^{m}\lambda_{i}(t)$2.2Loeser-Vaqui\’e formulaand let us consider
Let $[X, Y, Z]$ be homogenous coordinates ofwith aﬃne coordinates $(x, y)=$the aﬃne space$(X/Z, X/Y)$ . Let $f(x, y)$ be the defining polynomial of . Let Sing (C)and let $P\in$ Sing(C) be a singular point.
be the singular locus of,...,
be theof $(C, P)$ , and let
Consider a resolution :exceptional divisors of . Let $(u, v)$ be a local coordinate system centered
be respective order of zero of the canonical two form
at $P$ and and$\mathbb{P}^{2}$$\mathbb{C}^{2}=\mathbb{P}^{2}\backslash \{Z=0\}$

$C$

$C$

$\pi$

$\pi$

$k_{i}$$m_{i}$$\tilde{U}arrow U$$E_{1}$$E_{s}$

23
andis defined by$\pi^{*}(du\wedge dv)$$\mathcal{O}_{P}$$\pi^{*}f$along the divisor$E_{i}$. The adjunction ideal$\mathcal{J}_{P,k,d}=\{\phi\in \mathcal{O}_{P}|(\pi^{*}\phi)\geq\sum_{i}(\lfloor km_{i}/d\rfloor-k_{i})E_{i}\},$$\mathcal{J}_{P,k,d}$of$k=1$ , . . . , $d-1$

where $L*\rfloor=\max\{n\in \mathbb{Z}|n\leq*\}$ and we call it the floor function.
Let $0(j)$ be the set of polynomials inwhose degree is less than orequal to . We consider the canonical mapping :and its restriction:$x,$ $y$$j$$\sigma$$\mathbb{C}[x, y]arrow\oplus_{P\in Sing(C)}\mathcal{O}_{P}$$\sigma_{k}:O(k-3)arrow\bigoplus_{P\in Sing(C)}\mathcal{O}_{P}.$Put $V_{k}(P)$$:=\mathcal{O}_{P}/\mathcal{J}_{P,k,d}$surjection$C$is given as follows:

and denote the composition of and the naturalby . Then the Alexander polynomial of$\sigma_{k}$$\oplus \mathcal{O}_{P}arrow\oplus V_{k}(P)$$\overline{\sigma}_{k}$Theorem 1. ([5, 6, 1, 3]) The reduced Alexander polynomial
given by the product$\triangle_{C}(t)\sim=\prod_{k=1}^{d-1}\triangle_{k}(t)^{\ell_{k}},$$\ell_{k}$$:=\dim$ coker$\overline{\sigma}_{k}$$\triangle_{C}(t)\sim$

is(1)

where
$\triangle_{k}(t)=(t-\exp(\frac{2k\pi i}{d}))(t-\exp(-\frac{2k\pi i}{d}))$The Alexander polynomial$\Delta_{C}(t)$is given as$\triangle c(t)=(t-1)^{r-1}\triangle c(t)\sim$

where

2.3

$r$

is the number of irreducible components of $C.$

The adjunction ideal for non-degenerate singularities

In general, the computation of the ideal
requires an explicit computation of the resolution of the singularity $(C, P)$ . However for the case$\mathcal{J}_{P,k,d}$24
can be obtained combinaof non-degenerate singularities, the ideal
torially by a toric modification. Let $(u, v)$ be a local coordinate systemcentered at such that $(C, P)$ is defined by a function germ $f(u, v)$ and,...,
bethe Newton boundary $\Gamma(f;u, v)$ is non-degenerate. Letof,...,
the primitive weight vectors which correspond to the faces$\Gamma(f;u, v)$ .
be the canonical toric modification and let:Let. Recall that thebe the exceptional divisor corresponding toalong the divisor
order of zeros of the canonical two form
where $|Q_{i}|=p+q$ for a weight vector
is simply given by(see [8]). For a function germ $g(u, v)$ , let $m(g, R_{i})$ be the. Thenon
multiplicity of the pull-back$\mathcal{J}_{P,k,d}$$P$$R_{s}$$R_{1}$$\triangle_{s}$$\triangle_{1}$$\pi$$\tilde{U}arrow U$$\hat{E}(R_{\eta}\cdot)$$R_{\eta}\cdot$$\pi^{*}(du\wedge dv)$$\hat{E}(R_{\eta}\cdot)$$|R_{\eta}\cdot|-1$$R_{\eta}\cdot=t(p_{i}, q_{i})$$\hat{E}(R_{\eta}\cdot)$$(\pi^{*}g)$is contained in the ideal
Lemma 1 ([2, 9 A function germ
if and only if satisfies following condition:$9\in \mathcal{O}_{P}$$\mathcal{J}_{P,k,d}$$g$$m(g, R_{i}) \geq\lfloor\frac{k}{d}m(f, R_{i})\rfloor-|R_{i}|+1,$

The idealditions.

$\mathcal{J}_{P,k,d}$$i=1$ , . . . , $s.$

is generated by the monomials satisfying the above con-Example 1. Let $C$ be a plane curve of degree $2n+4$ such thatsingularities. Let be a singular point.
onlyand$A_{1}$

$C$

has

$P$

$A_{3}$(1) Assume that $P$ is an$A_{1}$

singularity. Then the adjunction ideal is$\mathcal{J}_{P,k,2n+4}=\langle u^{a}v^{b}|a+b\geq\lfloor\frac{k}{n+2}\rfloor-1\rangle=\mathcal{O}_{P},$for all $k=3$ , . . . , $2n+3$ . Hence singularity do not contribute tois for allcomputations of Alexander polynomials because$A_{1}$$V_{P}(k)$(2) Assume that $P$ is an$A_{3}$$0$$k.$

singularity. Then the adjunction ideal is$\mathcal{J}_{P,k,2n+4}=\langle u^{a}v^{b}|a+2b\geq\lfloor\frac{2k}{n+2}\rfloor-2\rangle$$=\{\begin{array}{l}\mathcal{O}_{P}, 3\leq k<\lceil\frac{3}{2}(n+2)\rceil,m_{P}, \lceil\frac{3}{2}(n+2)\rceil\leq k\leq 2n+3\end{array}$where $\lceil*\rceil=\min\{n\in \mathbb{Z}|*\leq n\}$ and we call it the ceil function.

25

2.4

The Alexander polynomials of subconfigurationsof reduced curves$B=B_{1}+\cdots+B_{r}$ be its irreducible
Let $B$ be a reduced curve ondecomposition. Let be a non-empty subset of the index $J:=\{1, . . . , r\}$and. We define Alex (B) as follows:$\mathbb{P}^{2},$$I$$B_{I}$$:= \sum_{i\in I}B_{i}$Alex$(B)=(\triangle_{B_{I}}(t))_{I\in 2^{J}\backslash \{\emptyset\}}.$

Clearly, Alex (B) is a topological invariant of. We also defineAlex (B) as the set of the reduced Alexander polynomials of subconfigurations of . We consider subsets of:$(\mathbb{P}^{2}, B)$$B$$\overline{A1ex}(B)$$\overline{A1ex}(B)_{s}:=(\triangle_{B_{I}}(t))_{\# I=s}\sim , 1\leq s\leq r.$

We say that
Alexis 1.
For our curves,$\overline{A1ex}(B)_{s}$is trivial if any reduced Alexander polynomial in$(B)_{S}$andare trivial.
Now we consider Alex
where $s\geq 3$ . Let be a non-empty subsetof $\{1, . . . , n+1\}$ . We correspond $n+1$ to the quartic . If $n+1\not\in I,$thenas
has onlysingularities. Hence we consider theAlexander polynomial ofwhere contains $n+1.$
To determine the Alexander polynomial of , we consider the adjunction ideals and the map: $O(k-3)arrow V(k)$ . The adjunction ideals
for each singular point are computed in Example 1. Now we consider themultiplicity in the formula (1) of Loeser-Vaqui\’e which is given as$\overline{A1ex}(B)_{1}$$\overline{A1ex}(B)_{2}$$I$$(B)_{s}$$Q$$\triangle_{\mathcal{B}_{I}}(t)\sim=1$$B_{I}$$A_{1}$$I$$B_{I}$$B_{I}$$\overline{\sigma}_{k}$$\ell_{k}$$\ell_{k}=\dim$coker$\overline{\sigma}_{k}=\rho(k)+\dim ker\overline{\sigma}_{k}.$where $\rho(k)=\sum_{P\in Sing(B_{I})}\dim V_{k}(P)-\dim O(k-3)$ . For fixed , theinteger
is determined by only the adjunction ideal. Hence we should
consider the dimension of
By Lemma 1, if, then $V(k)=$ O. That is $\ell_{k}=0$ . For
other cases, $V(k)=\mathbb{C}^{4n}$ andif and only if $\{g=0\}$ passes
through all
singular point $P$ . Hence we investigate the linear series. In general, the dimension ofis greater than or equal
to $N$ $:= \frac{(k-2)(k-1)}{2}-4n$ . Note that if, then$k$$\rho(k)$$ker\overline{\sigma}_{k}.$$k< \lceil\frac{3}{2}(n+2)\rceil$$g\in ker\overline{\sigma}_{k}$$A_{3}$$\mathcal{N}_{k-3}(\mathcal{P})$$\mathcal{N}_{k-3}(\mathcal{P})$$\dim \mathcal{N}_{k-3}(\mathcal{P})=N$

Lemma 2. If $k=2n+3$ , then$\dim \mathcal{N}_{2n}(\mathcal{P})=N.$$\ell_{k}=0.$

26. We take $4(n-r)-3$ distinct points
for $r=0$ , . . . , $n-1$ . Putonand. Then
where. AsHence we can take a non-zero element
we have $I(D, C_{n})\geq 4+4n-3=2\cdot 2n+1$ . Hence. By the same argument for $r=$
andwhere. But, . . . , $n-2$ , then $D$ is contained in. This is a contradiction.
because

Proof. Assume$\mathcal{Q}_{r}$$\dim \mathcal{N}_{2n}(\mathcal{P})\geq N+1$$\mathcal{Q}$$C_{n-r}\backslash \mathcal{P}_{n-r}$$\dim \mathcal{N}_{2n}(\mathcal{P}, \mathcal{Q})\geq \mathcal{N}_{2n}(\mathcal{P})-N=1.$$\#\mathcal{Q}=N$$R\not\in C_{1}$$:=\mathcal{Q}_{0}\cup\cdots\cup \mathcal{Q}_{n-1}\cup\{R\}$$D\in \mathcal{N}_{2n}(\mathcal{P}, \mathcal{Q})$$\mathcal{Q}_{0}\subset C_{n}\backslash \mathcal{P}_{n},$$D\in C_{n}\mathcal{N}_{2n-2}(\mathcal{P}’, \mathcal{Q}’)$$\mathcal{Q}’=\mathcal{Q}\backslash \mathcal{Q}_{0}$$\mathcal{P}’=\mathcal{P}\backslash \mathcal{P}_{n}$$C_{n}C_{n-1}\cdots C_{2}\mathcal{N}_{2}(\mathcal{P}_{1}, \mathcal{Q}_{n-1}, R)$$1$$\square$$R\not\in C_{1}$$\mathcal{N}_{2}(\mathcal{P}_{1}, \mathcal{Q}_{n-1}, R)=\{O\}$

Lemma 3. If $n\geq 3$ and $k=2n+2$ , then

Proof. We assume$\dim \mathcal{N}_{2n-1}(\mathcal{P})=N.$. We divide our considerations
for some
for all $(i,j)$ or$\dim \mathcal{N}_{2n-1}(\mathcal{P})\geq N+1$

into two cases$(i,j)$ . The first case is proved by the same argument of Lemma 2.
Now we consider the second case. We may assume that $(i, j)=(1,2)$. We take $4n-9$ distinct points
and we take a non-zero conic
for $r=$on
and $4(n-r)-9$ distinct points
onand
where, . . . , $n-3$ . Put
andare not collinear. Then. By the same
. Hence we can take a non-zero element. But
argument, $D$ is inare not collinear.
andbecause$\dim \mathcal{N}_{2}(\mathcal{P}_{ij})=0$$\dim \mathcal{N}_{2}(\mathcal{P}_{ij})\geq 1$$D_{2}\in \mathcal{N}_{2}(\mathcal{P}_{12})$$\mathcal{Q}_{0}$$\mathcal{Q}_{r}$$D_{2}\backslash \mathcal{P}_{12}$$C_{n-r+1}\backslash \mathcal{P}_{n-r+1}$$\mathcal{Q}=\mathcal{Q}_{0}\cup\cdots\cup \mathcal{Q}_{n-3}\cup\{R_{1}, R_{2}, R_{3}\}$$1$$\#\mathcal{Q}=N$$R_{1},$ $R_{2}$$R_{3}$$\dim \mathcal{N}_{2n-1}(\mathcal{P}, \mathcal{Q})\geq \mathcal{N}_{2n-1}(\mathcal{P})-N=$$D\in \mathcal{N}_{2n-1}(\mathcal{P}, \mathcal{Q})$$1$$D_{2}C_{n}\cdots C_{3}\mathcal{N}_{1}(R_{1}, R_{2}, R_{3})$$R_{1},$ $R_{2}$$\mathcal{N}_{1}(R_{1}, R_{2}, R_{3})=\{O\}$$R_{3}$

口Corollary 1.

$\overline{A1ex}(C)_{4}$is trivial.

参考文献[1] E. Artal Bartolo. Sur les couples des Zariski. J. Algebraic Geometry,
3:223-247, 1994.
[2] A. I. Degtyarev. Alexander polynomial of a curve of degree six.
Knot Theory Ramifications, 3:439-454, 1994.

$J.$[3] H. Esnault. Fibre de Milnor d’un c\^one sur une courbe plane singuli\‘ere. Invent. Math., $68(3):477-496$ , 1982.

27[4] M. Kawashima and M. Oka. On Alexander polynomials of certain$(2, 5)$ torus curves. J. Math. Soc. Japan, $62(1):213-238$ , 2010.
[5] A. Libgober. Alexander invariants of plane algebraic curves. In Singularities, Part 2 (Arcata, Calif., 1981), volume 40 of Proc. Sympos.
Pure Math., pages 135-143. Amer. Math. Soc., Providence, RI, 1983.
[6] F. Loeser and M. Vaqui\’e. Le polyn\^ome d’Alexander d’une courbe
plane projective. Topology, $29(2):163-173$ , 1990.

[7] M. Namba and H. Tsuchihashi. On the fundamental groups of Galoiscovering spaces of the projective plane. Geom. Dedicata, 104:97-117,2004.
[8] M. Oka. Non-degenerate complete intersection singularity. Hermann,Paris, 1997.
[9] M. Oka. Alexander polynomial of sextics. J. Knot Theory Ramifications, $12(5):619-636$ , 2003.

[10] M. Oka. A survey on Alexander polynomials of plane curves. Singularites Franco-Japonaise, Seminaire et congr\‘es, 10:209-232, 2005.