Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 43 (2002), No. 1, 209-216.

Weierstrass Gaps and Curves on a Scroll

Cícero Carvalho

Departmento de Matemática, Universidade Federal de Uberlândia av. J. N. de Ávila 2160, 38408-100 Uberlândia-MG, Brazil e-mail: cicero@ufu.br

Abstract. The aim of this paper is to study the Weierstrass semigroup of ramified points on non-singular models for curves on a rational normal scroll. We find properties of this semigroup and determine it in some special cases, finding also a geometrical interpretation for some of the Weierstrass gaps. MSC 2000: 14H55 (primary), 14H50 (secondary)

Introduction

The Weierstrass gap sequences at ramification points of a (non-singular) trigonal curve have been determined by Coppens in [4] and [5]. These sequences also appeared in a work by Stöhr and Viana (cf. [12]), where they were both obtained by a method based on the fact that trigonal curves are canonically immersed on a rational normal scroll (Coppens had already used this fact in [5]). On the other hand, Weierstrass gap sequences at non-singular points of a singular plane curve (or, more precisely, at the inverse image of the non-singular point by the normalization morphism over the curve) have been studied in recent papers (e.g. [6], [7], [2]), specially when the non-singular point is ramified with respect to some morphism over the projective line. In the present work, we study the Weierstrass gap sequences at non-singular ramification points of possibly singular curves on a rational normal scroll, generalizing the results in [4] and [5] (the ramification being with respect to the morphism over the projective line defined by a ruling of the scroll). Also, we obtain a geometrical interpretation for some gaps, when the singularity locus of the curve is contained in the directrix of the scroll, and contains only simple cusps or simple nodes.

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 2002 Heldermann Verlag

1. Divisors on curves on a scroll

A rational normal scroll S_{m n} ⊂ ℙ^{m+n+1}(k) defined over an algebraically closed field k is a surface which after a suitable choice of projective coordinates is given by

\begin{matrix}S_{m n} := \{ (x_{0} : …: x_{m+n+1}) ∈ ℙ^{m+n+1}(k) | \\ rank ( \begin{matrix}x_{0} & ⋯ & x_{n−1} & x_{n+1} & ⋯ & x_{n+m} \\ x_{1} & ⋯ & x_{n} & x_{n+2} & ⋯ & x_{n+m+1} \end{matrix} ) < 2 \} \end{matrix}

where the positive integers m and n are such that m ≤ n.

S_{m n} has a ruling given by the union of the disjoint lines

L_{b/a} := \overline{(a^{n} : a^{n−1}b : …: b^{n} : 0 : …: 0) , (0 : …: 0 : a^{m} : a^{m−1}b : …: b^{m})},

where b/a ∈ ℙ^{1}(k) = k ∪ {∞}, which join points of the non-singular rational curves

D := {(a^{n} : a^{n−1}b : …: b^{n} : 0 : …: 0) ∈ ℙ^{m+n+1}(k) | (a : b) ∈ ℙ^{1}(k)} and
E := {(0 : …: 0 : a^{m} : a^{m−1}b : …: b^{m}) ∈ ℙ^{m+n+1}(k) | (a : b) ∈ ℙ^{1}(k)}.

Following [12] we cover S_{m n} with four affine open sets, all isomorphic to 𝔸^{2}(k) and defined by

\begin{matrix}U_{0} := S_{m n} \ (L_{∞} ∪ E) = \\ \{(a^{0} : …: a^{n} : a^{0}b : …: a^{m}b) ∈ ℙ^{m+n+1}(k) | (a, b) ∈ 𝔸^{2}(k)\}, \end{matrix}

\begin{matrix}U_{n} := S_{m n} \ (L_{0} ∪ E) = \\ \{(a^{n} : …: a^{0} : a^{m}b : …: a^{0}b) ∈ ℙ^{m+n+1}(k) | (a, b) ∈ 𝔸^{2}(k)\}, \end{matrix}

\begin{matrix}U_{n+1} := S_{m n} \ (L_{∞} ∪ D) = \\ \{(a^{0}b : …: a^{n}b : a^{0} : …: a^{m}) ∈ ℙ^{m+n+1}(k) | (a, b) ∈ 𝔸^{2}(k)\}, \end{matrix}

\begin{matrix}U_{n+m+1} := S_{m n} \ (L_{0} ∪ D) = \\ \{(a^{n}b : …: a^{0}b : a^{m} : …: a^{0}) ∈ ℙ^{m+n+1}(k) | (a, b) ∈ 𝔸^{2}(k)\}. \end{matrix}

Associating to each affine curve in 𝔸^{2}(k) the Zariski closure of its image in U_{0} under the isomorphism (a : b) ↦ (a^{0} : …: a^{n} : a^{0}b : …: a^{m}b) we get a bijection between affine plane curves and the projective curves on S_{m n} that do not have L_{∞} or E as a component (we do not assume that a curve is irreducible, unless explictly stated).

We deal in this paper with (possibly) singular curves and divisors on them, following in this matter [11] (cf. also [9]). Thus let C be an integral curve defined over k and let k(C) be its function field, a divisor D on C is a non-zero coherent fractional ideal sheaf of C, which we denote by the product of its stalks D = ∏_{P ∈C} D_{P}. We denote by O the structure sheaf of C. The local degree at P ∈ C of D is the integer deg_{P} (D) defined by requiring that deg_{P} (O) = 0 and that deg_{P} (D) − deg_{P} (E) = dim_{k}(D_{P} /E_{P}) whenever D_{P} ⊇ E_{P}. The degree of D is the integer deg(D) := ∑_{P ∈C} deg_{P} (D). The divisor of a rational function h ∈ k(C)^{∗} is defined by div h := ∏_{P ∈C}(1/h)O_{P}. If F is a (Cartier) divisor on S_{m n} and C ⊂ S_{m n} is not a component of F then we define the intersection divisor of C and F as C · F := ∏_{P ∈C}(1/f_{P})O_{P}, where F is locally defined by f_{P} on a open set containing P. We observe that the local degree at P of C · F coincides with the intersection number i(C, F; P) of C and F at P as divisors on S_{m n}.

We also note that the divisors on a singular curve are not necessarily locally principal, i.e. of the form D = ∏ d_{P} O_{P}, where d_{P} ∈ k(C)^{∗} for all P ∈ C (cf. [9, Ex. 1.6.1] or [3, Ex. 2.4]) and they do not form a group under the operation defined by D ∗ E := ∏ D_{P} E_{P}. Nevertheless, the locally principal divisors do form a commutative group under this operation and since the divisors on C appearing on this paper are all (intersection divisors and hence) locally principal we will denote this operation as a sum, thus ∏ d_{P} O_{P} + ∏ e_{P} O_{P} = ∏(d_{P} e_{P})O_{P} and ∏ d_{P} O_{P} − ∏ e_{P} O_{P} = ∏(d_{p}/e_{P})O_{P}. Accordingly, instead of ∏ d_{P} O_{P} ⊇ ∏ O_{P} we write ∏ d_{P} O_{P} ≥ 0 and say that ∏ d_{P} O_{P} is a non-negative divisor. Two divisors D and E on C are linearly equivalent if D − E = div h for some h ∈ k(C)^{∗} and the set |K| of all non-negative divisors linearly equivalent to a canonical divisor K on C is called the canonical linear series of C.

Now let C be a curve on S_{m n} that does not have E or L_{∞} as a component and let c_{ℓ}(X)Y^{ℓ} + c_{ℓ−1}Y^{ℓ−1} + ⋯ + c_{0}(X) = 0 be the equation of the affine curve that corresponds to C ∩ U_{0} under the isomorphism 𝔸^{2}(k) ≃ U_{0} described above. Then deg(C · L_{a}) = ℓ for all a ∈ k ∪ {∞}, deg(C · E) = d_{ℓ} and deg(C · D) = d_{ℓ} + ℓ(n − m), where d_{ℓ} is the smallest integer such that deg c_{i}(X) ≤ d_{ℓ} + (ℓ − i)(n − m) for all i ∈ {0, …, ℓ} (and hence the equality holds for some i). The Picard group of S_{m n} is the free group generated by the classes of D and a line L, and the canonical divisor of S_{m n} is linearly equivalent to −2D + (n − m − 2)L (cf. [1, page 121]). From this we may deduce that C ∼ ℓD + d_{ℓ}L, where ∼ denotes the linear equivalence of divisors on S_{m n} and, if C is irreducible, from the adjunction formula 2g−2 = C ·(C +(n−m−2)L−2D) (cf. [10, page 75]) we get g = (ℓ−1)(2d_{ℓ}+ℓ(n−m)−2)/2, where g is the arithmetic genus of C. In what follows L will always denote a line of the ruling on S_{m n}. We recall that any two lines of the ruling on S_{m n} are linearly equivalent and we also have E ∼ D − (n − m)L (cf. [12]).

Theorem 1.1. The divisors of the canonical linear series of an irreducible curve C ∈ S_{m n} are exactly the intersections of C with curves linearly equivalent to (ℓ−2)E +(d_{ℓ} +(ℓ−1)(n− m) − 2)L.

Proof. Let x and y be the rational functions defined on C ∩ U_{0} by (a^{0} : …: a^{n} : a^{0}b : …: a^{m}b) ↦ a and (a^{0} : …: a^{n} : a^{0}b : …: a^{m}b) ↦ b, respectively and let K := (ℓ−2)C ·E +(d_{ℓ} + (ℓ−1)(n−m)−2)C·L_{∞}. We have div x = C·L_{0}−C·L_{∞} and div y = C·D−C·E−(n−m)C·L_{∞}, thus {x^{i}y^{j} | 0 ≤ j ≤ ℓ − 2, 0 ≤ i ≤ d_{ℓ} + (ℓ − 1 − j)(n − m) − 2} ⊂ H^{0}(K). The degree of K is (ℓ − 2)d_{ℓ} + (ℓ − 1)ℓ(n − m) + (d_{ℓ} − 2)ℓ = 2g − 2 hence the set of the g linearly independent elements x^{i}y^{j} form a basis for H^{0}(K) and K is canonical divisor of C. Now let f := ∑_{j=0}^{ℓ−2} ∑_{i=0}^{d_{ℓ}+(ℓ−1−j)(n−m)−2} a_{i j}x^{i}y^{j} be a non-zero element of H^{0}(K), let r be the greatest integer such that a_{i r} ≠ 0 for some i and let e_{r} be the least non-negative integer satifying max{i | a_{i j} ≠ 0; i = 0, …, d_{ℓ} +(ℓ−1−j)(n−m)−2} ≤ e_{r} +(r −j)(n−m) for all j = 0, …, r such that a_{i j} ≠ 0 for some i. Then 0 ≤ e_{r} ≤ d_{ℓ} +(ℓ−1−r)(n−m)−2, a_{i j} = 0 if j > r or i > e_{r} +(r−j)(n−m) and let F be the curve on S_{m n} whose correspondent curve on 𝔸^{2}(k) ≃ U_{0} is ∑_{j=0}^{r} ∑_{i=0}^{e_{r}+(r−j)(n−m)} a_{i j}X^{i}Y^{j} = 0. We claim that div(∑_{j=0}^{r} ∑_{i=0}^{e_{r}+(r−j)(n−m)} a_{i j}x^{i}y^{j})+K is the intersection divisor of C and G := (ℓ−2−r)E+(d_{ℓ}+(ℓ−1−r)(n−m)−2−e_{r})L_{∞}+F. In fact, if P ∈ C ∩ U_{0} then (C · G)_{P} = (1/ ∑_{j=0}^{r} ∑_{i=0}^{e_{r}+(r−j)(n−m)} a_{i j}x^{i}y^{j})O_{P} and the claim holds because K_{P} = O_{P}. Suppose now that P ∈ C∩U_{n+m+1} and let x̃ and ỹ be the rational functions defined on C ∩ U_{n+m+1} by (a^{n}b : …: a^{0}b : a^{m} : …: a^{0}) ↦ a and (a^{n}b : …: a^{0}b : a^{m} : …: a^{0}) ↦ b respectively, we have x = 1/x̃ and y = 1/(x̃^{(n−m)}ỹ) on C ∩ U_{0} ∩ U_{n+m+1}. Let (a^{n}b : …: a^{0}b : a^{m} : …: a^{0}) ↦ (a, b) be an isomorphism between U_{n+m+1} and 𝔸^{2}(k) and let X̃ and Ỹ be the affine coordinates in 𝔸^{2}(k), then F ∩U_{n+m+1}, E ∩U_{n+m+1} and L_{∞}∩U_{n+m+1} correspond to the plane curves given by ∑_{j=0}^{r} ∑_{i=0}^{e_{r}+(r−j)(n−m)} a_{i j}X̃^{e_{r}+(r−j)(n−m)−i}Ỹ^{r−j} = 0, Ỹ = 0 and X̃ = 0 respectively. Now it is easy to check that (C · G)_{P} = (div(∑_{j=0}^{r} ∑_{i=0}^{e_{r}+(r−j)(n−m)} a_{i j}x^{i}y^{j}) + K)_{P} = (1/ ∑_{j=0}^{r} ∑_{i=0}^{e_{r}+(r−j)(n−m)} a_{i j} x̃^{d_{ℓ}+(l−1−j)(n−m)−2−i} ỹ^{ℓ−2−j})O_{P}. The proof of the claim for P ∈ U_{n} and P ∈ U_{n+1} is similar. Thus any divisor in |K| is the intersection of C and a curve linearly equivalent to (ℓ − 2)E + (d_{ℓ} + (ℓ − 1)(n − m) − 2)L.

Conversely, if H is a curve linearly equivalent to (ℓ−2)E +(d_{ℓ} +(ℓ−1)(n−m)−2)L then we may write H = sE + tL_{∞} + G, with s and t non-negative integers, G a curve that does not have E or L_{∞} as a component, and G ∼ (ℓ − 2 − s)E + (d_{ℓ} + (ℓ − 1)(n − m) − 2 − t)L ∼ (ℓ−2−s)D+(d_{ℓ} +(s+1)(n−m)−2−t)L. Thus G∩U_{0} is an affine curve given in 𝔸^{2}(k) ≃ U_{0} by an equation of the form ∑_{j=0}^{ℓ−2−s} ∑_{i=0}^{d_{ℓ}+(s+1)(n−m)−2−t} a_{i j}X^{i}Y^{j} = 0, and as above one may check that div (∑_{j=0}^{ℓ−2−s} ∑_{i=0}^{d_{ℓ}+(s+1)(n−m)−2−t} a_{i j} x^{i} y^{j}) + K = C · H. This completes the proof of the theorem. □

2. Weierstrass gaps at ramification points

From now on C will always denote an irreducible curve on S_{m n}. Let η : C̃ → C be the normalization of C, let P̃ ∈ C̃ and let K̃ be a canonical divisor on C̃. The set of positive integers W G(P̃) := {1 + dim_{k} DP̃/OP̃ | D ∈ |K̃|} is called the Weierstrass gap sequence at P̃. The cardinality of this set is equal to the genus of C̃ and its complementary in the set of the non-negative integers is called the Weierstrass semigroup at P̃ (cf. [13]). Let P ∈ C be a non-singular point and let P̃ = η^{−1}(P). In this case we will refer to the set W G(P̃) as the Weierstrass gap sequence at P and write W G(P). Also, if T is the line of the ruling passing through P and r := i(C, T; P) then we say that P is an r-ramification point of C. We want to determine W G(P) at r-ramification points of C ∼ ℓD + d_{ℓ}L for r = ℓ, ℓ − 1 (observe that r ≤ deg(C · L) = ℓ). Let’s begin with the case where C is non-singular.

Theorem 2.1. Let C be a non-singular curve on a scroll S_{m n} such that C ∼ ℓD + d_{ℓ}L. Let P ∈ C be an r-ramification point with r ≥ 2 and let W G(P) be the Weierstrass gap sequence at P.

a) If P ∉ E then {ir +j +1 | j = 0, 1, …, ℓ−2 ; i = 0, 1, …, d_{ℓ} +(ℓ−1−j)(n−m)−2} ⊆ W G(P) and equality holds when r ∈ {ℓ, ℓ − 1}.

b) If P ∈ E then {ir+ℓ−1−j | j = 0, 1, …, ℓ−2 ; i = 0, 1, …, d_{ℓ}+(ℓ−1−j)(n−m)−2} ⊆ W G(P) and equality holds when r ∈ {ℓ, ℓ − 1}.

Proof. Let T be the line of the ruling through P. After a suitable automorphism of S_{m n} we may assume that P = T ∩ D, if P ∉ E (cf. [12, Prop. 1.2]) and of course P = T ∩ E, if P ∈ E. Since i(C, T; P) ≥ 2 we have i(C, D; P) = 1, if P ∉ E or i(C, E; P) = 1, if P ∈ E. Let L ≠ T be another line of the ruling and hence P ∉ L. From Theorem 1.1 we get that

W G(P) ⊇ {1 + i(C, jD + (ℓ − 2 − j)E + (d_{ℓ} + (ℓ − 1 − j)(n − m) − 2 − i)L +
iT; P) | 0 ≤ j ≤ ℓ − 2, 0 ≤ i ≤ d_{ℓ} + (ℓ − 1 − j)(n − m) − 2}.

The right hand side set is equal to {ir+j+1 | 0 ≤ j ≤ ℓ−2, 0 ≤ i ≤ d_{ℓ}+(ℓ−1−j)(n−m)−2} if P ∉ E, or is equal to {ir+ℓ−1−j | 0 ≤ j ≤ ℓ−2, 0 ≤ i ≤ d_{ℓ}+(ℓ−1−j)(n−m)−2} if P ∈ E. Moreover, if r ∈ {ℓ, ℓ − 1} these sets have cardinality equal to (ℓ − 1)(2d_{ℓ} + ℓ(n − m) − 2)/2 which is the genus of C and hence equality holds in either case. □

From now on we do not supppose that C is a smooth curve. Let F be the conductor divisor on C defined by F_{P} = (O_{P} : Õ_{P}) for all P ∈ C, where Õ_{P} is the integral closure of O_{P} in k(C). We call a divisor F on S_{m n} an adjoint curve if F ∼ (ℓ−2)E +(d_{ℓ} +(ℓ−1)(n−m)−2)L and Õ_{P} ⊆ (F · C)_{P} F_{P} for all P ∈ C. If Q is a singular point of C then F_{Q} ⊂ M_{Q}, where M_{Q} is the maximal ideal of O_{Q}, and if f_{Q} defines an adjoint curve F locally in an open set of S_{m n} containing Q we get Õ_{Q}f_{Q} ⊂ F_{Q} ⊂ M_{Q}, thus F intersects C at Q. Exactly as in the case of plane curves one may show that the divisors of the canonical series of C̃ are the scheme theoretic inverse image under η of the divisors ∏(F · C)_{P} F_{P}, where F is an adjoint curve. At a non-singular point P ∈ C we have (F · C)_{P} F_{P} = (F · C)_{P} since F_{P} = O_{P}, thus from the preceeding theorem we obtain the following result.

Lemma 2.2. Let P ∈ C ∼ ℓD + d_{ℓ}L be an r-ramification point, where r ∈ {ℓ, ℓ − 1}.

a) If P ∉ E then W G(P) ⊂ {ir + j + 1 | j = 0, 1, …, ℓ − 2 ; i = 0, 1, …, d_{ℓ} + (ℓ − 1 − j)(n − m) − 2}.

b) If P ∈ E then W G(P) ⊂ {ir + ℓ − 1 − j | j = 0, 1, …, ℓ − 2 ; i = 0, 1, …, d_{ℓ} + (ℓ − 1 − j)(n − m) − 2}.

The next result shows that the so called Namba’s Lemma holds for curves on S_{m n}.

Lemma 2.3. Let C, C_{1} and C_{2} be curves on a scroll S_{m n} and let P ∈ S_{m n} be a non-singular point of C. Then i(C_{1}, C_{2}; P) ≥ min{i(C, C_{1}; P), i(C, C_{2}; P)}.

Proof. Let F = 0, G_{1} = 0 and G_{2} = 0 be local equations for C, C_{1} and C_{2} respectively, in an open affine subset of S_{m n} isomorphic to 𝔸^{2}(k). For i ∈ {1, 2} we get i(C, C_{i}; P) = dim_{k} O_{𝔸^{2}(k),P}/(F, G_{i}) = dim_{k} O_{C,P} /(g_{i}) = ord_{P} (g_{i}) where g_{i} ∈ k(C) is the rational function determined by the polynomial G_{i}. Then i(C_{1}, C_{2}; P) = dim_{k} O_{𝔸^{2}(k),P}/(G_{1}, G_{2}) ≥ dim_{k} O_{𝔸^{2}(k),P}/(F, G_{1}, G_{2}) = dim_{k} O_{C,P} /(g_{1}, g_{2}) = min{ord_{P} (g_{1}), ord_{P} (g_{2})} = min{i(C, C_{1}; P), i(C, C_{2}; P)}. □

Theorem 2.4. Let P ∈ C ⊂ S_{m n} be a non-singular r-ramification point and let T be the line of the ruling passing through P. If F ∼ sE + tL is a divisor of S_{m n} such that s < r and i(C, F; P) ≥ r then F = T + G and G ∼ sE + (t − 1)L.

Proof. From the above Lemma i(F, T; P) ≥ min{i(C, T; P), i(C, F; P)} ≥ r but deg(F ·T) = s deg(E · T) + t deg(L · T) = s < r. Then (cf. [8, page 360]) F and T must have a common irreducible component so F = T + G for some divisor G ⊂ S_{m n} and G ∼ sE + (t − 1)L. □

Corollary 2.5. Let P ∈ C ⊂ S_{m n} be an r-ramification point of C ∼ ℓD + d_{ℓ}L where r ∈ {ℓ, ℓ − 1}. If ir + s is a Weierstrass gap at P, with s and i positive integers, then {(i − 1)r + s, …, r + s, s} are also Weierstrass gaps at P.

Proof. Since ir + s is a Weierstrass gap at P there exists an adjoint curve F such that i(C, F; P) = ir + s − 1. Let T be the line of the ruling passing through P, from the above Theorem we get that F = T + G where G ∼ (ℓ − 2)E + (d_{ℓ} + (ℓ − 1)(n − m) − 3)L and i(C, G; P) = (i−1)r+s−1. From deg(C ·T) = ℓ and r ∈ {ℓ, ℓ−1} we get that T intersects C at most at another non-singular point so if T′ is a line of the ruling not containing P then the curve G + T′ is an adjoint curve. From i(C, G + T′; P) = i(C, G; P) we get that the integer (i − 1)r + s is a Weierstrass gap at P. Thus the corollary follows from repeated applications of the above theorem. □

In view of the above proof, if ir + s ∈ W G(P) one would expect iT to be a component of an adjoint curve that yields this gap. The result below shows that if m < n then the curve E is also a component of many adjoint curves.

Theorem 2.6. Let C ∼ ℓD + d_{ℓ}L be a singular curve on S_{m n}, where m < n and let T be the line of the ruling passing through the r-ramification point P, with r ∈ {ℓ, ℓ − 1}. If F is an adjoint curve such that i(C, F; P) = ir+j or i(C, F; P) = ir+ℓ−2−j, where j ∈ {0, …, ℓ−3} and i ∈ {d_{ℓ} + (n − m) − 1, …, d_{ℓ} + (ℓ − 1 − j)(n − m) − 2}, then F = iT + sE + H, where s is the integer satisfying d_{ℓ} + s(n − m) − 2 < i ≤ d_{ℓ} + (s + 1)(n − m) − 2 and H is an effective divisor of S_{m n}.

Proof. Let F be an adjoint curve such that i(C, F; P) = ir + j or i(C, F; P) = ir + ℓ − 2 − j, with i and j as in the theorem. After successive applications of Theorem 2.4 we get F = iT + G, where G ∼ (ℓ − 2)E + (d_{ℓ} + (ℓ − 1)(n − m) − 2 − i)L ∼ (ℓ − 2)D + (d_{ℓ} + (n − m) − 2 − i)L. As we remarked in Section 1, a curve that does not have E or L_{∞} as a component is linearly equivalent to a divisor aD + bL of S_{m n}, with a ≥ 0 and b ≥ 0. Since d_{ℓ} + (n − m) − 2 − i < 0 the curve G must have E as a component, so G = E + G_{1} where G_{1} ∼ (ℓ − 3)D + (d_{ℓ} + 2(n − m) − 2 − i)L. We repeat this argument s times to obtain G = sE + G_{s} with G_{s} ∼ (ℓ − 2 − s)D + (d_{ℓ} + (s + 1)(n − m) − 2 − i)L. □

Taking into account that C_{Sing} is contained in every adjoint curve, we may interpret geometrically the greatest possible integers in W G(P) as follows.

Corollary 2.7. Let C ∼ ℓD + d_{ℓ}L be a singular curve on S_{m n} and let P be a non-singular r-ramification point, where r ∈ {ℓ, ℓ − 1}.

a) If m < n and (d_{ℓ}+(ℓ−1)(n−m)−2)r+1 ∈ W G(P) or (d_{ℓ}+(ℓ−1)(n−m)−2)r+ℓ−1 ∈ W G(P) then C_{Sing} ⊂ E.

b) If m = n and (d_{ℓ} − 2)r + ℓ − 1 ∈ W G(P) then C_{Sing} is contained in a curve linearly equivalent to D that also contains P.

Proof. To prove (a) we take i = (d_{ℓ} + (ℓ − 1)(n − m) − 2) and j = 0 in the above theorem and get s = ℓ − 2. Thus F = (d_{ℓ} + (ℓ − 1)(n − m) − 2)T + (ℓ − 2)E and we must have C_{Sing} ⊂ E. To prove (b) we use Theorem 2.4 to obtain an adjoint curve F = (d_{ℓ} − 2)T + G where G ∼ (ℓ − 2)D and C_{Sing} ⊂ G. Since n = m the curves linearly equivalent to D are exactly E and the curves given in 𝔸^{2}(k) ≃ U_{0} by the equations Y − b = 0, with b ∈ k; it is easy to check that these curves do not intersect each other. Since i(C, G; P) = ℓ − 2 we must have G = (ℓ − 2)D′, where D′ is a curve linearly equivalent to D that contains P and C_{Sing}. □

The following result determines W G(P) for curves satisfying certain restrictions on the singularities.

Proposition 2.8. Let C ∼ ℓD + d_{ℓ}L be a curve of singularity degree δ on the scroll S_{m n}. Let P ∈ C be an r-ramification point, where r ∈ {ℓ, ℓ−1}. Suppose that C_{Sing} ⊂ E if m < n, or that C_{Sing} is contained in a curve linearly equivalent to D, if m = n. Suppose also that the singularities of C are either simple nodes or simple cusps.

a) If m < n and P ∉ E, or if m = n and P and C_{Sing} are not contained in a curve linearly equivalent to D, then W G(P) = {ir + j + 1 | 0 ≤ j ≤ ℓ − 3; 0 ≤ i ≤ d_{ℓ} + (ℓ − 1 − j)(n − m) − 2} ∪ {ur + ℓ − 1 | 0 ≤ u ≤ d_{ℓ} + (n − m) − 2 − δ}.

b) If m < n and P ∈ E, or if m = n and P and C_{Sing} are contained in a curve linearly equivalent to D, then W G(P) = {ir + ℓ − 1 − j | 0 ≤ j ≤ ℓ − 3; 0 ≤ i ≤ d_{ℓ} + (ℓ − 1 − j)(n − m) − 2} ∪ {(d_{ℓ} + (n − m) − 2 − δ − u)r + 1 | 0 ≤ u ≤ d_{ℓ} + (n − m) − 2 − δ}.

Proof. Let F be the conductor divisor of C. We recall that C is a Gorenstein curve, for it lies on a surface, and thus the degree of singularity of a point Q ∈ C is equal to dim_{k}(O_{Q}/F_{Q}). From F_{Q} ⊂ M_{Q}, where M_{Q} is the maximal ideal of O_{Q}, and the hypothesis on the singularities we get F_{Q} = M_{Q} for all singular points of C (thus M_{Q}, as F_{Q}, is not only an O_{Q}-module but also an Õ_{Q}-module). If F is a curve intersecting C at a singular point Q and f_{Q} defines F locally on an open set of S_{m n} containing Q then f_{Q} ∈ M_{Q} and hence Õ_{P} f_{Q} ⊂ F_{Q}, i.e. Õ_{P} ⊂ (F · C)_{Q}F_{Q}. This shows that any curve F ∼ (ℓ − 2)E + (d_{ℓ} + (ℓ − 1)(n − m) − 2)L passing through all the singular points of C is an adjoint curve.

If m = n then there exists an automorphism of the scroll taking a given curve linearly equivalent to D onto E (cf. [12]), so we may assume that C_{sing} ⊂ E. Let T be the line of the ruling that contains P. If P ∉ E, let D′ be a curve linearly equivalent to D containing P, then P = T ∩ D′; if P ∈ E then P = T ∩ E. Let L_{1}, …, L_{δ} be the lines of the ruling passing through the points in C_{Sing}. Let L be a line of the ruling different from T. To obtain the Weierstrass gaps listed in the theorem it suffices to calculate the local degree at P of the intersection divisor of C and the adjoint curves iT + (ℓ − 2 − j)E + jD′ + (d_{ℓ} + (ℓ − 1 − j)(n − m) − 2 − i)L, where 0 ≤ j ≤ ℓ − 3, 0 ≤ i ≤ d_{ℓ} + (ℓ − 1 − j)(n − m) − 2 and (ℓ−2)D′+(d_{ℓ}+(n−m)−2−δ−u)T +(u+1)L_{1}+L_{2}+⋯+L_{δ}, where 0 ≤ u ≤ d_{ℓ}+(n−m)−2−δ. Using r ∈ {ℓ, ℓ−1} one may check that we get (ℓ−1)(2d_{ℓ}+ℓ(n−m)−2)/2−δ = g−δ distinct numbers (where g is the arithmetic genus of C) and this is the cardinality of W G(P). □

The next result follows from the above proposition and Corollary 2.7.

Corollary 2.9. Let C ∼ ℓD + d_{ℓ}L be a curve on S_{m n}, whose singularities are only simple nodes or simple cusps. Let P be a non-singular r-ramification point of C, where r ∈ {ℓ, ℓ−1}.

a) If m < n and P ∉ E then C_{Sing} ⊂ E if and only if (d_{ℓ} + (ℓ − 1)(n − m) − 2)r + 1 is a Weierstrass gap at P.

b) If m < n and P ∈ E then C_{Sing} ⊂ E if and only if (d_{ℓ} + (ℓ − 1)(n − m) − 2)r + ℓ − 1 is a Weierstrass gap at P.

c) If m = n then P and C_{Sing} are on a curve that is linearly equivalent to D if and only if (d_{ℓ} − 2)r + ℓ − 1 is a Weierstrass gap at P.

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Received July 10, 2000