MATEMATIQKI VESNIK

58 (2006), 71–75

UDK 511.311

originalni nauqni rad research paper

COVERING OF CURVES, GONALITY, AND SCROLLAR INVARIANTS

E. Ballico

Abstract. Let f: X → Y be a degree k covering of smooth and connected projective curves with p_{a}(Y) > 0. Here we continue the study of the Brill-Noether theory of divisors on X.

1. Introduction

Let X (resp. Y) be a smooth and connected curve of genus g (resp. genus q) and f: X → Y a degree k covering, k ⩾ 2. Thus g ⩾ kq−k+1 (Riemann-Hurwitz). Let u: X → P^{1} be a degree z morphism computing the gonality gon(X) of X. We always have z ⩽ k · gon(Y) and if z = k · gon(Y), then at least one degree z pencil X → P^{1} factors through f. By Brill-Noether theory we have gon(X) ⩽ ⌊(g + 3)/2⌋ and gon(Y) ⩽ ⌊(q + 3)/2⌋. Hence z ⩽ min{⌊(g + 3)/2⌋, k · ⌊(q + 3)/2⌋}. If z ⩽ (g − kq)/(k − 1), then u factors through f by Castelnuovo-Severi inequality [5]. In the first part we will consider several examples in which u does not factor through f and study their scrollar invariants in the sense of [3]. To state our first result we need the following notation/observation.

Remark 1. Let f: X → Y be a finite morphism between smooth and connected projective curves and D = ∑ n_{i}P_{i} any divisor on X. Set f_{!}(D) := ∑ n_{i}f(P_{i}). A key property of rational equivalence says that if D and D′ are linearly equivalent divisors on X, then f_{!}(D) and f_{!}(D′) are linearly equivalent divisors on Y; here the smoothness of Y is essential, because it implies that rational equivalence and linear equivalence are the same on Y. Hence for any d ∈ Z the map f_{!} induces a map f_{!} : Pic^{d}(X) → Pic^{d}(Y) such that h^{0}(Y, f_{!}(L)) ⩾ h^{0}(X, L) for all L ∈ Pic^{d}(X). Furthermore, if L is base point free, then f_{!}(L) is base point free.

A modification of the proof of [2], Th. 1, (see section 2) will give the following result.

AMS Subject Classification: 14 H 51; 14 H 50

Keywords and phrases: Covering of curves; gonality; scrollar invariant; hyperelliptic curve; Brill-Noether theory; ruled surface.

The author was partially supported by MIUR and GNSAGA of INdAM (Italy).

Theorem 1. Fix integers d, q, k, g such that q > 0, k ⩾ 2, g ⩾ kq − k + 1 and kd − d − k + kq + 1 − (⌊d/2⌋ + 1 − q) · (⌊k/2⌋ + 1) ⩽ g ⩽ kd − d − k + kq + 1. Set a := kd − d − k + kq + 1 − g and let δ(a) be the maximal integer t such that (k−1−t)(d+q−1) ⩾ a. Let Y be a smooth and connected curve of genus q such that there is a base point free M ∈ Pic^{d}(Y). Then there exist a smooth and connected genus g curve X, a degree k covering f: X → Y and a base point free L ∈ Pic^{d}(X) such that f_{!}(L) ≅ M and there is no base point free R ∈ Pic(Y) such that f^{∗}(R) = L and h^{0}(Y, R) = h^{0}(X, L). Furthermore, e_{d−1}(L) ⩾ max{e_{d−1}(M), δ(a) + 1}. If (d + k − 1)(t + 1 − k) > (t + 1)q, then e_{d−1}(L) ⩽ k − 2.

Obviously, in the statement of Theorem 1 we have δ(a) ⩽ k − 1. Notice that δ(a) = k − 1 if a = 0 and that δ(a) = k − 2 if a ⩽ d + q − 1.

Remark 2. Take the notation of the statement of Theorem 1. Obviously, we have gon(X) ⩽ d and gon(Y) ⩽ d. Hence, if gon(Y) > d/k, then the gonality of X is computed by a pencil not coming from Y, while if gon(Y) = d/k, then there is at least one pencil on X computing gon(X), but not coming from Y. If either 1 ⩽ q ⩽ 2 or q ⩾ 3 and Y has general moduli, then gon(Y) = ⌊(q + 2)/2⌋. Hence if ⌊(g +3)/2⌋ < k·⌊(q +3)/2⌋} (resp. ⌊(g +3)/2⌋ = k·⌊(q +3)/2⌋}, then we may apply the first part of this remark. Hence either 1 ⩽ q ⩽ 2 or q ⩾ 3 and Y has general moduli, then there is a small, but non empty, interval of integers g for which both Theorem 1 and the first part of this remark may be applied.

In section 3 we will continue [1] and study the rank k − 1 vector bundle E_{f} := f_{∗}(O_{X})/O_{Y}.

We work over an algebraically closed field K with char(K) = 0.

2. Proof of Theorem 1

Remark 3. Let Y be a smooth and connected projective curve. Set q := p_{a}(Y) and S := Y × P^{1}. Hence h^{1}(S, O_{S}) = q. Let π_{1} : S → Y and π_{2} : S → P^{1} denote the two projections. For any R ∈ Pic(S) there are unique M ∈ Pic(Y) and k ∈ Z such that R ≅ π_{1}^{∗}(M) ⊗ π_{2}^{∗}(O_{P^{1}}(k)). Set O_{S}(M, k) := π_{1}^{∗}(M) ⊗ π_{2}^{∗}(O_{P^{1}}(k)). If k < 0, then h^{0}(S, O_{S}(M, k)) = 0 and h^{1}(S, O_{S}(M, k)) = (−k − 1) · h^{0}(Y, M) (Künneth formula). If k ⩾ 0, then h^{0}(S, O_{S}(M, k)) = (k + 1) · h^{0}(Y, M) and h^{1}(S, O_{S}(M, k)) = (k + 1) · h^{1}(Y, M) (Künneth formula). Furthermore, if M is spanned and k ⩾ 0, then O_{S}(M, k) is spanned, while if M is (birationally) very ample and k > 0, then O_{S}(M, k) is (birationally) very ample. Fix integers k ⩾ 2 and d > 0 and M ∈ Pic^{d}(Y) such that |M| has no base point. Let C ⊂ S be an integral curve in the linear system |O_{S}(M, k)| and ν: X → C the normalization map. Set A(C) := Sing(C) and let B(C) ⊂ S the conductor of ν. We recall that B(C)_{red} = A(C) and that B(C) = A(C) if each singular point of C is either an ordinary double point or an ordinary cusp. For any A ∈ Pic(Y) and any integer x set O_{C}(A, x) := O_{S}(A, x)|C and O_{X}(A, x) := ν^{∗}(O_{C}(A, x)). We will also write O_{C}(0, x) (resp. O_{X}(0, x), resp. O_{S}(0, x)) instead of O_{C}(O_{Y} , x) (resp. O_{X}(O_{Y} , x), resp. O_{S}(O_{Y} , x)). Notice that O_{C}(A, x) is a line bundle of degree k · deg(A) + x · deg(M). The morphism π_{1} ∘ ν: X → Y is a degree k covering between smooth and projective curves. Since ω_{S} ≅ O_{S}(ω_{Y} , −2), then ω_{C} ≅ O_{C}(M ⊗ ω_{Y} , k − 2) (adjunction formula). Thus p_{a}(C) = kd − d − k + kq + 1.

Remark 4. Use the set-up of Remark 3. Notice that h^{0}(S, ω_{S}(O_{Y} , −t)) = 0 for all t ⩾ 0 and h^{1}(S, ω_{S}(O_{Y} , −t)) = h^{1}(S, O_{S}(O_{Y} , t) = q(t + 1) for all t ⩾ −1. Consider the exact sequence

0 → O_{S}(−M, t − k) → O_{S}(0, t) → O_{C}(0, t) → 0  (1)

Since O_{S}(M, k − t) is ample for all t < k, we have h^{1}(S, O_{S}(−M, −k + t) = 0 for all t < k (Kodaira’s vanishing). Obviously, h^{0}(S, O_{S}(−M, −k + t) = 0 for all t < k. Hence from (1) we get h^{0}(C, O_{C}(0, t) = t + 1 for −1 ⩽ t < k. Now assume t ⩾ k. Since h^{1}(S, O_{S}(0, t)) = (t + 1)q and h^{1}(S, O_{S}(−M, t − k) = h^{1}(S, O_{S}(M + ω_{Y} , k − t − 2)) = (d + q − 1)(t + 1 − k) (Riemann-Roch and Serre duality), the long cohomology exact sequence of the exact sequence (1) gives h^{0}(C, O_{C}(0, t)) ⩾ t+1+(d+q −1)(t+1−k)−(t+1)q. Hence if (d+k −1)(t+1−k) > (t+1)q, then e_{d−1}(O_{C}(0, 1)) = k and hence e_{d−1}(L) ⩽ k−2, where L := ν^{∗}(O_{C}(1, 0)) ∈ Pic^{d}(X).

Remark 5. Take the set-up of Remarks 1 and 3. Notice that f_{!}(L) ≅ M. Let e_{i}(M), 1 ⩽ i ⩽ d − 1 (resp. e_{i}(L), 1 ⩽ i ⩽ d − 1) denote the scrollar invariants of M (resp. L) [3]. Thus e_{1}(L) ⩾ ⋯ ⩾ e_{d−1}(L) ⩾ 0, e_{1}(M) ⩾ ⋯ ⩾ e_{d−1}(M) ⩾ 0, e_{1}(L) + ⋯ + e_{d−1}(L) = g − d + 1, and e_{1}(M) + ⋯ + e_{d−1}(M) = q + d − 1. We only need that the integer m := e_{d−1} + 2 is characterized by the property h^{0}(X, R^{⊗(m−1)}) = m and h^{0}(X, R^{⊗m}) ⩾ m + 2. Fix any integer t ⩾ 1 such that h^{0}(Y, M^{⊗t}) = t + 1. Hence h^{0}(X, L^{⊗t}) ⩽ h^{0}(Y, f_{!}(L^{⊗t})) = h^{0}(Y, M^{⊗t}) = t + 1. Hence h^{0}(X, L^{⊗t}) = t + 1. Thus e_{d−1}(L) ⩾ e_{d−1}(M).

Remark 6. Take the set-up of Remarks 3 and 5, but assume that C is nodal (and hence B(C) = A(C)) and that its singular points are general in S. Set A := Sing(C) and a := ♯(A). Thus a = p_{a}(C) − p_{a}(X) = kd − d − k + kq + 1 − g. Let δ(a) be the maximal integer t such that h^{0}(S, O_{S}(M + ω_{Y} , k − 2 − t)) ⩾ a, i.e. (k − 1 − t)(d + q − 1) ⩾ a. Obviously, δ(a) only depends from Y, M, k, a. By the generality of A we have h^{0}(S, I_{A}(M + ω_{Y} , k − 2 − t)) = h^{0}(S, O_{S}(M + ω_{Y} , k − 2 − t)) − a for all t ⩽ δ(a). Thus by adjunction theory and Riemann-Roch we have h^{0}(X, L^{⊗t}) = h^{0}(C, O_{C}(0, t)) for all t ⩽ δ(a). By Remark 4 we get e_{d−1}(L) ⩾ min{δ(a) − 1, k − 2}.

Proof of Theorem 1. Set a := kd − d − k + 1 − g. Let A ⊂ S := Y × P^{1} be a general subset with ♯(A) = a. The proof of [2], Th. 1, gives the existence of an integral nodal curve C ∈ |O_{S}(M, k)| such that A = Sing(C). Let ν: X → C be the normalization map. Apply Remarks 3, 5 and 6.

3. The vector bundle E_{f} := f_{∗}(O_{X})/O_{Y}

In this section we fix the following set-up. Fix positive integers k, d. Let Y be a smooth and connected projective curve, C an integral projective curve and u: C → Y a degree k morphism. Let ν : X → C be the normalization map. Set f: u ∘ ν, q: = p_{a}(Y), γ := p_{a}(C), and g := p_{a}(X). Since Y is smooth, C and X are locally Cohen-Macaulay and u, f are finite, u and f are flat ([4], Prop. III.9.7). Furthermore, every torsion free finite rank sheaf on a one-dimensional regular local ring is free. Thus u_{∗}(O_{C}) and f_{∗}(O_{X}) are locally free. Since char(K) = 0, the trace map shows that O_{Y} is in a natural way a direct factor of u_{∗}(O_{C}) and f_{∗}(O_{X}). Hence there are rank k locally free sheaves E_{u} and E_{f} on Y such that u_{∗}(O_{C}) ≅ O_{Y} ⊕ E_{u} and f_{∗}(O_{X}) ≅ O_{Y} ⊕ E_{f}. Following [1] we will say that E_{u} (resp. E_{f}) is the bundle associated to u (resp. f). Since X and C are integral, we have h^{0}(X, O_{X}) = h^{0}(C, O_{C}) = 1. Since f and u are finite, we have R^{1}f_{∗}(O_{X}) = R^{1}u_{∗}(O_{C}) = 0. Thus the Leray spectral sequences of f and u give χ(E_{u}) = χ(u_{∗}(O_{C})) − χ(O_{Y}) = q − γ and χ(E_{f}) = χ(f_{∗}(O_{X})) − χ(O_{Y}) = q − g. Thus by Riemann-Roch we have deg(E_{u}) = χ(E_{u}) + (k − 1)(q − 1) = kq − k − γ + 1 and deg(E_{f}) = χ(E_{f}) + (k − 1)(q − 1) = kq − k − g + 1.

Castelnuovo-Severi inequality gives a strong restriction on the cohomological properties of the associated bundle E_{f} when f does not factor non-trivially through another smooth curve, i.e. when there are no (Y′, f_{1}, f_{2}) such that Y′ is a smooth curve, f_{1} : X → Y′, f_{2} : Y′ → Y, f = f_{2} ∘ f_{1} and 1 < deg(f_{1}) < k. Notice that this is always the case if k is prime.

Theorem 2. Let f: X → Y be a degree k ⩾ 2 covering between smooth projective curves which does not factor non-trivially through another smooth curve. Set g := p_{a}(X) and q := p_{a}(Y). Then there exists no effective divisor D on Y such that deg(D) ⩽ (g − kq)/k(k − 1), h^{0}(Y, O_{Y} (D)) = 1, and h^{0}(Y, E_{f}(D)) > 0.

Proof. Set L := f^{∗}(O_{Y} (D)). By the projection formula we have h^{0}(X, L) = h^{0}(Y, O_{Y} (D))+h^{0}(Y, E(D)) ⩾ 2. Let B the base point of |L|. Hence deg(L(−B)) ⩽ deg(L) < (g−kq)/(k−1). Hence the morphism induced by |L(−B)| factors through f. Hence L(−B) ≅ f^{∗}(O_{Y} (D′)) for some D′ ⊆ D such that the linear system |D′| induces a non-constant morphism. Since h^{0}(Y, O_{Y} (D)) = 1, this is absurd.

Motivated by Theorem 2 we now introduce the following invariants of a covering f: X → Y of smooth curve. Fix an integer z ⩾ 1. Let ϵ(f, z) denote the minimal integer t ⩾ 0 such that h^{0}(Y, E_{f}(D)) ⩾ z for some effective degree t divisor D on Y. Set ϵ(f) := ϵ(f, 1). Let η(f, z) denote the minimal integer t such that that h^{0}(Y, E_{f} ⊗ R) ⩾ z for some R ∈ Pic^{t}(Y). Set η(f) := η(f, 1). Hence η(f, z) ⩽ ϵ(f, z). The connectedness of X is equivalent to the inequality ϵ(f) > 0. In many cases of coverings consideeed in [1] it is quite easy to compute these invariants and the divisors D (resp. line bundles R) which are “extremal”, i.e. which compute ϵ(f, z) (resp. η(f, z)). For instance, in the case considered in [1], Th. 1.4, we have η(f) = η(f, k − 1) = ϵ(f) = ϵ(f, k − 1) = b(γ − k + 1)/(k − 1).

REFERENCES

[1] E. Ballico, Cohomological properties of multiple coverings of smooth projective curves, Rocky Mountain J. Math. 33, 4 (2003), 1205–1222.

[2] E. Ballico, C. Keem and D. Shin, Pencils on coverings of a given curve whose degree is larger than the Castelnuovo-Severi lower bound, preprint.

[3] M. Coppens, Existence of pencils with prescribed scrollar invariants of some general type, Osaka J. Math. 36, 4 (1999), 1049–1057.

[4] R. Hartshorne, Algebraic Geometry, Springer, Berlin, 1977.

[5] E. Kani, On Castelnuovo’s equivalent defect, J. Reine Angew. Math. 352 (1984), 24–70.

(received 08.09.2005, in revised form 12.11.2005)

Dept. of Mathematics, University of Trento, 38050 Povo (TN), Italy

E-mail: ballico@science.unitn.it