数理解析研究所講究録 1233 巻 2001 年 82-89

82

Rational unicuspidal plane curves with$\overline{\kappa}=1$

埼玉大学理学部数学科 戸野恵太 (Keita Tono)
Department of Mathematics, Faculty of Science, Saitama University

1

Introduction. Asingular point of is said to be acusp
Let be acurve on
if it is alocally irreducible singular point. We say that is cuspidal if has
only cusps as its singular points. For acusp of , we denote the multiplicity, or simply. We use the abbreviation
for asequence of $(C, P)$ bymeans anconsisting of consecutive ra’s. For example,
subsequence ofoflogarithmic
bydimension
singularity. We denote
theKodaira$C’$. Let
the complement
be the strict transform of arational unicuspidal
plane curve via the minimal embedded resolution of the cusp of . We define$n(C):=-(C’)^{2}$ . By [Y],if and only if $n(C)<2$ . By [Tsl,. Thus2],
Proposition
there exist no rational cuspidal plane curves with
if and only if $n(C)\geq 2$ .
$C$

$C$

$\mathrm{P}^{2}=\mathrm{P}^{2}(\mathrm{C})$

$C$

$C$

$P$

$\overline{m}_{P}(C)$$C$$m_{k}$$\overline{m}_{P}$$k$$\overline{m}_{P}$$A_{2k}$$(2_{k})$$\overline{\kappa}(\mathrm{P}^{2}\backslash C)$$\mathrm{P}^{2}\backslash C$

$C$

$C$

$\overline{\kappa}(\mathrm{P}^{2}\backslash C)=-\infty$$\overline{\kappa}=0$$\overline{\kappa}(\mathrm{P}^{2} \langle C)\geq 1$, thenTheorem 1. If $C$ is a rational unicuspidal plane curve withsatisfying the following four conditions.
there exists a unique pencil Aon$\overline{\kappa}(\mathrm{P}^{2}\backslash C)=1$$\mathrm{P}^{2}$(i) The cusp $P$ of $C$ is a unique base point of A.
(ii) The pencil Ahas a unique reducible member $C+n(C)B$ . Here $B$ is a line
or an irreducible conic such that $(CB)_{P}=(\deg B)(\deg C)-1$ .

(iii) The pencil Ahas exactly two multiple members,,
are integers with$\mu_{A},\mu c$$\geq 2$

$A\backslash \{P\}\cong \mathrm{C}^{*}$$\mu_{A}A$,$\mu_{G}G$, where$G\backslash \{P\}\cong \mathrm{C}$.

$\mu_{A}$,$\mu c$(iv) The complement of $\{P\}$ to every member other than $\mu_{A}A_{f}\mu cG$ and $C+$
$n(C)B$ is isomorphic to.
$\mathrm{C}^{*}$. We sayLet
be arational unicuspidal plane curve withthat is of type (resp. type $II$ ) if the curve $B$ in Theorem 1(ii) is aline (resp.
an irreducible conic
$C$

$C$

$\overline{\kappa}(\mathrm{P}^{2}\backslash C)=1$

$I$

83
Theorem 2. Let $C$ be a rational unicuspidal plane curve with$\mathrm{D}utn:=n(C)$ . Let $P$ be the cusp
of $C$ .
(i) Type I. There eists an integer$\overline{\kappa}(\mathrm{P}^{2}\backslash C)=1$.

with $s\geq 2$ such that $\deg C=(n+1)^{2}(s-$$1)+1,\overline{m}_{P}=(n(n+1)(s-1), ((n+1)(s-1))_{2n+1}$ , $(n+1)_{2(s-1)})$ , $\mu_{A}=n+1$
,
with
such
and $\mu_{G}=(n+1)(s-1)+1$ . There exist ,that is projectively equivalent to the curve:$s$$a_{2}$$\ldots$$a_{s}\neq 0$$a_{s}\in \mathrm{C}$$C$,$((f^{s-1}y+ \sum_{i=2}^{s}a_{i}f^{s-i}x^{(n+1)i-n})^{\mu A}-f^{\mu G})/x^{n}=0$where $f=x^{n}z+y^{n+1}$ . $Co$ nversely, for arbitrary integers , with $n\geq 2$ ,, the above equation defines a rational
,
with
and ,
. The multiplicity sequence of
unicuspidal plane curve with.
the cusp is equal to$n$$s\geq 2$$a_{2}$$\ldots$$s$$a_{s}\neq 0$$a_{s}\in \mathrm{C}$$\overline{\kappa}(\mathrm{P}^{2}\backslash C)=1$$\overline{m}_{P}$(ii) Type $II$ and $\deg C=((4n+1)^{2}+1)/2$ . We have $\overline{m}_{P}=((n(4n+1))_{4},$ $(4n+$, $3n+1$ ,, $\mu_{A}=4n+1$ and $\mu c=2n+1$ . The curve $C$ is projectively
equivalent to the curve:$1)_{2n}$$n_{3})$,$((g^{n}y+x^{2n+1})^{\mu A}-(g^{2n}z+2x^{2n}yg^{n}+x^{4n+1})^{\mu G})/g^{n}=0$where $g=xz-y^{2}$ . $Co$ nversely, for an arbitrary integer with $n\geq 2$ , theabove equation defines a rational unicuspidal plane curve with.
1. The multiplicity sequence of the cusp is equal to$n$$\kappa-(\mathrm{P}^{2}\backslash C)=$$\overline{m}_{P}$(iii) Type $II$ and $\deg C\neq((4n+1)^{2}+1)/2$ . There exists a positive integer suchthat, by setting $m:=4n+1$ and $t:=4s-1$ , we have $\deg C=(m^{2}t+1)/2$ ,$s$, $(m)_{3},3n+1$ ,, $(sm)_{3}$ , $(s-1)m$ , $m_{2(s-1)}$ , $3n+1$ ,$((3mn)_{4}, (3m)_{2n}$$\overline{m}_{P}=\{$$((tmn)_{4}, (\mathrm{t}\mathrm{m})2\mathrm{n}$$n_{3})$$=2(\mathrm{m}5-n)$ .
,There eist ,andsuch that $C$ is projectively equivalent to the cume:$\mu_{A}=m$$\mu c$$a_{1}$$\ldots$$n_{3})$$a_{s}\in \mathrm{C}$if $s=1$ ,if $s>1$ ,with $a_{s}\neq 0$,$((h^{2s-1}(g^{n}y+x^{2n+1})+ \sum_{i=1}^{s}a_{i}h^{2(s-i)}g^{mi-n})^{\mu_{A}}-h^{\mu G})/g^{n}=0$where $h=g^{2n}z+2x^{2n}yg^{n}+x^{m}$ . Conversely, for an arbitrary integer, the above,
withwith $n\geq 2$ , a positive integer and ,.
equation defines a rational unicuspidal plane curve ettith.
The multiplicity sequence of the cusp is equal to
$n$

$s$

$a_{1}$$\ldots$$a_{s}\in \mathrm{C}$$a_{s}\neq 0$$\overline{\kappa}(\mathrm{P}^{2}\backslash C)=1$$\overline{m}_{P}$84
Aplane curve is said to be of type $(d, \nu)$ if the degree of is and the
maximal multiplicity of is . If is arational cuspidal curve of type $(d, \nu)$ ,then the inequality $d<3\nu$ holds true ([MS]). See also [0].
$C$

$C$

$C$

$\nu$

$d$

$C$

Corollary 1. Let $C$ be a rational unicuspidal plane curve of type $(d, \nu)$ with$\overline{\kappa}(\mathrm{P}^{2}\backslash C)=1$.

(i) Type I. We have $1<d/\nu\leq 5/3$ . The equality holds if and only if $C$ isprojectively equivalent to a curve in Theorem 2(i) with $n=s=2$ .
(ii). We have $2<d/\nu\leq 41/18$ . The equality holds if and only if$Rpe$ $II$

$C$

is

projectively equivalent to the curve in Theorem 2(ii) with $n=2$ .

Corollary 2. Let be a rational unicuspidal plane curve. Then
if and only if the multiplicity sequence of the cusp is one of those in Thiorem 2.
$C$$\overline{\kappa}(\mathrm{P}^{2\backslash }{}_{\backslash }C)=1$

Corollary 3. Let $C$ be a rational unicuspidal plane curve. Then
if and only if $n(C)\geq 2$ and the multiplicity sequence of the cusp is none of thosein Theorem 2.

$\overline{\kappa}(\mathrm{P}^{2}\backslash C)=2$

Remark 4. In [Tsl], Tsunoda claimed to have obtained the defining equationsof rational unicuspidal plane curves with. Comparing the degrees of his
with ours, it seems that the equations he obtained are those of type , $s=2$ in
Theorem 2(i).
$\overline{\kappa}=1$$\mathrm{I}$2Proof of Theorem 1
Let $C$ be arational unicuspidal curve on
with. Let :
be the composite of the shortest sequence of blowing-ups over $P$ such that the
reduced total transform $D$ of $C$ is anormal crossing divisor. Let $C’$ be thestrict transform of $C$ . Put $D’:=D-C’$ . We remark that every irreducible
components of $D’$ is asmooth rational curve, whose self-intersection number isless thm -1. Let
denote the exceptional curve of the last blowing-up of .
$D$
The dual graph of has the following shape.
$\mathrm{P}^{2}$$\overline{\kappa}(\mathrm{P}^{2}\backslash C)=1$$\sigma$$D_{0}$$Varrow \mathrm{P}^{2}$$\sigma$$C’$85
As aconvention,
contains the exceptional curve of the first blowing-up. Let
in the above figure. In the course ofdenote the leftmost component of$D’$
by , $A_{g}+D_{0}+B_{g}$ is contracted a(-1)-curve $E$ andthe contraction of, where$A_{g-1}+E+B_{g-1}$ to a(-1)-curve, and so on. Write$E$
contracts $A_{g-1}+E+B_{g-1}$
contracts $A_{g}+D_{0}+B_{g}$ to a(-1)-curve ,iscalledsprouting if it is donea(-1)-curve,
Ablowing-upoftoand so on.
at asmooth point of the exceptional curve of the preceding blowing-ups. As
denote theis not sprouting. Let
aconvention, the first blowing-up of
number of sprouting blowing-ups in .
of $(V, D)$ . We
Following [FZ], we consider astrictly minimal model
successively contract (-1)-curves $E$ such that $E\subset D$ and $(D-E)E\leq 2$ , orand $DE\leq 1$ . After afinite number of contractions, we have no (-1)denote the composite of the contractions.
curves to contract. Let :. It is clear that is adivisor with only
, writeFor adivisor. By [Ka, Theorem 2.3] and the fact
simple normal crossings andthat
is affine, we have the following:whose general fiber $F$ is
Lemma 5. There exists a fib ration :$\tilde{D}F=2$.
and-bundle :is obtained from afibration over
It is known that a. We have the. Put
by successive blowing-ups :following commutative diagram.
$A_{1}$$A_{i1}$$A_{i}$$\sigma$$\sigma=\sigma_{1}\circ\cdots\circ\sigma_{g}$$\sigma_{g}$$\sigma_{g-1}$$\sigma_{i}$$s_{i}$$\sigma_{1}$$\sigma_{i}$$(\tilde{V},\tilde{D})$$E\not\subset D$$\pi$$Varrow\tilde{V}$$\tilde{D}$$\tilde{\Delta}=\pi_{*}(\Delta)$$\Delta\subset V$$\overline{\kappa}(\tilde{V}\backslash \tilde{D})=1$$\tilde{V}\backslash \tilde{D}$$\tilde{V}arrow \mathrm{P}^{1}$$\mathrm{P}^{1}$$\tilde{p}$$\mathrm{P}^{1}$$\mathrm{P}^{1}$$\mathrm{P}^{1}$$p\wedge$$\Sigmaarrow \mathrm{P}^{1}$$\tilde{V}arrow\Sigma$$\tilde{\pi}$$p=\tilde{p}\circ\pi$isFollowing [FZ], we use the following terminology: The triple
is called horizontal if the image of it
called a-triple. Acomponent ofbe the sum of the horizontal components ofunder is 1-dimensional. Let
is irreducible;
is called of trnisted type if. The C’-triple
otherwise it is called of untwisted type. By [Kiz, Theorem 3], our C’-triple isof untwisted type. (See also [M2, Theorem 4.7.1, Lemma 4.10.3].) Thus
. Suppose has asingular fiber.
consists of two irreducible components Hi,
The dual graph of the sum of the singular fiber and the horizontal components, Lemma 5.5]).
has the following shape (cf.
$(\tilde{V},\tilde{D},\tilde{p})$$\tilde{D}$$\tilde{H}$$\tilde{p}$$(\tilde{V},\tilde{D},\tilde{p})$$\tilde{H}$$(\tilde{V},\tilde{D},\tilde{p})$$\tilde{H}$$H_{2}$$[\mathrm{F}\mathrm{Z}$$\tilde{p}$86Here E is $(-1)$ -curve, which is not contained in D. The curves F.,
areconnected components of D $-(H.+H_{2})$ . The fiber is contracted by to afiberof . By using [FZ, Theorem 5.8 and 5.11], we have the following lemma. (Thecase (B2) in [FZ, Theorem 5.8] does not occur.)$F_{2}$$\ovalbox{\tt\small REJECT}$$\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}$$7\ovalbox{\tt\small REJECT}$

Lemma 6. The C’-triple$(\tilde{V},\tilde{D},\tilde{p})$has the following properties.

(i) The fibration has exactly one smooth fiber (? contained insingular fibers, whereis the (-1)-curve in(resp.
).

$\tilde{D}$$\tilde{p}$$\tilde{F}_{A}=\tilde{A}_{1}+\tilde{E}_{A}+\tilde{B}_{g},\tilde{F}_{B}=\tilde{B}_{1}+\tilde{E}_{B}+\tilde{C}’$$\tilde{E}_{B})$$\tilde{F}_{A}$(ii) The curves$\tilde{D}_{0}$and two$\tilde{E}_{A}$(resp.

$\tilde{F}_{B}$, A21 are the horizontal components.

We can verify that$\pi$

has the following properties.

Lemma 7. The following assertions hold true.
and every subsequent blowing-down
first contracts a(-1) curveisthecontraction
component
of
of a
of .
(ii) The curve
is a component of. Every blowing-up of is performed at a point on the total transform of .
(i)

$E_{G}\not\subset\tilde{D}$$\pi$$\tilde{D}$$\pi$$E_{G}$$\mathrm{v}\mathrm{r}^{-1}(\tilde{G})$$\pi$$\tilde{G}$,Letdenote the strict transforms ofin , respectively.
Write $A=\sigma(E_{A})$ , $B=\sigma(E_{B})$ and $G=\sigma(E_{G})$ . Let,
anddenote$p^{*}(p(E_{A}))$$p^{*}(p(E_{B}))$$p^{*}(p(E_{G}))$
the coefficients of
,
,
and
in
,
and
respectively. We have $\mu_{B}=n(C)$ by [ , Proposition 4.8]. Since does not, it follows that is smooth and rational with self-intersection number
change. Thus is aline $(s_{1}=1)$ or an irreducible conic $(s_{1}=4)$ . Now it is clear thatthe pencil spanned byandsatisfies the whole condition in Theorem 1.
The uniqueness of the pencil follows from [ , Theorem 3].
$E_{A}$$E_{B}$$\mu_{A}$$E_{A}$$E_{B}$$s_{1}$$\mu_{B}$$\mu c$$E_{G}$$\mathrm{F}$$\tilde{F}_{B}$$V$$\tilde{E}_{A},\tilde{E}_{B}$

$\pi$

$B$

$B$

$\mu_{A}A$$\mu_{G}G$$\mathrm{I}$3Proof of Theorem 2
In order to prove Theorem 2, we determine the weighted dual graph of $D+$
$E_{A}+E_{B}+E_{G}$ . By using the properties of ,and , we obtain the diagram
1,$n=n(C)$in Figure wheremdmeans a(-1)-curve(resp. (-2)curve). In Theorem 2, we set $s=s_{3}$ . The curves in Theorem 2(ii) correspondto those of type II with $g=2$ and the curves in (iii) to those of type II with$\sigma$$*(\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{p}.

\bullet)$$\pi$$\tilde{p}$

87

-n

Type ITyp e Ij g $=2$
$D_{0}$

Type$\mathrm{I}\mathrm{I}$, $g=3$ , $s_{3}=1$

Type$\mathrm{I}\mathrm{I}$-n-1, $g=3$ , $s_{3}>1$

Figure 1: The weighted dual graph of $D+E_{A}+E_{B}+E_{G}$88Remark 8. Our fibration belongs to the class (D) in the sense of [Kiz]. The
last two graphs in [Kiz, Figure 54] coincide with those of type Iand type II with
$g=3$ , $s_{3}>1$ .
The multiplicity sequence of the cusp can be calculated from the weighted
dual graph of $D’$ (cf.
, p.516, Theorem 12]). The degree of $C$ is calculatedfrom. We calculate ,by using [ , Proposition 4.8]. The proof of theassertion for the defining equation of $C$ is based on the following fact. Let,
and $fc$ be the defining polynomials of , $B$ and , respectively. Then, since’$C+\mu_{B}B$ is amember of the pencil Ain Theorem 1, there exists
suchthat $C$ is defined by the equation.
Acknowledgment. The author would like to express his thanks to Professor Fumio Sakai for his valuable advice, guidance and encouragement.
$p$$[\mathrm{B}\mathrm{K}$$\overline{m}_{P}$$\mu_{A}$$\mathrm{F}$$\mu_{G}$$f_{A}$$A$$f_{B}$$G$$t\in \mathrm{C}$$(f_{A}^{\mu A}+tf_{G}^{\mu G})/f_{B}^{\mu B}=0$

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