数理解析研究所講究録 1222 巻 2001 年 58-63

58
Weierstrass semigroups of a pair of points whose first
non-gaps are three
Jiryo Komeda (米田二良)(Collaboration with Seon Jeong Kim)
Kan\’agawa Institute of Technology (神奈川工科大学)

Introduction

1

Let be the additive semigroup of non-negative integers. Let be acomplete
nonsingular curve of genus $g\geq 2$ over an algebraically closed field of characterthe field of rational functions
istic 0which is called acurve in this talk, and$C$$\mathrm{N}$$k$$\mathrm{K}(C)$

on

$C$.

Definition 1.1. For apoint $P$ of$H(P):=$ {a$\in \mathrm{N}|$$C$, we set

there exists f$\in \mathrm{K}(C)$with$(f)_{\infty}=\alpha P$},

which is called the Weierstrass semigroup of the point P. An integer n is called
the first non-gap of P if it is the minimum positive integer in $H(P)$ .
Definition 1.2. For distinct points$H(P, Q):=${$(\alpha,\beta)\in \mathrm{N}\cross \mathrm{N}|$$P$

and$Q$

there exists fof , we set$C$$\in \mathrm{K}(C)$with $(f)_{\infty}=\alpha P+\beta Q$ },which is called the Weierstrass semigroup of the pair (P, Q) of points.
Fact 1.3. (Kim [2]) If C is ahyperelliptic curve, i.e., adouble covering of the
projective Hue, then the semigroup $H(P,$Q) is determined explicitly.
Fact 1.4. (Kim-Komeda [4]) If C is of genus 3, then the semigroup $H(P,$Q) isdetermined explicitly.
first non-gaps
Aim 1.5. Let $P$ and be distinct points of $C$ of genus $g\geq 4$ . If$P$$H(P,Q)$
Moreover,explicitly.
three,semigroupof and are
we determine thewe give examples of two pointed curves $(C, P, Q)$ whose semigroups are the give
$Q$

$Q$

$\mathrm{n}$59Possible Weierstrass semigroups of genus

2First, let us review some Kim’s results. Let $C$ be acurve of genus$g$point.

$\geq 5$and $P$ its

Definition 2.1. We set$G(P):=\mathrm{N}\backslash H(P)=\{l_{1}<l_{2}<\ldots<l_{g}\}$

The integer$l_{g}$.

is called the last gap at $P$ .

Let be another point of which is distinct from $P$ . For each $l\in G(P)$ , theinteger $\min\{\beta|(l, \beta)\in H(P,Q)\}$ must be equal to some element in $G(Q)$ , say,and this correspondence gives abijective map between the sets $G(P)$ and $G(Q)$ .
$C$

$Q$

$\sigma(l)$$\sigma$

Definition 2.2. We set$\Gamma(P,$Q) $:=\{(l, \sigma(l))|l\in G(P)\}$ .

Fact 2.3. (Kim [2]) The semigroup $H(P, Q)$ is completely determined by thebijective correspondence , i.e.,$\sigma$$G(P, Q)= \bigcup_{l\in G(P)}(\{(l, \beta)|\beta=0,1, \ldots, \sigma(l)-1\}$$\cup\{(\alpha, \sigma(l))|\alpha=0,1, \ldots, l-1\})$

where we set$G(P, Q)=\mathrm{N}\cross$$\mathrm{H}(\mathrm{P}, Q)$

Thus, it suffices to determine the graph $\Gamma(P, Q)$ of$H(P, Q)$ .

$\sigma$,.

for describing the semigroup

We consider the case where the first non-gaps of distinct points
three.
Theorem 2.4. $\sigma(l_{g})=1$ or $\sigma(l_{g})=2$ , $i.e.$ , $(l_{g}, 1)\in\Gamma(P, Q)$ or

Proof. Assume that$(l_{g}, \beta)\in\Gamma(P, Q)$$(\alpha_{1},1)\in\Gamma(P, Q)$

,

with

$1\leq\alpha_{1}<l_{g}$$\beta\geq 4$$P$

and$Q$

are$(l_{g}, 2)\cdot\in\Gamma(P, Q)$.

. Thenand $(\alpha_{2},2)\in\Gamma(P, Q)$ ,$1\leq\alpha_{2}<l_{g}$.

Now$i\mathit{3}\equiv i\mathrm{m}\mathrm{o}\mathrm{d} 3$

Since

for some $i=1,2$ and$(\alpha_{\dot{l}}, i)\in \mathrm{F}(\mathrm{P}, Q)$$(l_{g}, \beta)\in\Gamma(P, Q)$

and$\alpha_{\dot{1}}$.

$(\alpha_{\dot{1}},\beta)=(\alpha:,i)+(0,3k)$

and $(0, 3k)\in$ $H(P, Q)$ , we have
. This contradicts Fact 2.3.
$<l_{g}$for some .
$k$

. But
Q.E.D

$(\alpha:,\beta)\in \mathrm{H}(\mathrm{P}, Q)$

60
Prom now on we assume that $g\geq 5$ . By the theory of curves we can find some
integer with$n$$\frac{g-1}{3}\leq n\leq\frac{g}{2}$such that$S(H(P))=\{3,3n+2,3(g-n)+1\}$or$S(H(P))=\{3,3n+1,3(g-n)+2\}$ ,where $S(H(P))=\{3, s_{1}, s_{2}\}$ , which is called the standard basis for $H(P)$ , with$s:={\rm Min}\{s\in H(P)|s\equiv i\mathrm{m}\mathrm{o}\mathrm{d} 3\}$

for $i=1,2$ .

Moreover, since there exists $f\in \mathrm{K}(C)$ such that $(f)=3P-3Q$ , the standard
basis $S(H(Q))$ must be one of the above two.

Definition 2.5. The point $P$ is said to be of the -th kind if$n$$S(H(P))=\{3,3n+2,3(g-n)+1\}$or$S(H(P))=\{3,3n+1,3(g-n)+2\}$ ,Definition 2.6. The point $P$ of the -th kind is said to be of type$n$$n \neq\frac{g}{2}$

We note that If$I$(resp. $II$) if

and $3n+2\in S(H(P))$ (resp. $3n+1\in S(H(P))$ ).

$n= \frac{g}{2}$, then $S(H(P))=\{3,3n+1,3n+2\}$ .

Using the types of the points P and Q we can determine whether$(l_{g},1)\in\Gamma(P,Q)$

or $(l_{g},2)\in\Gamma(P,Q)$ .

Theorem 2.7. Let and be two distinct points of the -th kind andi) If $P$ and are of type $II$, then
$P$

$Q$

$n$

$Q$

$(l_{g}, 2)\in\Gamma(P, Q)$\"u) If$P$(resp.

$Q$) is of type$I$

(resp. $II$), then$(l_{g}, 1)\in\Gamma(P, Q)$\"ui) If$P$

and$Q$

and $(3n-2,1)\in\Gamma(P, Q)$ .

and $(3n-2,2)\in\Gamma(P, Q)$ .

are of type , then$I$$(l_{g}, 1)\in\Gamma(P, Q)$

and $(3n-1,2)\in\Gamma(P, Q)$ .

$n \neq\frac{g}{2}$.

61
Moreover, if$(\alpha_{1},1)$and$(\alpha_{2},2.)$

belong to$\Gamma(P,$Q), then$\Gamma(P, Q)=\{(\alpha_{1}-3k, 1+3k)|0\leq k<\frac{\alpha_{1}}{3}\}\cup\{(\alpha_{2}-3k, 2+3k)|0\leq k<\frac{\alpha_{2}}{3}\}$

In some case with n$= \frac{g}{3}$.

we have no candidate of the semigroup $H(P,$Q).

Proposition 2.8. Let $P$ and be points of the -th kind. If they are of type $II$,
then $P=Q$ . In this case, $H(P)$ is generated by 3and $g+1$ withmod 3.
$Q$$\frac{g}{3}$$g\equiv 0$

There are two possibilities in the case nProposition 2.9. Let $P$ and$Q$$= \frac{g}{2}$.

be distinct points of the$\frac{g}{2}$-th kind, $i.e.$ ,$S(H(P))=S(H(Q))=\{3,3n+1,3n+2\}$ .

Then$(3n-1,1)\in\Gamma(P, Q)$ or $(3n-1,2)\in \mathrm{r}(\mathrm{P}, Q)$

If $(3n-1,1)\in\Gamma(P, Q)$ (resp. $(3n-1,2)\in\Gamma(P,$$Q)$.

), then$\Gamma(P, Q)=\{(\alpha, 3n-\alpha)|\alpha\in G(P)\}$(resp. $\{(3k-2, (3n-1)-(3k-2))|k=1, \ldots, n\}$$\cup\{(3k-1, (3n+1)-(3k-1))|k=1, \ldots, n\})$.

3The existence of two pointed curves
In the previous section we determined the possible Weierstrass semigroups $H$ ofapair of points on acurve of genus
whose first non-gaps are three. In thissection for each such asemigroup $H$ we give two pointed curves $(C, P, Q)$ suchthat.
$\geq 5$$\mathrm{H}(\mathrm{P}, Q)=H$

Let

$C$

equationbe the curve whose function field $\mathrm{K}(C)=k(x, y)$ is defined by the$y^{3}=(x-c_{1})\cdots(x-\mathrm{q}_{1}.)(_{X-\mathrm{G}_{1}+1}.)^{2}\cdots(x-c_{t_{1}+\dot{1}2})^{2}$,,
where ,are distinct elements of andis not divisible by 3. We$C$note that the genus of the curve is $i_{1}+i_{2}-1$ by Riemann-Hurwitz formula.
Let :be the morphism corresponding to the inclusion,i.e., $\pi(P)=(1:x(P))$ , where
denotes the projective line. We set$c_{1}$$\ldots$$k$$c_{i_{1}+:_{2}}$$i_{1}+2i_{2}$

$g$

$\pi$

$Carrow \mathrm{P}^{1}$$k(x)\subset \mathrm{K}(C)$$\mathrm{P}^{1}$$\{P_{\infty}\}=\pi^{-1}$(0 : 1) and $\{P_{f}\}=\pi^{-1}(1 : c_{s})$ for $s=1$ ,$\ldots$,$i_{1}+i_{2}$.

62Then we have[3]). If$S(H(P\ovalbox{\tt\small REJECT}.\ovalbox{\tt\small REJECT}))\ovalbox{\tt\small REJECT}$

1mod 3,$i_{\mathit{1}}+2\mathrm{i}_{2}\ovalbox{\tt\small REJECT}$$S(H(P_{l}))=\{$

If$\ovalbox{\tt\small REJECT}_{1}+2\ovalbox{\tt\small REJECT}_{2},2\mathrm{i}_{1}+\mathrm{i}_{2}\}$(For example, see Kim-Komeda

if
if $i_{1}+1\leq s\leq i_{1}+i_{2}$$1\leq s\leq i_{1}$$\{3, i_{1}+2i_{2}+1,2\mathrm{t}\mathrm{i}+i_{2}-1\}$$S(H(P_{\infty}))=\{3, i_{1}+2i_{2},2i_{1}+i_{2}\}$$i_{1}+2i_{2}\equiv 2\mathrm{m}\mathrm{o}\mathrm{d} 3$$S(H(P_{s}))=\{${3,,$S(H(P_{\infty}))=\{3, i_{1}+2\mathrm{i}22i_{1}+i_{2}\}$$\{3, i_{1}+2i_{2}-1,2i_{1}+i_{2}+1\}$

Prom now on we assume that g
examples :$\geq 5$if
if $i_{1}+1\leq s\leq i_{1}+i_{2}$$1\leq s\leq i_{1}$. By the above formula we get the following. $\mathrm{I}\mathrm{f}:_{1}=2g-3n+1>n+1$ and $\mathrm{i}_{2}=3n-g>0$ ,Example 3.1. i) Letand $P_{l}(:_{1}+1\leq s\leq i_{1}+:_{2})$ are of the -th kind of type .
then the points$\frac{g}{3}<n<\frac{g}{2}$$n$$P_{\infty}$\"u) Let$\mathrm{I}\mathrm{I}$. If $i_{1}=2g-3n+1>n+3$ and $i_{2}=3n-g\geq 0$ , then theare of theis of the -th kind of type 1I and the points P.

$\frac{g}{3}\leq n\leq\frac{g-1}{2}$points-th kind of type I.
$P_{\infty}$$(1 \leq s\leq:_{1})$

$n$

$n$

$..\ovalbox{\tt\small REJECT}.\mapsto$$\frac{g-1}{3}-\cdot\leq ae\leq\frac{g-1}{2}arrow B\dot{4}--\ovalbox{\tt\small REJECT}-\mathit{3}n\geq ae$

the points$P_{\infty}$. A then$+\mathit{2}\mathrm{A}\dot{4}_{l}=3n-g+1\geq$

and P. $(1 \leq s\leq i_{1})$ are of the -th kind of type I.
$n$In the case of Proposition 2.9 we get the following examples:
Example 3.2. Let $n\geq 3$ . If $i_{1}=n+1$ and $i_{2}=n$ , then $g=2n$ and the points
and P. are of the -th kind. Then $S(H(P_{\infty}))=S(H(P_{l}))=\{3,3n+1,3n+2\}$ .
Moreover,$P_{\infty}$$\frac{g}{2}$$\Gamma(P_{\infty}, P_{l})\ni(3n-1,1)$

and$\Gamma(P_{\infty},P.)\ni(3n-1,2)$

for 15$s\leq i_{1}$for $:_{1}+1\leq s\leq i_{1}+:_{2}$ .

4Weierstrass semigroups of genus 4
In this section we treat the curves C of genus 4with point P whose first non-gap
is 3. Then we have $S(H(P))=\{3,$ 5,10} or {3, 7, 8}.
Remark 4.1. Let $S(H(P))=\{3,5,10\}$ . If is another point of $C$ whose firstsuch that
non-gap is three, then $S(H(Q))=\{3,5,10\}$ and there exists$(f)=3P-3Q$ .
$Q$$f\in \mathrm{K}(C)$63

anProposition 4.2. Let P
C be two distinct points such that$\{(1,$ 7), (2,2), (4,4), (7,. For example,{3,5,10}. Thensuch pointed curves are given by$S(H(Q))\ovalbox{\tt\small REJECT}$$S(H(P))\ovalbox{\tt\small REJECT}$$1)\}$$I^{\ovalbox{\tt\small REJECT}}(P^{1}Q)\ovalbox{\tt\small REJECT}$$y^{3}=(x-c_{1})\cdots$ $(x-c_{5})$, $P$.

$=P_{\infty}$

and$Q=P_{\iota}$

for $s=1$ ,$\ldots$,5

where we use the notations in Section 3.
Proposition 4.3. Let $S(H(P))=\{3,7,8\}$ and another point of $C$ whose firstnon-gap is three. Suppose that there exists
such that $(f)=3P-3Q$ .
$Q)$i)
or.
$(5,1)\in\Gamma(P,Q)$$\Gamma(P,Q)=\{(5,1), (4,2), (2,4), (1,5)\}$ . For example,,thenIfsuch pointed curves are given by$Q$$f\in \mathrm{K}(C)$$(5, 1)\in \mathrm{T}(\mathrm{P},\mathrm{Q})$$(5, 2)\in \mathrm{T}\{\mathrm{P},$$\mathrm{i}\mathrm{i})$$y^{3}=(x-c_{1})(x-c_{2})(x-c_{3})(x-c_{4})^{2}(x-c_{5})^{2}$,P$=P_{\infty}$

and Q$=P_{s}$for s $=1,2,$ 3.

iii) If (5,$2)\in\Gamma(P,$ Q), then $\Gamma(P, Q)=\{(5,$ 2), (4,1), (1,4), (2, $5)\}$ . Such pointed
curves are given by the same equations as above, P
and Q $=P_{l}$ for s $=4,$ 5.
$=P_{\infty}$

Proposition 4.4. Let $S(H(P))=\{3,7,8\}$ and another point of whose first
non-gap is three. Suppose that there is no $f\in \mathrm{K}(C)$ such that $(f)=3P-3Q$ .
Then $(5, 1)\in\Gamma(P, Q)$ and $\Gamma(P, Q)=\{(5,1), (4,4), (2,2), (1,5)\}$ . Such curves
are also given by the equations
$C$

$Q$

$C$

$y^{3}=(x-c_{1})(x-c_{2})(x-c_{3})(x-c_{4})^{2}(x-c_{5})^{2}$

Using the result of Kato [1] we get our desired points $P$ and$Q$.

.

References[1] T. Kato, On Weierstrass points whose first non-gaps are three. J. Reine
Angew. Math. 316 (1980), 99-109.

[2] S.J. Kim, On the index of the Weierstrass semigroup of a pair of.points oncurve. Arch. Math. 62 (1994), 73-82.

$a$[3] S.J. Kim and J. Komeda, Numerical semigroups which cannot be realized as
semigroups of Galois Weierstrass points. To appear in Arch. Math.

[4] S.J. Kim and J. Komeda, The Weierstrass semigroup of a pair and moduli in. To appear in Bol. Soc. Bras. Mat$\mathcal{M}_{3}$