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. 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 sub-semigroups 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] (Brill-Nöther 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 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 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. 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, 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 → 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 2-sheeted 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 h-hyperelliptic 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 base-point-free 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 h-hyperelliptic 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 base-point-free 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 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 *( 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 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 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 . 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 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 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 pull-back 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 h-hyperelliptic 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 Riemann-Roch 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.   ☐