100

数理解析研究所講究録
第2008巻 2016年 100-106

Numerical semigroups attained by double covers of plane
1curves of degree six

神奈川工科大学・基礎・教養教育センター

米田二良

Jiryo Komeda
Center for Basic Education and Integrated Learning
Kanagawa Institute of Technology

Abstract
We investigate Weierstrass semigroups of ramification points on double covers of plane
curvesof degree 6. We determine all the Weierstrass semigroups, especially when thegenus of the covering curve is greater than 29 and the ramification point is on a totalflex.

Notations and terminologies

1Let \mathrm{N}_{0} be the additive monoid of non‐negative integers. A submonoid H of \mathb {N}_{0} is calledanumerical semigroup if the complement \mathrm{N}_{0}\backslash H is finite. The cardinality of \mathrm{N}_{0}\backslash H iscalled the genus of H denoted by g(H)In this paper H always stands for a numericalsemigroup. A curve means a complete non‐singular irreducible algebraic curve over analgebraically closed field k of characteritic O. For a pointed curve (C, P) we set,.

H(P)= {n\in \mathrm{N}_{0}|\exists f\in k(C) such that (f)_{\infty}=nP},
where k(C) is the field of rational functions on C.

Then H(P) is a numerical semigroup,Here g(H(P)) is equal to the genus g(C) of the curve C.

Let C be a plane curve of degree 6 and P_{1} its total inflection point, i.e., T_{P_{1}}.C=6P_{1},where T_{P_{1}} is the tangent line at P_{1} on C Then we have H(P_{1})=\langle 5 6), where for positive.

integers a_{1} ,.

.

.

,a_{s} .we denote by\{a_{1},.

.

,

.

the monoid generated by a_{1} ,, a_{s}\rangle

If P_{2} is a point of C with T_{P_{2}}.C=5P_{2}+Q,

.

.

.

,a_{s}.

Then we have H(P_{2})=Q\neq P_{2}\{5 9, 13, 17, 21\rangle We set d_{2}(H)=\{h'\in \mathrm{N}_{0}|2h'\in H\} which is a numerical semigroup.
Let $\pi$ : \tilde{C}\rightarrow C be a double covering of curves with a ramification point \tilde{P} Then weobtain d_{2}(H(\tilde{P}))=H( $\pi$(\tilde{P})) For example, if C=\mathbb{P}^{1} then we have H(\tilde{P})=\langle 2, 2g+1 )
,.

.

,.

.

with,g=g(\tilde{C}) and d_{2}(H(\tilde{P}))=\mathrm{N}_{0} We pose the following problem:

DCPHurwitz Problem.

.

Let C be a plane curve of degree d anddouble covering with a ramification point \tilde{P}.

Then determine

H(\tilde{P})$\pi$:\tilde{C}\rightarrow C\mathrm{a}.

This paper is an extended abstract and the details will appear elsewhere.
This work is a collaboration with Seon Jeong Kim.
The author is partially supported by Grant‐in‐Aid for Scientific Research (15\mathrm{K}04830)
for the Promotion Science.

,Japan Society

101i) In the case of d=1 2 the result is classical (Hyperelliptic case).
ii) In the case d=3 the result is also classical (Bielliptic case).
iii) In the case d=4 DCPHurwitz Problem is solved ([6], [1], [2], [3])iv) Let d=5 If g(\tilde{C})\geqq 15 and \mathrm{o}\mathrm{r}\mathrm{d}_{ $\pi$(P^{-})}(T_{ $\pi$(P^{-})}.C)=5 or 4, then DCPHurwitz Problem issolved ([4]) where for a line L in \mathb {P}^{2} and P\in C we denote by \mathrm{o}\mathrm{r}\mathrm{d}_{P}(L.C) the multiplicity
Fact.

,.

of the intersection of L and C at P.

We consider DCPHurwitz Problem in the case d=6.

The following are our results:

Theorem. Let C be a plane curve of degree 6 and $\pi$ : \tilde{C}\rightarrow C a double covering withramification point P. Assume that g(\tilde{C})\geqq 30.
i) If \mathrm{o}\mathrm{r}\mathrm{d}_{ $\pi$(P^{-})}(T_{ $\pi$(P^{-})}.C)=6 we determine H(\tilde{P}) If we explain it in detail, any numericalsemigroup H of genus g>29 with d_{2}(H)=\{5, 6\rangle is given by the Weierstrass semigroupof a ramification point on a double cover of a plane curve of degree 6.
ii) If \mathrm{o}\mathrm{r}\mathrm{d}_{ $\pi$(P^{-})}(T_{ $\pi$(P^{-})}.C)=5 there are eighty two candidates of the numerical semigroupsa.

,,H(\tilde{P})We determine whether seventy four candidates are attained by ramificationondouble covers of plane curves of degree 6 or not. We do not know whether thepoints.

remaining eight candidates are attained or not.

The candidates of Weierstrass semigroups of dou‐

2

bleof planecoversof degree 6 over total

curvesinflection points
Let H be a numerical semigroup with d_{2}(H)=\langle 5, 6\rangle.

We consider the following dia‐gram:(\rightarrow+2) (\mathrm{n}+2) (\mathrm{n}+4) (\mathrm{n}+6) (\mathrm{n}+8)\times(n)\times\times\times\downarow\times\times\times+10(\mathrm{n}+12)\searrow+12 (\mathrm{n}+24)\times\times\times(\mathrm{n}+36)(\mathrm{n}+48)

which is called the triangle \triangle associated to the semigroup 2\langle 5, 6\rangle+n\mathrm{N}_{0}' '\times\geqq 21means that.

The symbol

it is a non‐element of the numerical semigroup larger than n if n is odd

n+38-10i-2j with 0\leqq i\leqq j\leqq 3,
belongsemigroup 2\{5, 6\rangle+n\mathrm{N}_{0} We associates thecoordinate (i,j) to the odd number n+38-10i-2j The coordinate (k, l) generates
the parallelogram.

The triangle \triangle consists of the odd integers

which do notto the numerical.

.

P(k, l)=\{(i,j)|0\leqq i\leqq k, 0\leqq j\leqq l\}.
For example, P(3,2) is the following, where

denotes n+38-10*3-2*2.

102(\rightarrow+2) (\mathrm{n}+2) (\mathrm{n}+4) (\mathrm{n}+6) (\mathrm{n}+8)\times \times \downarrow\times(n)0(\mathrm{n}+12)\searrow+12 (\mathrm{n}+24)0\times00100(\mathrm{n}+36)(\mathrm{n}+48)

We can associate to the numerical semigroup H the polygon

P(H)=\{(i,j)\in $\Delta$|n+38-10i-2j\in H\}t(H) be the minimum number of generators for H which are odd and distinct
Then we have 0\leqq t(H)\leqq 4 If t(H)=0 then P(H)=\emptyset If t(H)=4,then P(H)=\triangle If t(H)=1 there are 10 kinds of P(H) i.e., H For example, if
H=2\langle 5, 6\rangle+\langle n, n+4\rangle Then we have P(H)=P(2,3) Ler r(H) be the number of theodd elements of H which are larger than n and less than n+40 Then 0\leqq r(H)\leqq 10.
If r(H)=0 then P(H)=\emptyset If r(H)=10 then P(H)= $\Delta$ If r(H)=9 there are 4
kinds of P(H) i.e., H For example, if H=2\langle 5, 6\rangle+\{n, n+2, n+4, n+6\rangle then P(H)Let

from n.

.

.

.

,,.

,.

.

.

,.

,,.

,.

,

is the following:(\rightarrow+2) (\mathrm{n}+2) (\mathrm{n}+4) (\mathrm{n}+6) (\mathrm{n}+8)⑦(n)0(\mathrm{n}+12)\searrow+12 (\mathrm{n}+24)④\times\downarow

00+10000(\mathrm{n}+36)(\mathrm{n}+48)

The following table shows the number of H\mathrm{s} with d_{2}(H)=\langle 5, 6\rangle having fixed

and r(H)3t(H).

The candidates of Weierstrass semigroups of dou‐

ble covers of plane curves of degree 6 over almost
total inflection points
Let H be a numerical semigroup with d_{2}(H)=\{5 , 9, 13, 17, 21\ranglelowing diagram:.

We consider the fol‐

103(\rightarrow+2)(n)(\mathrm{n}+2)(\mathrm{n}+4)(\mathrm{n}+6)(\mathrm{n}+8)\times\times\times\times\times\times\times\times\times(\mathrm{n}+26)\times(\mathrm{n}+42)(\mathrm{n}+34)\downarow+10(\mathrm{n}+18)\swarrow+8(\downarrow+10)

which is called the triangle \triangle_{a} associated to the semigroup 2\langle 5 , 9, 13, 17, 21\rangle+n\mathbb{N}_{0}.

Thesymbol
The triangle \triangle_{a} consists of the odd integers n+32-10i+2j with 0\leqq j\leqq i\leqq 3 which
do not belong to the numerical semigroup 2\langle 5 9, 13, 17, 21\rangle+n\mathrm{N}_{0} We associates thecoordinate (i,j) to the odd number n+32-10i+2j The coordinate (k, l) generates
the parallelogramis a non‐element of the numerical semigroup larger than n if n is odd \geqq 21.

\times,.

,.

P_{a}(k, l)=\{(i,j)|0\leqq i\leqq k-2(l-j), 0\leqq j\leqq l\}.
For example, P_{a}(3,2) is the following, where(\rightarrow+2)(n)denotes n+32-10*3+2*2.

(\mathrm{n}+2)(\mathrm{n}+4)(\mathrm{n}+6)(\mathrm{n}+8)\times\times④\times1\times\times\mathrm{o}-+10\times0-0(\mathrm{n}+42)(\mathrm{n}+34)(\mathrm{n}+26)(\mathrm{n}+18)\swarrow+8(\downarrow+10)

We can associate to the numerical semigroup H the polygon

P_{a}(H)=\{(i,j)\in\triangle|n+32-10i+2j\in H\}.
Let t(H) be the minimum number of generators for H which are odd and distinct fromn.

Let r(H) be the number of the odd elements of H which are larger than n and lessthan n+34.

For example, if H=2\langle 5 9, 13, 17, 21\rangle+\langle n, n+4, n+8, n+16 ), then P_{a}(H),is the following:(\rightarrow+2)(\mathrm{n}+2)(\mathrm{n}+4)\times(n)(\mathrm{n}+8)\downarow\times\times0000(\mathrm{n}+42)(\mathrm{n}+6)(\mathrm{n}+34)④-(\mathrm{n}+26)+10-(\mathrm{n}+18)\swarrow+8(\downarrow+10)

Hence, we have t(H)=3 and r(H)=7.

The following table shows the number of H\mathrm{s} with d_{2}(H)=\langle 5

fixedt(H) and r(H).

,9, 13, 17, 21\rangle having

104

Weierstrasssemigroups attained by ramification
points on double covers of plane curves ofdegree 64Theorem ([6]).
h is odd\}\geqq 21

Letd_{2}(H)=\langle 5 6} or\langle 5 9, 13, 17, 21\rangle,,We set r=r(H) and t=t(H).

.

H=2d_{2}(H)+\{n, n+2l_{1},Suppose that there exists a pointed curve (C, P)
Q_{1}
Q_{r} of C distinct from P such that,.

.

.

,.

Assumen=min \{h\in H|Let.

.

.

,n+2l_{t}\rangle.

with H(P)=d_{2}(H) and r points,i) h^{0}(Q_{1}+\cdots+Q_{r})=1,ii) nP-2(Q_{1}+\cdots+Q_{r}) is linearly equivalent to a reduced divisor not containing P,\mathrm{m}h^{0}(K-l_{i}P-Q_{1}-\cdots-Q_{r})=h^{0}(K-(l_{i}-1)P-Q_{1}-\cdots-Q_{r}) for any i\in\{1, 2, . . . , t\}where K is a canonical divisor on C.

Then H is attained byover P.

H(\tilde{P}) where \tilde{P} isramification point on a double cover of C

aIf n issufficiently large, for example, larger than 2r+20 then the condition ii)
A canonical divisor K on the non‐singular plain curve C of degree 6
is the intersection of a cubic curve and C
For points P_{1}
P_{s} of C we see thatisthedimensionofthecubics
theP_{s}.
throughpoints P_{1}h^{0}(K-P_{1}-\cdots-P_{s}),is sastisfied.

.

,.

.

.

,,Let H=2\langle 5, 6\}+\{n, n+6\rangle.

Thenr(H)=6.

(\rightarrow+2) (\mathrm{n}+2) (\mathrm{n}+4) (\mathrm{n}+6) (\mathrm{n}+8)(n)\times\times④\times\downarow\bullet\times00+10(\mathrm{n}+12)\searrow+12 (\mathrm{n}+24)000(\mathrm{n}+36)(\mathrm{n}+48)

It suffices to show that h^{0}(Q_{1}+\cdots+Q_{6})=1 andh^{0}(K-3P-Q_{1}-\cdots-Q_{6})=h^{0}(K-2P-Q_{1}-\cdots- Q6)

.

.

.

.

,105

We need to find a pointed curve (C, P) satisfying the above two conditions. Let C be
the curve defined by the equationz^{3}(yz^{2}-y^{3})+ax^{3}(yz^{2}-x^{3})+by^{3}(yz^{2}+x^{3}-2y^{3})=0
where a and b are general elements of k and P=
for i=1 ,.

.

.

6 where,$\zeta$ isa(0 : 0 : 1)

primitive 6‐th root of 1.

Let Q_{i}=($\zeta$^{i} : $\zeta$^{3i} : 1).

We introduce the two cubics

C31: yz^{2}-x^{3}=0 and C32: yz^{2}+x^{3}-2y^{3}=0 which are irreducible. Then we haveC_{31}.C_{32}=3P+Q_{1}+\cdots+Q_{6} C.C31 \geqq 3P+Q_{1}+\cdots+Q_{6} and C. C_{32}\geqq 3P+Q_{1}+\cdots+Q_{6}.
The irreducibility of C31 and C32 means h^{0}(Q_{1}+\cdots+Q_{6})=1 By Caylay‐Bacharach
Theorem which we show below we get,,.

h^{0}(K-2P-Q_{1}-\cdots-Q_{6})=h^{0}(K-3P-Q_{1}-\cdots- Q6).

Theorem of Caylay‐Bacharach. Let X_{1} and X_{2} be two plane curves of degree dand e resp., meeting in a collection $\Gamma$ of de points. Let C be a curve of degree d+e-3

containing all but one point of $\Gamma$.

Then C contains all of $\Gamma$.

Apply d=e=3 to our case.
A numerical semigroup H is said to be DCP if there exists a double cover \tilde{C} of aplane curve with a ramification point \tilde{P} such that H=H(\tilde{P})Theorem 1([5]). Let H be a numerical semigroup with d_{2}(H)=\{5, 6\rangle

that the minimum odd integer n is larger than 2r(H)+20
But H=2\langle 5 , 9, 13, 17, 21)+\langle n, n+16\rangle withwere.

.

.

AssumeThen H is DCP.

r(H)=1 is not DCP. Assume that H

DCP. Then h^{0}(K-8P-Q)=h^{0}(K-7P-Q)

that C_{3}.C\geqq 7P+Q and C_{3}.C\not\geqq 8P+Q.

.

But, there exists a cubic C3 such

This is a contradiction.

Theorem 2 ([5]). Let H be a numerical semigroup with d_{2}(H)=\langle 5 , 9, 13, 17, 21).
Assume that the minimum odd integer n is larger than 2r(H)+20.

Then we have thefollowing table:

Remark.

Recently we succeeded in determining whether the remaining eight un‐

known semigroups in the above table are DC or not ([5]).

References[1] T. Harui and J. Komeda, Numerical semigroups of genus eight and double coveringsof curves of genus three, Semigroup Forum. 89 (2014) 571‐581.

106[2] T. Harui and J. Komeda, Numerical semigroups of genus seven and double cover‐ings of curves of genus three, Semigroup Forum. 90 (2015) 491‐502.
[3] T. Harui and J. Komeda, Numerical semigroups of genus six and double coveringsof curves of genus three, Semigroup Forum. 91 (2015) 601‐610.
[4] S.J. Kim and J. Komeda, Weierstrass semigroups on double covers of plane curvesof degree 5, Kodai Mathematical Journal. 38 (2015) 270‐288.

[5] S.J. Kim and J. Komeda, WeieTstrass semigroups on double covers of plane curvesof degree 6, In preparation.

[6] J. Komeda, On Weierstrass semigroups of double coverings of genus three curves,Semigroup Forum. 83 (2011) 479‐488.