ON DEGENERATIONS OF FIBRE SPACES OF CURVES OF GENUS ≥ 2

HURŞİT ÖNSİPER AND SİNAN SERTÖZ

Abstract. In this note, we show that for surfaces admitting suitable fibrations, any given degeneration X /Δ is bimeromorphic to a fiber space over Δ and we apply this result to the study of the degenerate fiber.

This note is concerned with the problem of studying the degenerations of fibered surfaces via the degenerations of the base curve and the fibers. We consider a surface of general type X admitting a fibration X → S with base genus ≥ 2 and show that for a weakly projective degeneration Π : X → Δ of such a surface, satisfying a mild condition on monodromy, the components of the singular fiber are of the following types:

(i) a rational surface, (ii) a ruled surface, or (iii) a surface fibered over a curve obtained from a degeneration of the base curve S.

Throughout the paper we work over ℂ, and adapt the following notation : Δ = {z ∈ ℂ : |z| < 1}. Δ^{∗} = Δ − 0. X is a compact surface admitting a fibration X → S with general fiber F and genera g(S), g(F) ≥ 2. χ(?) denotes the holomorphic Euler characteristic of ?. K_{?} is the canonical class of ?. p_{g}(?), q(?) denote the geometric genus and the irregularity of ?, respectively. X is a smooth threefold, Π : X → Δ is a flat proper holomorphic map with X_{t_{0}} = X for some t_{0} ∈ Δ^{∗} and the only singular fiber X_{0} is a divisor with normal crossings and smooth components.

We further assume that (i) each fiber X_{t}, t ∈ Δ^{∗} admits a fibration X_{t} → S_{t} of the same type as the fibration on X and (ii) the monodromy action on H^{1}(X_{t}, ℂ) leaves the image of H^{1}(S_{t}, ℂ) invariant (cf. Lemma 3 and the discussion preceding Lemma 3, for some examples of degenerations satisfying these hypothesis). We recall that under these assumptions we have the following result ([4], Proposition 1 and Proposition 3).

1991 Mathematics Subject Classification. Primary: 14E15; Secondary: 14J10, 14J15, 14J29.

To appear in Archiv der Mathematik.

Theorem 1. Possibly after restricting to a smaller disk Δ′ around 0 ∈ Δ, we can find a degeneration Π′ : X′ → Δ′ and a relative curve p : Y → Δ′ such that :

(i) we have a bimeromorphic map X′/Δ′ → X /Δ′ which is an isomorphism over Δ′^{∗}, (ii) Π′ factors through p, X′ → Y is surjective and (iii) for each t ∈ Δ^{′∗}, Xt^{′} → Y_{t} is the fibration X_{t} → S_{t}.

Theorem 1 reduces the problem of understanding the structure of the given degeneration to the study of the degenerations of the curves S_{t} and of the fibres of X → S. In this direction, without any projectivity assumptions on X /Δ, we have

Lemma 1. If a component X_{i} of X0^{′} maps onto a component Y_{i} of Y_{0}, then it is algebraic.

Proof : If X_{i} → Y_{i} is a fibration of fiber genus 0 or ≥ 2 then clearly X_{i} is algebraic. So we assume that X_{i} → Y_{i} is an elliptic fibration. Since a surface with algebraic dimension = 0 has only a finite number of curves, X_{i} has algebraic dimension = 1 or 2. As the fibers of X0^{′} → Y_{0} are curves of genus ≥ 2 (except at a finite number of points where the map is not flat), we see that on X_{i} lie other curves obtained from intersection with the components of X0^{′} containing the rest of the fibers of X0^{′} → Y_{0} over Y_{i}. Clearly, such curves will not lie in the fibers of the elliptic fibration on X_{i} and this is impossible unless X_{i} has algebraic dimension 2. This proves the lemma. □

Lemma 2. Let X_{j} be a component of X0^{′} mapped to a point p ∈ Y_{0}. If X_{j} intersects some component X_{i} as in Lemma 1 along a smooth curve C with no triple points, then X_{j} is algebraic.

Proof : Let C_{i}(resp.C_{j}) denote the curve C on X_{i}(resp.X_{j}). If C_{i} is the fiber of X_{i} over p, then C_{i}^{2} = 0. Therefore, as there are no triple points on C we have C_{j}^{2} = −C_{i}^{2} = 0. Moreover g(C) = g(F) ≥ 2 and using the adjunction formula on X_{j}, we get K_{j}.C_{j} + C_{j}^{2} = 2(genus(C) − 1) ≥ 2. Hence (K_{j} + nC_{j})^{2} > 0 for large enough n and therefore X_{j} is algebraic. On the other hand, if C_{i} is a component of the fiber over p, then C_{i}^{2} < 0 and the equality C_{j}^{2} = −C_{i}^{2} gives C_{j}^{2} > 0, again proving the algebraicity of X_{j}. □

These two lemmata clearly fall short of proving the algebraicity of all components of the singular fiber (cf. [5], conjecture on p. 83). However, combining Lemma 1 with the flatification technique of ([2]) we get

Theorem 2. X /Δ′ is bimeromorphic to a degeneration X′′/Δ′ in which all components of the singular fiber X0^{′′} are algebraic.

Proof : To prove this result we first remove those components of X0^{′} where the map X′ → Y fails to be flat. For this purpose, we will apply flatification as described in ([2]). More precisely, we blow up points p_{1}, .., p_{k} ∈ Y_{0} over which our map is not flat, to get a new relative curve Y′/Δ′ and then in the complex space X^{∗} = X′ ×_{Y} Y′ we take the smallest closed analytic subspace X^{∗∗} containing X^{∗} − ∪Π^{∗−1}(p_{j}) where Π^{∗} is the composite map X^{∗} → Y′ → Y. Then X^{∗∗} → Y′ is flat. Finally, resolving the singularities of X^{∗∗} and of the components of the singular fiber, we get the required degeneration X′′/Δ′. □

Corollary 1. If X /Δ is weakly projective, then each component of X_{0} is either a fibration over a curve of genus ≤ g(S) with fiber genus ≤ g(F) or a ruled or rational surface.

Proof : Since X is weakly projective, so is X′′ of Theorem 2 and we apply ([5], Corollary 3.1.4). □

Next we address to the question of when the hypothesis of Theorem 1 are satisfied for a degeneration X → Δ. As to the first condition we have

a) if X → S is a smooth fibration with both fiber genus and base genus ≥ 2, then any deformation of X admits a fibration of the same type ([3], Lemma 7.1), and

b) if the fibration is a consequence of a relation among some deformation invariants, then trivially the first condition of the hypothesis holds. One notable example of this case is degenerations of minimal surfaces with K^{2} < 3χ, q ≥ 2. With this inequality satisfied, the given surface admits a fibration of fiber genus 2 or 3, the base curve being the image of the albanese map ([1], Theorem 2.6). The condition on monodromy is trivially satisfied if h^{1}(|Γ|) = 0 where Γ is the dual graph of the singular fiber. For more general degenerations we have

Lemma 3. In the following cases the hypothesis on monodromy is satisfied :

(a) X → S is a smooth fibration with g(S), g(F) ≥ 2,

(b) X is minimal and K^{2} < 3χ(X), q(X) ≥ 2.

Proof : (a) By ([3], Lemma 7.1), we have a deformation Φ : S → Δ^{∗} of S, varying continuously with t ∈ Δ^{∗}, such that Π|_{Δ^{∗}} factors through Φ. Therefore, we have an exact sequence 0 → R^{1}Φ_{∗}(ℂ) → R^{1}Π_{∗}(ℂ). Hence, the correspondence between representations of π_{1}(Δ^{∗}) and flat vector bundles on Δ^{∗} shows that, for each t ∈ Δ^{∗}, H^{1}(S_{t}, ℂ) is an invariant subspace of H^{1}(X_{t}, ℂ) under the monodromy action.

(b) In this case the fibration Ψ_{t} being the albanese fibration we have H^{1}(X_{t}, ℂ) = H^{1}(S_{t}, ℂ) and the conclusion follows trivially. □

References

[1] F. Catanese, Canonical rings and “special” surfaces of general type, in Proc. Sym. Pure Math. Volume 46, AMS, 1987.

[2] H. Hironaka, Flattening theorem in complex analytic geometry, Amer. J. of Math. 97 (1975), 503 - 547.

[3] J. Jost, S. T. Yau, Harmonic mappings and Kähler manifolds, Math. Ann. 262 (1983), 145 - 166.

[4] H. Önsiper, A note on degenerations of fibered surfaces, to appear in Indag. Math.

[5] U. Persson, On Degenerations of Algebraic Surfaces, Memoirs of the Amer. Math. Soc. 189 (1977).

Department of Mathematics, Middle East Technical University, 06531 Ankara, Turkey

   E-mail address: hursit@rorqual.cc.metu.edu.tr

Department of Mathematics, Bilkent University, 06533 Ankara, Turkey

   E-mail address: sertoz@fen.bilkent.edu.tr