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WEIERSTRASS SEMIGROUPS OF PAIRS ON H-HYPERELLIPTIC CURVES
WEIERSTRASS SEMIGROUPS OF PAIRS ON H-HYPERELLIPTIC CURVES
Journal of the Korean Society of Mathematical Education Series B The Pure and Applied Mathematics. 2015. Nov, 22(4): 403-412
Copyright © 2015, Korean Society of Mathematical Education
  • Received : November 02, 2015
  • Accepted : November 20, 2015
  • Published : November 30, 2015
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EUNJU KANG

Abstract
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.
Keywords
1. INTRODUCTION AND PRELIMINARIES
Let C be a nonsingular complex projective curve of genus
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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
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by
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where ( f ) means the divisor of poles of f . Indeed, these sets form sub-semigroups of ℕ 0 and
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, respectively. The cardinality of the set G ( P ) = ℕ 0 \ H ( P ) is exactly g . The set G ( P , Q ) =
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\ 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
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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 βα = min{ β | ( α , β ) ∈ H ( P , Q )}. Then α = min{ γ | ( γ , βα ) ∈ H ( P , Q )}. Moreover, we have
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Proof . See [7] .   ☐
Let G ( P ) = { p 1 < p 2 < ··· < pg } and G ( Q ) = { q 1 < q 2 < ··· < qg }. Above lemma implies that the set H ( P , Q ) defines a permutation σ = σ ( P , Q ) satisfying that ( pi , 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
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by
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which means nothing but
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. Clearly
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is a bijection. We use the following notations;
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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
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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
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defined as
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and the least upper bound of two elements ( α 1 , β 1 ), ( α 2 , β 2 ) is defined as
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Lemma 1.2. (1) The subset H ( P , Q ) of
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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
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. (3) The set G ( P , Q ) =
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\ H ( P , Q ) is expressed as
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Proof . See [7] and [8] .   ☐
We say a pair ( α , β ) ∈
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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
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We also need the following two theorems in [1] .
Theorem 1.3 ( [ 1 , p.10] (Brill-Nöther Reciprocity)) . Let C be a curve of genus g ≥ 2. If two linear series
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and
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and on C are complete and residual to each other, i.e.,
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where K is the canonical series, then n − 2 r = m − 2 s . This implies that if P is a base point of
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then |
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+ P | does not have P as a base point, this means that dim
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.
We use the following well-known lemmas to prove our theorems in this paper.
Lemma 1.4 ( [1] (The Inequality of Castelnuovo-Severi)) . Let C, C 1 and C 2 be curves of respective genera g, g 1 and g 2 . Assume that ϕi : C Ci, i = 1, 2 are di-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 r . Then the genus of C satisfies the inequality
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where
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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,
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and for | D | special
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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 ONh-HYPERELLIPTIC CURVES
Recall that a curve C is called h -hyperelliptic if it admits a double covering map π : C Ch where Ch 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, 0-hyperelliptic curves and 1-hyperelliptic curves are said to be hyperelliptic and bi-elliptic, 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 h-hyperelliptic curve for some h ≥ 0 with a double covering map π : C Ch. If a linear series
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is base point free and not compounded of π, then k > g − 2 h .
Proof . The k -sheeted map
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and 2-sheeted map π : C Ch induce a birational map
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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 h-hyperelliptic curve of genus g ≥ 6 h + 2 with a double covering map π : C Ch. Let P , Q C be distinct points and π ( P ) = π ( Q ) = P' . Then
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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
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be a linear subseries of | αP + βQ | which is base-point-free and not necessarily complete. If α β ,
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is not compounded of π . If α = β and α H ( P' ), let H ( P' ) ≤2h = { n 0 = 0, n 1 , ··· , nh = 2 h }. For some i , ni < α < n i+1 and dim | ni ( P + Q )| < dim | α ( P + Q )| by the assumption on α . Also dim | ni ( P + Q )| ≥ dim | αP' | = i so we have dim | αP' | < dim | α ( P + Q )|. Thus | α ( P + Q )| and
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is not compounded of π again. Now by Lemma 2.1,
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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
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with β ≥ 1 [ resp. α ≥ 1]. Then
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if and only if there exists
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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
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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
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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
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Then C is h-hyperelliptic with the double covering map ϕ : C Ch for some Ch . 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,
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Let s + 1 = dim |(3 h + 1)( P + Q )| and let’s denote |(3 h + 1)( P + Q )| by
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. Consider a rational map ϕ : C → ℙ s+1 defined by
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.
Claim: s = 2 h .
Suppose that s ≥ 2 h + 1. If ϕ is birational, then
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So by Lemma 1.5, we get
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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 base-point-free linear series
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. Since
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, we have t = 2. Thus C is a double covering of the curve C' and we have a complete linear series
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. 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
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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 pull-back 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 *( ( 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
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and a rational map ϕ : C → ℙ 2h+1 induced from
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. 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 h-hyperelliptic 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
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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 base-point-free linear series
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. Since
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, we have t = 2 or t = 3.
If t = 2, then we have
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on C' . Since
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, 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 pull-back 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
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on C' . By Lemma 1.6 again, this linear series is nonspecial, and the genus of C' is
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. 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 pull-back of any divisor on C' . Thus we have
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Now
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is a complete linear series on C' of degree
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. Since
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so V is base point free. Then
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which is obtained from the pullback of V is also base point free and we have
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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
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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)
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, (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 h-hyperelliptic with double covering map π : C Ch with
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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
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, ( n , n ) ∈ H ( P , Q ) and
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and ϕn : C → ℙ nh be a rational map defined by
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.
Claim 1: ϕn is not birational if n ≥ 2 h + 2.
Suppose that ϕn : C → ℙ n−h is birational. Then using the Castelnuovo bound, the genus of C satisfies the inequality
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, where
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and є = d − 1 − m ( r − 1). In this theorem, m satisfies
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or 3. If m = 2 and є = 2 h + 1 then
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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 ϕn is not birational if n ≥ 2 h + 2.
Let deg ϕn = t ≥ 2. Since ϕn is nondegenerate,
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so deg ϕn = 2 or deg ϕn = 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 ϕn = 2 and g ( ϕn ( C )) = h .
If t = 3, then 2 n is a multiple of 3 and there is a complete and nonspecial
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on C' = ϕn ( C ). Hence the genus of C' is
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. If ϕn ( P ) = ϕn ( Q ), then
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and the pullback of a multiple of ϕ ( P ) can not be n ( P + Q ). Thus we have ϕn ( P ) ≠ ϕn ( Q ) and hence
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Since
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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 ϕn = t = 2 and there is a complete, nonspecial
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on C' = ϕn ( C ). Hence the genus of C' is h and C is h -hyperelliptic with double covering map π = ϕn : C C' = Ch .
Claim 3: π ( P ) = π ( Q ) = P' and { k | ( k , k ) ∈ H ( P , Q )} = H ( P' )
Case 1: n is odd.
Since π = ϕn is a double covering map by Claim 2, there is a complete, nonspecial
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. By Riemann-Roch Theorem, g ( C' ) = k − ( k h ) = h . Since n ( P + Q ) is a pullback of some divisor D on C' = Ch , i.e., n ( P + Q ) = π * ( D ) and n is odd, we get π ( P ) = π ( Q ).
Case 2: n is even.
Suppose that ϕn ( P ) ≠ ϕn ( 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
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. 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
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on C'' = ϕ n−1 ( C ). So
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. Then the 3:1 map
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and the 2:1 map ϕn : C Ch induce a map
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which is birational onto its image. By Lemma 1.4,
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which is a contradiction. Thus deg ϕ n−1 = 2 and there is a complete, nonspecial
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on ϕ n−1 ( C ). In this case g ( ϕ n−1 ( C )) = h . Let
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. Since ϕn ( P ) ≠ ϕn ( Q ) and
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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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