Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 38 (1997), No. 1, 95-98.

On the Generalized Theta Divisor

Georg Hein^{1}

Institut für Reine Mathematik, Humboldt-Universität Berlin Ziegelstr. 13A, D-10099 Berlin hein@mathematik.hu-berlin.de

Abstract. We denote by SU_{X}(n) the moduli space of semistable rank n bundles with trivial determinant on a complex curve X. Let L be the ample generator of the Picard group of SU_{X}(n). In this paper we show the base point freeness of the linear system |L^{⊗k}|, for any integer k > (n − 1)^{2}(g − 1).

1. Introduction

Let X be a smooth complete complex curve of genus g. We consider the moduli space SU_{X}(n) of semistable rank n vector bundles with determinant O_{X}, which is a projective scheme. This space was constructed in [9] by C. S. Seshadri. If we fix an arbitrary line bundle L ∈ Pic_{X}^{g−1} and define Θ_{L} = {E ∈ SU_{X}(n) | h^{0}(L ⊗ E) > 0}, then we have Pic(SU_{X}(n)) = ℤO(Θ_{L}) (cf. [3]). The linear system |kΘ_{L}| is base point free, for k >> 0. But can we give concrete bounds? For the case n = 2 it is well known, that |Θ_{L}| is base point free (see [8]). We give a new proof of this fact in §3. For n > 2, |Θ_{L}| may have base points. But up to now there are no examples of base points for k = 2 (see [1]). The first affirmative answer to this problem gave J. Le Potier in [6]. He showed that for k > \frac{n^{3}}{4} (g − 1) the linear system |kΘ_{L}| is base point free. We will show in §2, that this bound can be replaced by k > (n − 1)^{2}(g − 1).

We use the following idea of J. Le Potier [6] to construct sections of O(Θ_{L})^{⊗k}: From a vector bundle F of rank k that satisfies H^{∗}(E ⊗ F) = 0, for one E ∈ SU_{X}(n), we obtain a canonical section (called Theta-function) in O(Θ_{L})^{⊗k}. Its divisor of zeros Θ_{F} is supported on the set {E ∈ SU_{X}(n) | h^{0}(E ⊗ F) > 0}. Consequently a vector bundle E is not a base point of |kΘ_{L}|, if there exists a vector bundle F of rank k such that H^{∗}(E ⊗ F) = 0. This implies, moreover, that F must be semistable of degree k(g − 1).

1Supported by the Graduiertenkolleg ”Geometrie und Nichtlineare Analysis” at Humboldt-Universität zu Berlin 0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 1997 Heldermann Verlag

G. Faltings gives in §1 of [4] a construction of the moduli space SU_{X}(n) which is based on Theta divisors. In particular this construction does not need the Geometric Invariant Theory.

I would like to thank the referee for some valuable remarks.

2. A bound for base point freeness

In this section we need the following lemma due to H. Lange (see [5]). We need it in the following form (see [2]).

Lemma 2.1. If E is a general point in the moduli space U_{X}(n, d), then for any F ⊂ E holds:

μ(F) ≤ μ(E) − \frac{rk(E) − rk(F)}{rk(E)}(g − 1) .

We need the following technical fact:

Lemma 2.2. Let S be a reduced scheme, s_{0} a geometric point of S and s̃ ≅ spec(k[ε]/ε^{2}) an infinitesimal neighbourhood of s_{0} in S. Let, moreover, E be a vector bundle over X × S with h^{0}(E_{s}) = m, for all s ∈ S. Then h^{0}(Es̃) = 2m.

Proof. We denote the projection of X × S to its second factor by q. By [7] Cor. II.5.2 it follows that q_{∗}E is a vector bundle on S. Now the assertion follows from the base change theorem. □

Theorem 2.3. Let E ∈ U_{X}(n, 0) be a stable bundle, and k > (n − 1)^{2}(g − 1) an integer. Then there exists a stable F ∈ U_{X}(k, k(g − 1)), such that h^{0}(E ⊗ F) = 0.

Proof. We use the isomorphism H^{0}(E ⊗ F) = Hom(E^{∨}, F). Step 1: Here we show that for a general F ∈ U(k, k(g − 1)) (in the sense of 2.1) any nontrivial morphism φ : E^{∨} → F is an embedding. So let F be a general bundle and suppose φ : E^{∨} → F were not an embedding. We set M = imφ. Since E^{∨} is stable, we obtain deg M ≥ 1. From rkM ≤ n − 1 and 2.1 we conclude:

\frac{1}{rkM} ≤ μ(M) ≤ g − 1 − \frac{k−rkM}{k} (g − 1)
and so k  ≤ (n − 1)^{2}(g − 1) .

But this contradicts our assumption on k.

Step 2: Let now U ⊂ U(k, k(g − 1)) be the open set of general stable bundles. This set is stratified into locally closed subsets U_{m}, defined by U_{m} = {F ∈ U | h^{0}(F ⊗ E) = m}. We show, that the codimension of U_{m} in U is at least m.

We consider a bundle F ∈ U_{m} and η ∈ Ext^{1}(F, F) corresponding to an extension 0 → F → F̃ → F → 0. If η is in the tangent space T_{U_{m},F}, then by lemma 2.2 we have h^{0}(E ⊗ F̃) = 2m.

In particular the boundary map ∂ : Hom(E^{∨}, F) → Ext^{1}(E^{∨}, F) is the zero homomorphism. Thus for any nontrivial φ ∈ Hom(E^{∨}, F) the element η is in the kernel of π : Ext^{1}(F, F) → Ext^{1}(E^{∨}, F). By step 1, any nontrivial φ is an embedding an d hence π is surjective. This shows that T_{U_{m},F} ⊂ ker(π) and codim(T_{U_{m},F} , T_{U,F}) ≥ dim(Ext^{1}(E^{∨}, F)) = m. □

Corollary 2.4. Let Θ_{L} be a Theta divisor on SU_{X}(n). Then the linear system |kΘ_{L}| is base point free, for k > (n − 1)^{2}(g − 1).

Proof. Given a semistable bundle E. If E is not stable, then we have a Jordan-Hölder filtration 0 ⊂ E_{1} ⊂ … ⊂ E_{k} = E, such that the quotients E_{l}/E_{l−1} are stable bundles. By theorem 2.3 there exist semistable vector bundles F_{l} on X with H^{∗}(F_{l} ⊗ (E_{l}/E_{l−1})) = 0, rk(F_{l}) = k and deg(F_{l}) = k(g − 1), for l = 1, …, k. We denote by U_{X}(k, k(g − 1)) the moduli space of rank-k vector bundles on X of degree k(g − 1). The subsets Θ_{E_{l}/E_{l−1}} = {F ∈ U_{X}(k, k(g − 1)) | H^{∗}((E_{l}/E_{l−1}) ⊗ F) ≠ 0} are properly closed subsets. Hence we can find a vector bundle F which is not in their union. For this bundle F holds H^{∗}(E ⊗ F) = 0. □

By the same method we prove:

Theorem 2.5. Let E be a semistable bundle of rank n and degree d. If (k, m) is a pair of integers with k > \frac{n(n−1)^{2}(g−1)}{(n,d)} and kd + nm = kn(g − 1), then there exists a stable bundle F of degree m and rank k such that h^{0}(F ⊗ E) = 0.

Corollary 2.6. Let U_{X}(n, L) be the moduli space of semistable rank n bundles on X with determinant L. We denote by h the ample generator of its Picard group. Then the linear system |h^{⊗k}| is base point free for k > (n − 1)^{2}(g − 1).

Corollary 2.7. Let E be a vector bundle of rank n and degree d. Then the following conditions are equivalent.

1. E is semistable.

2. For any r = \frac{n}{(n,d)}k with k > (n − 1)^{2}(g − 1), there exists a vector bundle F of rank r, such that H^{∗}(E ⊗ F) = 0.

3. There exists a vector bundle F, such that H^{∗}(E ⊗ F) = 0.

3. The case SU_{X}(2)

Theorem 3.1. Let F be a stable bundle of rank 2 and deg F = 0. Then there exists a line bundle L ∈ Pic_{X}^{g−1}, such that h^{0}(F ⊗ L) = 0.

Proof. We define U_{F}^{k} = {L ∈ Pic_{X}^{g−1+k} | h^{1}(L ⊗ F) = 0}, for any k ≥ 0. These are open subsets of the Jacobians. For k ≥ g − 1 they are nonempty. We set m = min{k | U_{F}^{k} ≠ ∅}.

If m = 0, then we are done, so we suppose m > 0. We take a L ∈ U_{F}^{m}. We have H^{0}(F ⊗ L) ≅ Hom(F^{∨}, L). We consider two linearly independent elements φ_{1} and φ_{2} of Hom(F^{∨}, L). If ker φ_{1} ≠ ker φ_{2}, then we choose a point P ∈ X, such that: ker φ_{1}(P) ≠ ker φ_{2}(P) and P ∉ supp(L/imφ_{i}).

Then the short exact sequence 0 → L(−P) → L → k(p) → 0 yields a surjection Hom(F^{∨}, L) → Hom(F^{∨}, k(p)). Hence L(−P) ∈ U_{F}^{m−1}. So if U_{F}^{m−1} = ∅, this would imply, that there exists a L_{0} ⊂ F with ker φ = L_{0}, for all nonzero φ ∈ Hom(F^{∨}, L). So we obtain an exact sequence 0 → L_{0} → F^{∨} → L_{1} → 0 and an isomorphism Hom(L_{1}, L) ≅ Hom(F^{∨}, L).

We set k = deg L_{1} and write L = L_{1}(D), where D is an effective divisor of degree g − 1 + m − k. From the above isomorphism we obtain h^{0}(O_{X}(D)) = 2m. But this must hold for almost all effective divisors of degree g − 1 + m − k. This implies m + k = 0 which contradicts the semistability of F. □

Corollary 3.2. The Theta divisor on SU_{X}(2) is base point free.

References

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Received October 23, 1995