WEAK APPROXIMATION AND R-EQUIVALENCE OVER FUNCTION FIELDS OF CURVES

MIKE ROTH AND JASON MICHAEL STARR

Abstract. For a rationally connected fibration over a complex curve, and for a closed point of the curve, we prove that every power series section of the fibration near the point is approximated to arbitrary order by polynomial sections provided the “Laurent fiber”, i.e., the deleted power series neighborhood of the fiber, as a variety over the Laurent series field is R-connected – the analogue of rational connectedness when working over a non-algebraically closed field such as Laurent series. In other words, we prove the Hassett-Tschinkel conjecture when the Laurent fiber is R-connected. For varieties over the fraction field of a complete DVR, we introduce a “continuous variant” of R-connectedness called pseudo R-connectedness and we prove pseudo R-connectedness of the Laurent fiber also implies the Hassett-Tschinkel conjecture. Our theorem implies all of the known cases of the Hassett-Tschinkel conjecture, and we also prove some new cases. The key is a new object, a “pseudo ideal sheaf”, which generalizes Fulton’s notion of effective pseudo divisor.

1. Introduction sec-int

Disclaimer. This is the current, working draft of this article. This draft is incomplete, but has a complete list of the results. An earlier, complete draft has been posted to the arXiv preprint server.

Let π : X → B be a surjective morphism from a smooth, projective, complex variety to a smooth, projective, complex curve. Considered differently, π is a 1parameter family {X_{b}}_{b∈B} of projective varieties. The morphism π satisfies weak approximation if for every finite sequence (b_{1}, …, b_{m}) of distinct closed points of B, for every sequence (ŝ_{1}, …, ŝ_{m}) of elements ŝ_{i} in X (Ô_{B,b_{i}}), i.e., formal power series section of π near b_{i}, and for every positive integer N, there exists a regular (i.e., polynomial) section s of π which is congruent to ŝ_{i} modulo 𝔪_{B,b_{i}}^{N} for every i = 1, …, m.

Hassett observed that if π satisfies weak approximation then every sufficiently general fiber of π is rationally connected, i.e., every pair of points in the fiber is in the image of a morphism from ℙ^{1} to the fiber. Hassett and Tschinkel conjectured the converse.

Conjecture 1.1 (Hassett-Tschinkel conjecture). [HT06, Conjecture 2] If a general conj-HT fiber of π is rationally connected, then π satisfies weak approximation.

We prove this conjecture assuming an extra condition on the “Laurent fibers” of π. For every closed point b of B, the fraction field K = Frac(Ô_{B,b}) of the completed local ring is isomorphic to the Laurent series field ℂ \begin{matrix}( \\ ( & t & ) \\ ) \end{matrix}. The Laurent fiber is the projective scheme X_{K} := X ⊗_{O_{B}} K. This is an algebraic version of the “deleted tubular neigborhood” of the closed fiber X_{b} := π^{−1}(b). Even when X_{b} is singular, the K-scheme X_{K} is smooth. Two K-points s and t of the Laurent fiber are directly R-equivalent if there exists a K-morphism from ℙ_{K}^{1} sending 0 to s and sending ∞ to t,

Date: August 1, 2009.

f : (ℙ_{K}, 0, ∞) → (X ⊗_{O_{B}} K, s, t).

The Laurent fiber is R-connected if all K-points are directly R-equivalent. When X_{b} = π^{−1}(b) is smooth, the Laurent fiber X_{K} is R-connected if and only if the closed fiber X_{b} is rationally connected. (The usual definition of R-connected is a bit different. But Kollár proved the equivalence of the two definitions when the geometric generic fiber is separably rationally connected, as it is here.)

We prove Conjecture 1.1 when the Laurent fibers of π are R-connected. In fact the proof requires a weaker, “continuous” variant of R-connnectedness which is technical but useful. As above, the underlying hypothesis is that the geometric generic fiber of π is rationally connected. Denote by Ô the complete local ring Ô_{B,b}, and let u be a uniformizer of Ô. This is a Henselian (even a complete) DVR. Denote by XÔ the base change Spec (Ô)×_{B} X. The “bidisk”, DÔ, is the Ô-scheme

DÔ := Spec (Ô \begin{matrix}[ \\ [ & v & ] \\ & ] \end{matrix}).

The “punctured bidisk” is the complement of the closed point,

D∗ ; Ô := DÔ \ {⟨u, v⟩}.

Denote by Δ_{v} the divisor in DÔ

Δ_{v} := Spec (Ô \begin{aligned}[ \\ [ & v & ] \\ & ] \end{aligned} /⟨v⟩) ≅ Spec (Ô).

And for every positive integer N, denote by Δ_{u^{N}} the divisor in DÔ

Δ_{u^{N}} := Spec (Ô \begin{aligned}[ \\ [ & v & ] \\ & ] \end{aligned} /⟨u^{N}⟩) ≅ Spec ((Ô/⟨u^{N}⟩) \begin{matrix}[ \\ [ & v & ] \\ & ] \end{matrix}).

Finally denote the intersections with D∗ ; Ô by Δ_{v}^{∗} = Δ_{v} ∩ D∗ ; Ô, Δ_{u^{N}}^{∗} = Δ_{u^{N}} ∩ D∗ ; Ô.

Let ŝ be an Ô-point of XÔ. The restriction of ŝ to Spec (Frac(Ô)) determines an Ô-morphism from Δ_{v}^{∗} → XÔ via the identification Δ_{v}^{∗} = Spec (Frac(Ô)). Similarly, the restriction of ŝ to Spec (Ô/⟨u^{N}⟩) composed with the “projection”

Spec ((Ô/⟨u^{N}⟩) \begin{matrix}( \\ ( & v & ) \\ & ) \end{matrix}) → Spec (Ô/⟨u^{N}⟩)

determines an Ô-morphism Δ_{u^{N}}^{∗} → XÔ. The morphisms Δ_{v}^{∗} → XÔ and Δ_{u^{N}}^{∗} → XÔ defined in this way are the associated morphisms of ŝ.

defn-psiRprelim Definition 1.2. Two Ô-points ŝ and t̂ of XÔ are directly pseudo R-equivalent if for every positive integer N there exist Ô-morphisms

q_{N}, r_{N} : D∗ ; Ô → XÔ

such that the restriction of q_{N}, resp. r_{N}, to Δ_{v}^{∗} is the morphism associated to ŝ, resp. the morphism associated to t̂, and such that the restriction of q_{N}, resp. r_{N}, to Δ_{u^{N}}^{∗} is the morphism associated to t̂, resp. the morphism associated to ŝ. And XÔ is pseudo R-connected if every pair of Ô-points is directly pseudo R-equivalent.

In fact pseudo R-connectedness depends only on the Laurent fiber X_{K} = Spec K×_{Spec Ô} XÔ. Also R-connectedness implies pseudo R-connectedness. Here is our main theorem.

thm-first

Theorem 1.3. The morphism π satisfies weak approximation if the Laurent fiber of π over every closed point of B is pseudo R-connected, e.g., if it is R-connected.

Pseudo ideal sheaves. The key is a new sheaf-theoretic object which we call a “pseudo ideal sheaf”. Let S be a scheme or an algebraic space and let Y be a flat, projective S-scheme or a flat, proper, algebraic space over S.

defn-pisprelim

Definition 1.4. For every S-scheme T, denoting Y_{T} = T ×_{S} Y, a pseudo ideal sheaf on Y/S parameterized by T is a pair (F, ϕ) of a T -flat, locally finitely presented, quasi-coherent O_{Y_{T}} -module F together with an O_{Y_{T}} -module homomorphism, ϕ : F → O_{Y_{T}}, such that the induced O_{Y_{T}} -module homomorphism,

ϕ′ : ⋀^{2} F → F, f_{1} ∧ f_{2} ↦ ϕ(f_{1})f_{2} − ϕ(f_{2})f_{1},

is the zero homomorphism. A morphism from a pseudo ideal sheaf (F_{1}, ϕ_{1}) to a pseudo ideal sheaf (F_{2}, ϕ_{2}) is an isomorphism of O_{Y_{T}} -modules, ψ : F_{1} → F_{2}, such that ϕ_{2} ∘ ψ equals ϕ_{1}. The identity morphisms and composition of morphisms are defined in the obvious way. For an S-morphism g : T′ → T and a pseudo ideal sheaf (F, ϕ) parameterized by T, the pullback by g is (g^{∗}F, g^{∗}ϕ).

As proved later, this category is an Artin stack Pseudo_{Y/S}. And the Hilbert scheme Hilb_{Y/S} (or “Hilbert algebraic space”, more generally) is an open substack of Pseudo_{Y/S}. Most importantly, when Y is a Cartier divisor in a flat, projective S-scheme X (or a flat, proper algebraic space over S), there is a restriction morphism

ι_{Y} : Hilb_{X} → Pseudo_{Y}

associating to a flat family of closed subschemes of X with ideal sheaf I, the restriction I ⊗_{O_{X}} O_{Y} with its natural map to O_{Y}. For a geometric point s of S and for an ideal sheaf I on the fiber X_{s} which is generated by a regular sequence, the morphism ι_{Y} is smooth at [I] if H^{1}(X, O_{X}(−Y) · Hom_{O_{X}} (I/I^{2}, O_{X}/I)) equals 0. And in this case, the Zariski tangent space of the fiber of ι_{Y} at [I] is canonically isomorphic to H^{0}(X, O_{X}(−Y) · Hom_{O_{X}} (I/I^{2}, O_{X}/I)). Therefore, given a closed subscheme of X and given an infinitesimal deformation of the intersection of the scheme with a “tubular neighborhood” of a Cartier divisor in X, this result allows us to find an infinitesimal deformation of the closed subscheme of X whose intersection with the tubular neighborhood is as prescribed. We will usually apply this when S equals Spec (ℂ), when X equals X, and when Y is an infinitesimal neighborhood of a fiber X_{b} of π.

R-connectedness and strong rational connectedness. As mentioned, X_{K} is R-connected if X_{b} is smooth and rationally connected. More generally, X_{K} is Rconnected if the smooth locus (X_{b})_{smooth} is strongly rationally connected, i.e., if every pair of closed points of (X_{b})_{smooth} is connected by a (complete) rational curve in (X_{b})_{smooth}. Thus Theorem 1.3 gives a new proof of a theorem of Hassett and Tschinkel, [HT06]: π satisfies weak approximation if the smooth locus of every fiber X_{b} is strongly rationally connected.

Additionally, we prove a structural result about strong rational connectedness using pseudo ideal sheaves. Let X be a normal, projective, complex variety and let D be a closed subset of X containing the singular locus. Assume there exists a very free curve in X which is disjoint from D. Denote by U the dense open subset of X which is the union of all very free curves which are disjoint from D, i.e., the largest open subset of X \ D which is strongly rationally connected. Then every connected component of X \ U intersects D. In particular, (X \ D) \ U has no isolated points. This generalizes one of the steps in Chenyang Xu’s proof that log Del Pezzo surfaces are strongly rationally connected, [Xu08]. Since Xu’s theorem is an important advance in this area, we present our own interpretation of his proof in an appendix to this article.

Special cases of R-connectedness. There are many other cases where the Laurent fibers X_{K} are known to be R-connected, cf. [CT08, Section 10]:

(i) X_{K} is a compactification of a connected linear group [CTS77, Corollaire 6], (ii) X_{K} is a conic bundle over ℙ^{1} with discriminant divisor of degree 4 [CTS87] (such surfaces include Del Pezzo surfaces of degree 4 and Chatelet surfaces),

(iii) X_{K} is a smooth complete intersection of type (2, 2) in ℙ^{n} with n ≥ 5

[CTSSD87, Theorem 3.27(ii)], and (iv) X_{K} is a smooth cubic hypersurface in ℙ^{n} with n ≥ 5 [Mad08, Footnote, p.

927]. So if the generic fiber of π satisfies one of (i) – (iv), then Theorem 1.3 implies π satisfies weak approximation. We generalize (i) below. In fact (ii) is one example of a fibration, which we consider below. We will also say more about Del Pezzo surfaces. Case (iv) improves a result of Hassett and Tschinkel, [HT], who proved weak approximation for cubic hypersurfaces in ℙ^{6}.

R-connectedness and rational simple connectedness. Hassett proved π satisfies weak approximation if the geometric generic fiber satisfies a suitably strong version of “rational simple connectedness”. Using [GHS03] the Laurent fibers are Rconnected if they satisfy a weak version of rational simple connectedness discussed below. This refines Hassett’s theorem. In particular, de Jong and the second author proved that every smooth complete intersection of type (d_{1}, …, d_{c}) in ℙ^{n} satisfies the weak version of rational simple connectedness provided that

n + 1 ≥ d_{1}^{2} + ⋯ + d_{c}^{2}.

We will review the proof of this quickly below. Thus π satisfies weak approximation if the geometric generic fiber of π is a complete intersection as above.

Pseudo R-connectedness and fibrations. A fibration over ℂ(B) is a surjective morphism ρ : X → Y of smooth, projective ℂ(B)-schemes whose geometric generic fiber is connected. If Y satisfies weak approximation, and if the fiber of ρ over each sufficiently general ℂ(B)-point of Y satisfies weak approximation, then also X satisfies weak approximation. The analogous statement for R-connectedness is probably false: the analogue would contradict the Fano conjecture that some conic bundles over ℙ_{ℂ}^{2} are non-unirational. However, pseudo R-connectedness does satisfy the analogous property for a fibration of K-schemes.

Pseudo R-connectedness of homogeneous spaces. Because of the fibration property, conic bundles as in (ii) are pseudo R-connected. This also applies to a compactification of a homogeneous space G/H under a connected, linear algebraic group G. When H is connected, i.e., when the connected component of the identity H_{0} equals all of H, then H is (geometrically) rationally connected. Hence Rconnectedness of G/H follows from R-connectedness of G. For general H, we do not know if G/H is R-connected. However, it is pseudo R-connected. The Deligne-Mumford stack B(π_{0}H) is R-connected in a suitable sense. Thus considering G/H as fibered over B(π_{0}H) with fibers of the form G/H_{0} (more correctly, G/H̃_{0} where H̃_{0} is a twist of H_{0}), it follows that every compactification of G/H is pseudo Rconnected. By Theorem 1.3, if the generic fiber of π is a compactification of G/H then π satisfies weak approximation. This gives a new proof of a theorem of Colliot-Thélène and Gille, [CTG04].

The Hassett-Tschinkel Conjecture. In fact the main importance of Theorem 1.3 is a new perspective on the Hassett-Tschinkel conjecture. We believe this conjecture fails as stated. As we understand it from conversations with experts, there is an a priori obstruction to weak approximation defined using notions of 𝔸^{1}-homotopy theory. The two authors do not know how to compute whether or not this obstruction is zero. But we are quite hopeful that this will be possible in the near future, based on the work of Asok, Morel and Bhatt relating rational connectedness to notions of 𝔸^{1}-homotopy theory. Our understanding is that the obstruction is analogous to the Brauer-Manin obstruction to weak approximation for varieties defined over number fields. Just as the “Brauer pairing” is actually an obstruction to R-connectedness of p-adic fibers – and even to pseudo R-connectedness of p-adic fibers – so too any obstruction defined using a cohomology theory with a “homotopy axiom” should factor through R-equivalence. And if the cohomology theory has a “continuous nature”, then the obstruction should even factor through pseudo R-equivalence. Thus R-connectedness and pseudo R-connectedness of Laurent fibers is really a stand-in for vanishing of these cohomological obstructions, at the moment beyond our reach. Each positive result listed above supporting the Hassett-Tschinkel conjecture has a hypothesis implying R-connectedness or pseudo R-connectedness of Laurent fibers. Therefore the conjecture which these results more properly support is that the only obstruction to weak approximation for a rationally connected fibration is a cohomological obstruction. This is an analogue for ℂ(B) of a well-known conjecture of Colliot-Thélène for number fields, [CT03].

New questions and other fields. This perspective leads to an array of enticing questions. Is weak approximation preserved by finite base change? Is it an open condition in families? Is it a closed condition? Is it preserved by “twists”? Is it preserved under quotients by finite groups? Moreover, what can one say about the set of formal sections of π which are approximable, i.e., approximated to arbitrary order by rational sections? We will discuss these questions further at the end of the article.

Of course all of these results remain valid when ℂ is replaced by any algebraically closed field of characteristic 0. We will discuss which results remain valid in positive characteristic, and the issues in extending the other results.

Cubic surfaces with integral special fibers. Finally we prove a new weak approximation result for Del Pezzo surfaces which is elementary and which has nothing to do with Theorem 1.3. If every fiber X_{b} of π is an integral scheme whose dualizing sheaf ω_{X_{b}} is antiample with degree c_{1}(ω_{X_{b}})^{2} ≥ 3, then weak approximation holds. In particular this implies all previously known weak approximation results for cubic surfaces. For Del Pezzo surfaces of degrees 2 and 1, the results of Amanda Knecht and the results of Chenyang Xu give the best current results. In an appendix we present our interpretation of the proof of Xu’s theorem.

Overview of this paper. [This still needs to be written]

Acknowledgments. We are very grateful to Universita’ Roma Tre, Queen’s University, Stony Brook University and Columbia University where this work was carried out. We are also very grateful to A. J. de Jong who made major contributions to this work, particularly regarding the notion of pseudo R-equivalence, but who preferred not to be a coauthor.

2. Extension of two results of Grothendieck

sec-G

This section proves technical results which are useful in the following section. But this section is not strictly necessary for understanding the main ideas. The reader can safely skip this section by assuming all proper morphisms of algebraic spaces in later sections are actually projective morphisms of schemes.

One of Grothendieck’s results which is frequently useful in studying coherent sheaves on proper schemes is [Gro63, Corollaire 7.7.8], stated below. The original proof includes a hypothesis which Grothendieck refers to as “surabondante” (Grothendieck’s emphasis). This hypothesis has subsequently been removed by Lieblich, cf. [Lie06, Proposition 2.1.3]. We would like to briefly explain another way to remove these hypotheses based on the representability of the Quot functor. Martin Olsson proved a very general result about representability of the Quot functor. This leads to the following version of Grothendieck’s result.

thm-Hom Theorem 2.1. [Gro63, Corollaire 7.7.8], [Lie06, Proposition 2.1.3] Let Y be an algebraic space. Let f : X → Y be a separated, locally finitely presented, algebraic stack over the category (Aff/Y) of affine Y -schemes. Let F and G be locally finitely presented, quasi-coherent O_{X} -modules. Assume that G is Y -flat and has proper support over Y. Then the covariant functor of quasi-coherent O_{Y} -modules,

T (M) := f_{∗}(Hom_{O_{X}} (F, G ⊗_{O_{Y}} M)),

is representable by a locally finitely presented, quasi-coherent O_{Y} -module N, i.e., there is a natural equivalence of functors

T (M) ⟶^{≅} Hom_{O_{Y}} (N , M).

Grothendieck deduced his version of this result as a corollary of another theorem. In fact the theorem also follows from the corollary.

cor-Hom Corollary 2.2. [Gro63, Théorème 7.7.6] Let f : X → Y be as in Theorem 2.1. Let G be a locally finitely presented, quasi-coherent O_{X} -module which is Y -flat and has proper support over Y. Then there exists a locally finitely presented, quasi-coherent O_{Y} -module Q and a natural equivalence of covariant functors of quasi-coherent O_{Y} -modules M,

f_{∗}(G ⊗_{O_{Y}} M) ⟶^{≅} Hom_{O_{Y}} (Q, M).

Grothendieck in turn deduces [Gro63, Théorème 7.7.6] as a special case of a more general representability result regarding the hyperderived pushforwards of a bounded below complex of locally finitely presented, quasi-coherent O_{X} -modules which are Y -flat and have proper support over Y. Undoubtedly this general representability result extends to the setting of stacks. However, to deduce [Gro63, Corollaire 7.7.8] from [Gro63, Théorème 7.7.6], Grothendieck requires a hypothesis that F has a presentation by locally free O_{X} -modules. This is quite a restrictive hypothesis.

Instead one can try to prove Theorem 2.1 directly using Artin’s representability theorems. This is precisely what Lieblich does in [Lie06, Proposition 2.1.3]. Here we point out that Theorem 2.1 also follows from from representability of the Quot functor as proved by Olsson, [Ols05, Theorem 1.5].

lem-Hom1

Lemma 2.3. Let f : X → Y be as in Theorem 2.1. Let F and G be locally finitely presented, quasi-coherent O_{X} -modules. And let

ϕ : F → G

be a homomorphism of O_{X} -modules. Assume G has proper support over Y, resp. F and G have proper support over Y and G is Y -flat. Then there exists an open subspace U, resp. V, of S with the following property. For every morphism g : Y′ → Y of algebraic spaces, the pullback morphism of sheaves on Y′ ×_{Y} X,

(g, Id_{X})^{∗}ϕ : (g, Id_{X})^{∗}F → (g, Id_{X})^{∗}G

is surjective, resp. an isomorphism, if and only if g factors through U, resp. V.

Proof. Since Coker(ϕ) is a locally finitely presented, quasi-coherent sheaf supported on Supp(G), Supp(Coker(ϕ)) is closed in Supp(G). Since Supp(G) is proper over Y, f(Supp(Coker(ϕ))) is a closed subset of Y. Define U to be the open complement. For a morphism g : Y′ → Y, (g, Id_{X})^{∗}ϕ is surjective if and only if Coker((g, Id_{X})^{∗}ϕ) is zero, i.e., if and only if the support of Coker((g, Id_{X})^{∗}ϕ) is empty. Formation of the cokernel is compatible with pullback, i.e.,

Coker((g, Id_{X})^{∗}ϕ) ≅ (g, Id_{X})^{∗}Coker(ϕ).

Thus the support of Coker((g, Id_{X})^{∗}ϕ) is empty if and only if g(Y′) is disjoint from f(Supp(Coker(ϕ))), i.e., if and only if g(Y′) is contained in U.

Next assume that F, G each have proper support over Y and G is Y -flat. Since G is Y -flat, the kernel of ϕ is locally finitely presented by [Gro67, Lemme 11.3.9.1] (the property of being locally finitely presented can be checked locally in the fppf topology on X, thus reduces to the case of a morphism of schemes). The support of the kernel is contained in the support of the kernel of F. Thus Ker(ϕ) is a locally finitely presented, quasi-coherent sheaf on the support of F, which is proper over Y. So the support of Ker(ϕ) is also proper over Y. Thus its image under f is a closed subset of Y. Define V to be the open complement of this closed subset in U.

For every morphism g : Y′ → U, since G is Y -flat, the following sequence is exact

0 → (g, Id_{X})^{∗}Ker(ϕ) → (g, Id_{X})^{∗}F → (g, Id_{X})^{∗}G → 0.

Thus (g, Id_{X})^{∗}ϕ is an isomorphism if and only if (g, Id_{X})^{∗}Ker(ϕ) is zero, i.e., if and only if g(Y′) is contained in V. □

lem-Hom2

Lemma 2.4. Let f : X → Y, F and G be as in Theorem 2.1. There exists a locally finitely presented, separated morphism of algebraic spaces h : Z → Y and a morphism of quasi-coherent sheaves on Z ×_{Y} X

ϕ : (h, Id_{X})^{∗}F → (h, Id_{X})^{∗}G

which represents the contravariant functor associating to every morphism g : Y′ → Y the set of morphisms of quasi-coherent sheaves on Y′ ×_{Y} X

ψ : (g, Id_{X})^{∗}F → (g, Id_{X})^{∗}G.

Proof. By [Ols05, Theorem 1.5], there exists a locally finitely presented, separated morphism i : W → Y of algebraic spaces and a quotient

θ : (i, Id_{X})^{∗}(F ⊕ G) → H

representing the Quot functor of flat families of locally finitely presented, quasicoherent quotients of the pullback of F ⊕ G having proper support over the base. Denote by

θ_{G} : (i, Id_{X})^{∗}G → H

the composition of the summand

e_{G} : (i, Id_{X})^{∗}G ↪ (i, Id_{X})^{∗}(F ⊕ G)

with θ.

By Lemma 2.3, there is an open subspace Z of W such that for every morphism j : Y′ → W, the pullback

(j, Id_{X})^{∗}θ_{G} : (i ∘ j, Id_{X})^{∗}G → (j, Id_{X})^{∗}H

is an isomorphism if and only if j(Y′) factors through Z. Denote by h : Z → Y the restriction of i to Z. If j(Y′) factors through Z, then (j, Id_{X})^{∗}θ equals the composition

(i ∘ j, Id_{X})^{∗}(F ⊕ G) ⟶^{(ψ,Id)} (i ∘ j, Id_{X})^{∗}G ⟶^{j^{∗}θ_{G}} (j, Id_{X})^{∗}H

for a unique morphism of quasi-coherent sheaves

ψ : (i ∘ j, Id_{X})^{∗}F → (i ∘ j, Id_{X})^{∗}G.

In particular, applied to Id_{Z} : Z → Z, this produces the homomorphism ϕ. And it is straightforward to see that the natural transformation associating to every morphism j : Y′ → Z the homomorphism ψ is an equivalence of functors, i.e., (h : Z → Y, ϕ) is universal. □

Proof of Theorem 2.1. Let h : Z → Y and ϕ be as in Lemma 2.4. The zero homomorphism 0 : F → G defines a Y -morphism z : Y → Z. Since Z is separated over Y, the Y -morphism z is a closed immersion. Since Z is locally finitely presented over Y, the ideal sheaf I of this closed immersion is a locally finitely presented, quasi-coherent O_{Z}-module. Thus z^{∗}I is a locally finitely presented, quasi-coherent O_{Y} -module. Denote this O_{Y} -module by N.

Consider the closed subspace Z_{1} of Z with ideal sheaf I^{2}. The restriction of h to Z_{1} is a finite morphism, thus equivalent to the locally finitely presented O_{Y} -algebra h_{∗}O_{Z_{1}}. Of course this fits into a short exact sequence

0 → z^{∗}I → h_{∗}O_{Z_{1}} → O_{Y} → 0

where the injection is an ideal sheaf and the surjection is a homomorphism of O_{Y} - algebras. And the morphism z defines a splitting of this surjection of O_{Y} -algebras. Thus, as an O_{Y} -algebra, there is a canonical isomorphism

h_{∗}O_{Z_{1}} ≅ O_{Y} ⊕ z^{∗}I.

The restriction of ϕ to Z_{1} together with adjunction of h^{∗} and h_{∗} defines a homomorphism of O_{Y} -modules

F → G ⊗_{O_{Y}} h_{∗}O_{Z_{1}}.

Using the canonical isomorphism, this homomorphism is of the form

F ⟶^{(0,χ)} G ⊕ (G ⊗_{O_{Y}} N).

The homomorphism χ defines a natural transformation of covariant functors of quasi-coherent O_{Y} -modules M

Hom_{O_{Y}} (N , M) → f_{∗}(Hom_{O_{X}} (F, G ⊗_{O_{Y}} M).

By the same argument used to prove the equivalence of parts (a) and (d) of [Gro63, Théorème 7.7.5], this is an equivalence of functors and the induced Y -morphism

\underline{Spec}_{Y} (Sym^{•}(N)) → Z

is an isomorphism. □

Proof of Corollary 2.2. This follows from Theorem 2.1 by taking F to be O_{X}. □

3. Pseudo ideal sheaves sec-pis

Given a fibration over a curve, given a fiber, and given a section, the N-jet of the section at the fiber is simply the intersection of the section and the N^{th} infinitesimal neighborhood of the fiber, considered as a closed subscheme of the infinitesimal neighborhood. The rule associating to a section its N-jet gives a morphism from the parameter scheme of sections to the Hilbert scheme of the N^{th} infinitesimal neighborhood of the fiber. In order to prove Theorem 1.3, it is necessary to extend this morphism to the locus parameterizing “combs” whose intersection with the fiber may not be transverse. This is done using “pseudo ideal sheaves”, a generalization of Fulton’s notion of effective pseudo divisor.

Let f : X → Y be a flat, locally finitely presented, proper morphism of algebraic spaces. For every morphism of algebraic spaces, g : Y′ → Y, denote by f_{Y′} : X_{Y′} → Y′ the basechange of f. Using results of Martin Olsson, the following definitions and results are still valid whenever X is a flat, locally finitely presented, proper algebraic stack over Y. But since our application is to algebraic spaces, we leave the case of stacks to the interested reader.

defn-pis

Definition 3.1. For every morphism g : Y′ → Y of algebraic spaces, a flat family of pseudo ideal sheaves of X/Y over Y′ is a pair (F, u) consisting of

(i) a Y′-flat, locally finitely presented, quasi-coherent O_{X_{Y ′}} -module F, and (ii) an O_{X_{Y ′}} -homomorphism u : F → O_{X_{Y ′}}

such that the following induced morphism is zero,

u′ : ⋀^{2} F → F, f_{1} ∧ f_{2} ↦ u(f_{1})f_{2} − u(f_{2})f_{1}.

For every pair g_{1} : Y′_{1} → Y, g_{2} : Y′_{2} → Y of morphisms of algebraic spaces, for every pair (F_{1}, u_{1}), resp. (F_{2}, u_{2}), of flat families of pseudo ideal sheaves of X/Y over Y′_{1}, resp. over Y′_{2}, and for every Y -morphism h : Y′_{1} → Y′_{2}, a pullback map from (F_{1}, u_{1}) to (F_{2}, u_{2}) over h is an isomorphism of O_{X_{Y ′_{1}}} -modules

η : F_{1} → h^{∗}F_{2}

such that h^{∗}u_{2} ∘ η equals u_{1}.

The category of pseudo ideal sheaves of X/Y, Pseudo_{X/S}, is the category whose objects are data (g : Y′ → Y, (F, u)) of an affine Y -scheme Y′ together with a flat family of pseudo ideal sheaves of X/Y over Y′, and whose Hom sets

Hom((g_{1} : Y′_{1} → Y, (F_{1}, u_{1})), (g_{2} : Y′_{2} → Y, (F_{2}, u_{2})))

are the sets of pairs (h, η) of a Y -morphism h : Y′_{1} → Y′_{2} together with a pullback map η from (F_{1}, u_{1}) to (F_{2}, u_{2}) over h. Identity morphisms and composition of morphisms are defined in the obvious manner. There is an obvious functor from Pseudo_{X/Y} to the category (Aff/Y) of affine Y -schemes sending every object (g : Y′ → Y, (F, u)) to (g : Y′ → Y) and sending every morphism (h, η) to h.

prop-algebraic Proposition 3.2. The category Pseudo_{X/Y} is a limit-preserving algebraic stack over the category (Aff/Y) of affine Y -schemes. Moreover, the diagonal is quasicompact and separated.

Proof. Denote by Coh_{X/Y} the category of coherent sheaves on X/Y, cf. [LMB00, (2.4.4)]. There is a functor G : Pseudo_{X/Y} → Coh_{X/Y} sending every object (g : Y′ → Y, (F, u)) to (g : Y′ → Y, F) and sending every morphism (h, η) to (h, η). This is a 1-morphism of categories over (Aff/Y). By [LMB00, Théorème 4.6.2.1] and [Sta06, Proposition 4.1], Coh_{X/Y} is a limit-preserving algebraic stack over (Aff/Y) with quasi-compact, separated diagonal. Thus to prove the proposition, it suffices to prove that G is representable by locally finitely presented, separated algebraic spaces.

Let Y′ be a Y -algebraic space and let F be a locally finitely presented, quasicoherent O_{X_{Y ′}} -module. Since f_{Y′} : X_{Y′} → Y′ is flat, locally finitely presented and proper, by Lemma 2.4, there exists a locally finitely presented, separated morphism h : Z → Y′ of algebraic spaces and a universal homomorphism

u : (h, Id_{X_{Y ′}})^{∗}F → O_{X_{Z}} .

In fact, as follows from the proof of Theorem 2.1, there is a locally finitely presented, quasi-coherent O_{Y′}-module N and a homomorphism of O_{X_{Y ′}} -modules

χ : F → O_{X_{Y ′}} ⊗_{O_{Y ′}} N

such that Z = \underline{Spec}_{Y′}(Sym^{•}(N)) and such that u is the homomorphism induced by χ.

Since F is a locally finitely presented, quasi-coherent O_{X_{Y ′}} -module which is Y′- flat and has proper support over Y′, the same is true of (h, Id_{X_{Y ′}})^{∗}F relative to Z. Thus, by Theorem 2.1, there exists a locally finitely presented, quasi-coherent O_{Z}-module N and a universal homomorphism

χ : ⋀^{2}(h, Id_{X_{Y ′}})^{∗}F → (h, Id_{X_{Y ′}})^{∗}F ⊗_{O_{Z}} N .

Thus there exists a unique homomorphism of O_{Z}-modules,

ζ : N → O_{Z}

such that the induced homomorphism of O_{X_{Z}} -modules

⋀^{2}(h, Id_{X_{Y ′}})^{∗}F ⟶^{χ} (h, Id_{X_{Y ′}})^{∗}F ⊗_{O_{Z}} N ⟶^{Id⊗ζ} (h, Id_{X_{Y ′}})^{∗}F

equals u′. Let P denote the closed subspace of Z whose ideal sheaf equals ζ(N). Chasing diagrams, the restriction morphism h|_{P} : P → Y′ together with the pullback of u to P is a pair representing Y′×_{Coh_{X/Y}} Pseudo_{X/Y} → Y′. Since Z → Y′ is a locally finitely presented, separated morphism of algebraic spaces (also schematic), and since P is a closed subspace of Y′ whose ideal sheaf is locally finitely generated (being the image of the locally finitely presented sheaf N), also P → Y is a locally finitely presented, separated morphism of algebraic spaces (also schematic). □

Denote by (g : Hilb_{X/Y} → Y, C) a universal pair of a morphism of algebraic spaces g and a closed subspace C of Hilb_{X/Y} ×_{Y} X which is flat, locally finitely presented, and proper over Hilb_{X/Y}, i.e., an object representing the Hilbert functor, cf. [Art69, Corollary 6.2]. Denote by

0 → I ⟶^{u} O_{Hilb_{X/Y} ×_{Y} X} → O_{C} → 0.

the natural exact sequence, where I is the ideal sheaf of C in Hilb_{X/Y} ×_{Y} X.

prop-Hilbopen

Proposition 3.3. The pair (I, u) is a family of pseudo ideal sheaves of X/Y over Hilb_{X/Y}. The induced 1-morphism

ι : Hilb_{X/Y} → Pseudo_{X/Y}

is representable by open immersions.

Proof. Since the kernel of a surjection of flat modules is flat, I is flat over Hilb_{X/Y}. By [Gro67, Lemme 11.3.9.1], I is a locally finitely presented, quasi-coherent sheaf. Since the homomorphism u is injective, to prove that u′ is zero it suffices to prove the composition u ∘ u′ is zero. This follows immediately from the definition of u′. Thus the pair (I, u) is a family of pseudo ideal sheaves of X/Y over Hilb_{X/Y}.

To prove that ι is representable by open immersions, it suffices to prove that it is representable by quasi-compact, étale monomorphisms of schemes. To this end, let Y′ be an affine Y -scheme and let (F, v) be a flat family of pseudo ideal sheaves of X/Y over Y′. Denote the cokernel of v by

w : O_{X_{Y ′}} → G.

By [OS03, Theorem 3.2], there is a morphism of algebraic spaces σ : Σ → Y′ such that (σ, Id_{X_{Y ′}})^{∗}G is flat over Σ and such that Σ is universal among Y′-spaces with this property. Moreover, Σ is a surjective, finitely-presented, quasi-affine monomorphism (in particular schematic). As an aside, please note that the remark preceding [OS03, Theorem 3.2] is incomplete – [OS03, Proposition 3.1] should be properly attributed to Laumon and Moret-Bailly, [LMB00, Théorème A.2].

By [Gro67, Lemma 11.3.9.1], Ker((σ, Id_{X_{Y ′}})^{∗}w) is a locally finitely presented, quasicoherent sheaf. Moreover, because it is the kernel of a surjection of sheaves which are flat over Σ, it is also flat over Σ. By Lemma 2.3, there is an open subscheme W of Σ such that a morphism S → Σ factors through W if and only if the pullback of

(σ, Id_{X_{Y ′}})^{∗}F → Ker((σ, Id_{X_{Y ′}})^{∗}w)

is an isomorphism. Chasing universal properties, it is clear that W → Y′ represents

Y′ ×_{Pseudo_{X/Y}} Hilb_{X/Y} .

Thus ι is representable by finitely-presented, quasi-affine monomorphisms of schemes.

It only remains to prove that ι is étale. Because ι is a finitely-presented it remains to prove that ι is formally étale. Thus, let Y′ = Spec A′ where A′ is a local Artin O_{Y} -algebra with maximal ideal 𝔪 and residue field κ. And let

0 → J → A′ → A → 0

be an infinitesimal extension, i.e., 𝔪J is zero. Let (F, u) be a pseudo ideal sheaf of X/Y over Y′, and assume the basechange to Spec A is an ideal sheaf with A-flat cokernel. Since ι is a monomorphism, formal étaleness for ι precisely says that Y′ → Pseudo_{X/Y} factors through ι, i.e., u is injective and Coker(u) is A′-flat.

To prove this use the local flatness criterion, e.g., as formulated in [Gro67, Proposition 11.3.7]. This criterion is an equivalence between the conditions of

(i) injectivity of u is injective and A′-flatness of Coker(u)

(ii) and injectivity of

u ⊗_{A′} κ : F ⊗_{A′} κ → O_{X_{Y ′}} ⊗_{A′} κ

By hypothesis, (i) holds after basechange to A. Thus (ii) holds after basechange to A. But since A′/𝔪 equals A/𝔪, (ii) for the original family over A′ is precisely the same as (ii) for the basechange family over A. Thus also (i) holds over A′. □

The significance of pseudo ideal sheaves has to do with restriction to Cartier divisors. Let D be an effective Cartier divisor in X, considered as a closed subscheme of X, and assume D is flat over Y. Denote by I_{D} the pullback

I_{D} := I ⊗_{O_{X}} O_{D}

on Hilb_{X/Y} ×_{X} D. And denote by

u_{D} : I_{D} → O_{Hilb_{X/Y} ×_{X}D}

the restriction of u.

prop-flat Proposition 3.4. The locally finitely presented, quasi-coherent sheaf I_{D} is flat over Hilb_{X/Y}. Thus the pair (I_{D}, u_{D}) is a flat family of pseudo ideal sheaves of D/Y over Hilb_{X/Y}.

Proof. Associated to the Cartier divisor D there is an injective homomorphism of invertible sheaves

t′ : O_{X}(−D) ⟶^{O}_{X} .

This induces a morphism of locally finitely presented, quasi-coherent sheaves

t : I ⊗_{O_{X}} O_{X}(−D) → I.

The cokernel of t is I_{D}. By the local flatness criterion, [Gro67, Proposition 11.3.7], to prove that t is injective and I_{D} is flat over Hilb_{X/Y}, it suffices to prove that the “fiber” of t over every point of Hilb_{X/Y} is injective. Thus, let κ be a field, let

y : Spec κY be a morphism, and let I_{y} ⟶^{O}_{X_{y}} be an ideal sheaf. Since D is Y -flat, the homomorphism of locally free sheaves

t′_{y} : O_{X}(−D) ⊗_{O_{Y}} κ → O_{X} ⊗_{O_{Y}} κ

is injective, thus a flat resolution of O_{D}⊗_{O_{Y}} κ. In particular, Tor_{2}^{O_{Xy}} (O_{X_{y}}/I_{y}, O_{D_{y}}) equals zero because there is a flat resolution of O_{D_{y}} with amplitude [−1, 0]. By the long exact sequence of Tor associated to the short exact sequence

0 → I_{y} → O_{X_{y}} → O_{X_{y}}/I_{y} → 0,

there is an isomorphism

Tor_{1}^{O_{Xy}} (I_{y}, O_{D_{y}}) ≅ Tor_{2}^{O_{Xy}} (O_{X_{y}}/I_{y}, O_{D_{y}}) = 0.

But this Tor sheaf is precisely the kernel of

t_{y} : I_{y} ⊗_{O_{X}} O_{X}(−D) → I_{y}.

Thus t_{y} is injective, and so I_{D} is flat over Hilb_{X/Y}. □

notat-iotaD

Notation 3.5. Denote by

ι_{D} : Hilb_{X/Y} → Pseudo_{D/Y}

the 1-morphism associated to the flat family (I_{D}, u_{D}) of pseudo ideal sheaves of D/Y over Hilb_{X/Y}. This is the divisor restriction map.

Since Hilb_{X/Y} and Pseudo_{D/Y} are both locally finitely presented over Y, ι_{D} is locally finitely presented. Since Hilb_{X/Y} is an algebraic space, ι_{D} is representable (by morphisms of algebraic spaces). Since the diagonal morphism of Pseudo_{D/Y} over Y is separated, and since Hilb_{X/Y} is separated over Y, ι_{D} is separated.

ssec-inf

3.1. Infinitesimal study of the divisor restriction map. Let A′ be a local Artin O_{Y} -algebra with maximal ideal 𝔪 and residue field κ. And let

0 → J → A′ → A → 0

be an infinitesimal extension, i.e., 𝔪J is zero. Denote by X_{A′}, resp. X_{A}, X_{κ}, the fiber product of X → Y with Spec A′ → Y, resp. Spec A → Y, Spec κ → Y.

Let (F_{A′}, u_{A′}) be a pseudo ideal sheaf of D/Y over Spec A′. Denote by (F_{A}, u_{A}), resp. (F_{κ}, u_{κ}), the restriction of (F_{A′}, u_{A′}) to A, resp. to κ. Let I_{A} be the ideal sheaf of a flat family C_{A} of closed subschemes of X/Y over Spec A. Denote by I_{κ}, resp. C_{κ}, the restriction of I_{A} to κ, resp. of C_{A} to κ. And assume that ι_{D} sends I_{A} to (F_{A}, u_{A}).

prop-inf

Proposition 3.6. Let n be a nonnegative integer. Assume that C_{κ} is a regular immersion of codimension n in X_{κ}, cf. [Gro67, Définition 16.9.2] (since X_{κ} is an algebraic space, in that definition one must replace the Zariski covering by affine schemes by an étale covering by affine schemes).

(i) The morphism ι_{D} is locally unobstructed at C_{κ} in the following sense. For every étale morphism Spec R_{A′} → X_{A′} such that the pullback I_{κ} of I_{κ} in R_{κ} is generated by a regular sequence, there exists an ideal I_{A′} in R_{A′} whose restriction I_{A} to R_{A} equals the pullback of I_{A} and whose “local pseudo ideal sheaf” (I_{A′} ⊗_{O_{X}} O_{D}, v) equals the pullback of (F_{A′}, u_{A′}). Moreover, the set of such ideals I_{A′} is naturally a torsor for the R_{κ}-submodule

J ⊗_{κ} O_{X}(−D) · Hom_{R_{κ}}(I_{κ}, R_{κ}/I_{κ})

of

J ⊗_{κ} Hom_{R_{κ}}(I_{κ}, R_{κ}/I_{κ})

(here O_{X}(−D)· denote multiplication by the inverse image ideal of O_{X}(−D)).

(ii) There exists an element ω in

J ⊗_{κ} H^{1}(C_{κ}, O_{X}(−D) · Hom_{O_{Cκ}} (I_{κ}/I_{κ}^{2}, O_{C_{κ}}))

which equals 0 if and only if there exists a flat family C_{A′} of closed subschemes of X/Y over Spec A′ whose restriction to Spec A equals C_{A} and whose image under ι_{D} equals (F_{A′}, u_{A′}). When it equals 0, the set of such families C_{A′} is naturally a torsor for the κ-vector space

J ⊗_{κ} H^{0}(C_{κ}, O_{X}(−D) · Hom_{O_{Cκ}} (I_{κ}/I_{κ}^{2}, O_{C_{κ}})).

In particular, if h^{1}(C_{κ}, O_{X}(−D)·Hom_{O_{Cκ}} (I_{κ}/I_{κ}^{2}, O_{C_{κ}})) equals 0, then ι_{D} is smooth at [C_{κ}].

Proof. (i) Let v_{(A,1)}, …, v_{(A,n)} be a regular sequence in R_{A} generating I_{A}. Denote by v_{A} the R_{A}-module homomorphism

v_{A} : R_{A}^{⊕n} → R_{A}, v_{A}(e_{i}) = v_{(A,i)}.

Denote the associated R_{A}-module homomorphism by

v′_{A} : R_{A}^{⊕\binom{n}{2}} → R_{A}^{⊕n}, v′_{A}(e_{i} ∧ e_{j}) = v_{(A,i)}e_{j} − v_{(A,j)}e_{i}.

Since (v_{(A,1)}, …, v_{(A,n)}) is a regular sequence, the following sequence is exact

R_{A}^{⊕\binom{n}{2}} ⟶^{v′_{A}} R_{A}^{⊕n} → I_{A} → 0.

Thus there is an exact sequence

(R_{A} ⊗_{O_{X}} O_{D})^{⊕\binom{n}{2}} ⟶^{v′_{A}⊗Id} (R_{A} ⊗_{O_{X}} O_{D})^{⊕n} → (I_{A} ⊗_{O_{X}} O_{D}) → 0.

Denote by F_{A′} the R_{A′}-module whose associated quasi-coherent sheaf is the pullback of F_{A′}. Since F_{A} equals I_{A} ⊗_{O_{X}} O_{D}, there is an isomorphism

F_{A′}/JF_{A′} ≅ I_{A} ⊗_{O_{X}} O_{D}

compatible with the maps u_{A′} and u_{A}. Since (R_{A′} ⊗_{O_{X}} O_{D})^{⊕n} is a projective (R_{A′} ⊗_{O_{X}} O_{D})-module, there exists an (R_{A′} ⊗_{O_{X}} O_{D})-module homomorphism

a : (R_{A′} ⊗_{O_{X}} O_{D})^{⊕n} → F_{A′}

whose restriction to A is the surjection above. So, by Nakayama’s lemma, this map is also surjective. Since both the source and target of the surjection are A′-flat, also the kernel is A′-flat. Thus there is also a lifting of the set of generators of the kernel, i.e., there is an exact sequence of (R_{A′} ⊗_{O_{X}} O_{D})-modules

(R_{A′} ⊗_{O_{X}} O_{D})^{⊕\binom{n}{2}} ⟶^{b} (R_{A′} ⊗_{O_{X}} O_{D})^{⊕n} ⟶^{a} F_{A′} → 0

whose restriction to A is the short exact sequence above.

The composition of the surjection with u_{A′} defines an (R_{A′} ⊗_{O_{X}} O_{D})-module homomorphism

w_{A′} : (R_{A′} ⊗_{O_{X}} O_{D})^{⊕n} → (R_{A′} ⊗_{O_{X}} O_{D})

whose restriction to A equals v_{A} ⊗ Id. There is an associated map

w′_{A′} : (R_{A′}⊗_{O_{X}} O_{D})^{⊕\binom{n}{2}} → (R_{A′}⊗_{O_{X}} O_{D})^{⊕n}, w′_{A′}(e_{i}∧e_{j}) = w_{A′}(e_{i})e_{j}−w_{A′}(e_{j})e_{i}.

Since (F_{A′}, u_{A′}) is a pseudo ideal sheaf, the induced map u′_{A′} equals 0. Therefore the image of w′_{A′} is contained in the kernel of a. Since (R_{A′} ⊗_{O_{X}} O_{D})^{⊕\binom{n}{2}} is a free (R_{A′} ⊗_{O_{X}} O_{D})-module, there is a lifting

c : (R_{A′} ⊗_{O_{X}} O_{D})^{⊕\binom{n}{2}} → (R_{A′} ⊗_{O_{X}} O_{D})^{⊕\binom{n}{2}}

such that w′_{A′} = b ∘ c. In particular, the restriction of c to A is an isomorphism. Thus, by Nakayama’s lemma, c is surjective. A surjection of free modules of the same finite rank is automatically an isomorphism. Thus there is a presentation

(R_{A′} ⊗_{O_{X}} O_{D})^{⊕\binom{n}{2}} ⟶^{w′_{A′}} (R_{A′} ⊗_{O_{X}} O_{D})^{⊕n} ⟶^{a} F_{A′} → 0.

Because both R_{A′} and R_{A′} ⊗_{O_{X}} O_{D} are A′-flat, there is a commutative diagram of exact sequences

0 ⟶  J ⊗_{κ} R_{κ}  ⟶  R_{A′}  ⟶  R_{A}  ⟶ 0
↓  ↓  ↓
0 ⟶ J ⊗_{κ} R_{κ} ⊗_{O_{X}} O_{D} ⟶ R_{A′} ⊗_{O_{X}} O_{D} ⟶ R_{A} ⊗_{O_{X}} O_{D} ⟶ 0

where the vertical maps are each surjective. By the snake lemma, the induced map

R_{A′}/(JO_{X}(−D) · R_{A′}) → R_{A} ×_{(R_{A}⊗_{O_{X}} O_{D})} (R_{A′} ⊗_{O_{X}} O_{D})

is an isomorphism. Thus, for every integer i = 1, …, n, there exists an element v_{(A′,i)} in R_{A′} whose image in R_{A} equals v_{(A,i)} and whose image in R_{A′} ⊗_{O_{X}} O_{D} equals w_{A′}(e_{i}). Moreover, the set of all such elements is naturally a torsor for J ⊗_{κ} (O_{X}(−D) · R_{κ}). In other words, there is an R_{A′}-module homomorphism

v_{A′} : R_{A′}^{⊕n} → R_{A′}

whose restriction to A equals v_{A} and such that v_{A′} ⊗ Id equals w_{A′}.

Since (v_{(A,1)}, …, v_{(A,n)}) is a regular sequence in R_{A}, also (v_{(A′,1)}, …, v_{(A′,n)}) is a regular sequence in R_{A′}. One way to see this is to tensor the Koszul complex K^{•}(R_{A′}, v_{A′}) of v_{A′} with the short exact sequence of A′-modules

0 → J → A′ → A → 0.

Since the terms in the Koszul complex are free R_{A′}-modules, and since R_{A′} is A′-flat, the associated sequence of complexes is exact

0 → J ⊗_{A} K^{•}(R_{A}, v_{A}) → K^{•}(R_{A′}, v_{A′}) → K^{•}(R_{A}, v_{A}) → 0.

Thus there is a long exact sequence of Koszul cohomology

⋯ → J⊗_{A}H^{n}(K^{•}(R_{A}, v_{A})) → H^{n}(K^{•}(R_{A′}, v_{A′})) → H^{n}(K^{•}(R_{A}, v_{A})) → J⊗_{A}H^{n+1}(K^{•}(R_{A}, v_{A})) → …

Since v_{A} is regular, H^{n−1}(K^{•}(R_{A}, v_{A})) is zero, which then implies H^{n−1}(K^{•}(R_{A′}, v_{A′})) by the long exact sequence above. Thus also v_{A′} is regular. Moreover, this gives a short exact sequence

0 → J ⊗_{κ} (R_{A}/I_{A}) → R′_{A}/Image(v_{A′}) → R_{A}/I_{A} → 0

from which it follows that R_{A′}/Image(v_{A′}) is A′-flat. Denote by I_{A′} the image of v_{A′}.

Both the pseudo ideal I_{A′} ⊗_{O_{X}} O_{D} and F_{A′} equal the cokernel of w′_{A′}. Thus there is a unique isomorphism between them compatible with w′_{A′}. Moreover, since the compositions

(R_{A′} ⊗_{O_{X}} O_{D})^{⊕n} ⟶^{a} F′_{A} ⟶^{u_{A′}} (R_{A′} ⊗_{O_{X}} O_{D})

and

(R_{A′} ⊗_{O_{X}} O_{D})^{⊕n} ⟶^{v_{A′} ⊗Id} I′_{A} ⊗_{O_{X}} O_{D} → (R_{A′} ⊗_{O_{X}} O_{D})

both equal w_{A′}, the isomorphism above is compatible with the maps to (R_{A′} ⊗_{O_{X}} O_{D}). Thus it is an isomorphism of pseudo ideal sheaves. Therefore there exists I_{A′} satisfying all the conditions in the proposition.

For every lift I_{A′} of I_{A} which is A-flat, the surjection v_{A} : R_{A}^{⊕n} → I_{A} lifts to a surjection R_{A′}^{⊕n} → I_{A′}. Composing this surjection with the injection I_{A′} ↪ R_{A′}, it follows that every lift I_{A′} arises from a lift v_{A′} of v_{A}. As mentioned previously, the set of lifts v_{A′} whose restriction to A equals v_{A} and with v_{A′} ⊗ Id equal to w_{A′} is naturally a torsor for

J ⊗_{κ} Hom_{R_{A′}} (R_{A′}^{⊕n}, O_{X}(−D) · R_{κ}).

But the translate of a lift by a homomorphism with image in I_{κ} gives the same ideal I_{A′} (just different surjections from R_{A′}^{⊕n} to the ideal). Thus, the set of lifts I_{A′} of I_{A} whose pseudo ideal sheaf is the pullback of (F_{A′}, u_{A′}) is naturally a torsor for

J ⊗_{κ} Hom_{R_{A′}} (R_{A′}^{⊕n}, O_{X}(−D) · (R_{κ}/I_{κ})) = J ⊗_{κ} Hom_{R_{κ}}(I_{κ}, R_{κ}/I_{κ}).

(ii) Let Γ be an indexing set and let (Spec R_{A′}^{γ} → X_{A′})_{γ∈Γ} be an étale covering such that for every γ, either I_{κ}^{γ} ⊂ R_{κ}^{γ} is generated by a regular sequence of length n or else equals R_{κ}^{γ}. By the hypothesis that C_{κ} is a regular immersion of codimension n, there exists such a covering.

Suppose first that I_{κ}^{γ} equals R_{κ}^{γ}. Then also (F_{κ}^{γ}, u_{κ}^{γ}) is isomorphic to R_{κ}^{γ}⊗_{O_{X}} O_{D} ⟶^{=} R_{κ}^{γ} ⊗_{O_{X}} O_{D}. It is straightforward to see that the only deformations over A′ of

R_{κ}^{γ} ⟶^{=} R_{κ}^{γ}, resp. R_{κ}^{γ} ⊗_{O_{X}} O_{D} ⟶^{=} R_{κ}^{γ} ⊗_{O_{X}} O_{D}, as pseudo ideals are R_{A′}^{γ} ⟶^{=} R_{A′}^{γ},

resp. R_{A′}^{γ} ⊗_{O_{X}} O_{D} ⟶^{=} R_{A′}^{γ} ⊗_{O_{X}} O_{D}. Thus there is a lifting I_{A′}^{γ} of I_{A}^{γ}, and it is unique.

On the other hand, if I_{κ}^{γ} is generated by a regular sequence of length n, by (i) there exist liftings I_{A′}^{γ} and the set of all liftings is a torsor for

J ⊗_{κ} O_{X}(−D) · Hom_{R_{κ}^{γ}}(I_{κ}^{γ}, R_{κ}^{γ}/I_{κ}^{γ}).

For a collection of liftings (I_{A′}^{γ})_{γ∈Γ}, for every γ_{1}, γ_{2} ∈ Γ, the basechanges of I_{A′}^{γ_{1}} and I_{A′}^{γ_{2}} to

R_{A′}^{γ_{1},γ_{2}} := R_{A′}^{γ_{1}} ⊗_{O_{X}} R_{A′}^{γ_{2}}

differ by an element ω^{γ_{1},γ_{2}} in

J ⊗_{κ} O_{X}(−D) · Hom_{R_{κ}^{γ_{1},γ_{2}}} (I_{κ}^{γ_{1},γ_{2}}, R_{κ}^{γ_{1},γ_{2}}/I_{κ}^{γ}).

It is straightforward to see that (ω^{γ_{1},γ_{2}})_{γ_{1},γ_{2}∈Γ} is a 1-cocycle for

O_{X}(−D) · Hom_{O_{Cκ}} (I_{κ}/I_{κ}^{2}, O_{C_{κ}})

with respect to the given étale covering. Moreover, changing the collection of lifts (I_{A′}^{γ}) by translating by elements in

(J ⊗_{κ} O_{X}(−D) · Hom_{R_{κ}^{γ}}(I_{κ}^{γ}, R_{κ}^{γ}/I_{κ}^{γ}))_{γ∈Γ}

precisely changes (ω^{γ_{1},γ_{2}})_{γ_{1},γ_{2}∈Γ} by a 1-coboundary. Therefore, the cohomology class

ω ∈ J ⊗_{κ} H^{1}(C_{κ}, O_{X}(−D) · Hom_{O_{Cκ}} (I_{κ}/I_{κ}^{2}, O_{C_{κ}}))

is well-defined and equals 0 if and only if there is a lifting I_{A′} as in the proposition. And in this case, the set of liftings is a torsor for the set of compatible families (t^{γ})_{γ∈Γ} of elements t^{γ} in

J ⊗_{κ} O_{X}(−D) · Hom_{R_{κ}^{γ}}(I_{κ}^{γ}, R_{κ}^{γ}/I_{κ}^{γ}),

i.e., it is a torsor for the set of elements t in

J ⊗_{κ} H^{0}(C_{κ}, O_{X}(−D) · Hom_{O_{Cκ}} (I_{κ}/I_{κ}^{2}, O_{C_{κ}})).
□

4. Pseudo R-equivalence sec-2q

Overview. The definition of R-equivalence in the introduction is valid over any field. But in applications to number theory and geometry, R-equivalence is most often applied when the field is the fraction field of a Henselian DVR. (Although it is interesting to see what can be proved over other fields.) For varieties over a DVR there is another notion, “pseudo R-equivalence”, which follows from R-equivalence and which captures the “continuous nature” of the known cohomological obstructions to weak approximation.

Let Ô be a DVR. In later sections Ô will equal the complete local ring of a smooth, closed point of a curve, i.e., k \begin{matrix}[ \\ [ & u & ] \\ ] \end{matrix} for an algebraically closed field k. Let Y be a flat, quasi-projective Ô-scheme whose generic fiber is smooth. Let ŝ and t̂ be Ô-points of Y and consider the corresponding Frac(Ô)-points of the generic fiber. Since Y is separated, t̂ specializes to ŝ only if t̂ equals ŝ. However, if ŝ and t̂ are R-equivalent then ŝ is the specialization of a family of sections of Y which agree with t̂ to arbitrary order. We say that t̂ “pseudo specializes” to ŝ. The same holds with ŝ and t̂permuted. Thus we say that ŝ and t̂are “pseudo R-equivalent”. Pseudo specialization is invariant under modification of the closed fiber of Y. If Ô is Henselian and if t̂ pseudo specializes to ŝ, then both t̂ and ŝ determine the same “Brauer pairing”,

Br(Y_{Frac(Ô)}) → Br(Frac(Ô)).

Finally, pseudo R-equivalence satisfies the fibration property from the introduction which is not currently known for (usual) R-equivalence. For all these reasons we develop these “pseudo” notions in this section. Some of the arguments are technical. The reader who wants to skip this section may safely replace “pseudo R-equivalence” by “R-equivalence” wherever it occurs in the later sections.

Definitions. As above, let Ô be a DVR, and denote by u a generator of the maximal ideal. Let Y be a flat, quasi-projective Ô-scheme whose generic fiber is smooth. The bidisk DÔ is the power series ring,

DÔ := Spec (Ô \begin{matrix}[ \\ [ & v & ] \\ & ] \end{matrix}).

The punctured bidisk is the scheme

D∗ ; Ô := DÔ \ \{⟨u, v⟩\} = Spec (Ô \begin{matrix}[ \\ [ & v & ] \\ & ] \end{matrix}) \ \{⟨u, v⟩\}.

The v-divisor Δ_{v} is the closed subscheme of DÔ with defining ideal ⟨v⟩,

Δ_{v} := Spec (Ô \begin{aligned}[ \\ [ & v & ] \\ & ] \end{aligned} /⟨v⟩).

Similarly, define Δ_{v}^{∗} to be the intersection of Δ_{v} with D∗ ; Ô. Also for every integer N the u^{N}-divisor Δ_{u^{N}} is the closed subscheme of DÔ with defining ideal ⟨u^{N}⟩,

Δ_{u^{N}} = Spec (Ô \begin{aligned}[ \\ [ & v & ] \\ & ] \end{aligned} /⟨u^{N}⟩).

Similarly, define Δ_{u^{N}}^{∗} to be the intersection of Δ_{u^{N}} with D∗ ; Ô.

Observe that Δ_{v} equals Spec Ô and Δ_{v}^{∗} equals Spec Frac(Ô) as Ô-schemes. Also Δ_{u^{N}} equals Spec (Ô/u^{N}) \begin{matrix}[ \\ [ & v & ] \\ & ] \end{matrix} and Δ_{u^{N}}^{∗} equals Spec (Ô/u^{N}) \begin{matrix}( \\ ( & v & ) \\ & ) \end{matrix} as Ô-schemes. In particular, the restriction of ŝ to Frac(Ô) gives an Ô-morphism from Δ_{v}^{∗} to Y. And the restriction of t̂ to Ô/u^{N} gives an Ô-morphism from Δ_{u^{N}}^{∗} to Y (which happens to be constant in v).

defn-psiR Definition 4.1. Let ŝ be a Ô-point of Y and let t̂_{N} be a Ô/u^{N}-point of Y. An N-jet specialization of t̂_{N} to ŝ is an Ô-morphism

r : D∗ ; Ô → Y

such that r|_{Δ_{v}^{∗}} equals the restriction of ŝ to Spec Frac(Ô) and r|_{Δ_{u^{N}}^{∗}} equals the

base change of t̂_{N}. When t_{N} equals the restriction of t̂ to Ô/u^{N}, we say that t̂ N-jet specializes to ŝ.

The Ô-point t̂ directly pseudo specializes to ŝ if for every nonnegative integer N, t̂ N-jet specializes to ŝ. The Ô-point t̂ pseudo specializes to ŝ if there exists a sequence of Ô-points t̂ = t̂_{0}, …, t̂_{n} = ŝ such that for every i = 1, …, n, t̂_{i−1} directly pseudo specializes to t̂_{i}. Elements ŝ and t̂ are pseudo R-equivalent, resp. directly pseudo R-equivalent, if each element pseudo specializes to the other, resp. directly pseudo specializes to the other. If every pair of Ô-points of Y are pseudo R-equivalent, then Y is pseudo R-connected.

Basic results. Although it is defined in terms of Ô-schemes, in fact pseudo Rspecialization depends only on the generic fiber, i.e., the fiber over Frac(Ô).

lem-model Lemma 4.2. Let f : Y′ → Y be a morphism of flat, quasi-projective O-schemes. Let s′ and t̂′ be Ô-points of Y′ mapping to ŝ and t̂.

(i) Every N-jet specialization of t̂′ to s′ in Y′ maps to an N-jet specialization of t̂ to ŝ in Y. Thus if t̂′ directly pseudo specializes to s′ in Y′, then t̂ directly pseudo specializes to ŝ in Y.

(ii) Assume that f is projective and is an isomorphism on Frac(Ô)-fibers. Then there exists a nonnegative integer c such that for every nonnegative integer N, every (N + c)-jet specialization of t̂ to ŝ in Y is the image of an N-jet specialization of t̂′ to s′ in Y′. Thus if t̂ directly pseudo specializes to ŝ in Y, then t̂′ directly pseudo specializes to s′ in Y′.

(iii) If t̂and ŝ are directly R-equivalent, then they are directly pseudo R-equivalent.

Proof. Item (i) is obvious. Next assume the hypotheses in Item (ii). Let

r : D∗ ; Ô → Y

be an M-jet specialization of t̂ to ŝ. By the valuative criterion of properness there exists a unique Ô-morphism

r′ : D∗ ; Ô → Y′

such that f ∘ r′ equals r. Since f is an isomorphism on Frac(Ô)-fibers and since r|_{Δ_{v}^{∗}} equals ŝ, also r′|_{Δ_{v}^{∗}} equals s′. It remains to prove that there exists an integer c such that for every nonnegative integer N, r′|_{Δ_{u^{N}}^{∗}} equals the restriction of t̂′ if r|_{Δ_{u^{N+c}}^{∗}} equals the restriction of t̂.

This is easiest to see in coordinates. Let V be an open affine in Y containing t̂ and let V′ be an open affine in f^{−1}(V) containing t̂′. Let y_{1}, …, y_{l} be generators for Γ(V′, O_{V′}) as a Ô-algebra such that the ideal of t̂′ is ⟨y_{1}, …, y_{l}⟩. There exists an integer c such that u^{c}y_{i} is in Γ(V, O_{V}) for every i = 1, …, l. And then u^{c}y_{i} is in the ideal of t̂ for every i. Let τ be a Ô-point of V′. If (f ∘ τ)^{∗}(u^{c}y_{i}) is divisible by u^{N+c}, then τ^{∗}(y_{i}) is divisible by u^{N+c}. Thus if f ∘ τ agrees with t̂ to order N + c, then τ agrees with t̂′ to order N. Applying this to the germ of r′ at Δ_{u^{N}}^{∗} shows that r′|_{Δ_{u^{N}}^{∗}} equals the restriction of t̂′ if r|_{Δ_{u^{N+c}}^{∗}} equals the restriction of t̂.

For (iii), first consider the special case when Y equals ℙ1 ; Ô, i.e., Proj Ô[Y_{0}, Y_{1}]. Also assume that ŝ, resp. t̂, equals [1, 0], resp. [0, 1]. For every nonnegative integer N, consider the morphism

r : D∗ ; Ô → ℙ1 ; Ô, r^{∗}[Y_{0}, Y_{1}] = [v, u^{N}].

This is clearly an N-jet specialization of t̂ to ŝ. Thus t̂ directly pseudo specializes to ŝ. By symmetry, t̂ and ŝ are directly pseudo R-equivalent.

Next consider the case when Y is an Ô-curve P whose generic fiber P_{Frac(Ô)} is isomorphic to ℙ_{Frac(Ô)}^{1}. Any two distinct rational points of ℙ_{Frac(Ô)}^{1} are projectively equivalent to [1, 0] and [0, 1]. So by the previous paragraph and by (ii), any two Ô-points of P are directly pseudo R-equivalent.

Finally consider the case of general Y. Let ŝ and t̂ be Ô-points which are directly R-equivalent, i.e., which are the images under a Ô-morphism

f : P → Y

of Ô-points of P, where P is a Ô-curve as in the previous paragraph. By the previous paragraph, every pair of Ô-points of P are directly pseudo R-equivalent. Thus, by (i), ŝ and t̂ are directly pseudo R-equivalent in Y. □

To prove the relation between pseudo specialization and Brauer equivalence, it is useful to prove a lemma about extending Brauer classes. We believe this is wellknown; we include the proof to illustrate the technique.

lem-unramified

Lemma 4.3. Let t̂ be an Ô-point of Y. Let α be a Brauer class on the generic fiber Y_{Frac(Ô)} such that the pullback t̂^{∗}α on Frac(Ô) extends to all of Ô. Then there

exists a projective, birational morphism f : Y′ → Y, an Ô-point t̂′ of Y′, and an open subscheme U of Y′ containing the generic fiber and containing Image(t̂′), such that α extends to all of U.

Proof. There is a first reduction. Denote by p : Y → Spec Ô the “structure morphism”. Consider β = α−p^{∗}t̂^{∗}α. Since p^{∗}t̂^{∗}α is defined on all of Y, for a morphism f and open set U as above, f^{∗}α extends to all of U if and only if f^{∗}β extends to all of U. So, up to replacing α by β, we may assume that t̂^{∗}α equals 0.

Since Y_{Frac(Ô)} is smooth and quasi-projective, there exists an integer n and a Brauer-Severi variety

π : P → Y_{Frac(Ô)}

with geometric fibers ℙ^{n−1} such that the associated Brauer class is α. Since t̂^{∗}α equals 0, t̂^{∗}P is isomorphic to ℙ_{Frac(Ô)}^{n−1}.

Denote by ω_{π} the relative dualizing sheaf. The associated sheaf ω_{π}^{∨} is π-relatively very ample and π_{∗}(ω_{π}^{∨}) is locally free of rank \binom{2n}{n}. Since Brauer classes on a regular scheme satisfy glueing for Zariski open covers, it suffices to prove the existence of f and U after replacing Y by any open affine neighborhood of t̂. So without loss of generality assume that Y is an affine neighborhood on which π_{∗}(ω_{π}^{∨}) is trivialized. More precisely, choose a free basis s_{1}, …, s_{M} for H^{0}(ℙ_{ℤ}^{n−1}, ω_{ℙ_{ℤ}^{n−1}}^{∨}) and then choose a trivialization

ϕ : O_{Y_{Frac(Ô)}}^{⊕M} → π_{∗}(ω_{π}^{∨})

whose pullback by t̂ has components (s_{1}, …, s_{M}). Since ω_{π}^{∨} is π-relatively very ample, this trivialization induces a closed immersion of Y-schemes,

ι : P ↪ ℙ_{Y_{Frac(Ô)}}^{M−1}.

Denote by h(d) the numerical polynomial

h(d) := \binom{n(d + 1)}{n}.

The closed immersion ι determines a Ô-morphism to the Hilbert scheme

ι̃: Y_{Frac(Ô)} → Hilb_{ℙ_{Y}^{M−1}/Y}^{h(d)}.

As Hilb_{ℙ_{Y}^{M−1}/Y}^{h(d)} is projective over Y, there exists a projective, birational morphism

f : Y′ → Y

such that the pullback of P over Y′ extends to a Y′-flat closed subscheme

P′ ↪ ℙ_{Y′}^{M−1}.

There exists an open subscheme U of Y′ such that for every Y′-scheme T, the pullback of P′ to T is smooth over T if and only if T factors through U; for the projection π′ : P′ → Y′, U is the complement of the image under π′ of the singular locus of π′. By construction, ι̃∘ t̂|_{Frac(Ô)} is the constant morphism corresponding to the anticanonical embedding of ℙ^{n−1} in ℙ^{M−1} by the basis [s_{1}, …, s_{M}]. Thus the restriction of P′ to the strict transform t̂′ of t̂ in Y′ equals the constant family [s_{1}, …, s_{M}](ℙ^{n−1}) over the dense open subscheme Spec Frac(Ô) of Spec Ô. By separatedness of the Hilbert scheme, the restriction of P′ over all of t̂′(Spec Ô) equals the constant family. In particular, t̂′(Spec Ô) is contained in U.

The restriction of P′ over U is a Brauer-Severi scheme whose associated Brauer class extends α. □

As promised, pseudo specialization implies Brauer equivalence.

prop-Brauereq

Proposition 4.4. Assume that Ô is Henselian. If t̂ pseudo specializes to ŝ, then the pullback maps,

t̂^{∗}, s^{∗} : Br(Y_{Frac(Ô)}) → Br(Frac(Ô))

are equal.

Proof. Of course it suffices to verify this when t̂ directly pseudo specializes to ŝ. The goal is to prove that the difference t̂^{∗} −s^{∗} equals the zero map, i.e., the kernel is the entire Brauer group. Of course the kernel contains the pullback p^{∗}Br(Frac(Ô)) since t̂^{∗}p^{∗} = (p ∘ t̂)^{∗} = Id^{∗} and likewise for s^{∗}p^{∗}. And the sum of p^{∗}Br(Frac(Ô)) and Ker(t̂^{∗}) is the entire Brauer group. Thus it suffices to prove that the kernel of t̂^{∗} is contained in the kernel of s^{∗}.

To this end, let α be a Brauer class on Y_{Frac(Ô)} such that t̂^{∗}α equals 0. By Lemma 4.3, there exists a projective, birational morphism f : Y′ → Y and an open subscheme U of Y′ containing both the generic fiber and the strict transform t̂′ of t̂ such that α extends to U. Denote by s′ the strict transform of ŝ. By Lemma 4.2(ii), t̂′ directly pseudo specializes to s′. In particular, there exists a 1-jet specialization

r : D∗ ; Ô → Y′

of t̂′ to s′. Since U contains both the generic fiber and the image of t̂′, U also contains the image of r. Consider r^{∗}α. By purity for Brauer classes, cf. [Gro68, Théorème 6.1(b), Corollaire 6.2], r^{∗}α extends to a Brauer class γ defined on all of DÔ.

Consider first the restriction of γ to Δ_{u^{1}} = Spec (Ô/u) \begin{matrix}[ \\ [ & v & ] \\ & ] \end{matrix}. Since (Ô/u) \begin{matrix}[ \\ [ & v & ] \\ & ] \end{matrix} is Henselian, every Brauer class γ on (Ô/u) \begin{matrix}[ \\ [ & v & ] \\ & ] \end{matrix} is the pullback of a Brauer class γ_{0} on Ô/u. But since r|_{Δ_{u^{1}}^{∗}} equals the restriction of t̂′, and since α is trivial on t̂′, the

restriction of γ to (Ô/u) \begin{matrix}( \\ ( & v & ) \\ & ) \end{matrix} equals 0. Therefore γ_{0} equals 0.

Consider next the restriction of γ to Δ_{v} = Spec (Ô). The further restriction to the closed point is γ_{0}, which equals 0 by the previous paragraph. Since Ô is Henselian, a Brauer class on Ô which is trivial on the closed point is trivial on all of Ô. Thus the restriction of γ to Δ_{v} equals 0. In other words, s^{∗}(α) equals 0. Therefore the kernel of t̂∗ is contained in the kernel of s^{∗}. □

Relation to weak approximation. Let k be an algebraically closed field. Let π : X → B be a flat, proper k-morphism from an algebraic space X to a smooth, projective k-curve B. Let b be a k-point of B. Denote by Ô the completion of the stalk of O_{B} at b. And denote by XÔ the base change of X to Ô. A Ô-point ŝ of XÔ is approximable if for every integer N there exists a section s_{N} of π such that s_{N} ≅ ŝ modulo 𝔪^{N}. If each s_{N} intersects the very free locus of π, then ŝ is approximable by sections intersecting the very free locus.

thm-second

Theorem 4.5. Assume that XÔ is regular. Let ŝ and t̂ be Ô-points of XÔ. If t̂ pseudo specializes to ŝ, and if ŝ is approximable by sections intersecting the very free locus, then also t̂ is approximable by sections intersecting the very free locus.

Of course one can deduce an analogue using a larger number of closed points by the Weil restriction technique of de Jong, cf. [GHS03]. Sections of π are parameterized by an algebraic space which is locally of finite type over k. But Ô-points of XÔ are not parameterized by an object which is locally of finite type over k. An important step in the proof of Theorem 4.5, and hence Theorem 1.3, is the introduction of a new parameter space which is locally of finite type over k and which serves a similar role to the set of Ô-points of XÔ. This is done in the next two sections.

[CONTINUE HERE: From this point on remains to be written / revised.]

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Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario, K7L 3N6

   E-mail address: mikeroth@mast.queensu.ca

Department of Mathematics, Stony Brook University, Stony Brook, NY 11794

   E-mail address: jstarr@math.sunysb.edu