Every rationally connected variety over the function field of a curve has a rational point

by A.J. de Jong, MIT and J. Starr, MIT

Introduction

Let k be an algebraically closed field and let V be a normal proper variety over k. We say that V is separably rationally connected if there exists a smooth rational curve C → V mapping to V which avoids the singularities of V such that T_{V} |_{C} is ample on C. This definition differs from [8, IV Definition 3.2] in Kollar’s book; it agrees with his definition if V has a resolution of singularities, see [8, IV Theorem 3.7]. Here is our main theorem; it is Conjecture IV 6.1.1 of [8] and it was proved by Graber, Harris and Starr in the case that k has characteristic 0, see [5].

Theorem. Let X be an irreducible nonsingular projective curve. Suppose that f : P → X is a proper flat morphism whose geometric generic fibre is a normal and separably rationally connected variety. Then f has a section.

The proof of this theorem is at the end of the paper. We briefly indicate the layout of this paper, and the main ideas.

Graber, Harris and Starr use a topological argument to specialize a family of ramified coverings Y_{t} → X to a map Y_{∞} → X so that Y_{∞} contains an irreducible component mapping isomorphically onto X. We replace this by an elementary construction which produces, starting with Y → X, a family Y_{t} → X specializing in the desired manner so that Y_{0} is a nodal curve whose normalization is Y. This is done in Section 1.

We advise that the reader skip Section 2 on a first reading. Here we show that a proper flat family P → X of varieties over X with reduced general fibre has a normalized pullback all of whose fibres are reduced. The pullback morphism Y → X can be taken to be finite generically étale. This is used in Section 3 to show that it suffices to produce a section of f after such a normalized base change. In Section 4 we produce the section in this case; here the idea, if not the execution, is similar to that in [5].

Definitions 4.2.1 and 4.2.2 are taken from [5], and so are the main lines of proof, as is the conviction that such a theorem can be proven (in characteristic p).

Acknowledgements. Part of this research was conducted by A.J. de Jong for the Clay Mathematics Institute. A.J. de Jong was partially supported by an NSF grant.

1. Making families of curves

Let k be an algebraically closed field. Suppose we are given the following data:

1. A finite morphism π : Y → X of irreducible smooth projective curves over k. We assume that π is generically étale. The degree of this morphism is denoted d.

2. A finite set of points S ⊂ X(k). We assume that S contains all branch points of the morphism π.

1.1. Proposition. Given data as above there exist a projective surface W and flat morphisms f : W → ℙ^{1}, g : W → X with the following properties:

(a) The fibre W_{0} = f^{−1}(0) is a nodal curve, whose normalization is the curve Y; the map Y → W_{0} ⟶^{g} X is equal to π. The image g(w) of any node w ∈ W_{0} is not in S. The surface W is nonsingular along W_{0}.

(b) The general fibre W_{t} = f^{−1}(t) is smooth, and the morphism π_{t} : W_{t} → X induced by g has degree d.

(c) The formal local structure of the morphism π_{t} : W_{t} → X in a neighbourhood of π_{t}^{−1}(x) is isomorphic to the formal local structure of the morphism π : Y → X in a neighbourhood of π^{−1}(x) for all x ∈ S and all t ∈ ℙ^{1}(k), t ≠ ∞.

(d) There is an irreducible component X′ of W_{∞} = f^{−1}(∞) which is mapped birationally to X under g. The surface W is generically smooth along X′ and X′ has multiplicity 1 in W_{∞}. (In other words f is generically smooth along X′.)

Proof. We choose an invertible O_{Y} -module L of sufficiently high degree. Choose a pair of sections s_{0}, s_{1} of L which generate L and let h be the morphism φ_{(s_{0},s_{1})} : Y → ℙ^{1}. Let \overline{Y} ⊂ X × ℙ^{1} be the (reduced) image closed subscheme.

1.2. Lemma. (a) If deg(L) > 2g_{Y} − 2 + 2d then for general choice of (s_{0}, s_{1}) the curve \overline{Y} has at worst ordinary double points, and Sing(\overline{Y}) ∩ S × ℙ^{1} = ∅.

(b) The divisor class of \overline{Y} on X × ℙ^{1} is the divisor class of the invertible sheaf pr_{1}^{∗}(Nm_{π}(L)) ⊗ pr_{2}^{∗}(O(d)).

Proof. For a point x ∈ X(k) consider the closed subscheme Y_{2,x} = π^{−1}(Spec O_{X,x}/𝔪_{x}^{2}) of Y. The bound on the degree implies that the map Γ(Y, L) → Γ(Y_{2,x}, L) is surjective. For x ∈ S we choose the pair (s_{0}, s_{1}) such that the associated morphism Y_{2,x} → ℙ^{1} is a closed immmersion. For x ∉ S the differential dπ identifies the tangent spaces T_{y} := T_{y}(Y_{2,x}) = T_{y}Y with T_{x}X (for all y ∈ Y_{2,x}(k)). We choose the pair (s_{0}, s_{1}) such that the morphism Y_{2,x} → ℙ^{1} is either a closed immersion, or identifies at worst two points y, y′ of Y_{2,x} but does not induce the same map on the tangent spaces T_{y} = T_{x}X = T_{y′}. The reader checks that these conditions define nonempty open sets U_{x} (in the space of all pairs), and that the complement of the open U_{x} for x ∉ S has codimension 2. Hence there is a pair satisfying all of these conditions. It is straightforward to check that such a pair works in (a).

To verify (b) we have to compute \overline{Y} ∩ \{x\} × ℙ^{1} as a divisor on ℙ^{1} and \overline{Y} ∩ X × \{t\} as a divisor on X. The first divisor clearly has degree d as desired. The second is the image in X of a fibre of the map φ : Y → ℙ^{1}. The divisor class of the fibre is c_{1}(L) and the divisor class of the image is c_{1}(Nm_{π}(L)). □

Let D ⊂ X be the divisor D = ∑_{x∈S} x. Pick a large number N divisible by d and let L = O_{Y} (\frac{N}{d} π^{−1}(D)). Then Nm_{π}(L) ≅ O_{X}(ND). By the lemma we have following rational equivalence of divisors on X × ℙ^{1}:

[\overline{Y}] ≡ ∑_{j=1}^{d}[X × \{σ_{j}\}] + ∑_{x∈S} N[\{x\} × ℙ^{1}].

Here σ_{1}, …, σ_{d} ∈ ℙ^{1}(k) are pairwise distinct points chosen such that X × \{σ_{j}\} ∩ Sing(\overline{Y}) = ∅ for all j = 1, …, d.

The above rational equivalence implies that there exists a pencil of curves D_{t} on X × ℙ^{1} with the following properties: (i) D_{0} = \overline{Y} has at worst nodes as singularities, (ii) D_{∞} = ∑ X × \{σ_{j}\} + ∑_{x∈S} N\{x\} × ℙ^{1} misses the nodes of D_{0}. Let W be the blow up of the surface X × ℙ^{1} in the zero dimensional closed subscheme D_{0} ∩ D_{∞}. Then we obtain the right half of the following diagram (the left half will be explained below)

∐_{x∈S} Z_{x} × ℙ^{1} ⟶ W ⟶^{f} ℙ^{1}
π|_{Zx} ↓  ^{g}↓
X  ⟶ X

such that D_{t} = f^{−1}(t) (scheme theoretically) and such that g is the composition W → X × ℙ^{1} → X. Properties (i), (ii) imply that the general fibre of f is smooth and that W is smooth along D_{0} = f^{−1}(0) = \overline{Y}. Further, the morphism f is also smooth generically along the irreducible components X × {σ_{j}} of the fibre D_{∞}. Note that these components are mapped isomorphically to X under the morphism g. Thus (a), (b) and (d) are satisfied; it remains to show (c).

For x ∈ S we let Z_{x} ⊂ Y be the scheme

Z_{x} = π^{−1}(Spec O_{X,x}/𝔪_{x}^{N}) = Y ×_{X} Spec (O_{X,x}/𝔪_{x}^{N}).

The morphism (π × h) : Y → \overline{Y} induces a closed immersion of Z_{x} into \overline{Y}. Our choice of D_{∞} = ∑ X × {σ_{j}} + ∑_{x∈S} N{x} × ℙ^{1} insures that (π × h)(Z_{x}) ⊂ D_{∞} and hence we may think of Z_{x} as a subscheme of D_{t} for all t ∈ ℙ^{1}(k). We conclude there exists a morphism Z_{x} × ℙ^{1} → W which for every t ∈ ℙ^{1} induces the closed immersion Z_{x} → D_{t} we just described. It fits into the commutative diagram displayed above.

Note that for t ≠ ∞ the morphism π_{t} = g|_{D_{t}} : D_{t} → X is finite. Upon comparing lengths we see that

Z_{x} = π_{t}^{−1}(Spec O_{X,x}/𝔪_{x}^{N}).

In particular this implies that D_{t} is smooth along Z_{x} (by looking at tangent spaces). Thus Lemma 1.4 (a) below applies and we deduce that the formal local structure of the morphism π_{t} : D_{t} → X in a neighbourhood of π_{t}^{−1}(x) is isomorphic to the formal local structure of the morphism Y → X in a neighbourhood of π^{−1}(x) for all x ∈ S and all t ∈ ℙ^{1}(k), t ≠ ∞. □

1.3. Corollary. Suppose that π : Y → X is a finite generically étale morphism of projective nonsingular curves over k and that S ⊂ X(k) is a finite set of points. There exists a projective surface W and morphisms f : W → C, g : W → X such that the following conditions hold:

(a) f is semi-stable family of curves with smooth general fibre.

(b) There is a point 0 ∈ C(k) such that W_{0} = f^{−1}(0) has the following description: W_{0} = Y ∪ ⋃ L_{i}, where g|_{Y} = π, g(L_{i}) is a point of X not in S, L_{i} ≅ ℙ^{1} and Y ∩ L_{i} = {y_{i1}, y_{i2}} where (of course) π(y_{i1}) = π(y_{i2}) in X.

(c) There is a point ∞ ∈ C(k) and an irreducible component X′ of the fibre W_{∞} which is mapped isomorphically to X under g.

Proof. Enlarge S so that it contains the branch points of the morphism π and apply the proposition. Apply base change by the map δ : ℙ^{1} → ℙ^{1}, s ↦ t = s^{2}/(s + 1) and blow up the resulting ordinary double points of W ×_{ℙ^{1},δ} ℙ^{1} above s = 0 to obtain the lines L_{i}. Finally, apply the semi-stable reduction theorem to make f semistable. Details left to the reader. □

We end this section by stating the lemma that was used above. The lemma is basically a restatement of Krasner’s lemma, see [3, Section 3.4.2].

1.4. Lemma. (a) Let π : Y → X be a finite morphism of smooth curves and assume that π is generically étale. Let x ∈ X. There exists an integer N_{0} such that for every N ≥ N_{0} we have the following: If Y′ → X is another finite morphism of smooth curves and if there exists an isomorphism

Y ×_{X} Spec O_{X,x}/𝔪_{x}^{N} ≅ Y′ ×_{X} Spec O_{X,x}/𝔪_{x}^{N}

over O_{X,x} then there exists an isomorphism

Y ×_{X} Spec Ô_{X,x} ≅ Y′ ×_{X} Spec Ô_{X,x}

over O_{X,x}.

(b) Let X be a smooth curve over k and let Ô_{X,x} ⊂ R be a finite generically étale extension of complete discrete valuation rings. There exists an integer N_{0} such that for every N ≥ N_{0} we have: If Y → X is a finite morphism of smooth curves, and y ∈ Y (k) is over x such that

O_{Y,y}/𝔪_{x}^{N}O_{Y,y} ≅ R/𝔪_{x}^{N}

over O_{X,x} then Ô_{Y,y} ≅ R over O_{X,x}.

Proof. Part (a) follows from part (b) by writing Y′ ×_{X} Spec Ô_{X,x} as a disjoint union of branches. Part (b) follows from Krasner’s Lemma in the following way. We can write R = Ô_{X,x}[T]/(f(T)). For any pair (y, Y′) as in (b) we can choose an isomorphism Ô_{Y′,y} = Ô_{X,x}[T′]/(f′(T′)) such that f′(T) − f(T) ∈ 𝔪_{x}^{N}. In other words, the class of T′ in Ô_{Y′,y} is a solution of f(X) = 0 up to order N. Krasner’s Lemma says that, provided N_{0} is large enough, this implies that there is a root of f(X) = 0 in Ô_{Y′,y}. The corresponding map of R to Ô_{Y′,y} is the desired isomorphism. □

2. Obtaining reduced fibres after normalized base change

Suppose that f : P → X is a flat morphism of a normal variety P to an irreducible smooth projective curve X over k. We will assume that the following equivalent conditions hold: (a) the geometric generic fibre is reduced, (b) the general fibre is reduced, and (c) the smooth locus of the morphism f is dense in P.

We note that for x ∈ X(k) the fibre P_{x} = f^{−1}(x) is a scheme satisfying S_{1} (every local ring of dimension ≥ 1 had depth ≥ 1). Thus P_{x} is reduced if and only if P_{x} is generically reduced, i.e., all local rings of P_{x} at generic points are fields.

Let π : Y → X be a finite morphism of another irreducible smooth curve Y to X. Set

P_{Y} = (P ×_{X} Y)^{normalized}.

In this section we present a technical result that is similar to Lemma 2.3 in [7] and which follows in a straightforward manner from the results in [4]. Namely we prove there is at least one finite generically étale morphism Y → X such that P_{Y} → Y has reduced fibres, i.e., such that the smooth locus of P_{Y} → Y is dense in every fibre.

2.1. Proposition. With the assumptions and notations as above. There exists a π : Y → X such that all fibres of P_{Y} → Y are reduced and such that π is generically étale.

Proof. First we examine this question when X = Spec R is the spectrum of a complete discrete valuation ring R with algebraically closed residue field k. Indeed, let P be an irreducible scheme, and let P → Spec R be a flat morphism of finite type with geometrically reduced generic fibre. It follows by unwinding [4, Theorem 2.1’, page 368] that there exists a finite generically étale extension R ⊂ R′ and a commutative diagram

P′ ⟶ P ⊗_{R} R′ ⟶ P
↓  ↓  ↓
Spec R′ ⟶ Spec R′ ⟶ Spec R.

where P′ → P ⊗_{R} R′ is finite and an isomorphism over the generic point of Spec R′, and where the morphism P′ → Spec R′ is flat with reduced geometric fibres. This implies that the normalization (P ⊗_{R}R′)^{normalized} of P ⊗_{R}R′ dominates P′. Since the special fibre of P′ → Spec R′ is reduced we see that (P ⊗_{R}R′)^{normalized} → P′ is an isomorphism above the generic points of the special fibre of P′ and hence we see that (P ⊗_{R}R′)^{normalized} has reduced special fibre by the discussion before the proposition as desired. We note that any further extension R ⊂ R′ ⊂ R′′ (assumed generically étale) will do the job as well.

Let x_{1}, …, x_{r} ∈ X(k) be the points of X such that P_{x_{i}} = f^{−1}(x_{i}) is not reduced. Let R_{i} be the complete local ring of x_{i} on X. Choose R_{i} ⊂ R′_{i} as above. By the remark at the end of the last paragraph we way choose the extensions R_{i} ⊂ R′_{i} such that the field extensions K_{i} = f.f.(R_{i}) ⊂ f.f.(R′_{i}) = L_{i} have a degree d independent of i.

Let K = k(X). We claim that there exists a separable field extension K ⊂ L of degree d such that L ⊗_{K} K_{i} is isomorphic to L_{i}. To see this we write L_{i} = K_{i}(α_{i}) and we let f_{i} ∈ K_{i}[X] be the minimal polynomial of α_{i} over K_{i}. We choose a polynomial f(X) ∈ K[X] which approximates f_{i} simultaneously for all i and we set L = K[X]/(f). By Krasner’s lemma we’ll have L ⊗_{K} K_{i} ≅ L_{i} if the approximation is good enough, see our Lemma 1.4 (b). □

2.2. Remark. In the preprint version of this paper an alternative proof of this result was given using Lemma 2.3 in [7], avoiding the reference [4]. The strategy there was to show that the result with π possibly inseparable actually implies the a priori stronger result with π generically separable. We refer the interested reader to the web page of the first author for a web-based version of this argument.

3. The reduction to the case of reduced fibres

Suppose that f : P → X is a proper flat morphism of a normal scheme P to an irreducible smooth projective curve X over k. We will assume that the following equivalent conditions hold: (a) the geometric generic fibre is reduced, (b) the general fibre is reduced, and (c) the smooth locus of the morphism f is dense in P. Consider the question: Does f have a section? In this section we will show that the answer is affirmative if and only if f has a section after normalized base change P_{Y} to those Y → X such that P_{Y} has reduced fibres over Y.

3.1. Theorem. Assumptions and notations as above.

(i) In case k uncountable. Assume that P_{Y} → Y has a section whenever π : Y → X has the following two properties: (a) π is generically étale, and (b) P_{Y} → Y has reduced fibres. Then P → X has a section. (ii) In case k countable. Suppose that for some uncountable algebraically closed extension K of k the assumptions of (i) hold for P_{K} → X_{K} over K. Then P → X has a section.

Proof. Case (ii) can be deduced from (i) by a specialization argument. Namely, by (i) we obtain a section σ_{K} : X_{K} → P_{K} over some (uncountable) extension k ⊂ K. Clearly σ_{K} can be defined over a finitely generated extension of k, i.e., over the function field of some variety T over k. By shrinking T we may assume we have a section of P × T → X × T and by choosing a k-rational point in T we obtain our section of P → X.

Next we prove (i). By Proposition 2.1 there is at least one π : Y → X generically étale such that P_{Y} → Y has reduced fibres. Let π_{t} : W_{t} → X be the family of morphisms of curves constructed in Proposition 1.1 using as S = {x_{1}, …, x_{r}} the set of branch points of the morphism Y → X. By construction, for t ≠ ∞ the formal local structure of the morphism π_{t} : W_{t} → X in a neighbourhood of π_{t}^{−1}(x_{i}) is isomorphic to the formal local structure of the morphism π : Y → X in a neighbourhood of π^{−1}(x_{i}). This clearly implies that the normalization P_{t} of P ×_{X} W_{t} has reduced fibres.

By assumption each of the morphisms P_{t} → W_{t} has a section σ_{t}, and so P ×_{X} W_{t} → W_{t} has a section σ_{t}. The relative Hilbert scheme Hilb of P ×_{X} W over ℙ^{1} is locally of finite type over k and has a countable number of irreducible components. Since k is uncountable, we see that infinitely many of the points σ_{t}(W_{t}) of Hilb will be in the same irreducible component Z ⊂ Hilb which dominates ℙ^{1}. We deduce that there is a section σ of P ×_{X} W_{L} → W_{L} where k(t) ⊂ L is a finite extension (by taking an L-valued point of Z). Let C be the smooth projective curve over k whose function field is L and let c ∈ C(k) be a point which lies over t = ∞ (under the map C → ℙ^{1} coming from k(t) ⊂ L). Note that there is a component X′ of the fibre of

W ×_{f,ℙ^{1}} C ⟶ C

over c which is mapped birationally to X under the composite W ×_{ℙ^{1}} C → W → X, and along which W ×_{ℙ^{1}} C is generically smooth (this corresponds to the component X′ of Proposition 1.1 (d)). Think of σ as a rational map fitting into the following commutative diagram:

C ⟵ W ×_{ℙ^{1}} C ⟶^{σ} P
↓  ↓  ↓
ℙ^{1} ⟵^{f} W  ⟶^{g} X

By construction σ extends to a neighbourhood of the generic point of X′ and this gives a k(X′) = k(X)-valued point of P over X as desired. □

4. The case of reduced fibres

Let X be an irreducible nonsingular projective curve over k, and let P → X be a flat morphism. We will assume the following assumptions hold:

4.0.1. P is a normal projective variety. 4.0.2. All fibres of f are reduced and connected. 4.0.3. The following equivalent conditions hold: (a) the geometric generic fibre of f is normal, (b) there is a nonempty Zariski open set U ⊂ X such that for each x ∈ U(k) the fibre P_{x} = f^{−1}(x) is normal, and (c) there is a nonempty Zariski open set U ⊂ X such that the nonsmooth locus of P_{U} → U has codimension ≥ 2 in the fibres. 4.0.4. There exists a nonempty Zariski dense open set V ⊂ P contained in the smooth locus of f with the following property: for every point x ∈ X(k) and every finite set of closed points p_{1}, …, p_{n} ∈ V ∩ f^{−1}(x), there exists a smooth rational curve ℙ^{1} ≅ C → V ∩ f^{−1}(x) = V_{x} mapping to V_{x} containing each p_{i} and such that T_{P/X}|_{C} is an ample vector bundle.

This last condition means that the general fibre P_{x} of f is rationally connected. The version of this we give above is equivalent to the following apparently weaker condition:

4.0.4’. There exists a closed point x ∈ X and a smooth rational curve ℙ^{1} → P_{x} mapping into the smooth locus of P_{x} = f^{−1}(x) such that T_{P/X}|_{C} is an ample vector bundle.

It follows from the proof of [8, Theorem IV.3.9.4] that, under the hypothesis 4.0.4’, a Zariski dense open subset V ⊂ P as in 4.0.4 exists. Thus 4.0.4 and 4.0.4’ are equivalent conditions, but 4.0.4 is the form which is more useful to us. Namely, it will be used in the proof of the following theorem.

4.1. Theorem. In the situation above f has a section.

We do not immediately start the proof of the theorem. Instead we first give a definition. Note that the open subset V ⊂ P of condition 4 will be fixed throughout the discussion.

4.2. Definition. Let Y be an irreducible nonsingular projective curve over k. 4.2.1. A morphism Φ : Y → P is pre-flexible (relative to f) if Φ(Y) is contained in the smooth locus of f, Φ(Y) ∩ V ≠ ∅, and f ∘ Φ : Y → X is generically étale. 4.2.2. A pre-flexible curve Φ : Y → P is flexible if H^{1}(Y, Φ^{∗}T_{P/X}) = 0. 4.2.3. A flexible curve with links is a morphism Φ : Y → P from a connected, projective, nodal curve Y = Y_{0} ∪ L_{1} ∪ … ∪ L_{r} such that Φ|_{Y_{0}} is flexible, such that each link L_{i} is a smooth rational curve, the links L_{i} are pairwise disjoint, and each L_{i} intersects Y_{0} in two distinct smooth points p_{i}, q_{i} ∈ Y_{0}, and such that Φ|_{L_{i}} is a closed immersion of L_{i} as a rational curve in a fibre of P → X such that T_{P/X}|_{L_{i}} is ample.

4.3. Lemma. There exists a pre-flexible curve Φ : Y → P.

Proof. Conditions 2 and 3 imply that the complement of the smooth locus of f has codimension ≥ 2 in P. Let p ∈ V be a point. Let Y be a connected component of the intersection of dim P − 1 general hyperplanes in P which pass through p. By Bertini’s theorem and the codimension estimate Y is a smooth projective curve contained in the smooth locus of f. It clearly meets V and the generic étaleness follows by choosing the hyperplanes suitably. □

4.4. Lemma. There exists a flexible curve Φ : Y → P.

Proof. By the lemma above we know there exists a pre-flexible curve Φ_{0} : Y_{0} → P. Let n_{1} ≥ 0 be an integer such that for every invertible sheaf L on Y_{0} with deg(L) ≥ n_{1}, we have H^{1}(Y, Φ_{0}^{∗}T_{P/X} ⊗ L) = 0. Let n_{2} = h^{1}(Y, Φ_{0}^{∗}T_{P/X}). Choose an integer n ≥ n_{1} + n_{2} and choose distinct points p_{1}, …, p_{n} ∈ Φ_{0}^{−1}(V). Let C_{1}, …, C_{n} ⊂ V be curves as in condition (4) which contain p_{1}, …, p_{n}. Form the comb Y_{0} ∪ C_{1} ∪ … ∪ C_{n} where each tooth C_{i} is connected to the handle Y_{0} at the point p_{i}, see [8, II Definition 7.7] for the definition of the concept “comb”. Define Ψ : (Y_{0} ∪ C_{1} ∪ … ∪ C_{n}) → P by Ψ|_{Y_{0}} = Φ_{0} and Ψ|_{C_{i}} is the inclusion map. By [8, III Theorem 7.9], there exists a subcomb, say Y_{0} ∪ C_{1} ∪ … C_{m} and a smoothing of the restriction of Ψ such that m ≥ n − n_{2} ≥ n_{1}. The condition that a curve be contained in the nonsingular locus of f and intersect V is open in the parameter space, so for a general member Φ : Y → P of the 1-parameter smoothing of Ψ : Y_{0} ∪ C_{1} ∪ … ∪ C_{m} → P, Φ is pre-flexible. By [8, III Lemma 7.16.1], if Φ is a general deformation, then H^{1}(Y, Φ^{∗}T_{P/X}) = 0. Thus Φ : Y → P is a flexible curve. □

4.5. Definition. Suppose that π : Y → X is a morphism from a connected, projective, nodal curve to X. A W-diagram is a diagram:

X_{∞}, Y ⟶ W ⟶^{g} X
↓  ^{h}↓
b_{∞}, b ⟶ B

where B is a smooth curve, h is a proper, flat family of nodal curves with smooth general fibre such that h^{−1}(b) ≅ Y for some b ∈ B(k), g|_{Y} = π, and there is a point b_{∞} ∈ B(k) and an irreducible component X_{∞} ⊂ h^{−1}(b_{∞}) with g|_{X_{∞}} : X_{∞} → X an isomorphism.

4.6. Lemma. There exists a morphism Φ : Y → P which is a flexible curve with links and such that there exists a W-diagram for π = f ∘ Φ.

Proof. By Lemma 4.4, we know that there exists a flexible curve Φ_{0} : Y_{0} → P. By Proposition 1.1, we know that there exists a curve with links Y = Y_{0} ∪ L_{1} ∪ … ∪ L_{r} such that there exists a W-diagram for π : Y → X where π|_{Y_{0}} = f ∘ Φ_{0} and where each π|_{L_{i}} is a constant morphism. Moreover, we can choose Y so that the sets L_{i} ∩ Y_{0} all miss the finitely many points of Y_{0} − Φ_{0}^{−1}(V). By condition (4), we can find embeddings Φ_{i} : L_{i} → P as in condition (4) passing through Φ_{0}(p_{i}) and Φ_{0}(q_{i}). Define Φ : Y → P by Φ|_{Y_{0}} = Φ_{0} and each Φ|_{L_{i}} = Φ_{i}. □

4.7. Lemma. Suppose C is a smooth curve, α : A → C, β : B → C are projective, flat morphisms and γ : A → B is a morphism of C-schemes which is an isomorphism over some point c ∈ C. Then there exists a Zariski open set c ∈ U ⊂ C such that γ : α^{−1}(U) → β^{−1}(U) is an isomorphism.

Proof. Even though this result is well-known, we could not find a reference. Therefore we will include a proof. Without loss of generality we may assume that C is affine. Choose an ample invertible sheaf L on B. The pullback γ^{∗}L is ample on β^{−1}(c). By [EGA, III, part 1, Theorem 4.7.1], γ^{∗}L is ample when we shrink C. There exists an integer N such that for n ≥ N, we have that both L^{n} and γ^{∗}L^{n} are very ample and have no higher cohomologies. By cohomology and basechange, the graded algebras S(A, L^{n}) and S(B, γ^{∗}L^{n}) are flat O_{B}-algebras. Moreover the pullback map γ^{∗} : S(B, L) → S(A, γ^{∗}L^{n}) is an isomorphism when we tensor with O_{C}/m_{c}O_{C}. Thus by Nakayama’s lemma applied to the finitely-generated graded pieces (and using the fact that both algebras are generated and presented in finite degrees), we can find an open subset U ⊂ C such that γ^{∗} is an isomorphism over U. So γ is an isomorphism over U. □

Proof of Theorem 4.1. By Lemma 4.6, we know there exists a Φ : Y → P which is a flexible curve with links such that there is a W-diagram for π : Y → X. We use the notation of Lemma 4.6 for Y and the notation of Definition 4.5 for the W-diagram. Let Z ⊂ W ×P be the scheme theoretic preimage of the graph Γ_{g} ⊂ W × X under (1, f) : W × P → W × X. We think of Z as a B-scheme using h ∘ pr_{1} : Z → B. Then the graph Γ_{Φ} ⊂ Z_{b} can be considered as a point in the relative Hilbert scheme [Γ_{Φ}] ∈ Hilb(Z/B)(k) over the point b ∈ B(k). Since f : P → X is smooth along Φ(Y), we see that (1, f) : W × P → W × X is smooth along Γ_{Φ}, and that Z_{b} → (Γ_{g})_{b} = Γ_{π} is smooth along Γ_{Φ}. Since Γ_{Φ} is a section of the last mentionned morphism, it is a locally complete intersection. By [8, I Lemma 2.12.1], Γ_{Φ} ⊂ Z_{b} is locally unobstructed. Here is a diagram to illustrate the above; all squares but the right one are cartesian.

Γ_{Φ} ⟶ Z_{b} ⟶ Z ⟶ W × P ⟶ P
≅↓  ↓  ↓  ↓  ^{f}↓
Γ_{π} ⟶^{≅} (Γ_{g})_{b} ⟶ Γ_{g} ⟶ W × X ⟶ X

By [8, I Proposition 2.14.2], the obstruction group of Γ_{Φ} ⊂ Z_{b} is H^{1}(Y, \underline{Hom}_{Y} (I/I^{2}, O_{Y})) where we identify Γ_{Φ} with Y and where I/I^{2} is the conormal bundle of Γ_{Φ} ⊂ Z_{b}. From the diagram we infer that \underline{Hom}_{Y} (I/I^{2}, O_{Y}) is isomorphic to Φ^{∗}T_{P/X}. Consider the short exact sequence

0 ⟶ ⊕_{i=1}^{r}Φ^{∗}T_{P/X}|_{L_{i}}(−p_{i} − q_{i}) ⟶ Φ^{∗}T_{P/X} ⟶ Φ_{0}^{∗}(T_{P/X}) ⟶ 0.

Because Φ_{0} : Y_{0} → P is flexible, H^{1}(Y_{0}, Φ_{0}^{∗}T_{P/X}) = 0. Because each Φ^{∗}T_{P/X} is ample on L_{i}, every line bundle summand has degree ≥ 1, so that every line bundle summand of Φ^{∗}T_{P/X}(−p_{i} − q_{i}) has degree ≥ −1 and H^{1}(L_{i}, Φ_{i}^{∗}T_{P/X}(−p_{i} − q_{i})) = 0. Thus by the long exact sequence in cohomology associated to the short exact sequence above, we conclude that the obstruction space of Γ_{Φ} ⊂ Z_{b} is zero. Therefore the morphism Hilb(Z/B) → B is smooth at [Γ_{Φ}], see [8, I Theorem 2.10]. Therefore we can find a map D → Hilb(Z/B) of a smooth, connected curve D into the Hilbert scheme and a point d ∈ D mapping to [Γ_{Φ}] such that the composite map D → B is étale at d. Since the Hilbert scheme satisfies the valuative criterion of properness over B we may also assume that D → B is finite, hence there is a point d_{∞} ∈ D(k) which maps to b_{∞} in B. The base change by (D, d, d_{∞}) → (B, b, b_{∞}) of the W-diagram of π : Y → X leads to a W-diagram. Thus, by replacing the W-diagram of π : Y → X by the base-change, we may assume that Hilb(Z/B) → B has a section passing through [Γ_{Φ}], i.e., there is a closed subscheme Γ ⊂ Z flat over B with Γ_{b} = Γ_{Φ}. By Lemma 4.6, the composition Γ ↪ Z → W is an isomorphism over an open subset U, b ∈ U ⊂ B, i.e. Γ → W is birational. Let Γ′ → Γ be the normalization, so Γ′ → W is still birational. And the indeterminacy locus of a birational morphism of normal varieties has codimension ≥ 2. Therefore, X_{∞} is not contained in the indeterminacy locus, i.e. there exists a curve X′ ⊂ Γ′ which maps isomorphically to X_{∞}. By construction, the composite map Γ′ → Z → P → X equals the composite map Γ′ → W → X. Thus the image of X′ in P is a curve which maps isomorphically to X, i.e. f : P → X has a section. This ends the proof of Theorem 4.1. □

Proof of the main Theorem. Assumption and notations as in the theorem. First we normalize P to get a normal variety. So Theorem 3.1 applies. Hence it suffices to prove the theorem in those case where all fibres of P → X are reduced. (This reduction changes the field of definition, but no matter.) If the original P was projective then conditions 4.0.1, 4.0.2 and 4.0.3 above are satisfied. Condition 4.0.4’ follows from the existence of the rational curve in the geometric generic fibre. Hence Theorem 4.1 applies and we’re done.

The nonprojective case. The main theorem claims the same result for any proper P → X and the argument above only works if P is projective. Why do we not apply Chow’s lemma and reduce to the projective case?

Here is one reason: Suppose that P′ → P is a projective birational morphism such that P′ is projective and normal. To apply the previous argument we need the geometric generic fibre P ′ ; η̄ of P′ → X to be normal. Unfortunately we do not know how to produce P′ → P with this property.

Finally, here is our proof in the general proper case circumventing this difficulty. As before we first normalize P to get a normal variety. So Theorem 3.1 applies. Hence we may assume all fibres of P → X are reduced. Note that 4.0.1, 4.0.2, 4.0.3 and 4.0.4 hold except for the projectivity in 4.0.1. The only place this was used in the proof of Theorem 4.1 is in the proof of Lemma 4.3. Thus we give an additional argument to prove the existence of a pre-flexible curve in the proper nonprojective case. The idea is to use 4.0.4 to construct a surface Σ → X with rational fibres, a morphism Σ → P over X, and to construct the pre-flexible curve as a divisor on Σ. We will use the following notations: V ⊂ P denotes the open mentioned in 4.0.4, Sm(P/X) ⊂ P denotes the open subset of points of P where the morphism f : P → X is smooth. Note that Sm(P/X) is dense in every fibre of f.

We claim that there exists a 1-dimensional closed subscheme T ⊂ P with the following properties: (a) T is the scheme theoretic union of irreducible reduced curves flat and generically étale over X (in other words, T → X is finite, flat and generically étale), (b) each irreducible component of T meets V, and (c) for every x ∈ X(k), the intersection T_{x} ∩ Sm(P/X) is not empty. We leave it to the reader to construct T. (Hints: Start with some irreducible curve meeting V, generically étale over X, and not contained in a fibre of P → X. Then (a) and (b) hold and (c) holds except for a finite number of points x_{i} ∈ X(k). For each i add an irreducible component T_{i} passing through a nonsingular point of the fibre P_{x_{i}}, generically étale over X with T_{i} ∩ V ≠ ∅.)

Pick a point x ∈ X(k) such that T_{x} ⊂ V, and such that T_{x} is a reduced set of points (i.e., such that T → X is étale at all points of T_{x}). Pick a smooth rational curve C ⊂ V_{x} such that T_{x} ⊂ C and such that T_{P/X}|_{C} is ample. Think of T_{x} as a divisor on C. We claim we may assume that T_{P/X}|_{C} ⊗ O_{C}(−T_{x}) is ample. The proof of [8, IV Theorem 3.9.4] shows that T_{P/X}|_{C} ⊗ O_{C}(−N) can be made ample for any N. (Note that the open V corresponds to the open X^{0} in the statement of [8, IV 3.9.4].) Here is a direct argument. Pick general points c_{i}, i = 1, …, m, m >> 0 on C, and pick a rational curve C_{i} → P_{x} passing through c_{i} such that T_{P/X}|_{C_{i}} is ample. The union C ∪ ⋃ C_{i} is a comb. The results of [8, Section II 7] say that a subcomb can be smoothed fixing T_{x} and free over T_{x} union two auxiliary points. This freeness implies the claim (see [8, II 3.1]).

Let Hilb_{P/X} be the relative Hilbert scheme of P over X. This is in general just an algebraic space and not a scheme, see [2]. For a scheme Y over X a Y -valued point of Hilb_{P/X} corresponds to a closed subscheme Z ⊂ P ×_{X} Y flat and of finite presentation over Y. There is a closed algebraic subspace Hilb_{T ⊂P/X} which parametrizes only those closed subschemes Z ⊂ P ×_{X} Y which contain the closed subscheme T ×_{X} Y. The curve C we constructed above defines a point [C] ∈ Hilb_{T ⊂P/X}(k) lying over x ∈ X(k).

The ampleness above means that the morphism Hilb_{T ⊂P/X} → X is smooth at the point [C]. Namely, the obstruction space for the corresponding deformation problem is equal to H^{1}(C, N_{C}P_{x} ⊗ O_{C}(−T_{x})). This is a quotient of the cohomology group H^{1}(C, T_{P/X}|_{C} ⊗ O(−T_{x})) which is zero.

Thus we can find an (irreducible) étale neighbourhood (Y, y) of (X, x) and a map ψ : Y → Hilb_{T ⊂P/X} such that ψ(y) = [C]. However, since the Hilbert scheme satisfies the valuative criterion of properness, we can extend ψ to a morphism ψ̄ : \overline{Y} → Hilb_{T ⊂P/X} on a nonsingular projective completion \overline{Y} of Y. This means we have a closed subscheme Z ⊂ P ×_{X} \overline{Y} fitting into a commutative diagram:

T ×_{X} \overline{Y} ⟶ Z ⟶ P ×_{X} \overline{Y} ⟶ P
↓  ↓  ↓  ↓
\overline{Y}  ⟶ \overline{Y} ⟶ \overline{Y}  ⟶^{π} X

The morphism π is finite and generically étale. The morphism Z → \overline{Y} is flat and proper and the fibre of Z → \overline{Y} over y is C.

We choose a resolution of singularities Σ → Z; then Σ is a proper smooth surface over k hence projective (see [6, II 4.2]). Clearly, Σ → \overline{Y} is a ruled surface; its fibre over the point y is the curve C. Let U ⊂ Σ be the inverse image of Sm(P/X). By our choice of C we have: (a) the general fibre of Σ → \overline{Y} is contained in U. Namely, the fibre Σ_{y} = C is contained in U. By our choice of T we have: (b) all fibres of U → \overline{Y} are nonempty. Namely, for each \overline{y} ∈ \overline{Y} (k) there is some t ∈ T (k) over π(\overline{y}) which is in Sm(P/X).

We claim that there exists a finite generically étale morphism of irreducible nonsingular projective curves Ỹ → \overline{Y} and an \overline{Y} -morphism Ỹ → U ⊂ Σ. This is a generality on ruled surfaces and open subsets satisfying (a), (b) which follows easily from the lemma below. Granted this generality, we see that the composition Ỹ → Σ → P is the pre-flexible curve as desired. □

4.8. Lemma. Let Σ → X be a nonsingular ruled surface over an irreducible nonsingular projective curve. Assume given irreducible components C_{i}, i = 1, …, r of the singular fibres such that no fibre is covered completely by the C_{i}. Then there exists a projective surface Σ′ over X and a birational morphism Σ → Σ′ over X which contracts all the curves C_{i}.

Proof. We leave it to the reader that condition (b) of [1, Theorem 2.3] holds, and then [1, Theorem 2.3] implies the lemma. If the characteristic of k is p > 0 (which is the case of interest here) then one can also apply [1, Theorem 2.9] directly. □

References

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