A LOCAL-GLOBAL PRINCIPLE FOR WEAK APPROXIMATION ON VARIETIES OVER FUNCTION FIELDS

MIKE ROTH AND JASON MICHAEL STARR

Abstract. We present a new perspective on the weak approximation conjecture of Hassett and Tschinkel: formal sections of a rationally connected fibration over a curve can be approximated to arbitrary order by regular sections. The new approach involves the study of how ideal sheaves pullback to Cartier divisors.

1. Statement of results sec-intro

Let k be an algebraically closed field. Let B be a smooth, projective, connected curve over k.

notat-rings

Notation 1.1. For every closed point b of B, denote the Henselian local ring of B at b by O_{B,b}^{h} and denote its fraction field K(O_{B,b}^{h}) by K_{B,b}^{h}. Denote the completion of the local ring by Ô_{B,b}, and denote its fraction field by K̂_{B,b}.

More generally, for an effective Cartier divisor E in B with Supp(E) = {b_{1}, …, b_{N}}, denote by O_{B,E} the semilocal ring of B at {b_{1}, …, b_{N}}, denote by O_{B,E}^{h} the product O_{B,b_{1}}^{h} × ⋯ × O_{B,b_{N}}^{h}, and denote its total ring of fractions of O_{B,E}^{h} by K_{B,E}^{h} = K_{B,b_{1}}^{h} × ⋯ × K_{B,b_{N}}^{h}. Denote by Ô_{B,E} the product Ô_{B,b_{1}} × ⋯ × Ô_{B,b_{N}}, and denote by K̂_{B,E} its total ring of fractions K̂_{B,E} × ⋯ × K̂_{B,E}.

For each of the B-algebras, O = O_{B,E}, resp., O = O_{B,E}^{h}, O = Ô_{B,E}, the morphism

Spec O ×_{B} E → E

is an isomorphism. Thus we identify E with its inverse image in each of these affine B-schemes.

Let X be a normal, projective, connected k-scheme and let π : X → B be a surjective k-morphism whose geometric generic fiber is normal.

defn-comblike

Definition 1.2. Let s : B → X be a section of π. A comb-like curve with handle s in X is a connected, closed curve C in X such that C → X is a regular immersion and such that the intersection of C with the generic fiber of B equals s(η_{B}). In other words, s(B) is the only irreducible component of C dominating B, and C is reduced at the generic point of s(B).

rmk-comblike

Remark 1.3. Among all comb-like curves are the combs with handle s as defined by Kollár-Miyaoka-Mori, and the combs with broken teeth as defined by Hassett-Tschinkel. There are many comb-like curves which are neither combs nor combs with broken teeth, most of which are uninteresting. We are interested only in those comb-like curves which deform to smooth, connected curves. This imposes many local constraints which we have not attempted to characterize.

Date: January 21, 2008.

Definition 1.4. For an effective Cartier divisor E in B, an E-section of π, or just E-section, is a B-morphism s_{E} : E → X. A comb-like curve C in X deforms to section curves agreeing with s_{E} Zariski locally, resp. étale locally, formally locally, if for B′ = Spec O_{B,E}, resp. for B′ = Spec O_{B,E}^{h}, B′ = Spec Ô_{B,E}, there exists a DVR (R, 𝔫) with coefficient field k and an R-flat, closed subscheme C ⊂ Spec R ×_{Spec k} B′ ×_{B} X such that

(i) the fiber of C over the residue field Spec R/𝔫 equals the base change of C to B′ ×_{B} X, and

(ii) the fiber of C over the fraction field Spec K(R) equals the image of a section s_{η} : Spec K(R) ×_{Spec k} B′ → Spec K(R) ×_{Spec k} B′ ×_{B} X whose restriction to Spec K(R) ×_{Spec k} E equals the base change of s_{E}.

In other words, on a Zariski neighborhood, resp. étale neighborhood, formal neighborhood, of the fiber π^{−1}(E) in X, C′ deforms to section curves all of which contain

s_{E}(E). rmk-jet

Remark 1.5. Let E = E_{1} ⊔ ⋯ ⊔ E_{N} be the connected components of E. Then C deforms to section curves agreeing with s_{E} étale locally, resp. formally locally, if and only if for every i = 1, …, N, C deforms to section curves agreeing with s_{E}|_{E_{i}} étale locally, resp. formally locally. Thus the étale and formal local conditions can each be studied fiber-by-fiber.

thm-main Theorem 1.6. Let s : B → X be a section of π mapping the geometric generic point of B into the very free locus of the geometric generic fiber of π. (In particular, the very free locus is nonempty and π is a separably rationally connected fibration.) Let E be an effective Cartier divisor in B. Let s_{E} be an E-section of π. Let C be a comb-like curve with handle s. Assume C deforms to section curves agreeing with s_{E} étale locally or formally locally. Then C deforms to section curves agreeing with s_{E} Zariski locally. To be precise, after attaching to C sufficiently many very free teeth in fibers of π at general points of s(B) and with general normal directions, C deforms to comb-like curves all of which are section curves agreeing with s_{E} over a Zariski open neighborhood of E in B.

This reduces weak approximation to the problem of finding comb-like curves C with handle s deforming to section curves agreeing with s_{E} either étale locally or formally locally. There is a criterion due to Hassett guaranteeing existence of such curves. Originally formulated as a criterion in characteristic 0, the extension to positive characteristic poses some minor issue in characteristics 2, 3 and 5. In order to avoid these issues, we use a slight trick.

defn-conic Definition 1.7. Let B be a smooth, connected, projective k-curve and let E be an effective Cartier divisor in B. An E-conic is a smooth, projective k-surface S together with a birational morphism ν : P → ℙ^{1} ×_{k} B such that the restriction

ν : ν−1(ℙ^{1} ×_{k} (B − E)) → ℙ^{1} ×_{k} (B − E)

is an isomorphism. The zero section of P is the unique B-morphism

0_{P} : B → P

whose composition with ν is the zero section of ℙ^{1} ×_{k} B.

Remark 1.8. (i) Given a section s and an E-section s_{E}, suppose there exists an E-conic ν : P → X ×_{B} ℙ^{1} and a comb-like curve C in X ×_{B} ℙ^{1} with handle (s, 0_{P}) which étale locally or formally locally deforms to section curves agreeing with a given E-section (s_{E}, t_{E}). Then Theorem 1.6 applies to produce a section in X ×_{B} ℙ^{1} agreeing with (s_{E}, t_{E}). Thus the projection into X is a section of π agreeing with s_{E}. Therefore, to prove the existence of sections of π agreeing with s_{E}, it suffices to prove the local condition for X ×_{B} ℙ^{1}

(ii) Also,for a divisor E with support {b_{1}, …, b_{N}}, the modification ν : P → ℙ^{1}×_{k}B and the comb-like curve C can both be defined in the neighborhood of every fiber π^{−1}(b_{i}) and then these local pieces can be assembled into a global modification, resp. a global comb-like curve. Thus the existence of ν and C with the required properties can be verified near each fiber π^{−1}(b_{i}) separately.

prop-Hassett

Proposition 1.9 (Hassett). Let B be a smooth, connected, projective k-curve, let X be a normal, projective, connected k-scheme, and let π : X → B be a surjective k-morphism. Let s : B → X be a section of π whose image s(B) is contained in the smooth locus of π. Let E be an effective Cartier divisor in B supported at a single point b. Let O denote O_{B,b}^{h}, resp. Ô_{B,b}, and let K denote K_{B,b}^{h}, resp. K̂_{B,b}. Let s_{E} be an E-section of π.

Roughly stated, if the étale, resp. formal, germ of s over b gives a K-point of X(K) which is R-equivalent to the K-point of a germ agreeing with s_{E}, then for some Econic P, there exists a comb-like curve C in X ×_{B} P with handle (s, 0_{P}) which étale locally, resp. formally locally, deforms to section curves agreeing with an E-section (s_{E}, t_{E}).

Following is one precise formulation (certainly not the most general). Assume there exists a morphism of K-schemes

f : ℙ_{K}^{1} → Spec K ×_{B} X

whose closure S in Spec O ×_{B} X intersects the closed fiber π^{−1}(b) only at smooth points of X and sending the K-points 0, resp. ∞, of ℙ_{K}^{1}(K) to the K-point of the germ of s, resp. to a K-point whose unique extension

s′ : Spec O → Spec O ×_{B} X

agrees with s_{E}. Then there exists an E-conic P and a comb-like curve C in X ×_{B} P with handle (s, 0_{P}) which étale locally, resp. formally locally, deforms to section curves agreeing with an E-section (s_{E}, t_{E}).

Moreover, if f is an immersion and if the closed fiber of S is smooth, then there exists a comb-like curve C in X with handle s which étale locally, resp. formally locally, deforms to section curves agreeing with s_{E} (i.e., there is no need for the E-conic).

When combined with Proposition 1.9, Theorem 1.6 unifies and extends several known results regarding weak approximation. Most of these follow immediately; we content ourselves with the following.

Corollary 1.10. [HT06] With hypotheses as in Theorem 1.6, assume also that cor-HT s(B) is contained in the smooth locus of X. Let E be an effective Cartier divisor in B and let s_{E} be an E-section of π. If the relative dimension of π is ≥ 3 and if every fiber π^{−1}(b) in π^{−1}(E) is smooth and separably rationally connected, then there exists a comb-like curve C with handle s which Zariski locally deforms to section curves agreeing with s_{E}. If the relative dimension is ≤ 2 and if every fiber π^{−1}(b) in π^{−1}(E) is smooth and rationally connected, then for the E-conic P = B ×_{k} ℙ^{1} there exists a comb-like curve C in X ×_{B} P with handle (s, 0_{P}) which Zariski locally deforms to section curves agreeing with an E-section (s_{E}, t_{E}).

More generally, the two conclusions above hold if for every point b in Supp(E) there exists an immersed, resp. not necessarily immersed, very free rational curve in the smooth locus of π^{−1}(b) containing both s(b) and s_{E}(b) (possibly both the same point).

cor-rational Corollary 1.11. With hypotheses as in Theorem 1.6, assume also that s(B) is in the smooth locus of X. Let E be an effective Cartier divisor in B such that X is smooth along π^{−1}(E). Assume that for each b in Supp(E), either Spec K̂_{B,b} ×_{B} X is K̂_{B,b}-rational or K̂_{B,b}-stably rational. Then for every E-section s_{E} there exists an E-conic P over B and a comb-like curve C in X ×_{B} P with handle (s, 0_{P}) which Zariski locally deforms to section curves agreeing with an E-section (s_{E}, t_{E}).

In particular, for a smooth cubic surface over a local field specializing to a cubic surface with only rational double points and no more than 3 of these, the cubic surface is rational.

cor-cubic Corollary 1.12. [HT07] Let S be a smooth cubic surface over Spec k \begin{matrix}( \\ ( & t & ) \\ ) \end{matrix} extending to a flat, proper scheme over Spec k \begin{matrix}[ \\ [ & t & ] \\ ] \end{matrix} whose closed fiber is a cubic surface with only rational double points, and at most 3 of these. Then S is k \begin{matrix}( \\ ( & t & ) \\ ) \end{matrix}-rational.

Assume X is smooth and the geometric generic fiber of π is a smooth cubic surface. Let s be a section of π. Let E be a Cartier divisor in B such that for every point b in Supp(E) the basechange Spec K̂_{B,b} ×_{B} X extends to a flat, proper scheme over Spec Ô_{B,b} whose closed fiber is a cubic surface with only rational double points, and at most 3 of these. Then for every E-section s_{E}, there exists an E-conic P and a comb-like curve C in X ×_{B} P with handle (s, 0_{P}) which Zariski locally deforms to section curves agreeing with an E-section (s_{E}, t_{E}).

rmk-cubic Remark 1.13. In fact Hassett and Tschinkel prove this result for families of cubic surfaces whose singular fibers have rational double points, but possibly more than 3 such double points. There is only one cubic surface with 4 rational double points, the zeroset of X_{0}X_{1}X_{2} + X_{0}X_{1}X_{3} + X_{0}X_{2}X_{3} + X_{1}X_{2}X_{3} which has 4 ordinary double points at the vertices of the coordinate tetrahedron. The method in [HT07] uses strong rational connectedness, i.e., proving the smooth locus of each singular fiber is rationally connected, combined with Corollary 1.10. For the unique cubic surface with 4 ordinary double points, one can explicitly construct rational curves in the smooth locus connecting any pair of points.

2. Extension of two results of Grothendieck

sec-G

One of Grothendieck’s results which is frequently useful in studying coherent sheaves on proper schemes is [Gro63, Corollaire 7.7.8]. The original proof includes a hypothesis which Grothendieck refers to as “surabondante” (Grothendieck’s emphasis). This hypothesis has subsequently been removed, 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 scheme. Using a general such representability result of Olsson, this leads to the following formulation.

Theorem 2.1. [Gro63, Corollaire 7.7.8], [Lie06, Proposition 2.1.3] Let Y be an thm-Hom 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.

Corollary 2.2. [Gro63, Théorème 7.7.6] Let f : X → Y be as in Theorem 2.1. Let cor-Hom 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

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. Deforming comb-like curves sec-def

Let k be an algebraically closed field. Let B be a smooth, projective, connected curve over k. Let X be a normal, projective, connected k-scheme. And let π : X → B be a surjective k-morphism of relative dimension d. Assume that the geometric generic fiber of π is normal and contains a very free rational curve inside its smooth locus.

lem-comblike

Lemma 4.1. There is an open subscheme of Hilb_{X/k} precisely parametrizing the comb-like curves.

Proof. There is an open subscheme of Hilb_{X/k} parametrizing regular immersions of codimension d − 1. The degree of such a curve over B is a locally constant integer-valued function. Thus the open subscheme of comb-like curves is precisely the open subset on which the degree equals 1. □

prop-comblike

Proposition 4.2. Let s : B → X be a section of π mapping the geometric generic point of B into the very free locus of the geometric generic fiber of π. Let E be an effective Cartier divisor in B, and denote by D the inverse image Cartier divisor π^{−1}(E) in X. Let C be a comb-like curve with handle s which is contained in the scheme-theoretic union of s(B) and D. There exists an integer N with the following property. For every comb-like curve C′ obtained by attaching at least N very free curves in fibers of π to C at general points of s(B) and with general normal directions to s(B), the divisor restriction map, ι_{D} : Hilb_{X/k} → Pseudo_{D/k}, is smooth at [C′].

Proof. Since s maps the geometric generic point of B into the very free locus of π, with the exception of finitely many points, for every k-point b of B, π is smooth at s(b) and for every normal direction to s(B) in X at s(b), there exists a very free curve in π^{−1}(b) containing s(b) and having the specified normal direction. So long as the attachment points s(b) is a smooth point of C (which also holds with finitely many exceptions), the curve obtained by attaching this very free curve to C is again comb-like. The final exception is that we only attach very free curves at points b not contained in E.

Let C′ be a curve obtained from C by attaching a collection of very free rational curves in fibers of π at points of s(B) which are smooth points of C. Denote by C′′ the subcurve of C′ consisting of s(B) and all the attached very free curves in fibers of π. Since C′ → X is a regular immersion, the sheaf I_{C′/X}/I_{C′/X}^{2} is a locally free O_{C′}-module, where I_{C′/X} is the ideal sheaf of C′ in X. Denote by N_{C′/X} the dual sheaf,

N_{C′/X} := Hom_{O_{C′}} (I_{C′/X}/I_{C′/X}^{2}, O_{C′}).

The subsheaf O_{X}(−D)·N_{C′/X} is supported on C′′ since C is contained in the scheme theoretic union of s(B) and D. Moreover, O_{X}(−D) · N_{C′/X}|_{s(B)} is obtained from O_{X}(−D) · N_{C/X}|_{s(B)} by the construction from [GHS03, Lemma 2.6]. Thus, by the argument from [GHS03, p. 62], if C′ is obtained from C by attaching sufficiently many very free curves in fibers of π to general points of s(B) with general normal directions, then h^{1}(C′′, O_{X}(−D) · N_{C′/X}) equals 0. Thus, by Proposition 3.6, the divisor restriction map is smooth at [C′]. □

cor-comblike Corollary 4.3. With the same hypotheses as in Proposition 4.2, let F be a subdivisor of E, let s_{F} be a section of π over F, assume that the pseudo-ideal sheaf ι_{D}([C]) deforms to pseudo-ideal sheaves agreeing with s_{F}. Then there exists a section s_{∞} of π whose restriction to F equals s_{F}.

Proof. Since none of the very free curves attached to C intersect D, ι_{D}([C]) equals ι_{D}([C′]). Let (T, t) be as in Definition 1.2. Form the pullback morphism

ι_{D,T} : T ×_{ψD/k,ι_{D}} Hilb_{X/k} → T.

By Proposition 4.2, this morphism is smooth at (t, [C′]). By Lemma 4.1, there is an open neighborhood U of (t, [C′]) precisely parametrizing pairs for which the closed subscheme is a comb-like curve. The restriction of ι_{D} to this open subscheme is also smooth at (t, [C′]). Thus it is dominant. So there exists a k-point (t_{∞}, [C_{∞}]) in this open subscheme such that t_{∞} ≠ t. Therefore the pseudo-ideal sheaf of C_{∞} agrees with s_{F}. Let s_{∞} be the unique section of π such that s_{∞}(B) is an irreducible component of C_{∞}. Then the ideal sheaf of s_{∞}(E_{F}) is a subsheaf of the pseudo-ideal sheaf of C_{∞} over E_{F}. But two ideal sheaves of sections can be contained one in the other if and only if they are equal. Therefore the restriction of s_{∞} to E_{F} is the given section over E_{F}. In particular, the restriction of s_{∞} to F equals s_{F}.1 □

5. Proofs of the main theorems

sec-proofs Proof of Theorem 1.6. Let E′ be an effective divisor in B containing E and such that for the inverse image Cartier divisor D = π^{−1}(E′), C is contained in the scheme-theoretic union s(B)∪D. The étale local deformation of C to section curves agreeing with s_{E} in particularly restricts on D to a deformation of the pseudo-ideal sheaf of C to ideal sheaves of section curves agreeing with s_{E}. Thus the theorem follows from Corollary 4.3. □

Proof of Proposition 1.9. First we consider the case when the O-morphism

f : ℙ_{K}^{1} → Spec O ×_{B} X

does not necessarily extend to all of ℙ_{O}^{1}.

claim-Hassett1

Claim 5.1. There exists an E-conic ν : P → ℙ^{1} ×_{k} B such that the composition

f ∘ ν : Spec K ×_{B} P → ℙ_{K}^{1} → Spec O ×_{B} X

extends to an O-morphism

f_{P} : Spec O ×_{B} P → Spec O ×_{B} X.

To prove this, first form the closure Σ of the graph of f in ℙ_{O}^{1} ×_{B} X. This gives a possibly singular, 2-dimensional scheme whose projection onto ℙ_{O}^{1} is projective and birational. By resolution of singularities for 2-dimensional schemes, e.g. as in [Lip78], there exists a projective desingularization of Σ. The composition with projection onto ℙ_{O}^{1} is a projective, birational morphism, thus equals the blowing up of ℙ_{O}^{1} along a coherent ideal sheaf I_{O} supported on the closed fiber. Since every such ideal sheaf I_{O} equals the basechange of a coherent ideal sheaf I on ℙ^{1} ×_{k} B supported on ℙ^{1} ×_{k} {b}, the blowing up of ℙ_{O}^{1} along I_{O} equals Spec O ×_{B} P where P is the E-conic obtained by blowing up ℙ^{1} ×_{k} B along I. This proves Claim 5.1.

claim-Hassett2

Claim 5.2. Denote the graph of f_{P} by

Γ_{f_{P}} : Spec O ×_{B} P → Spec O ×_{B} (X ×_{B} P).

Each comb-like curve C in P which equals s(B) over B − {b} equals the isomorphic projection of a unique comb-like curve C_{Γ} in X×_{B}P whose basechange to Spec O×_{B} (X ×_{B} P) equals Γ_{f_{P}} (Spec O ×_{B} C).

There exists a nonnegative integer r such that C is contained in the scheme-theoretic union of s(B) and (r · b) ×_{B} P. Denote by Z the scheme-theoretic union of s(B) and (r · b) ×_{B} (X ×_{B} P). The pullback of O_{Z} to Spec O ×_{B} (X ×_{B} P) equals the scheme-theoretic union of the germ of s(B) and (r ·b)×_{B} (X ×_{B} P). The ideal sheaf I of Γ_{f_{P}} (Spec O ×_{B} C) in Spec O ×_{B} Z is a coherent subsheaf supported entirely on (r · b) ×_{B} (X ×_{B} P). Thus there exists a unique coherent ideal sheaf I_{C_{Γ}} in O_{Z} supported entirely on (r · b) ×_{B} (X ×_{B} P) whose pullback to Spec O ×_{B} (X ×_{B} P) equals I. The corresponding closed subscheme C_{Γ} of Z is the unique closed subscheme whose basechange to Spec O ×_{B} (X ×_{B} P) equals Γ_{f_{P}} (Spec O ×_{B} C). Moreover, because the projection of Γ_{f_{P}} (Spec O ×_{B} C) into Spec O ×_{B} P equals Spec O ×_{B} C, also the projection of C_{Γ} into P equals C.

Finally, since Spec O ×_{B} X is smooth along S, also Spec O ×_{B} (X ×_{B} P) is smooth along Γ_{f_{P}} (Spec O ×_{B} P). Thus Γ_{f_{P}} is a regular immersion. Since Spec O ×_{B} C is a Cartier divisor in Spec O ×_{B} P, also Γ_{f_{P}} (C) is regularly immersed in Spec O ×_{B} P. Thus C_{Γ} is regularly immersed étale locally, resp. formally locally, near {b} ×_{B} (X ×_{B} P). Now the property of being a regular immersion is a formal local property since the Koszul cohomology relative to the complete local ring of the images of a minimal set of generators of the stalk of the ideal equals the completion of the Koszul cohomology relative to the local ring of the minimal set of generators. Thus C_{Γ} is a regular immersion at every point of C_{Γ} in {b} ×_{B} (X ×_{B} P). Since also X ×_{B} P is smooth along (s, 0_{P}), C_{Γ} is regularly immersed at every point of C_{Γ}, i.e., C_{Γ} is a comb-like curve in X ×_{B} P. This proves Claim 5.2.

claim-Hassett3 Claim 5.3. If there exists a comb-like curve C in P with handle 0_{P} which étale locally deforms to section curves agreeing with the restriction ∞_{P} |_{E}, then there exists a comb-like curve C in X ×_{B} P with handle (s, 0_{P}) which étale locally deforms to section curves agreeing with an E-section (s_{E}, t_{E}).

There is a unique E-section of X ×_{B} P such that Γ_{f_{P}} ∘ ∞_{P} equals (s_{E}, t_{E}). By Claim 5.2, there exists a comb-like curve C_{Γ} in X ×_{B} P with handle (s, 0_{P}) whose basechange to Spec O ×_{B} (X ×_{B} P) equals Γ_{f_{P}} (Spec O ×_{B} C). If Spec O ×_{B} C deforms to a family of section curves D_{t} in Spec O ×_{B} P agreeing with ∞_{P} |_{E}, then Γ_{f_{P}} (Spec O ×_{B} C) deforms to the family of section curves Γ_{f_{P}} (D_{t}) in Spec O ×_{B} (X ×_{B} P) agreeing with Γ_{f_{P}} ∘ ∞_{P} = (s_{E}, t_{E}). This proves Claim 5.3.

claim-Hassett4 Claim 5.4. There exists a comb-like curve C in P with handle 0_{P} which Zariski locally deforms to section curves in P agreeing with ∞_{P} |_{E}.

On P consider the invertible sheaf L = O_{P} (∞_{P} (B) − 0_{P} (B) − π^{∗}E). Since the restriction to the generic fiber of P → B is a degree 0 invertible sheaf on ℙ^{1}, it has a 1-dimensional space of global sections. Thus the pushforward of L to B is a torsion-free, coherent O_{B}-module of rank 1, i.e., it is an invertible sheaf. Thus there exists an effective divisor Δ in B, not intersecting E such that O_{P} (∞_{P} (B) − 0_{P} (B) + π^{∗}Δ − π^{∗}E) has a global section. In other words, there is an effective divisor B in P, necessarily vertical, such that

∞_{P} (B) + π^{∗}Δ = 0_{P} (B) + π^{∗}E + B.

Define C to be 0_{P} (B)+π^{∗}E +B and let the curves D_{t} be the members of the pencil spanned by C and ∞_{P} (B)+π^{∗}Δ. Since ∞_{P} (B)+π^{∗}Δ is a section curve over E, all but finitely many members of this pencil are section curves over E. Since the base locus of the pencil contains ∞_{P} (B) ∩ π^{∗}E, these section curves agree with ∞_{P} (B) over E. This proves Claim 5.4, completing the proof of the proposition.

Of course if f extends to a closed immersion

f_{O} : Spec O ×_{k} ℙ^{1} → Spec O ×_{B} X,

then one can take P to be ℙ^{1} ×_{k} B and one can work with f_{O} instead of Γ_{f_{P}}. In this way one produces a comb-like curve in X which étale locally, resp. formally locally, deforms to section curves in X agreeing with s_{E}. □

Proof of Corollary 1.10. First of all, the results of the first paragraph follow from the results of the second, since every pair of points in a smooth, projective, separably rationally connected variety are connected by a very free rational curve, which may also be taken to be immersed if the dimension is ≥ 3. Next, if there is a very free rational curve in π^{−1}(b) containing s(b) and s_{E}(b), then deformations of the rational curve in fibers of π and intersecting both s and s_{E} are unobstructed. Thus there exist both étale local and formal local morphisms

f_{O} : Spec O ×_{k} ℙ^{1} → Spec O ×_{B} X

whose closed fiber is the given very free rational curve in π^{−1}(b) and whose 0-section, resp. ∞-section, is the basechange of s, resp. agrees with s_{E}. If the original very free rational curve in π^{−1}(b) is an immersion, then also f_{O} is an immersion. So the corollary follows from Theorem 1.6 and Proposition 1.9. □

Proof of Corollary 1.11. Let U be a dense Zariski open subset of ℙ_{Spec K̂_{B,b}}^{n} which

is isomorphic to a dense Zariski open subset of Spec K̂_{B,b} ×_{B} (X ×_{k} ℙ_{k}^{r}). Possibly after deforming s, we may assume the K̂_{B,b}-point of Spec K̂_{B,b} ×_{B} X determined by s is the image of a K̂_{B,b}-point of Spec K̂_{B,b} ×_{B} (X ×_{k} ℙ_{k}^{r}) which is contained in V. Similarly, there exists a formal section of Spec Ô_{B,b} ×_{B} X agreeing with s_{E} whose K̂_{B,b}-point is the image of a K̂_{B,b}-point of V. The isomorphism of U and V sends these two K̂_{B,b}-points of V two K̂_{B,b} points of U. Every pair of K̂_{B,b}- points of ℙ_{Spec K̂_{B,b}}^{n} are contained in a line which is isomorphic to ℙ_{Spec K̂_{B,b}}^{1}. The

composition with the isomorphism from U to V determines a K̂_{B,b}-rational map

f_{K} : ℙ_{Spec K̂_{B,b}}^{1} −−→ Spec K̂_{B,b} ×_{B} (X ×_{k} ℙ_{k}^{r})

which extends to all of ℙ_{Spec K̂_{B,b}}^{1} by the valuative criterion of properness. Now apply Theorem 1.6 and Proposition 1.9. □

Proof of Corollary 1.12. Let b be a point of B for which X_{b} is a cubic surface with rational double point singularities. The Galois group of K̂_{B,b} is topologically cyclic.

The action on the Néron-Severi lattice of the geometric generic fiber Spec \overline{K̂}_{B,b}×_{B}X sends a topological generator to an element in the automorphism group of the Néron-Severi lattice, namely W(E_{6}). The conjugacy class of this element in W(E_{6}) is well-defined independent of the choice of topological generator. We call it the monodromy class.

The possibilities for the monodromy class are listed in [Man86, Table 1, p. 176]. As explained in [Man86, 31.1(2), p. 174], the cubic surface Spec K̂_{B,b} ×_{B} X is minimal if and only if the monodromy class is in one of the first 5 rows of the table, and the minimal model is a Del Pezzo of degree ≥ 5 unless it is one of the first 11 rows. Since every smooth Del Pezzo of degree ≥ 5 is rational, Corollary 1.11 applies unless the monodromy class is in one of the first 11 rows.

claim-cubic1

Claim 5.5. If the cubic surface X_{b} has only rational double points, and no greater than 3 of these, then the monodromy class is not in one of the first 11 rows.

The 8^{th} column of each row lists the orbit decomposition of the 27 lines. Each orbit specializes to a line in the singular fiber whose multiplicity is at least as large as the size of the orbit. More precisely, the multiplicity of each line in the singular fiber equals the sum of the lengths of the orbits specializing to that line.

Every line in the singular fiber not containing a singular point necessarily has multiplicity 1 since the normal bundle of a line in the nonsingular locus of a cubic surface is isomorphic to the nonspecial invertible sheaf O_{ℙ^{1}}(−1). Thus it remains to consider the multiplicities of lines containing a rational double point. For each double point, the length of the scheme of lines in the cubic surface containing that double point is always 6. The multiplicity of a line joining two or more double points equals the sum of the contributions from each double point it contains. Thus, if there are m rational double points, then the sum of the multiplicities of the lines containing at least one double point equals 6m. Since all other lines have multiplicity 1, the number of orbits of size 1 must be at least 27 − 6m. Since every orbit type in the first 11 rows begins with 1^{e} for e ≤ 3, such a monodromy class may occur only if the cubic surface has 4 double points. This proves Claim 5.5. Finally apply Corollary 1.11 to complete the proof. □

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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