WEIERSTRASS SEMIGROUPS OF PAIRS ON H-HYPERELLIPTIC CURVES

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.

1. INTRODUCTION AND PRELIMINARIES

Let C be a nonsingular complex projective curve of genus denote the field of meromorphic functions on C and ℕ0 be the set of all nonnegative integers. For two distinct points P, Q ∈ C, we define the Weierstrass semigroup H(P) ⊂ ℕ0 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 ℕ0 and , respectively. The cardinality of the set G(P) = ℕ0 \ 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 βα = min{β | (α, β) ∈ H(P, Q)}. Then α = min{γ | (γ, βα) ∈ H(P, Q)}. Moreover, we have

Proof. See [7].   ☐

Let G(P) = {p1 < p2 < ··· < pg} and G(Q) = {q1 < q2 < ··· < qg}. Above lemma implies that the set H(P, Q) defines a permutation σ = σ(P, Q) satisfying that (pi , qσ(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 (α1, β1), (α2, β2) 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 − 2r = m − 2s. 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, C1 and C2 be curves of respective genera g, g1 and g2. Assume that ϕi : C → Ci, i = 1, 2 are di-sheeted coverings such that ϕ = ϕ1 × ϕ2 : C → C1 × C2 is birational onto its image. Then g ≤ (d1 − 1)(d2 − 1) + d1g1 + d2g2.

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 ℙr. 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 ON h-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 > 4h + 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 is base point free and not compounded of π, then k > g − 2h.

Proof. The k-sheeted map and 2-sheeted map π : C → Ch 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−2h.   ☐

Theorem 2.2. Let C be an h-hyperelliptic curve of genus g ≥ 6h + 2 with a double covering map π : C → Ch. 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 ≤ 2h + 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 = {n0 = 0, n1, ··· , nh = 2h}. For some i, ni < α < ni+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 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, ··· , 2h} and 2h, 2h + 1 ∈ H. Then H contains any integers k ≥ 2h.

Proof. First, we show that 2h + 2 ∈ H. The set I2h+1 = {1, 2, ⋯ , 2h, 2h + 1} has h + 1 elements of H. Consider a partition of I2h+1

If h + 1 ∈ H, then 2h + 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 2h + 2 ∈ H.

Next, we show that 2h + 3 ∈ H. The set I2h+2 = {1, 2, ··· , 2h, 2h + 1, 2h + 2} has h + 2 elements of H. Consider a partition of I2h+2

Then at least of one is contained in H and hence 2h + 3 ∈ H.

Repeating this process, we conclude that k ∈ H for all k ≥ 2h.   ☐

Theorem 2.5. Let C be a curve of genus g ≥ 6h + 4 and P, Q ∈ C. Assume that H(P, Q) contains exactly h terms in {(1, 1),(2, 2), ··· ,(2h, 2h)} and that

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 ≥ 2h. By Lemma 2.3,

Let s + 1 = dim |(3h + 1)(P + Q)| and let’s denote |(3h + 1)(P + Q)| by . Consider a rational map ϕ : C → ℙs+1 defined by .

Claim: s = 2h.

Suppose that s ≥ 2h + 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' = 3h − s < h. Here we have two possibilities

Subclaim: ϕ(P) = ϕ(Q).

If ϕ(P) ≠ ϕ(Q), then ϕ* (ϕ(P)) = 2P and ϕ* (ϕ(Q)) = 2Q, since the divisor (3h + 1)(P + Q) is the pull-back of some divisor on C' via ϕ. In this case, 3h + 1 must be even and hence h is odd. Consider a linear series |(3h + 2)(P + Q)| and let its dimension be u + 1. Then s + 2 ≥ u ≥ s + 1 ≥ 2h + 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 |(3h + 2)(P + Q)| is compounded of the latter map ϕ'. Since h is odd, 3h + 2 is also odd. Hence ϕ'*(ϕ'*(P)) = P + Q. Now ϕ × ϕ' is birational, and by Lemma 1.4, we have g ≤ 1 + 4h 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 = 2h.

Now we have a complete linear series and a rational map ϕ : C → ℙ2h+1 induced from . Suppose ϕ is birational. Then by Lemma 1.5, we get g(C) ≤ 6h + 3 which contradicts the assumption g ≥ 6h + 4.

Thus ϕ is a double covering map from C to ϕ(C) with g(ϕ(C)) = h. Therefore C is h-hyperelliptic. Since |(2h + 1)(P + Q)| and |2h(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 ≥ 6h + 5. Suppose that (2h, 2h), (2h + 1, 2h + 1) ∈ H(P, Q) and dim(2h, 2h) = h, dim(2h + 1, 2h + 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 → ℙh+1 defined by the linear series

If ϕ is birational, then g ≤ 6h + 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 2h + 1 is odd and the divisor (2h + 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 (4h + 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)) = 2P + Q or P + 2Q. Then (2h + 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 (2h, 2h) ∈ H(P, Q) by assumption, we have (2h + 1, 2h) ∈ H(P, Q) by Lemma 1.2. Thus

which contradicts the assumption dim(2h + 1, 2h + 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 ≥ 6h + 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 → Ch with

Proof. If n = 2h + 1, we already proved in Theorem 2.7. Now we assume n ≥ 2h + 2.

Let n be a number such that , (n, n) ∈ H(P, Q) and and ϕn : C → ℙn−h be a rational map defined by .

Claim 1: ϕn is not birational if n ≥ 2h + 2.

Suppose that ϕn : C → ℙn−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 = 2h + 1 then which is a contradiction. If m = 3 and є = −n + 3h + 2 then g ≤ 6h + 3 < g which is a contradiction again. Thus ϕn is not birational if n ≥ 2h + 2.

Let deg ϕn = t ≥ 2. Since ϕn is nondegenerate, 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 2n is a multiple of 3 and there is a complete and nonspecial on C' = ϕn(C). Hence the genus of C' is . If ϕn(P) = ϕn(Q), then and the pullback of a multiple of ϕ(P) can not be n(P + Q). Thus we have ϕn(P) ≠ ϕn(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 ϕn = t = 2 and there is a complete, nonspecial 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 . 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 ≥ 2h + 1 and n is even, n ≥ 2h + 2 and dim |(n−1)(P + Q)| = (n−1)−h and (n−1, n−1) ∈ H(P, Q) by the assumption on n. Consider ϕn−1 which is defined by . By Castelnuovo’s bound, ϕn−1 is not birational and deg ϕn−1 = 2 or 3. If deg ϕn−1 = 3, there is a complete, nonspecial on C'' = ϕn−1(C). So . Then the 3:1 map and the 2:1 map ϕn : C → Ch induce a map which is birational onto its image. By Lemma 1.4, which is a contradiction. Thus deg ϕn−1 = 2 and there is a complete, nonspecial on ϕn−1(C). In this case g(ϕn−1(C)) = h. Let . Since ϕn(P) ≠ ϕn(Q) and is birational onto its image. Again by Lemma 1.4, g(C) ≤ (2 − 1)(2 − 1) + 2h + 2h = 4h + 1 < g which is a contradiction.

Thus we have π(P) = π(Q) and the last assertion follows from Theorem 2.2.   ☐