double coverings of curves and non-weierstrass semigroups
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Double Coverings of Curves and Non-WeierstrassSemigroupsJiryo Komeda aa Department of Mathematics, Kanagawa Institute of Technology, Atsugi, JapanPublished online: 04 Jan 2013.
To cite this article: Jiryo Komeda (2013) Double Coverings of Curves and Non-Weierstrass Semigroups, Communications inAlgebra, 41:1, 312-324, DOI: 10.1080/00927872.2011.629324
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Communications in Algebra®, 41: 312–324, 2013Copyright © Taylor & Francis Group, LLCISSN: 0092-7872 print/1532-4125 onlineDOI: 10.1080/00927872.2011.629324
DOUBLE COVERINGS OF CURVES ANDNON-WEIERSTRASS SEMIGROUPS
Jiryo KomedaDepartment of Mathematics, Kanagawa Institute of Technology, Atsugi, Japan
We give a new method of constructing a non-Weierstrass semigroup H , which meansthat there is no smooth projective pointed curve over an algebracally closed fieldof characteristic 0 whose Weierstrass semigroup is H . This method depends on adescription of a pointed smooth projective curve such that there exists a double coveringof the curve ramified over the point with a certain condition on the genus of thecovering curve. Using this we find non-Weierstrass semigroups whose minimum positiveintegers are 8 and 12, respectively.
Key Words: Double coverings of curves; Non-Weierstrass semigroups; Symmetric numericalsemigroups; 8-semigroups; 12-semigroups.
2010 Mathematics Subject Classification: 14H55; 14H30; 20M14.
1. INTRODUCTION
Let �0 be the additive semigroup of non-negative integers. A submonoid Hof �0 is called a numerical semigroup if its complement �0\H is a finite set, whosecardinality is called the genus of H , denoted by g�H�. For a positive integer m anm-semigroup means a numerical semigroup whose minimum positive integer is m. Inthis article a curve means a complete smooth 1-dimensional algebraic variety overan algebraically closed field k of characteristic 0. For a pointed curve �C� P�, we set
H�P� = �n ∈ �0 � ∃f ∈ k�C� such that �f�� = nP��
which is called the Weierstrass semigroup of P where k�C� denotes the field ofrational functions on C and �f�� is the pole divisor of a rational function f . Fordistinct two points P and Q of C, we set
H�P�Q� = ��n� l� ∈ �0 ×�0 � ∃f ∈ k�C� such that �f�� = nP + lQ��
which is called the Weierstrass semigroup of a pair of points P and Q. Then H�P� isa numerical semigroup of genus g where g is the genus of the curve. A numericalsemigroup H is said to be Weierstrass if there exists a pointed curve �C� P� such
Received December 25, 2010; Revised August 21, 2011. Communicated by S. Kleiman.Address correspondence to Jiryo Komeda, Department of Mathematics, Kanagawa Institute of
Technology, Atsugi 243-0292, Japan; E-mail: [email protected]
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DOUBLE COVERINGS AND NON-WEIERSTRASS SEMIGROUPS 313
that H = H�P�. We know that for any positive integer m � 5 every m-semigroupis Weierstrass (see [6, 7, 11] in the cases m = 3, 4, and 5, respectively). But it isshowed that for any integer m � 13 not every m-semigroup is Weierstrass (see [1, 8]).Therefore, we have an open problem as follows:
For each 6 � m � 12 is every m-semigroup Weiersrass?
The aim of this article is to show the following theorem.
Main Theorem.
i) Let l � 2 and n an odd number with n � 16l+ 19 (resp., 16l+ 27). Then
�8� 12� 8l+ 2� 8l+ 6� n� n+ 4�(resp.�8� 12� 8l+ 6� 8l+ 10� n� n+ 4��
is a non-Weierstrass 8-semigroup.ii) Let l � 2 and n an odd number with n � 24l+ 27. Then
�12� 16� 20� 12l+ 2� 12l+ 6� 12l+ 10� n� n+ 4�
is a non-Weierstrass 12-semigroup.
In Main Theorem we use the following notation: For positive integersa1� a2� � � � � am, we denote by �a1� a2� � � � � am� the additive monoid generated bya1� a2� � � � � am. To prove Main Theorem, i.e., to find non-Weierstrass 8-semigroupsand 12-semigroups, we give a new method of constructing non-Weierstrasssemigroups. This method is different from those of Buchweitz [1] and Torres [12].It depends on a characterization of a double covering of a curve satisfying acertain relation between the genera of the base curve and the covering curve interms of Weierstrass semigroups of pairs of points. To explain the characterization,we need some terminologies. A numerical semigroup H is said to be symmetric ifc�H� = 2g�H� where c�H� = min�c ∈ �0 � c +�0 � H�. For an m-semigroup H theset S�H� = �m� s1� � � � � sm−1� denotes the standard basis for H where si = min�h ∈H �h ≡ i mod m� for each i with 1 � i � m− 1. We set smax = max�s1� s2� � � � � sm−1�.For a numerical semigroup H̃ , we denote by d2�H̃� the set consisting of h̃
2 with evenh̃ ∈ H̃ , which becomes a numerical semigroup. If � � C̃ −→ C is a double coveringof a curve with a ramification point P̃, then d2�H�P̃�� = H���P̃�� (for example, see[12]). A numerical semigroup H̃ is called the double covering type if there exists adouble covering � � C̃ −→ C with a ramification point P̃ such that H̃ = H�P̃�.
In [12], Torres proved the following statement:If H̃ is a Weierstrass semigroup with g�H̃� � 6g�d2�H̃��+ 4, then it is the double
covering type, hence d2�H̃� is Weierstrass.We are interested in the converse of the Torres’ result as follows.
Problem A. Let H̃ be a numerical semigroup with g�H̃� � 6g�d2�H̃��+ 4. Assumethat d2�H̃� is Weierstrass. Then is H̃ Weierstrass? That is to say, is H̃ the doublecovering type?
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314 KOMEDA
We solve Problem A negatively by giving non-Weierstrass 8-semigroups H̃with g�H̃� � 6g�d2�H̃��+ 4 and Weierstrass semigroups d2�H̃�. We get such 8-semigroups using the following characterization of numerical semigroups of doublecovering type with a certain condition.
Theorem B. Assume that an m-semigroup H is nonsymmetric. Let si = smax with noj such that sj = si + sk for some k. Then the following statements are equivalent:
i) There exists an odd integer n � 2c�H�− 1 with n = 2m− 1 such that H̃ = 2H +n�0 + �n+ 2�si −m���0 is the double covering type.
ii) For any odd n � 2c�H�− 1 with n = 2m− 1, the semigroup H̃ = 2H + n�0 +�n+ 2�si −m���0 is the double covering type.
iii) There exists a pointed curve �C� P� with H�P� = H and a point P1 = P such that�si −m� 1� ∈ H�P� P1�.
In Section 2 we study numerical semigroups H̃ satisfying g�H̃� = 2g�H�+n−12 − 1 where we set H = d2�H̃� and n is as in Theorem B. Especially, we prove
the same statements as those of Theorem B in the case of H̃ = 2H + n�0 + �n+2�smax −m���0. In Section 3, we consider the case where d2�H̃� is a symmetricsemigroup satisfying g�H̃� = 2g�d2�H̃��+ n−1
2 − 1. In this case, we show a resultsimilar to our previous result ([10]) on Weierstrass semigroups H̃ of double coveringtype satisfying g�H̃� = 2g�d2�H̃��+ n−1
2 . In Section 4 we prove Theorem B. UsingTheorem B we solve Problem A negatively in Section 5. Moreover, we showMain Theorem. Namely, non-Weierstrass 8-semigroups and non-Weierstrass 12-semigroups are given.
2. SEMIGROUPS OF DOUBLE COVERING TYPE
In [10] we have the following result.
Remark 2.1. Let H be an m-semigroup and H̃ a numerical semigroup withd2�H̃� = H . We set n = min�h̃ ∈ H̃ � h̃ is odd �. Assume that n � 2c�H�− 1 with n =2m− 1. Then the following statements are equivalent:
i) g�H̃� = 2g�H�+ n−12 ;
ii) H̃ = 2H + n�0.In this case, if H is Weierstrass, then H̃ is the double covering type.
We also have a description similar to Remark 2.1 in the case g�H̃� = 2g�H�+n−12 − 1 except the statement related to the double covering.
Lemma 2.2. Let H , H̃ and n be as in Remark 2.1. Then the following statements areequivalent:
i) g�H̃� = 2g�H�+ n−12 − 1;
ii) There exists i ∈ �1� 2� � � � � m− 1� with no j satisfying sj = si + sk for some k suchthat
H̃ = 2H + n�0 + �n+ 2�si −m���0�
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DOUBLE COVERINGS AND NON-WEIERSTRASS SEMIGROUPS 315
Proof. By [10] we have
S�2H + n�0� = �2m� 2s1� � � � � 2sm−1� n� n+ 2s1� � � � � n+ 2sm−1��
First, we prove that i) implies ii). By Remark 2.1 and the assumption i), we obtain
�0\H̃ = ��0\�2H + n�0��\�n+ 2�si −m��
for some i. Assume that there were j such that sj = si + sk for some k. Thenn+ 2sj − 2m = 2sk + n+ 2�si −m� ∈ H̃ , which implies that g�H̃� � 2g�H�+ n−1
2 −2. This is a contradiction.
Conversely, we assume that g�H̃� < 2g�H�+ n−12 − 1. Then there exists j
distinct from i such that n+ 2sj − 2m ∈ H̃ . Hence
n+ 2sj − 2m = c0 · 2m+m−1∑l=1
cl · 2sl + d0 · n+m−1∑l=1
dl�n+ 2sl�+ e�n+ 2�si −m��
with e � 1. We have
2�sj − si� = c0 · 2m+m−1∑l=1
cl · 2sl + d0 · n+m−1∑l=1
dl�n+ 2sl�+ �e− 1��n+ 2�si −m���
In view of n � 2c�H�− 1, we obtain d0 = dl = e− 1 = 0, which implies that
2�sj − si� = c0 · 2m+m−1∑l=1
cl · 2sl�
So, we get
sj = c0m+m−1∑l=1
clsl + si�
By the definition of sj , we must have sj = sk + si for some k, which contradicts theassumption ii).
Since we have no j such that sj = smax + sk for some k, by Lemma 2.2 we getthe following remark.
Remark 2.3. Let H be an m-semigroup and n an odd number � 2c�H�− 1 withn = 2m− 1. We set
H̃ = 2H + n�0 + �n+ 2�smax −m���0�
Then g�H̃� = 2g�H�+ n−12 − 1.
We can show the following, which means Theorem B in the case where H̃ isas in Remark 2.3.
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Proposition 2.4. Let H be an m-semigroup. Then the following statements areequivalent:
i) H is Weierstrass;ii) There exists an odd number n � 2c�H�− 1 with n = 2m− 1 such that H̃ = 2H +
n�0 + �n+ 2�smax −m���0 is the double covering type;iii) For any odd n � 2c�H�− 1 with n = 2m− 1 the semigroup H̃ = 2H + n�0 +
�n+ 2�smax −m���0 is the double covering type.
Proof. It is trivial that iii) implies ii).Assume that ii) holds. Then there exists a double covering � � C̃ −→ C of a
curve with a ramification point P̃ such that H�P̃� = H̃ . Hence, we get H = d2�H̃� =d2�H�P̃�� = H���P̃��, which implies that H is Weierstrass.
The main part of the proof is to show that i) implies iii). Assume that �C� P�is a pointed curve with H�P� = H . Let n be an odd number with n � 2c�H�− 1 andn = 2m− 1. We set
D = n+ 12
P − P1�
where P1 is a point of C distinct from P. In view of the conditions on n,we get deg�2D − P� � 2g�H�+ 1. In fact, if H = �m�m+ 1� � � � � m+m− 1�, thendeg�2D − P� = n− 2 � 2m+ 1− 2 � 2g�H�+ 1. Otherwise, we have
deg�2D − P� = n− 2 � 2c�H�− 3 � 2�g�H�+ 2�− 3 = 2g�H�+ 1�
Thus, the complete linear system �2D − P� is very ample, hence base-point free,which implies that
2D ∼ P +Q1 + · · · +Qn−2�
where the points P, Q1, � � � , Qn−2 are distinct. We set � = �C�−D�. Then there
is an isomorphism �⊗2� �C�−�P +Q1 + · · · +Qn−2�� ⊂ �C . Hence, the direct sum
�C ⊕� has an �C-algebra structure through . Let � � C̃ = Spec��C ⊕�� −→ C bea canonical morphism. We note that the genus g�C̃� of C̃ is 2g�H�+ n−1
2 − 1, becausethe branch divisor of � is P +Q1 + · · · +Qn−2. Let P̃ ∈ C̃ such that ��P̃� = P. ByProposition 2.1 in [9], we obtain
l��n− 1�P̃� = l
(n− 12
P
)+ l
(n− 12
P −D
)
= l
(n− 12
P
)+ l�P1 − P� = l
(n− 12
P
)�
because of P1 = P. Here, we denote dimk H0�X��X�D�� by l�D� for a divisor D on
a curve X. Moreover, we have
l��n+ 1�P̃� = l
(n+ 12
P
)+ l�P1� = l
(n+ 12
P
)+ 1�
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DOUBLE COVERINGS AND NON-WEIERSTRASS SEMIGROUPS 317
By the assumption n � 2c�H�− 1, we get n+12 � c�H�. Hence,
l
(n+ 12
P
)= l
(n− 12
P
)+ 1�
which implies that n ∈ H�P̃�. Thus, it suffices to show that
n+ 2�smax −m� ∈ H�P̃�
because of g�H�P̃�� = g�C̃� and Remark 2.3. We obtain
l��n+ 2�smax −m�− 1�P̃� = l
((smax −m+ n− 1
2
)P
)+ l��smax −m− 1�P + P1�
and
l��n+ 2�smax −m�+ 1�P̃� = l
((smax −m+ n− 1
2+ 1
)P
)+ l��smax −m�P + P1��
On the other hand, we have
l�K − �smax −m− 1�P� = l�K − �smax −m�P�+ 1 = 1�
because smax −m is the largest gap at P, where K is a canonical divisor on C. Hencethere exists a unique effective divisor E on C such that
E ∼ K − �smax −m− 1�P�
Let us take the point P1 < E. Then we have
l�K − �smax −m− 1�P − P1�� = 0�
which implies that
l��smax −m�P + P1� = l��smax −m− 1�P + P1�+ 1�
Hence, we get n+ 2�smax −m� ∈ H�P̃�.
3. SEMIGROUPS H̃ WITH SYMMETRIC d2�H̃�
In this section we will give a characterization of H̃ similar to Remark 2.1 in thecase where H = d2�H̃� is a symmetric m-semigroup, i.e., 2g�H�− 1 = smax −m, withg�H̃� = 2g�H�+ n−1
2 − 1. It is well-known that a necessary and sufficient conditionfor H to be symmetric is the following equation:
∈ �\H if and only if 2g�H�− 1− ∈ H�
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Lemma 3.1. If H is a symmetric m-semigroup, then we have that, if i ∈�1� 2� � � � � m− 1� satisfies the condition where there is no j such that sj = si + sk forsome k, then si = smax.
Proof. Since H is symmetric, we get
2g�H�− 1− �si −m� ∈ H for any i�
Moreover, we have
smax = 2g�H�− 1+m = 2g�H�− 1− �si −m�+ si�
Since smax belongs to S�H�, for si = smax, there is some k such that
2g�H�− 1− �si −m� = sk�
Thus, we have that, if si = smax, then smax = si + sk for some k. �
We want to prove the converse of Lemma 3.1 which we do in the followingtwo lemmas.
Lemma 3.2. Let H be an m-semigroup. Assume that we have sq = smax if q ∈�1� � � � � m− 1� has no j such that sj = sq + sk for some k. Then for any i with si = smax,we have smax = si + sk for some k.
Proof. We set
S�H�\�m� = �smax = s�1� > s�2� > · · · > s�m−1���
By the assumption there exists some k�2� such that sj2 = s�2� + sk�2� > s�2�, whichimplies that sj2 = smax. We prove the statement by induction on l with s�l�. Assumethat for any 2 � � � l, we have smax = s��� + sk��� . By the assumption sjl+1
= s�l+1� +sk�l+1� > s�l+1�. Hence, we have s��� = s�l+1� + sk�l+1� for some � with 1 � � � l. If s��� =smax, then by the induction hypothesis we have
smax = s��� + sk��� = s�l+1� + sk�l+1� + sk��� �
Hence, we must have smax = s�l+1� + sp for some p.
Lemma 3.3. Let H be an m-semigroup such that for any i ∈ �1� � � � � m− 1� with si =smax we have smax = si + sk for some k. Then H is symmetric.
Proof. Let smax = ml+ n with a positive integer n � m− 1. Now we have thefollowing equalities:
n−1∑i=1
[si + sn−i
m
]+
m−n−1∑i=1
([sn+i + sm−i
m
]− 1
)
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DOUBLE COVERINGS AND NON-WEIERSTRASS SEMIGROUPS 319
=n−1∑i=1
([sim
]+
[sn−i
m
])+
m−n−1∑i=1
([sn+i
m
]+
[sm−i
m
])
= 2m−1∑i=1
[sim
]− 2
[smax
m
]= 2g�H�− 2
[smax
m
]�
By the assumption, we obtain
n−1∑i=1
[si + sn−i
m
]+
m−n−1∑i=1
([sn+i + sm−i
m
]− 1
)
=n−1∑i=1
[smax
m
]+
m−n−1∑i=1
([smax
m
]− 1
)= �m− 2�
[smax
m
]−m+ n+ 1�
Hence,
2g�H� = m
[smax
m
]−m+ n+ 1 = smax − �m− 1� = c�H��
which implies that H is symmetric.
Thus, by Lemmas 3.1, 3.2, and 3.3, we get another characterization of asymmetric semigroup as follows:
Lemma 3.4. Let H be a numerical semigroup. The following statements areequivalent:
i) H is symmetric;ii) For any i, if si = smax, then si + sk ∈ S�H� for some k;iii) For any i, if si = smax, then si + sk = smax for some k.
By Lemmas 2.2 and 3.4, we have the following remark.
Remark 3.5. Let H be a non-symmetric m-semigroup and n an odd number� 2c�H�− 1 with n = 2m− 1. Then there exists a numerical semigroup H̃ withd2�H̃� = H and n = min�h̃ ∈ H̃ � h̃ is odd � such that
g�H̃� = 2g�H�+ n− 12
− 1 and H̃ = 2H + n�0 + �n+ 2�smax −m���0�
By Lemma 2.2, Proposition 2.4, and Lemma 3.4, we gain a result similar toRemark 2.1 in the case we discuss in this section.
Theorem 3.6. Let H be a symmetric m-semigroup and H̃ a numerical semigroup withd2�H̃� = H . We set n = min�h̃ ∈ H̃ � h̃ is odd �. Assume that n � 2c�H�− 1 with n =2m− 1.Then the following statements are equivalent:
i) g�H̃� = 2g�H�+ n−12 − 1;
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320 KOMEDA
ii) H̃ = 2H + n�0 + �n+ 2�smax −m���0. In this case, if H is Weierstrass, H̃ is thedouble covering type.
4. A CHARACTERIZATION OF A SEMIGROUP OF DOUBLE COVERINGTYPE BY A PAIR OF POINTS
In this section we give the proof of Theorem B. First, using the notation ofH�P�Q� we can put in another way the essential part of the proof of Proposition 2.4as follows.
Remark 4.1. Let H be a Weierstrass m-semigroup. Take a pointed curve �C� P�with H�P� = H . Let E be a unique effective divisor on C which is linearly equivalentto K − �smax −m− 1�P. If a point P1, distinct from P, satisfies P1 < E, then �smax −m� 1� ∈ H�P� P1�.
Proof. We note that �n� l� ∈ H�P�Q� with positive integers n and l if and only if
l�nP + lQ� = l��n− 1�P + lQ�+ 1 = l�nP + �l− 1�Q�+ 1�
Since smax −m is the last gap at P, we get
l��smax −m�P + P1� = l��smax −m�P�+ 1�
In view of P1 < E, we have
l��smax −m�P + P1� = l��smax −m− 1�P + P1�+ 1�
Hence, we obtain �smax −m� 1� ∈ H�P� P1�.
Here we prove Theorem B. In terms of Weierstrass semigroups of pairs ofpoints the theorem gives a characterization of an m-semigroup H such that H̃ =2H + n�0 + �n+ 2�si −m���0 is the double covering type.
The Proof of Theorem B. iii� ⇒ ii� We set D = n+12 P − P1. Using the method
as in the proof of Proposition 2.4 we construct a pointed curve �C̃� P̃� such that C̃ isa double covering of C with a ramification point P̃. In view of �si −m� 1� ∈ H�P� P1�we have
l��si −m�P + P1� = l��si −m− 1�P + P1�+ 1�
which, as seen in the proof of Proposition 2.4, is enough to conclude that H�P̃� �n+ 2�si −m�. Hence, we get H̃ = H�P̃�, i.e., H̃ is the double covering type.
ii)⇒ i) It is trivial.
i)⇒ iii) Let �C̃� P̃� be a pointed curve with H�P̃� = H̃ and � � C̃ −→ C adouble covering which is ramified at P̃. We set P = ��P̃� with H�P� = H . Then the
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DOUBLE COVERINGS AND NON-WEIERSTRASS SEMIGROUPS 321
double covering � can be constructed using a divisor D = n+12 P − P1 with P1 = P on
C in the usual way as in the proof of Proposition 2.4, because n ∈ H̃ implies that
l
(n+ 12
P −D
)= l
(n− 12
P −D
)+ 1 = 1�
Since n+ 2�si −m� ∈ H�P̃�, we must have
l��si −m�P + P1� = l��si −m− 1�P + P1�+ 1�
which implies that
l��si −m�P�+ 1 � l��si −m�P + P1� � l��si −m− 1�P�+ 1�
The integer si −m is a gap at P. Hence, using the above inequalities we get
l��si −m�P�+ 1 = l��si −m�P + P1��
which implies that �si −m� 1� ∈ H�P� P1�.
Using Theorem B we give numerical semigroups H̃ of double covering typewith d2�H̃� = �4� 5� 6� 7�.
Example 4.1. Let H = �4� 5� 6� 7�. Then
S�H� = �m = 4� s1 = 5� s2 = 6� smax = s3 = 7��
Let n be an odd number � 2c�H�− 1 = 7 with n = 2m− 1 = 7, i.e., n � 9.
i) By Proposition 2.4 we see that
H̃ = 2H + n�0 + �n+ 2�smax −m���0 = �8� 10� 12� 14� n� n+ 6�
is the double covering type.ii) Using Theorem B and the method in [5] we can show that
H̃ = 2H + n�0 + �n+ 2�s2 −m���0 = �8� 10� 12� 14� n� n+ 4�
is the double covering type. In fact, let C be a plane curve defined by
a�yz3 − x3z�+ b�xy3 + y2z2� = 0
where a and b are general constants. We set P = �0 � 1 � 0� which is a point of C.Then the tangent line at P is x = 0, and hence H�P� = �4� 5� 6� 7�. Moreover, wehave K ∼ 2P + P1 +Q where we set P1 = �0 � 0 � 1� and Q = �0 � −a � b�. Hence,we get l�K − 2P − P1� = 1, which implies that �2� 1� ∈ H�P� P1�. By Theorem B,H̃ is the double covering type.
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iii) Let C be a hyperelliptic curve of genus 3 and P an ordinary point. Take a uniqueordinary point P1 ∈ C with P + P1 ∼ g12. Then we have �1� 1� ∈ H�P� P1�. Hence,by Theorem B
H̃ = 2H + n�0 + �n+ 2�s1 −m���0 = �8� 10� 12� 14� n� n+ 2�is the double covering type.
5. NON-WEIERSTRASS 8-SEMIGROUPS AND NON-WEIERSTRASS12-SEMIGROUPS
In this section we solve Problem A negatively and prove Main Theorem. First,we prove the following lemma.
Lemma 5.1. Let l � 2. The semigroup
H = �4� 6� 4l+ 1� 4l+ 3� �resp.,�4� 6� 4l+ 3� 4�l+ 1�+ 1��cannot be attained by a pointed trigonal curve. In particular, if �C� P� is a pointed curvesuch that H�P� is the above semigroup H , then for any point P1 of C, distinct from P,we have �2� 1� ∈ H�P� P1�.
Proof. Let �C� P� be a pointed curve such that H�P� is the semigroup H as in thestatement. We will show that C is not a trigonal curve. Assume that C were trigonal.Then we have two morphisms �4P� and g13
from C to �1 determined by the linearsystems �4P� and g13, respectively. Hence, we see that the genus g�C� of C is lessthan or equal to 6. Since g�H� = 2l+ 1 (resp., 2l+ 2), we may assume that l = 2.ByCoppens’ assertions on p. 9 in [2] and page 11 in [3] and Kim’s assertion on page 3in [4] this is a contradiction. �
Using Lemma 5.1 and Theorem B we get the following:
Theorem 5.2. Let l � 2 and n an odd number � 8l− 1 (resp. 8l+ 3�. Then
H̃ = �8� 12� 8l+ 2� 8l+ 6� n� n+ 4� �resp��8� 12� 8l+ 6� 8l+ 10� n� n+ 4��is not the double covering type.
Proof. Assume that H̃ were the double covering type. By Theorem Bii� ⇒ iii�] there exists a pointed curve �C� P� with H�P� = �4� 6� 4l+ 1� 4l+ 3�
(resp., �4� 6� 4l+ 3� 4l+ 5�) such that �2� 1� ∈ H�P� P1�. This contradicts Lemma 5.1.�
Hence, if we take sufficiently large n, Problem A has been solved negatively.In fact, we have the following counterexamples.
Example 5.1. Let l � 2 and n an odd number with n � 16l+ 19 (resp., 16l+ 27�.We set
H̃ = �8� 12� 8l+ 2� 8l+ 6� n� n+ 4� �resp���8� 12� 8l+ 6� 8l+ 10� n� n+ 4���
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DOUBLE COVERINGS AND NON-WEIERSTRASS SEMIGROUPS 323
Then
d2�H̃� = �4� 6� 4l+ 1� 4l+ 3� �resp���4� 6� 4l+ 3� 4�l+ 1�+ 1��
is Weierstrass by [6]. Moreover, we have
g�H̃� = 2g�d2�H̃��+ n− 12
− 1 � 6g�d2�H̃��+ 4�
But by Theorem 5.2, H̃ is not the double covering type.
Theorem 5.3. Let l � 2 and n an odd number with n � 12l− 1. Then
H̃ = �12� 16� 20� 12l+ 2� 12l+ 6� 12l+ 10� n� n+ 4�
is not the double covering.
Proof. By [2, 3] and [4] the semigroup d2�H̃� = �6� 8� 10� 6l+ 1� 6l+ 3� 6l+ 5� ofgenus 3l+ 2 cannot be attained by a pointed trigonal curve. By Theorem B
i� ⇒ iii� we get the desired result. �
By Torres’ assertion on p. 2 in [12], Example 5.1, and Theorem 5.3, we canprove Main Theorem.
ACKNOWLEDGMENT
The author is partially supported by Grant-in-Aid for Scientific Research(21540052), Japan Society for the Promotion Science.
REFERENCES
[1] Buchweitz, R. O. (1980). On Zariski’s criterion for equisingularity and non-smoothablemonomial curves. Preprint 113, University of Hannover.
[2] Coppens, M. (1985). The Weierstrass gap sequences of the total ramification points oftrigonal coverings of �1. Indag. Math. 47:245–276.
[3] Coppens, M. (1986). The Weierstrass gap sequences of the ordinary ramification pointsof trigonal coverings of �1: Existence of a kind of Weierstrass gap sequences. J. PureAppl. Algebra 43:11–25.
[4] Kim, S. J. (1990). On the existence of Weierstrass gap sequences on trigonal curves.J. Pure Appl. Algebra 63:171–180.
[5] Kim, S. J., Komeda, J. (2001). The Weierstrass semigroup of a pair and moduli in �3.Bol. Soc. Bras. Mat. 32:149–157.
[6] Komeda, J. (1983). On Weierstrass points whose first non-gaps are four. J. reine angew.Math. 341:68–86.
[7] Komeda, J. (1992). On the existence of Weierstrass points whose first non-gaps arefive. Manuscripta Math. 76:193–211.
[8] Komeda, J. (1998). Non-Weierstrass numerical semigroups. Semigroup Forum57:157–185.
[9] Komeda, J., Ohbuchi, A. (2004). Weierstrass points with first non-gap four on a doublecovering of a hyperelliptic curve. Serdica Math. J. 30:43–54.
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by [
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Tec
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ogy]
at 1
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19
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ust 2
013
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[10] Komeda, J., Ohbuchi, A. (2007). On double coverings of a pointed non-singular curvewith any Weierstrass semigroup. Tsukuba J. Math. 31:205–215.
[11] Maclachlan, C. (1971). Weierstrass points on compact Riemann surfaces. J. LondonMath. Soc. 3:722–724.
[12] Torres, F. (1994). Weierstrass points and double coverings of curves with application:Symmetric numerical semigroups which cannot be realized as Weierstrass semigroups.Manuscripta Math. 83:39–58.
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013