DEGENERATIONS OF RATIONALLY CONNECTED VARIETIES AND PAC FIELDS

JASON STARR

Abstract. A degeneration of a separably rationally connected variety over a field k contains a geometrically irreducible subscheme if k contains the algebraic closure of its subfield. If k is a perfect PAC field, the degeneration has a k-point. This generalizes [FJ05, Theorem 21.3.6(a)]: a degeneration of a Fano complete intersection over k has a k-point if k is a perfect PAC field containing the algebraic closure of its prime subfield.

1. Statement of results

Recently, a number of fields long known to be C_{1} were proved to satisfy an a priori stronger property related to rationally connected varieties.

(i) Every rationally connected variety defined over the function field of a curve over a characteristic 0 algebraically closed field has a rational point, [GHS03]. (ii) Every separably rationally connected variety defined over the function field of a curve over an algebraically closed field of arbitrary characteristic has a closed point, [dJS03].

(iii) Every smooth, rationally chain connected variety over a finite field has a

rational point, [Esn03]. Moreover, in each of these cases, degenerations of these varieties also have rational points, at least under some mild hypotheses on the degeneration.

This article considers the same problem for perfect PAC fields containing an algebraically closed field. Such fields are known to be C_{1}, [FJ05, Theorem 21.3.6(a)]. The main theorem is the following.

Theorem 1.1. Let k be a perfect PAC field containing the algebraic closure of its prime subfield. Let X_{k} be a k-scheme which is the closed fiber of a proper, flat scheme over a DVR whose geometric generic fiber is separably rationally connected (in the sense of [dJS03]). Then X_{k} has a k-point.

Remark 1.2. This gives a new proof of [FJ05, Theorem 21.3.6(a)], i.e., every perfect PAC field containing an algebraically closed field is C_{1}.

This should be compared to the following theorems of Kollár and de Jong respectively. Theorem 1.3. [Kol05] Let k be a characteristic 0 PAC field. Let X_{k} be a kscheme whose base-change X ⊗_{k} \overline{k} is the closed fiber of a projective, flat scheme over a DVR whose geometric generic fiber is a Fano manifold. Then X_{k} has a k-point. In particular, k is C_{1}.

Theorem 1.4 (de Jong). Let k be a characteristic 0 field having a point in every rationally connected k-scheme and containing \overline{ℚ}. Let X_{k} be a k-scheme which is the closed fiber of a proper, flat scheme over a DVR whose geometric generic fiber is rationally connected. Then X_{k} has a k-point. In particular, k is C_{1}.

Theorem 1.1 is a consequence of the following more precise result.

Theorem 1.5. Let k be a field containing the algebraic closure of its prime subfield. Let X be a proper k-scheme which is the closed fiber of a proper, flat scheme over a DVR whose geometric generic fiber is separably rationally connected (in the sense of [dJS03]). There exists a closed subscheme Y of X such that Y ⊗_{k} \overline{k} is irreducible.

Acknowledgments. I thank A. J. de Jong for explaining his proof of Theorem 1.4 some years ago. I thank János Kollár for encouraging me to write-up this article and pointing out [FJ05, Theorem 21.3.6(a)].

2. Proofs

Definition 2.1. Let R be a DVR and let X_{R} be an R-scheme. A finite type model of (R, X_{R}) consists of a datum

((P, D) → (S, 𝔰), Spec R → P, X_{P})

of

(i) a Dedekind domain S of finite type over ℤ or over ℚ and a maximal ideal 𝔰 of S,

(ii) a normal, flat, projective S-scheme P with geometrically integral generic

fiber and an integral Weil divisor D of P contained in P_{𝔰}, (iii) a dominant morphism Spec R → P, (iv) and a P -scheme X_{P}

such that the inverse image of D equals 𝔪_{R} and the base change of X_{P} equals X_{R}.

Lemma 2.2. If X_{R} is a finite type R-scheme, there exists a finite type model such that X_{P} is of finite type over P.

If X_{R} is proper, resp. projective, there exists a finite type model such that X_{P} is proper, resp. projective.

Proof. This follows easily from [Gro63, §8], results about normalization, e.g., [Gro67, Scholie 7.8.3], and Nagata compactification, [Lüt93]. □

Let B be a scheme. As in [Jou83], for each integer e denote by Gr(e, N) the Grassmannian over B parametrizing codimension-e linear subspaces of fibers of ℙ_{B}^{N}. Denote by Λ_{e} the universal codimension-e linear subscheme of Gr(e, N) ×_{B} ℙ_{B}^{N}.

Notation 2.3. For every B-scheme T and B-morphism i : T → ℙ_{B}^{N}, denote by Z_{i,e} the fiber product T ×_{ℙ_{B}^{N}} Λ_{e}. If i is inclusion of a subscheme, denote this by Z_{T,e}. Observe that for a k-morphism j : U → T, Z_{i∘j,e} equals U ×_{T} Z_{i,e}. Denote by pr_{G} : Z_{i,e} → Gr(e, N) and pr_{T} : Z_{i,e} → T the 2 projections.

Denote by G_{i,e} the maximal open subscheme of Gr(e, N) such that G_{i,e} ×_{Gr(e,N)} Z_{i,e} → G_{i,e} is flat. Denote by Z_{i,e,G} → G_{i,e} the base change of pr_{G} to G_{i,e}.

Let k be a field and let I be a geometrically irreducible, locally closed subscheme of ℙ_{k}^{N}. Denote by d + 1 the dimension of I. Denote G_{I,d} by G and denote Z_{I,d,G} by Z. Denote by pr_{G} : Z → G and pr_{I} : Z → I the 2 projections.

Let H be a k-scheme and let f : H → G be a morphism. Denote by Z_{H} the fiber product H ×_{G} Z. Denote by pr_{H} : Z_{H} → H and pr_{Z} : Z_{H} → Z the 2 projections. Denote by p : Z_{H} → I the composition pr_{I} ∘ pr_{Z}.

Lemma 2.4. If the dimension of I is positive, if H is geometrically irreducible and if f is dominant, then Z_{H} is geometrically irreducible, and the geometric generic fiber of p is irreducible.

Proof. By definition of G, Z is flat over G of relative dimension 1. By [Jou83, Théorème I.6.10.2], the geometric generic fiber of Z → G is irreducible. Since f is dominant, Z_{H} → H is flat of relative dimension 1 with irreducible geometric generic fiber. Since H is also geometrically irreducible, Z_{H} is geometrically irreducible.

Because f is dominant and pr_{I} are dominant, also pr_{Z}, and thus the composition p, is dominant. Thus, to prove the geometric generic fiber of p is irreducible, it suffices to prove the geometric generic fiber of

pr_{1} : Z_{H} ×_{I} Z_{H} → Z_{H}

is irreducible.

The morphism pr_{H} ∘pr_{1} : Z_{H} ×_{I} Z_{H} → H is the base-change of Z_{p,d} → Gr(d, N) by H → Gr(d, N). The image of p is dense in I, thus has dimension d + 1. Thus, by [Jou83, Théorème I.6.10.2], the geometric generic fiber of pr_{G} : Z_{p,d} → Gr(d, n) is irreducible. Since H → Gr(d, N) is dominant, also the geometric generic fiber of pr_{H} ∘ pr_{1} is irreducible. Denote the geometric generic fiber by X.

Denote by Y the geometric generic fiber of pr_{H} : Z_{H} → H. By the argument above, Y is also irreducible. The morphism pr_{1} induces a dominant morphism q : X → Y. The geometric generic fiber of p is irreducible if and only if the geometric generic fiber of q is irreducible.

Denote by q̃: Ỹ → Y_{red} the integral closure of Y_{red} in the fraction field of X_{red}. By the Noether normalization theorem, q̃ is finite. And the geometric generic fiber of X_{red} → Ỹ is integral. Thus the geometric generic fiber of p is irreducible if and only if the geometric generic fiber of q̃ is irreducible.

Because X is irreducible, also Ỹ is irreducible. The diagonal morphism Δ : Z_{H} → Z_{H} ×_{I} Z_{H} gives a section of pr_{1}. This induces a section s : Y → X of q. This induces a section s̃: Y_{red} → Ỹ of q̃.

Since q̃ is finite, dim(Ỹ) equals dim(Y_{red}). The image of s̃ is an irreducible closed subscheme of Ỹ whose dimension equals dim(Y_{red}), i.e., dim(Ỹ). Therefore, the image is an irreducible component of Ỹ. Since Ỹ is irreducible, the image of s̃ is all of Ỹ. Thus s̃ is an inverse of q̃. Therefore the geometric generic fiber of q̃ is a point, which is irreducible. □

Proof of Theorem 1.5. Let R be a DVR whose residue field k contains the algebraic closure of its prime subfield. Let X_{R} be a proper R-scheme whose geometric generic fiber is separably rationally connected. By Lemma 2.2, there exists a finite type model with X_{P} proper. By hypothesis, k contains the algebraic closure of the residue field \overline{κ(𝔰)}.

If P has relative dimension 0 over S, then P equals S. Thus X_{k} is the base change of the proper κ(𝔰)-scheme X_{𝔰}. Since κ(𝔰) is a finite extension of the prime subfield of k, the algebraic closure \overline{κ(𝔰)} is contained in k. The base change Y of any \overline{κ(𝔰)}-point of X_{𝔰} is a geometrically irreducible subvariety of X_{k}.

Thus, assume the relative dimension of P over S is positive, d + 1. Let P ↪ ℙ_{S}^{N} be a closed immersion. Denote by G_{P,d} and Z_{P,d,G} the schemes over S from Notation 2.3. By definition, the geometric generic fiber of P over S is normal. In particular it is smooth in codimension 1 points. Therefore, by [Jou83, Théorème I.6.10.2], the geometric generic fiber of Z_{P,d,G} → G_{P,d} is smooth.

By [Art69, Theorem 6.1], there exists an algebraic space Π separated and locally of finite type over G_{P,d}, and a universal morphism

σ : Π ×_{G_{P,d}} Z_{P,d,G} → X_{P}

whose composition with projection to P equals the composition pr_{P} ∘ pr_{Z_{P,d,G}}. Since the geometric generic fiber of X_{R} is separably rationally connected, also the geometric generic fiber of X_{P} → P is separably rationally connected. Therefore, by [dJS03], the geometric generic fiber of Π → G_{P,d} is nonempty. Therefore some irreducible component Π_{i} of Π dominates G_{P,d}.

By [Knu71, Corollary 5.20], there exists a dense open subspace U of Π_{i} which is an affine scheme. There exists a dense open immersion of U in a projective G_{P,d}-scheme \overline{U}. The morphism σ induces a rational transformation

σ_{U} : \overline{U} ×_{G_{P,d}} Z_{P,d,G} −−→ X_{P} .

Denote by \overline{V} the normalization of \overline{U} ×_{G_{P,d}} Z_{P,d,G}. Denote by V the maximal open subscheme of \overline{V} over which σ_{U} extends to a regular morphism. By the valuative criterion of properness, the complement of V has codimension 2. In particular, some irreducible component of V_{𝔰} dominates (G_{P,d})_{𝔰}.

Since k contains \overline{κ(𝔰)} and the function field κ(D) of D, it contains the function field κ(I) of one of the irreducible components I of D ⊗_{κ(𝔰)} \overline{κ(𝔰)}. Let H denote an irreducible component of V_{𝔰} ⊗_{κ(𝔰)} \overline{κ(𝔰)} dominating (G_{P,d})_{𝔰} ⊗_{κ(𝔰)} \overline{κ(𝔰)}. Let G, Z, etc., be as in Lemma 2.4. Denote by Z_{k} the base change Z ⊗_{O_{I}} k. Note that the base change X_{I} ⊗_{O_{I}} k equals X_{k} by definition of a finite type model.

The transformation σ_{U} determines a morphism

σ_{H} : Z_{H} → X_{I}.

The base change to k is a morphism of k-schemes,

σ_{k} : Z_{k} → X_{k}.

Denote the closure of the image by Y. By Lemma 2.4, Z_{H} is geometrically irreducible and the geometric generic fiber of Z_{H} → I is irreducible. Thus Z_{k} ⊗_{k} \overline{k} is irreducible. Therefore Y is a closed subscheme of X_{k} such that Y ⊗_{k} \overline{k} is irreducible. □

Proof of Theorem 1.1. If k is a perfect PAC field, then every geometrically irreducible k-scheme has a k-point. In particular, Y has a k-point. Therefore X_{k} has a k-point. □

Every field k is the closed fiber of a DVR R whose generic fiber has characteristic 0. Every complete intersection in ℙ_{k}^{n} is the closed fiber of a complete intersection in ℙ_{R}^{n} whose generic fiber is smooth. If the complete intersection satisfies the C_{1} inequality, the generic fiber is a Fano manifold. By [KMM92], [Cam92], a Fano manifold in characteristic 0 is rationally connected. Therefore Theorem 1.1 implies the complete intersection in ℙ_{k}^{n} has a k-point if k is a perfect PAC field containing the algebraic closure of its prime subfield. In other words, every perfect PAC field containing an algebraically closed field is C_{1}, cf. [FJ05, Theorem 21.3.6(a)].

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