Kato
[6]
and Torres
[9]
characterized the Weierstrass semigroup of ramification points on
h
hyperelliptic curves. Also they showed the converse results that if the Weierstrass semigroup of a point
P
on a curve
C
satisfies certain numerical condition then
C
can be a double cover of some curve and
P
is a ramification point of that double covering map. In this paper we expand their results on the Weierstrass semigroup of a ramification point of a double covering map to the Weierstrass semigroup of a pair (
P
,
Q
). We characterized the Weierstrass semigroup of a pair (
P
,
Q
) which lie on the same fiber of a double covering map to a curve with relatively small genus. Also we proved the converse: if the Weierstrass semigroup of a pair (
P
,
Q
) satisfies certain numerical condition then
C
can be a double cover of some curve and
P
,
Q
map to the same point under that double covering map.
1. INTRODUCTION AND PRELIMINARIES
Let
C
be a nonsingular complex projective curve of genus
denote the field of meromorphic functions on
C
and ℕ
_{0}
be the set of all nonnegative integers. For two distinct points
P
,
Q
∈
C
, we define the Weierstrass semigroup
H
(
P
) ⊂ ℕ
_{0}
of a point and the Weierstrass semigroup of a pair of points
by
where (
f
)
_{∞}
means the divisor of poles of
f
. Indeed, these sets form subsemigroups of ℕ
_{0}
and
, respectively. The cardinality of the set
G
(
P
) = ℕ
_{0}
\
H
(
P
) is exactly
g
. The set
G
(
P
,
Q
) =
\
H
(
P
,
Q
) is also finite, but its cardinality is dependent on the points
P
and
Q
. In
[7]
, the upper and lower bound of such sets are given as
We review some basic facts concerning the Weierstrass semigroups at a pair of points on a curve (
[4]
,
[7]
).
Lemma 1.1.
For each α
∈
G
(
P
),
let β_{α}
= min{
β
 (
α
,
β
) ∈
H
(
P
,
Q
)}.
Then α
= min{
γ
 (
γ
,
β_{α}
) ∈
H
(
P
,
Q
)}.
Moreover, we have
Proof
. See
[7]
. ☐
Let
G
(
P
) = {
p
_{1}
<
p
_{2}
< ··· <
p_{g}
} and
G
(
Q
) = {
q
_{1}
<
q
_{2}
< ··· <
q_{g}
}. Above lemma implies that the set
H
(
P
,
Q
) defines a permutation
σ
=
σ
(
P
,
Q
) satisfying that (
p_{i}
,
q
_{σ(i)}
) ∈
H
(
P
,
Q
). Homma
[4]
obtained the formula for the cardinality of
G
(
P
,
Q
) using the cardinality of the set of pairs (
i
,
j
) which are reversed by
σ
. Also we define
by
which means nothing but
. Clearly
is a bijection. We use the following notations;
The above set Γ(
P
,
Q
) is called the
generating subset
of the Weierstrass semigroup
H
(
P
,
Q
). For given distinct two points
P
,
Q
, the set Γ(
P
,
Q
) determines not only
but also the sets
H
(
P
,
Q
) and
G
(
P
,
Q
) completely, as described in the lemma below. To state the lemma we use the natural partial order on the set
defined as
and the least upper bound of two elements (
α
_{1}
,
β
_{1}
), (
α
_{2}
,
β
_{2}
) is defined as
Lemma 1.2.
(1) The subset H
(
P
,
Q
)
of
is closed under the lub(least upper bound) operation. (2) Every element of H
(
P
,
Q
)
is expressed as the lub of one or two elements of the set
.
(3) The set G
(
P
,
Q
) =
\
H
(
P
,
Q
)
is expressed as
Proof
. See
[7]
and
[8]
. ☐
We say a pair (
α
,
β
) ∈
is
special
[ resp.
nonspecial
,
canonical
] if the corresponding divisor
αP
+
βQ
is special [ resp. nonspecial, canonical ]. We denote dim(
α
,
β
) the dimension of complete linear series 
αP
+
βQ
 and use the notations
We also need the following two theorems in
[1]
.
Theorem 1.3
( [
1
, p.10] (BrillNöther Reciprocity))
.
Let C be a curve of genus g
≥ 2.
If two linear series
and
and
on C are complete and residual to each other, i.e.,
where K is the canonical series, then n
− 2
r
=
m
− 2
s
.
This implies that if P is a base point of
then

+
P

does not have P as a base point, this means that
dim
.
We use the following wellknown lemmas to prove our theorems in this paper.
Lemma 1.4
(
[1]
(The Inequality of CastelnuovoSeveri))
.
Let C, C
_{1}
and C
_{2}
be curves of respective genera g, g
_{1}
and g
_{2}
.
Assume that ϕ_{i}
:
C
→
C_{i}, i
= 1, 2
are d_{i}sheeted coverings such that ϕ
=
ϕ
_{1}
×
ϕ
_{2}
:
C
→
C
_{1}
×
C
_{2}
is birational onto its image. Then g
≤ (
d
_{1}
− 1)(
d
_{2}
− 1) +
d
_{1}
g
_{1}
+
d
_{2}
g
_{2}
.
Lemma 1.5
([
2
, p.116] (Castelnuovo’s Bound))
.
Let C be a smooth curve that admits a birational mapping onto a nondegenerate curve of degree d in
ℙ
^{r}
.
Then the genus of C satisfies the inequality
where
and є
=
d
− 1 −
m
(
r
− 1).
Lemma 1.6
([
2
, p.251] (Clifford’s Theorem))
.
For any two effective divisors on a smooth curve C,
and for

D

special
with equality holding only if D
= 0,
D
=
K, or C is hyperelliptic.
In Section 2, we study the Weierstrass semigroups of pairs on
h
hyperelliptic curves.
2. SEMIGROUPS ONhHYPERELLIPTIC CURVES
Recall that a curve
C
is called
h
hyperelliptic if it admits a double covering map
π
:
C
→
C_{h}
where
C_{h}
is a curve of genus
h
, or equivalently, if there is an automorphism of order two on
C
which is defined by interchanging of the two sheets of this covering. Such
π
is unique if
g
> 4
h
+ 1
[3]
, which we can prove easily using above Lemma 1.4. Usually, 0hyperelliptic curves and 1hyperelliptic curves are said to be hyperelliptic and bielliptic, respectively. The results in this section was motivated by
[6]
and
[9]
, where the authors studied ordinary Weierstrass semigroups of points on
h
hyperelliptic curves.
Lemma 2.1.
Let C be a curve of genus g. Suppose that C is an hhyperelliptic curve for some h
≥ 0
with a double covering map π
:
C
→
C_{h}. If a linear series
is base point free and not compounded of π, then k
>
g
− 2
h
.
Proof
. The
k
sheeted map
and 2sheeted map
π
:
C
→
C_{h}
induce a birational map
onto its image. By Lemma 1.4,
g
≤ (
k
−1)(2−1)+
k
·0+2·
h
so we get
k
>
g
−2
h
. ☐
Theorem 2.2.
Let C be an hhyperelliptic curve of genus g
≥ 6
h
+ 2
with a double covering map π
:
C
→
C_{h}. Let P
,
Q
∈
C be distinct points and π
(
P
) =
π
(
Q
) =
P' . Then
Proof
. Suppose that there exists an element (
α
,
β
) ∈
H
(
P
,
Q
)
_{≤(2h+1,2h+1)}
not contained in {(
k
,
k
) 
k
∈
H
(
P'
),
k
≤ 2
h
+ 1}. Let
be a linear subseries of 
αP
+
βQ
 which is basepointfree and not necessarily complete. If
α
≠
β
,
is not compounded of
π
. If
α
=
β
and
α
∉
H
(
P'
), let
H
(
P'
)
_{≤2h}
= {
n
_{0}
= 0,
n
_{1}
, ··· ,
n_{h}
= 2
h
}. For some
i
,
n_{i}
<
α
<
n
_{i+1}
and dim 
n_{i}
(
P
+
Q
) < dim 
α
(
P
+
Q
) by the assumption on
α
. Also dim 
n_{i}
(
P
+
Q
) ≥ dim 
αP'
 =
i
so we have dim 
αP'
 < dim 
α
(
P
+
Q
). Thus 
α
(
P
+
Q
) and
is not compounded of
π
again. Now by Lemma 2.1,
which contradicts the choice of (
α
,
β
) ∈
H
(
P
,
Q
)
_{≤(2h+1,2h+1)}
. ☐
Each of the following two theorems is a converse of Theorem 2.2 in a different view point. For the next theorem, we need two lemmas.
Lemma 2.3.
Let
(
α
,
β
)
be an element in
with β
≥ 1 [ resp.
α
≥ 1]. Then
if and only if there exists
with 0 ≤
γ
≤
α
[ resp. 0 ≤
δ
≤
β
].
Proof
. See
[7]
.
Lemma 2.4.
Let H
⊂ ℕ
be a semigroup. Assume that H contains h terms in
{1, 2, ··· , 2
h
}
and
2
h
, 2
h
+ 1 ∈
H
.
Then H contains any integers k
≥ 2
h
.
Proof
. First, we show that 2
h
+ 2 ∈
H
. The set
I
_{2h+1}
= {1, 2, ⋯ , 2
h
, 2
h
+ 1} has
h
+ 1 elements of
H
. Consider a partition of
I
_{2h+1}
If
h
+ 1 ∈
H
, then 2
h
+ 2 ∈
H
since
H
is a semigroup. If
h
+ 1 ∉
H
, then at least one of the sets other than {
h
+ 1} is contained in
H
, and hence we have 2
h
+ 2 ∈
H
.
Next, we show that 2
h
+ 3 ∈
H
. The set
I
_{2h+2}
= {1, 2, ··· , 2
h
, 2
h
+ 1, 2
h
+ 2} has
h
+ 2 elements of
H
. Consider a partition of
I
_{2h+2}
Then at least of one is contained in
H
and hence 2
h
+ 3 ∈
H
.
Repeating this process, we conclude that
k
∈
H
for all
k
≥ 2
h
. ☐
Theorem 2.5.
Let C be a curve of genus g
≥ 6
h
+ 4
and P
,
Q
∈
C. Assume that H
(
P
,
Q
)
contains exactly h terms in
{(1, 1),(2, 2), ··· ,(2
h
, 2
h
)}
and that
Then C is hhyperelliptic with the double covering map ϕ
:
C
→
C_{h} for some C_{h}
.
Moreover ϕ
(
P
) =
ϕ
(
Q
)
and H
(
ϕ
(
P
)) = {
k
 (
k
,
k
) ∈
H
(
P
,
Q
)}.
Proof
. By Lemma 2.4, (
k
,
k
) ∈
H
(
P
,
Q
) for all
k
≥ 2
h
. By Lemma 2.3,
Let
s
+ 1 = dim (3
h
+ 1)(
P
+
Q
) and let’s denote (3
h
+ 1)(
P
+
Q
) by
. Consider a rational map
ϕ
:
C
→ ℙ
^{s+1}
defined by
.
Claim:
s
= 2
h
.
Suppose that
s
≥ 2
h
+ 1. If
ϕ
is birational, then
So by Lemma 1.5, we get
which contradicts our bound of genus. Let
t
be the degree of
ϕ
and
C'
be a normalization of
ϕ
(
C
). Then
C'
admits a complete basepointfree linear series
. Since
, we have
t
= 2. Thus
C
is a double covering of the curve
C'
and we have a complete linear series
. By Clifford’s theorem, it is a complete nonspecial linear series on
C'
, hence the genus of
C'
is
h'
= 3
h
−
s
<
h
. Here we have two possibilities
Subclaim:
ϕ
(
P
) =
ϕ
(
Q
).
If
ϕ
(
P
) ≠
ϕ
(
Q
), then
ϕ
* (
ϕ
(
P
)) = 2
P
and
ϕ
* (
ϕ
(
Q
)) = 2
Q
, since the divisor (3
h
+ 1)(
P
+
Q
) is the pullback of some divisor on
C'
via
ϕ
. In this case, 3
h
+ 1 must be even and hence
h
is odd. Consider a linear series (3
h
+ 2)(
P
+
Q
) and let its dimension be
u
+ 1. Then
s
+ 2 ≥
u
≥
s
+ 1 ≥ 2
h
+ 2. Through the similar steps as above, we conclude that
C
is a double covering of another curve
C''
of genus
h''
≤
h
− 1, and the series (3
h
+ 2)(
P
+
Q
) is compounded of the latter map
ϕ'
. Since
h
is odd, 3
h
+ 2 is also odd. Hence
ϕ'
*(
ϕ'
*(
P
)) =
P
+
Q
. Now
ϕ
×
ϕ'
is birational, and by Lemma 1.4, we have
g
≤ 1 + 4
h
contrary to our assumption. Therefore we proved the Subclaim
ϕ
(
P
) =
ϕ
(
Q
).
Since
k
(
P
+
Q
) =
ϕ
*(
kϕ
(
P
)) for any integer
k
, we have (
k
,
k
) ∈
H
(
P
,
Q
) for
k
∈
H
(
ϕ
(
P
)). Then the cardinality of the set {(
k
,
k
)  (
k
,
k
) ∉
H
(
P
,
Q
),
k
≥ 1} is less than
h
, which is a contradiction to our assumption. Thus we proved the Claim
s
= 2
h
.
Now we have a complete linear series
and a rational map
ϕ
:
C
→ ℙ
^{2h+1}
induced from
. Suppose
ϕ
is birational. Then by Lemma 1.5, we get
g
(
C
) ≤ 6
h
+ 3 which contradicts the assumption
g
≥ 6
h
+ 4.
Thus
ϕ
is a double covering map from
C
to
ϕ
(
C
) with
g
(
ϕ
(
C
)) =
h
. Therefore
C
is
h
hyperelliptic. Since (2
h
+ 1)(
P
+
Q
) and 2
h
(
P
+
Q
) is also compounded of
ϕ
, we conclude that
ϕ
(
P
) =
ϕ
(
Q
). ☐
Remark 2.6.
The above theorem is a modification of Theorem A in
[9]
.
Theorem 2.7.
Let C be a curve of genus g
≥ 6
h
+ 5.
Suppose that
(2
h
, 2
h
), (2
h
+ 1, 2
h
+ 1) ∈
H
(
P
,
Q
)
and
dim(2
h
, 2
h
) =
h
, dim(2
h
+ 1, 2
h
+ 1) =
h
+ 1.
Then C is an hhyperelliptic curve. Moreover, P and Q have same image under the double covering map
.
Proof
. Consider the rational map
ϕ
:
C
→ ℙ
^{h+1}
defined by the linear series
If
ϕ
is birational, then
g
≤ 6
h
+ 4 by Lemma 1.5. Thus
ϕ
is not birational. Let
t
be the degree of
ϕ
and
C'
be a normalization of
ϕ
(
C
). Thus
C'
admits a complete basepointfree linear series
. Since
, we have
t
= 2 or
t
= 3.
If
t
= 2, then we have
on
C'
. Since
, this series is nonspecial by Lemma 1.6 and the genus of
C'
is exactly
h
. Since 2
h
+ 1 is odd and the divisor (2
h
+ 1)(
P
+
Q
) is also a pullback of some divisor via a double covering map
ϕ
, we conclude that
ϕ
(
P
) =
ϕ
(
Q
).
Now it remains to show that the case
t
= 3 can not occur. If
t
= 3, then (4
h
+ 2) is a multiple of 3 and we have a complete
on
C'
. By Lemma 1.6 again, this linear series is nonspecial, and the genus of
C'
is
. If
ϕ
(
P
) =
ϕ
(
Q
), then
ϕ
* (
ϕ
(
P
)) = 2
P
+
Q
or
P
+ 2
Q
. Then (2
h
+ 1)(
P
+
Q
) can not be a pullback of any divisor on
C'
. Thus we have
Now
is a complete linear series on
C'
of degree
. Since
so
V
is base point free. Then
which is obtained from the pullback of
V
is also base point free and we have
Since (2
h
, 2
h
) ∈
H
(
P
,
Q
) by assumption, we have (2
h
+ 1, 2
h
) ∈
H
(
P
,
Q
) by Lemma 1.2. Thus
which contradicts the assumption dim(2
h
+ 1, 2
h
+ 1) =
h
+ 1. Hence the case
t
= 3 can not occur. ☐
Remark 2.8.
In Theorem 2.7, we assume the existence of only two elements in
H
(
P
,
Q
) and their dimensions without assuming the sequence of elements in
H
(
P
,
Q
).
We state a generalized version of Theorem 2.7.
Theorem 2.9.
Let C be a curve of genus g
≥ 6
h
+
a
,
a
≥ 5.
Suppose that there exists an integer n satisfying that
(i)
, (ii) dim 
n
(
P
+
Q
) =
n
−
h
and
(
n
,
n
) ∈
H
(
P
,
Q
)
and
(iii) dim (
n
−1)(
P
+
Q
) = (
n
−1)−
h
and
(
n
−1,
n
−1) ∈
H
(
P
,
Q
).
Then C is hhyperelliptic with double covering map π
:
C
→
C_{h} with
Proof
. If
n
= 2
h
+ 1, we already proved in Theorem 2.7. Now we assume
n
≥ 2
h
+ 2.
Let
n
be a number such that
, (
n
,
n
) ∈
H
(
P
,
Q
) and
and
ϕ_{n}
:
C
→ ℙ
^{n−h}
be a rational map defined by
.
Claim 1:
ϕ_{n}
is not birational if
n
≥ 2
h
+ 2.
Suppose that
ϕ_{n}
:
C
→ ℙ
^{n−h}
is birational. Then using the Castelnuovo bound, the genus of
C
satisfies the inequality
, where
and
є
=
d
− 1 −
m
(
r
− 1). In this theorem,
m
satisfies
or 3. If
m
= 2 and
є
= 2
h
+ 1 then
which is a contradiction. If
m
= 3 and
є
= −
n
+ 3
h
+ 2 then
g
≤ 6
h
+ 3 <
g
which is a contradiction again. Thus
ϕ_{n}
is not birational if
n
≥ 2
h
+ 2.
Let deg
ϕ_{n}
=
t
≥ 2. Since
ϕ_{n}
is nondegenerate,
so deg
ϕ_{n}
= 2 or deg
ϕ_{n}
= 3.
Claim 2: If (
n
,
n
),(
n
−1,
n
−1) ∈
H
(
P
,
Q
), dim 
n
(
P
+
Q
) =
n
−
h
and dim (
n
− 1)(
P
+
Q
) = (
n
− 1) −
h
, then deg
ϕ_{n}
= 2 and
g
(
ϕ_{n}
(
C
)) =
h
.
If
t
= 3, then 2
n
is a multiple of 3 and there is a complete and nonspecial
on
C'
=
ϕ_{n}
(
C
). Hence the genus of
C'
is
. If
ϕ_{n}
(
P
) =
ϕ_{n}
(
Q
), then
and the pullback of a multiple of
ϕ
(
P
) can not be
n
(
P
+
Q
). Thus we have
ϕ_{n}
(
P
) ≠
ϕ_{n}
(
Q
) and hence
Since
is base point free, (
n
,
n
− 3) ∈
H
(
P
,
Q
). Then dim 
nP
+
nQ
 = dim (
n
−1)
P
+ (
n
−1)
Q
+2 which is a contradiction to our assumption.
Therefore we conclude deg
ϕ_{n}
=
t
= 2 and there is a complete, nonspecial
on
C'
=
ϕ_{n}
(
C
). Hence the genus of
C'
is
h
and
C
is
h
hyperelliptic with double covering map
π
=
ϕ_{n}
:
C
→
C'
=
C_{h}
.
Claim 3:
π
(
P
) =
π
(
Q
) =
P'
and {
k
 (
k
,
k
) ∈
H
(
P
,
Q
)} =
H
(
P'
)
Case 1:
n
is odd.
Since
π
=
ϕ_{n}
is a double covering map by Claim 2, there is a complete, nonspecial
. By RiemannRoch Theorem,
g
(
C'
) =
k
− (
k
−
h
) =
h
. Since
n
(
P
+
Q
) is a pullback of some divisor
D
on
C'
=
C_{h}
, i.e.,
n
(
P
+
Q
) =
π
* (
D
) and
n
is odd, we get
π
(
P
) =
π
(
Q
).
Case 2:
n
is even.
Suppose that
ϕ_{n}
(
P
) ≠
ϕ_{n}
(
Q
). Since
n
≥ 2
h
+ 1 and
n
is even,
n
≥ 2
h
+ 2 and dim (
n
−1)(
P
+
Q
) = (
n
−1)−
h
and (
n
−1,
n
−1) ∈
H
(
P
,
Q
) by the assumption on
n
. Consider
ϕ
_{n−1}
which is defined by
. By Castelnuovo’s bound,
ϕ
_{n−1}
is not birational and deg
ϕ
_{n−1}
= 2 or 3. If deg
ϕ
_{n−1}
= 3, there is a complete, nonspecial
on
C''
=
ϕ
_{n−1}
(
C
). So
. Then the 3:1 map
and the 2:1 map
ϕ_{n}
:
C
→
C_{h}
induce a map
which is birational onto its image. By Lemma 1.4,
which is a contradiction. Thus deg
ϕ
_{n−1}
= 2 and there is a complete, nonspecial
on
ϕ
_{n−1}
(
C
). In this case
g
(
ϕ
_{n−1}
(
C
)) =
h
. Let
. Since
ϕ_{n}
(
P
) ≠
ϕ_{n}
(
Q
) and
is birational onto its image. Again by Lemma 1.4,
g
(
C
) ≤ (2 − 1)(2 − 1) + 2
h
+ 2
h
= 4
h
+ 1 <
g
which is a contradiction.
Thus we have
π
(
P
) =
π
(
Q
) and the last assertion follows from Theorem 2.2. ☐
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