54
Diagrams of numerical semigroups whose general members arenon‐Weierstrass 1

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 lntegrated Learning
Kanagawa lnstitute of Technology
Abstract

We construct diagrams consisting of an infinite number of numerical semigroups throughdividing by two whose general members are non‐Weierstrass where the bottom of thediagram is some Weierstrass numerical semigroup.

1

Introduction

Let \mathb {N}\lceil_{0} be the additive monoid of non‐negative integers. A submonoid H of N_{0} is called anumerical semigroup if the complement \mathbb{N}_{0}\backslash H is finite. The cardinality of N_{0}\backslash H is called
the genus of H , denoted by g(H) . \ln this article H always stands for a numerical semi‐group. We setc(H)=m\dot{m}\{c\in \mathbb{N}_{0}|c+\mathbb{N}_{0}\subseteqq H\},

which is called the conductor of H. lt is known that c(H)\leqq 2g(H) . H is said to besymmetric if c(H)=2g(H) . H is said to be quasi‐symmetric if c(H)=2g(H)-1 . We areinterested in the case c(H)=2g(H)-2.
A curve means a complete non‐singular irreducible algebraic curve over an alge‐
braically closed field k of characteritic 0 . For a pointed curve (C, P) we set

H(P)= { n\in N_{0}|\exists fi\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 ofgenus g(C) where g(C) is the genus of C.
We set d_{2}(H)=\{h'\in \mathbb{N}\lceil_{0}|2h'\in H\} , which is a numerical semigroup. Let \pi : \tilde{C}arrow C
be a double covering of a curve with a ramification point \tilde{P} . Then d_{2}(H(\tilde{P}))=H(\pi(\tilde{P})) .
H is said to be Weierstrass if there exists a pointed curve (C, P) with H(P)=H. H issaid to be of double covering type (abbreviated to DC) if there exists a double covering\pi :Carrow C' with a ramification point P such that H=H(P) . If H is DC , then both Hand d_{2}(H) are Weierstrass. For positive integers a_{1} ,, a_{s} we denote by \langle a_{1} ,, a_{s}\rangle themonoid generated by a_{1} ,, a_{s} . For example, H=\langle 2,2g+1\rangle is DC with d_{2}(H)=B\nabla_{0}.
Indeed, let \pi be a double covering from a curve of genus g to the projective Iine \mathb {P}^{1} and P
be any ramification point. Then H(P)=H. The following is an open problem:

1This paper is an extended abstract and the details will appear elsewhere.
This work was supported by JSPS KAKENHI Grant Numberl 8 K03228.

55Problem. ([4] and [1]) レレh at is the proporfion of non‐Weierstrass numerical semigroups
in the whole set ofnumerical semigroups?
Our purpose in this article is to construct diagrams consisting of an infinite number ofnumerical semigroups through the map d_{2} whose general members in the diagram are
non‐Weierstrass where the bottom of the diagram is a Weierstrass numerical semigroup
H with c(H)=2g(H)-2.

2Towers of symmetric numerical semigroups

We set m(H)= \min\{h\in H|h>0\} , which is called the multiplicity of H.

Remark 2.1 (i) Let n be an odd integer. Then 2H+n\mathbb{N}_{0} is a numerical semigroup.
(II) Let n be an odd integer with n\geqq c(H)+m(H)-1 . Then we have d_{2}(2H+n\mathbb{N}_{0})=H.
We have the following result for the above numerical semigroups:
Th伽屋 rem 2.2 (Komeda‐Ohbuchi [5]) Let n be an odd integer withn \geqq\max\{c(H)+m(H)-1, 2g(H)+{\imath}\}.
If H is Weierstrass, then 2H+nN_{0} is DC, hence Weierstrass.

We have towers consisting of symmetric numerical semigroups which are DC.
Th伽屋 rem 2.3 Let H_{0} be asymmetric Weierstrass numerical semigroup. For each i\geqq 1
Iet us take an odd integer
ni \geqq max{c(Hj‐ı) + m(H‐l)—l, 2g(H_{i-1})+1 }where we set H_{i}=2H_{i-1}+n_{i}\mathbb{N}_{0} for i\geqq 1 . Then we have towers of symmetric numerICal
semigroups which are DC as follows:Hi ı+\downar ow d_{2}for i\geqq 1.

私3Towers of quasi‐symmetric numerical semigroups

Lemma 3.1 ([2]) Let H and \tilde{H} be quasi‐symmetric numerical semigroups with d_{2}(\tilde{H})=H.
Then we obtaing(\tilde{H})=2g(H)-1.

By the above lemma and Riemann‐Hurwitz Formula we get the following:Th伽屋 rem 3.2 Lef H and \tilde{H} be quasi‐symmetric numerical semigroups with d_{2}(\tilde{H})=H.
Then \tilde{H} is not DC.

56
Pr屋 p 屋牡 ition 3.3([2]) Let H' be aquasi‐symmetric numerical semigroup、We set

n= \min\{h'\in H'|h' is odd \} and s_{i}= \min\{h'\in H|h'\equiv i mod n\}
for all i=1 ,, n-1 . We set\{s_{1}, . . . s_{n-1}\}=\{s^{(1)}< <s^{(n-1)}\}andH=\langle n, 2s^{(1)}, 2s^{(\frac{n-3}{2})}, 2s^{(\frac{n-1}{2})}-n, 2s^{(n-1)} -n\rangle.
Then H is a quasi‐symmetric numerical semigroup of genus 2g(H')-1 with d_{2}(H)=H'.

Example. Let H_{0}=\langle 3,4,5\rangle . For each odd m\geqq ı (resp. even m\geqq 2 ) we set
H_{m}=\langle 3,3m+2,3\cdot 2m+1\rangle (resp. H_{m}=\langle 3,3m+1 , 3(2m-1)+2\rangle ).

Then we have towers of quasi‐symmetric numerical semigroups which are not DC as
follows:

H_{i+1}\downar ow d_{2}for i\geqq 0.

H_{i}

4

Diagrams of numerical semigroups withc(H)=2g(H)-2We setPF(H)= { \gamma\in \mathbb{N}_{0}\backslash H|\gamma+h\in H, all h\in H>0 },whose elements are called pseudo‐Frobenius numbers of H . We have c(H)-1\in PF(H) .
We set t(H)=\# PF(H) , which is called the type of H.
Remark 4.1 We have c(H)+t(H)\leqq 2g(H)+1 . (For example, see [6].)H is said to be almost symmetric if the equalityc(H)+t(H)=2g(H)+1 holds.

Remark 4.2 i) H is symmetric if and only if t(H)=1 . In this case H is almost symmetric.
i_{I}) If H is quasi‐symmetric, then t(H)=2. The converse does not hold. In this case H isalso almost symmetric.
iii) If c(H)=2g(H)-2 , then t(H)=2 or 3.
We set PF^{*}(H)=PF(H)\backslash \{c(H)-1\}.
P 『屋 p 屋牡 ition 4. 3([3])lfHis almost symmefric, then we have an automorphism ofPF^{*}(H)sending \gamma to c(H)-1-y.

57Corollary 4.4 Ifc(H)=2g(H)-2 and t(H)=3_{J} we have PF^{*}(H)=\{\gamma, 2g(H)-3-\gamma\} forsome \gamma\in \mathbb{N}_{0}\backslash H.
Example. Let H=\langle 4,6,4l+1 , 4l+3\rangle for l\geqq 1 . Then we have c(H)=4l=2g(H)-2 andPF^{*}(H)=\{2,4l-1-2\} , hence t(H)=3 , i.e., H is almost symmetric.
Example. Let H=\langle 4,41+ ı, 4(2l-1)+3\rangle for l\geqq 1 . Then we have c(H)=12l-4=2g(H)-2and PF^{*}(H)=\{4\cdot 2l-2\} , hence t(H)=2 , i.e., H is not almost symmetric.

Remark 4.5 Let n be an odd integer withn \geqq\max\{c(H)+m-1,2m\}.
Then we haveg(2H+n N_{0})=2g(H)+\frac{n-1}{2}wIthd_{2}(2H+nN_{0})=H.

By the definition of PF(H) we get the following:

Lemma 4.6 Let d_{2}(\tilde{H})=H and n= \min\{\tilde{h}\in\tilde{H}|hハ is 。dd \} . Then the f。ll。wing areequivalent:I)g(H)=2g( り+ \frac{n-1}{2}-1.

Ii)\tilde{H}=2H+\langle n, n+2f\rangle for some f\in PF(H) .
Th伽屋 rem 4.7 Assume fhatc(H)=2g(H)-2 and t(H)=3 . Let PF^{*}(H)=\{fi_{1}, f_{2}\} . We set
Then one of the following holds:\tilde{H}_{i}=2H+\langle n, n+2f_{i}\rangle fori) H is Weierstrass and \tilde{H}_{1},\tilde{H}_{2} are DC.
Ii)H is Weierstrass and renumbering 1 and 2 \tilde{H}_{1} is DC and \tilde{H}_{2} is not DC.
IiI)H is non‐Weierstrass.
If n>>0 , then both \tilde{H}_{1} and \tilde{H}_{2} are non‐Weierstrass.
i=1 , 2.

Proof. For i) and ii)see [2]. Applylng [7] we get iii).

ロ

For 1\leqq i\leqq m(H)-1 we define s_{i} by m i n\{h\in H|h\equiv imod m(H)\} . We set

S(H)=\{m(H)\}\cup\{s_{i}|i=1,m(功 -1\},

which is called the standard basis for H.

Remark 4.8 ([3]) We have PF(H)=\{s_{i}-m(H)|s_{i}+s_{j}\not\in S(H) for all j\}.
Th伽屋 rem 4. 9([2]) Assume thatc(H)=2g(H)-2. Leff=s_{i}-m(H)\in PF^{*}(H^{-}.) . Lef n bean odd number withn\geqq 4((2m(H)-1)(s_{i}-m(H))+1-g(H))+1.
We set \tilde{H}=2H+\langle n, n+2f). Then we have the following:I)c(\tilde{H})=2g(\tilde{H})-2 and t(\tilde{H})=3.

i/) Assume (2i+1, m(H))=1 . For odd N\geqq n+2(2g(r-3+m(H)) we obtain thatH^{\approx}=2\tilde{H}+\langle N, N+2(2s_{j}-2m(H))\rangleis not DC.

58\cup sing the above theorem we get our main result in this article.

Corollary 4.10 Let H be a numerical semigroup with c(H)=2g(H)-2 . Assume thatm(H) is a power of 2. Then we can construct a diagram of numerical semigroups whose
general members are non‐Weierstrass such that the bottom of the diagram is H. Here_{J}general members mean all members in the interior of the diagram except fnite ones.

References[1] N. Kaplan and L. Ye, The proportion of Weierstrass semigroups, J. Algebra 373(2013) 377‐391.

[2] J. Komeda, Non‐Weierstrass numerical semigroups of high conductor, ln prepara‐tion.

[3] J. Komeda, Pseudo‐Frobenius numbers of numerical semigroups with high conduc‐
tor, Research Reports of Kanagawa lnstitute of Technology B‐42 (2018) 41‐46.
[4] J. Komeda, Non‐Weierstrass numerical semigroups, Semigroup Forum 57 (1988)
157‐185.

[5] J. Komeda and A. Ohbuchi, On double coverings of a pointed non‐singular curvewith any Weierstrass semigroup, Tsukuba J. Math. Soc. 31 (2007) 205‐215.

[6] H. Nari, Symmetries on almost symmetric numerical semigroups, Semigroup Forum86 (2013) 140‐154.
[7] F. Torres, Weierstrass points and double coverings of curves with application: Sym‐
metric numerical semigroups which cannot be realized as Weierstrass semigroups,
Manuscripta Math. 83 (1994) 39‐58.
Department of Mathematics

Center for Basic Education and lntegrated Learning
Kanagawa lnstitute of Technology
Atsugi 243‐0292
JapanE‐mail address: komeda@gen.kanagawa‐it.ac.jp