Zero-cycles in families of rationally connected varieties

We study zero-cycles in families of rationally connected varieties. We show that for a smooth projective scheme over a henselian discrete valuation ring the restriction of relative zero cycles to the special fiber induces an isomorphism on Chow groups if the special fiber is separably rationally connected. We further extend this result to certain higher Chow groups and develop conjectures in the non-smooth case. Our main results generalise a result of Koll\'ar [31].

6 Conjectures in the non-smooth case and the case of rational surfaces 21

Introduction
In this article we study algebraic cycles in families of rationally connected varieties.Since their introduction in the 90s by Kollár, Miyaoka and Mori (see for example [32] and [30]), these varieties have played an important role in algebraic geometry.In order to be mindful of inseparability problems in positive characteristic, we work with the following definition.A smooth and proper variety X over a field k is called separably rationally connected if there is a variety U and a morphism F : U × P 1 → X such that the induced map F (−, (0 : 1)) × F (−, (1 : 0)) : U → X × X is dominant and separable.Being separably rationally connected implies being rationally connected and the converse is true if ch(k) = 0.If a proper smooth variety X over an algebraically closed field k is separably rationally connected, then any two distinct closed points p, q ∈ X may be joined by a rational curve, that is are in the image of a morphism P 1 → X [30, Ch.IV, Thm.3.9], which implies in particular that CH 0 (X) ∼ = Z.Rationally connected varieties enjoy further remarkable properties which distinguish them from general varieties.One of these features, on which we focus in this article, is that many results about algebraic cycles in families, which usually just hold with finite coefficients, hold integrally for rationally connected varieties.Our main theorem is the following: Theorem 1.1 (Thm.4.1).Let A be a henselian discrete valuation ring with residue field k.Let S = Spec(A) and X be a smooth projective scheme over S of relative dimension d.Assume that the special fiber X k of X is separably rationally connected.Let (X) d,g denote the set of codimension d points x of X such that {x} is flat over S. Let (X) d,t ⊂ (X) d,g denote the subset of points x for which in addition the intersection of {x} and X k is transversal.Then the following statements hold: (i) The restriction map CH d (X) → CH d (X k ) is an isomorphism.
(ii) Assume that the Gersten conjecture for Milnor K-theory and for higher Chow groups holds for henselian discrete valuation rings.Then for each j ≥ 1 and x ∈ (X k ) (d) the image of the map which arises from the local to global spectral sequence for higher Chow groups, is equal to the image of Ker[K M j (y) for some y ∈ Spec(O X,x ) (d),t .(iii) Let the assumptions be as in (ii) and assume that X k has a rational point.If we assume further that the degree map deg : CH d (X k ′ ) → Z is an isomorphism for every finite field extension k ′ /k (e.g.k = F p or C), then the vertical maps in the commutative diagram CH d+j (X, j) are isomorphisms for all j ≥ 0.Here K M j denotes the improved Milnor K-theory of a ring defined in [27].
The assumptions in (ii) on the Gersten conjecture are always satisfied if A is of equal characteristic.Furthermore, by Proposition 2.2(ii) we do not need to assume the Gersten conjecture if j ≤ 1 or even if j ≤ 2 if k is finite by Proposition A.2.
Corollary 1.2 (Cor.4.3).Let the notation be as in Theorem 1.1 and denote the generic fiber of X by X K .Then there is a a zigzag of isomorphisms Theorem 1.1 and Corollary 1.2 generalise a theorem of Kollár which was the main motivation for our article: in [31], Kollár proves that, under the same assumptions as in Theorem 1.1(i), the specialisation map sp : CH 0 (X K ) → CH 0 (X k ) is an isomorphism if the residue field k is perfect.Another motivation is that, under the same assumptions as in Theorem 1.1, but not assuming that the special fiber X k is separably rationally connected, there is an isomorphism CH d+j (X, j, Z/n) ∼ = − → CH d+j (X k , j, Z/n) for (n, ch(k)) = 1.For j = 0 this theorem is due to Saito and Sato [44,Cor. 9.5] if additionally k is finite or algebraically closed.A different proof was given by Bloch in [15,App.].In [28], Kerz, Esnault and Wittenberg showed that the assumptions on the residue field can be dropped and extended the result to the semi-stable case.The assumptions on the singularities of the special fiber were relaxed further by Binda and Krishna in [5].Finally, in [38] the author showed the isomorphism for all j ≥ 0 in the smooth case.
In a similar vein, we show the following theorem: Theorem 1.3 (Thm.5.2).Let the notation be as in Theorem 1.1.Then the restriction map Let us give an overview of the structure of the article.In Section 2 we recall the theory of Chow groups with coefficients in Milnor K-theory which is needed for Theorems 1.1(ii) and 1.3.For example, if X k is a variety of dimension d over a field k, then for all j ≥ 0 If k is a local field and X k proper, then the group CH d+1 (X k , 1) is often denoted by SK 1 (X k ) and appears in the study of higher dimensional local class field theory as the domain of the reciprocity map ρ X : SK 1 (X k ) → π ab 1 (X k ) (see for example [16]).We also set up a theory of correspondences and their action on Chow groups with coefficients in Milnor K-theory.This allows to prove the following theorem: Theorem 1.4 (Thm.2.26).If X k is smooth projective over k and rationally connected, then This generalises a theorem of Colliot-Thélène (for j = 0) [10,Prop. 11] to higher zero-cycles and leads us to extend the Bloch-Beilinson conjectures concerning filtrations on Chow groups to higher Chow groups (see Section 2.6).In Section 3 we study the case of relative (semi-stable) curves which is needed in Sections 4 and 5 in the proofs of our main theorems.The key idea, originally due to Kollár, is to use the deformation theory of rational curves in order to reduce to the case of relative (semi-stable) curves.
It was observed in [28] that when studying Chow groups in a family, more precisely on a regular flat and projective scheme X over a henselian discrete valuation ring A, the ordinary Chow group of the special fiber X k should be replaced by a cohomological one.This is related to the fact that there is a restriction map for K-groups K 0 (X ) → K 0 (X n ) induced by the inclusion X n ֒→ X, where X n = X × A A/π n is the thickened special fiber, π being a regular parameter.This map is natural in the sense that it comes from the restriction of locally free sheaves.Inspired by this, we conjecture that if the generic fiber X K of X is separably rationally connected, then, assuming the Gersten conjecture for Milnor K-theory for the first isomorphism, the restriction map is an isomorphism.In Section 6 we give some evidence for this conjecture in case that d = 2: Theorem 1.5.(Thm.6.8) Let X be as above, ch(k) = p and ch(K) = 0. Assume that the reduced special fiber X k of X is a simple normal crossing divisor.Assume further that the Gersten conjecture holds for the Milnor K-sheaf K M 2,X and that the induced map is an isomorphism.
In Appendix A we recollect some facts about the Gersten conjecture which we need in several places in the article.In Appendix B we propose a theory of cohomological higher Chow groups of zero cycles which generalise the Levine-Weibel Chow group [34].These groups already appear naturally in Section 3 when studying curves.In general we expect them to calculate the lowest filtered piece of Quillen's higher K-groups.
Notation 1.6.If A is an abelian group, and n a natural number, we denote by A[n] the n-torsion of A. If l is a prime number, we denote by A l−tors the l-primary torsion of A and by A tors the entire torsion subgroup of A.
Acknowledgement.I would like to thank Christian Dahlhausen, Salvatore Floccari and Stefan Schreieder for helpful comments and discussions and János Kollár and Kay Rülling for helpful e-mail correspondence.Furthermore, I would like to thank the participants of a seminar on zero-cycles at Hannover, which led to some of the ideas of the article.
2 Filtrations on higher zero-cycles

Chow groups with coefficients: the four basic maps
We introduce Chow groups with coefficients from [43] and recall some of their properties.Note that in [43] Rost assumes that the base is a field.We need his results for schemes over discrete valuation rings but we formulate them in the largest generality possible.In this section let S be an excellent1 noetherian scheme and X be a d-dimensional scheme of finite type over S. Let and let ) be the map induced by the tame symbol (see [43, (3.2)]).It can be shown that these maps fit into a complex which we call cycle complex with coefficients in Milnor K-theory.In fact this is just the Gersten complex for Milnor K-theory.We consider these complexes homologically with C p (X, n) in degree p and let By definition of the complex, there are isomorphisms Remark 2.1.One could also work with a cohomological indexing, setting But also some higher Chow groups (and therefore motivic cohomology groups) can be identified with Chow groups with coefficients in Milnor K-theory and the latter point of view can sometimes offer advantages in calculations.For the definition of higher Chow groups and their properties we refer to [7] or Section 2.6.The following proposition tells us in which range such an identification holds.
Proof.For the first part of (i) see [38,Prop. 2.3.1].For the first part of (ii) see [38,Rem. 2.4.2].Indeed, if we consider the spectral sequence of loc.cit.
then the map in line −j − 1 is always surjective if we assume (a) and the term CH j−1 (k(x), j) = 0 vanishes if we assume (b).That A 0 (X, j − 1) = 0 follows from the surjectivity of the map ∂ 0 .For the furthermore of (i) and (ii) see [36,Proof of Cor. 2.7].
Our next goal is to define an action of correspondences, i.e. of classical Chow groups of products of schemes, on Chow groups with coefficients in Milnor K-theory.For the definition we need to recall the following four standard (also called basic) maps defined by Rost: Definition 2.3.Let X and Y be schemes of finite type over S.
(i) ( [43, (3.4)])Let f : X → Y be an S-morphism of schemes.Then the pushforward is defined as follows: (ii) ([43, (3.5)])Let g : Y → X be an equidimensional S-morphism of relative dimension s.Then is defined as follows Here i : k(x) → k(y) is the inclusion on residue fields and i * the induced map on Milnor K-theory.
is defined by {a} x y (ρ) = {a} • ρ for x = y and zero otherwise.
(iv) ( [43, (3.7)])Let Y be a closed subscheme of X and U = X \ Y .Then the boundary map is defined by the tame symbol ∂ x y .
Notation 2.4.We denote by α : We state the following corollary to make it easier to keep track of the indexing.
Corollary 2.6.Let X and Y be schemes of finite type over S.
(i) Let f : X → Y be proper S-morphism.Then f * induces a map on homology (ii) Let g : Y → X be flat S-morphism which is equidimensional with constant fiber dimension s.Then Then g * induces a map on homology Then {a} induces a map on homology {a} : A p (X, n) → A p (X, n + 1).

More properties and calculation for P 1
The functorial properties of Rost's Chow groups with coefficients are those of a homology theory.
Proposition 2.7 (Localisation).([43, p.356]) Let X be a scheme and Y a closed subscheme with complement U .Then there is a long exact sequence Proof.This follows from the boundary triple (Y, i, X, j, U ) having the decomposition Proposition 2.8 (A 1 -invariance).([43, Prop.8.6]) Let π : V → X be an affine bundle of dimension t, then the pullback map is an isomorphism.
Proposition 2.9 (Mayer-Vietoris for closed covers).Let X = i X i be a union of pairwise different irreducible varieties.Let Y = i =j X i ∩ X j .Then there is a long exact sequence Proof.The statement follows from the short exact sequence of complexes (in which we set p = dim X) Proposition 2.10 (Descent for blow ups).Let X be a smooth k-scheme and Y be a smooth closed subscheme.Let π : X → X be the blow up of X along Y .Then there is a long exact sequence Proof.This follows from the short exact sequence of complexes (with rows indexed by p) Finally, we use the localisation sequence and A 1 -invariance to make some calculations for projective space.
Proposition 2.11.([18, Prop.8.2.6])Let k be a field, then Proof.The proposition is clear for d = 0.For d = 1 the localisation sequence of Proposition 2.7 gives an exact sequence The isomorphism on the left follows from reciprocity (see Remark 2.14 and Proposition 2.15) and A 0 (A 1 k , j) = 0 by Proposition 2.8.This implies the statement for d = 1.Next we proceed by induction on d.For d > 1 the localisation sequence gives the exact sequence Proposition 2.12.Let A be a discrete valuation ring, then assuming the Gersten conjecture for Milnor K-theory for the isomorphism on the right.
Proof.The proposition is clear for d = 0.For d = 1 the localisation sequence of Proposition 2.7 gives an exact sequence The isomorphism on the left follows from reciprocity (see Remark 2.14 and Proposition 2.15) applied to the generic fiber of P 1 A and the fact, that the sets of codimension 1 points of P 1 A and A 1 A contained in the special fiber coincide.The group on the right is zero since A 1 (A 1 A , j − 1) ∼ = A 0 (Spec(A), j) = 0 by Proposition 2.8 and 2.2(ii).This implies the statement for d = 1.Next we proceed by induction on d.For d > 1 the localisation sequence gives the exact sequence ) by Proposition 2.8 and 2.2(ii).

A higher degree map
Let X k be a proper k-scheme with structure map p : X k → Spec(k).Higher Chow groups are covariantly functorial for proper maps.In particular, p induces a push-forward map in which the vertical maps are proper push-forwards.For Milnor K-theory this is the norm map.Furthermore, if F is an arbitrary field extension of k, then the diagram in which the vertical maps are flat pullbacks, commutes.
Definition 2.13.We denote the kernel of p * : CH d+j (X k , j) → K M j (k) by CH d+j (X k , j) 0 .If j = 0, then we sometimes drop the j from the notation.

Remark 2.14 (Reciprocity). In view of the isomorphism
from Proposition 2.2(i), the existence of the proper push-forward p * implies Weil reciprocity for curves: ) be the tame symbol coming from the valuation on k(C) defined by P .Then for all α ∈ K M n (k(C)) we have If A is a discrete valuation ring with quotient field K and residue field k and X a proper A-scheme with structural map p : X → Spec(A), then p induces a push-forward map p * : CH d+j (X, j) → CH j (Spec(A), j).
Assuming the Gersten conjecture for Milnor K-theory and higher Chow groups for a henselian DVR, the group on the right is isomorphic to the j-th improved Milnor K-theory K M j (A) (see Propositions A.1 and A.2). Definition 2.16.We denote the kernel of p * : Proposition 2.17.Let F/k be a finite field extension of degree m.Let X k be a k-scheme and X F its base change to F .Then the composition Proof.This is [7,Cor. 1.4].Alternatively this follows from Proposition 2.2(ii) and the corresponding fact for Milnor K-theory.

Correspondences on Chow groups with coefficients
We define an action of classical correspondences on Chow groups with coefficients.The definition is probably well-known to the expert; for example in [13], Déglise shows that Chow groups with coefficients are Nisnevich sheaves with transfer.In order to define this action, we need the following proposition, which is one of the main results of Rost in [43], generalising the pullback map on algebraic cycles defined using the deformation to the normal cone (see [17]).
Proposition 2.18.Let f : Y → X be a morphism of schemes with X smooth over S and r = dim S X − dim S Y .Then there is a homomorphism of complexes which has all the expected properties of a pullback map of algebraic cycles.It is defined in terms of the four basic maps and therefore induces a map on homology In [43,Sec. 14] Rost defines a cross product and shows that it induces a product on homology Definition 2.19.Let δ : X → X × X be the diagonal map and dim X = d.We define an intersection product by the composition In particular, if r = −p and s = −q, then this coincides with the usual product of algebraic cycles.
This allows us to define the action of a correspondence.
Definition 2.20 (Action of correspondences).Let k be a field and X and Y be smooth k-varieties.
(ii) Let d = dim X and d ′ = dim Y .Let X furthermore be projective, which implies that the projection p 2 : X × Y → Y to the second component is proper (we denote the first projection by p 1 ).A correspondence Γ defines a morphism Note that for X = Y and q = dim Y , this simplifies to Remark 2.21.More generally, in Definition 2.20, we could have defined the action of higher correspondences Γ ∈ A p (X × Y, q) for arbitrary p, q ∈ Z.
Lemma 2.22.Let X be a regular integral scheme and Y a closed subscheme of X.Let U = X \ Y .Then the map Proof.Let y ∈ Y (0) be a closed point and α ∈ K M j (y).We pick a curve C, i.e. an integral closed subscheme of dimension 1, of X with the following properties: (1) The generic point This curve can be constructed as follows: we may assume that Y is a divisor which locally at y is defined by an element π ∈ O X,y .If dim X = d, then we pick d − 1 independent regular parameters x 1 , . . ., x d−1 ∈ O X,y such that x i O X,y πO X,y and let C := V (x 1 , . . ., x d−1 ) (cf. [44,Sublem. 7.4]).Let ρ : C → C be the normalisation of C and α ∈ K(C) = K( C) be a lift of α.By the Chinese remainder theorem we can pick a rational function f ∈ K(C) = K( C) which is congruent to 1 at all elements of the set of points ρ −1 (C ∩ Y \ y) and vanishes at the closed point ρ −1 (y).Then Proposition 2.23.Let X and Y be regular projective and birationally equivalent k-varieties.Then In particular, by Proposition 2.2(i), CH d+j (X, j) Proof.We follow the proof of [11,Prop. 6.3] and [17,Ex. 16.1.11],where the statement is proved for j = 0. Let Γ be the closure of the graph of the a birational map from X to Y .Then Γ and its inverse Γ ′ are correspondences from X to Y , resp.Y to X.We need to show that Γ ′ * • Γ * = id X and id Y = Γ * • Γ ′ * .But Γ ′ • Γ is the sum of the identity correspondence and correspondences whose projections to X are of codimension > 0 in X (and similarly for Y ).The statement now follows from Lemma 2.22.

Chow groups with coefficients in Milnor K-theory of rationally connected varieties of degree zero are of finite exponent
In this section we fix the following notation.Let k be a field.Let X k be a proper k-scheme.
Proof.By Proposition 2.2(i) we have that The assertion therefore follows from the definition of rational connectedness and Proposition 2.11.
Corollary 2.25.If X k is rationally connected, then CH d+j (X k , j) 0 is torsion.
Proof.Let α ∈ CH d+j (X k , j) 0 .It follows from the commutativity of the diagram and Proposition 2.24 that there is a finite field extension F of k such that π * (α) = 0 for π : Then, by Proposition 2.17, n • α = 0.
Proof.Let η be the generic point of X k and L = K(X k ).By a trace argument we may assume that is multiplication by m by Proposition 2.17 and therefore nm • CH d+j (X k , j) 0 = 0. Therefore let P ∈ X k (k).Let Ω be the algebraic closure of L. When base changed to X Ω , the points η and P are Requivalent in X(Ω) and therefore η = P Ω ∈ CH 0 (X Ω ).This equivalence already holds over some finite extension of L and therefore, again by a trace argument, there is an n > 0 such that n • (η where Z is a cycle supported on X k × Y and Y is a subscheme of codimension at least 1 in X.By Section 2.4 we get an action of the correspondence Lemma 2.22 implies that Z * = 0 and therefore n∆ Corollary 2.27.Let K be a p-adic field and X K be a smooth projective, geometrically integral variety over K. Then the group CH d+j (X K , j) 0 l−tors is a finite for l = p.
Proof.First assume that X K has a semistable model X whose special fiber X k is a simple normal crossing divisor.We recall that the étale cycle class map is an isomorphism for all j ≥ 0. For j = 0 this is due to Saito and Sato [44, Thm.0.6].For j = 1 this follows from the Kato conjectures proved by Kerz and Saito [29, Thm.9.3].For j > 1 this follows from cohomological dimension.By the localisation sequence for higher Chow groups the sequence is exact.The finiteness of H 2d+j ét (X, Z/l r (d + j)) and CH d+j−1 (X k , j − 1, Z/l r ) implies that the group CH d+j (X K , j, Z/l r ) is finitely generated.The case of a general X K now follows from Gabber's refined uniformisation.
Since by Theorem 2.26 the group CH d+j (X K , j) 0l−tors is of bounded exponent, the map is injective for r sufficiently large.Since, as we have seen above, the group CH d+j (X K , j, Z/l r ) is finitely generated, this implies that CH d+j (X K , j) 0l−tors is finite.
Remark 2.28.We expect that even more holds, i.e. that the group CH d+j (X K , j) 0 of Corollary 2.27 is a direct sum of a finite group and a p-primary group of finite exponent.For j = 0 this is shown in [44, Cor.0.5].

Bloch-Beilinson conjectures for higher Chow groups
Let X be a variety over a field.In [7] Bloch defines a generalisation of Chow groups, so called higher Chow groups, as a candidate for motivic cohomology.These groups are defined as the homology groups of a certain complex ) is the subset of codimension * cycles meeting all faces properly (for more details see loc.cit.).More precisely, In particular, CH r (X, 0) ∼ = CH r (X) for all r ≥ 0. Bloch's higher Chow groups have the following properties: let f : X → Y be a morphism of k-varieties.
(i) If f is flat, then there is a pullback map f * : CH p (Y, q) → CH p (X, q).
(ii) If f is proper, then there is a push forward map f * : CH p+d (X, q) → CH p (Y, q), where d is the relative dimension of X over Y .
(iii) There is a product CH p (X, q) ⊗ CH r (Y, s) → CH p+r (X × Y, q + s).If X = Y is smooth, then pulling back along the diagonal gives a product CH p (X, q) ⊗ CH r (X, s) → CH p+r (X, q + s).
These properties allow to define an action of correspondences on higher Chow groups: Definition 2.29.Let d = dim X and d ′ = dim Y .Let X furthermore be projective, which implies that the projection p 2 : X × Y → Y to the second component is proper (we denote the first projection by p 1 ).A correspondence Γ ∈ CH a (X × Y ) defines a morphism The following conjecture proposes a generalisation of conjectures due to Bloch and Beilinson for Chow groups (see for example [23,Conj. 2.1] and [3, Sec.5]) to higher Chow groups.
Conjecture 2.30.For every smooth projective variety X over a field k there exists a filtration

A cohomological theory of cycle complexes with coefficients in Milnor K-theory
In [8] Bloch and Ogus define the notion of a Poincaré duality theory with supports which consists of a (twisted) homology theory and a (twisted) cohomology theory with supports which coincide for regular schemes by Poincaré duality.We have already remarked that Rost's Chow groups with coefficients in Milnor K-theory have the properties of a Borel-Moore homology theory, i.e, the following: they are covariantly functorial for proper maps, contravariantly functorial for flat maps, there exists a long exact localisation sequence and they are homotopy invariant.In order to get a corresponding cohomological theory with supports of Chow groups with coefficients in Milnor K-theory, we suggest to replace the Gersten complex for Milnor K-theory by the Cousin resolution of Milnor K-theory.The cohomology groups of these Cousin complexes will reappear in Appendix B where we define cohomological higher Chow groups of zero-cycles.
Definition 2.32.Let X be a noetherian scheme and F a sheaf of abelian groups on X.The complex is called the Cousin complex of F and denoted by C • F (X).Let Z be a closed subscheme of X.The complex is called the Cousin complex of F on X supported on Z and denoted by C • F (X) Z .
By [22, Ch.IV, Prop.2.3] the Cousin complex depends functorially on F and is therefore contravariantly functorial for morphisms of schemes in the following sense: if f : X → Y is a morphism of schemes, F a sheaf on X and G a sheaf on Y and if F → f * G is a morphism of sheaves, then there is an induced morphism of Cousin complexes f * : The following proposition is immediate from is the definition.Proposition 2.33 (Localisation).Let X be a noetherian scheme and Y a closed subscheme of X.Let U = X \Y be the open complement.Let j : U ֒→ X denote the inclusion.Then the sequence of complexes Proposition 2.34 (Poincaré duality).Let X be a smooth variety over a field k.Then Proof.This follows from the Gersten conjecture and purity for Milnor K-theory.

Zero cycles in families of relative dimension 1
In this section we study the case of a family of genus zero curves.This will be an essential ingredient of the proof of our main theorem in the next section.Proposition 3.1.Let A be a henselian discrete valuation ring with local parameter π and residue field k.Let X be a regular scheme flat and projective over A. Let d be the relative dimension of X over A and X n := X × A A/π n .We also denote the special fiber X 1 by X k .
(i) If X is smooth over A, k = C and X k is rationally connected, then the restriction map is an isomorphism.Here the last group on the right is the Levine-Weibel Chow group (see Definition B.8).
Proof.We begin with some preliminaries for (i) and (ii).First note that Pic(X) Since O Xn is flat over A n , tensoring with ⊗ An O Xn gives the short exact sequence 0 (the second isomorphism is induced by mapping an element a ∈ O X1,x to an element aπ n−1 with a ∈ O Xn,x a lift of a) and in turn the short exact sequence Taking cohomology, we get the exact sequence In the situation of (i), the first and last group are zero, since X k is rationally connected and k = C.This implies the statement.
In the situation of (ii), H 2 (X 1 , O X1 ) = 0 for dimension reasons and H 1 (X 1 , O X1 ) = 0 by assumption.This implies that Pic(X) ∼ = Pic(X k ).Finally, the last isomorphism follows from Lemma B.9. Remark 3.2 (Relation to blow-ups of closed points).Proposition 3.1(ii) is closely related to the blow-up formula for the Picard group.Let the notation be as in Proposition 3.1(ii) and π : X → X be the blow up in a closed point x.Then the exceptional divisor E is isomorphic to P 1 k(x) and Pic( X) ∼ = Pic(X) ⊕ Pic(E) and Pic(E) ∼ = Z.The same holds for the Levine-Weibel Chow group of the special fiber: The main example we have in mind for Proposition 3.1(ii) is that in which the special fiber is a rational comb: Definition 3.3.A genus zero comb over k with n teeth is a reduced projective curve of genus zero (i.e. a curve C with h 1 (C, O C ) = 0) having n + 1 irreducible components over k and only nodes as singularities.One component, defined over k, is called the handle (say C 0 ).The other n components, C 1 , . . ., C n , are disjoint from each other and intersect C 0 in n distinct points.Every C i is smooth and rational.The curves C 1 , . . ., C n may not be individually defined over k.
Remark 3.4 (Residue fields of intersection points of combs).Being a comb implies that the handles C i are of the form P 1 k(pi) for some p i ∈ C 0 .This is important when constructing combs by attaching rational curves over nonclosed fields.Let Taking cohomology, we get the exact sequence Finally, the group H 1 (C i , O Ci (−p i )) is zero by standard calculations of the cohomology of projective space since p i is of degree one on C i .Proposition 3.5.Let the notation be as in Proposition 3.1(ii).Assume the Gersten conjecture for Milnor K-theory and for higher Chow groups holds for henselian discrete valuation rings.Let j ≥ 1.Let Z 1 , Z 2 ⊂ X be two horizontal subschemes which intersect the special fiber X k transversally in the same component and such that Frac(A) ∼ = K(Z 1 ) ∼ = K(Z 2 ).Let α Z1 ∈ K M j (K(Z 1 )) be supported on Z 1 and α Z2 ∈ K M j (K(Z 2 )) be supported on Z 2 .Let p : X → Spec(A) be the structure map and p * the proper push-forward defined in Section 2.3.Assume that p * (α Z1 − α Z2 ) = 0 and ∂ 0 (α Z1 − α Z2 ) = 0. Then Proof.By Proposition 3.1 there is an f ∈ K(X) such that div(f ) = Z 1 − Z 2 .We may further assume that f is congruent to 1 in the generic points of the special fiber of X (see the proof of Theorem 5.2).Let α ∈ K M j (Frac(A)) be the element given by α Z1 and α Z2 and α its image in Remark 3.6.We expect that the restriction map Indeed, if X is obtained by a finite number of successive blow ups of closed points of P 1 A , then this should follow from the Propositions 2.12, 2.9 and 2.10.
There is no easy way to formulate a generalisation of this expectation to higher (relative) dimension, unless we make the assumptions as in our main theorem 1.1(ii).

The smooth case
In this section let A be a henselian discrete valuation ring with function field K and residue field k.Let S = Spec(A) and X be a regular scheme flat and projective over S with generic fiber X K and special fiber X k .Let d be the relative dimension of X over S. The goal of this section is to prove the following theorem: Theorem 4.1.Assume that X is smooth over S and that X k is separably rationally connected.
(i) The restriction map CH d (X) → CH d (X k ) is an isomorphism.
(ii) Assume X k has a rational point that the degree map deg : CH d (X k ′ ) → Z is an isomorphism for every finite field extension k ′ /k and assume the Gersten conjecture for Milnor K-theory and for higher Chow groups for henselian discrete valuation rings.Then the vertical maps in the commutative diagram CH d+j (X, j) are isomorphisms for all j ≥ 0.
We begin by outlining the proof of Theorem 4.1.Let (X) d,g denote the set of codimension d points x such that {x} is flat over S, i.e. the cycles in good position.Let (X) d,t ⊂ (X) d,g denote the subset of points x for which in addition the intersection of {x} and O O γ e e ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ The idea of the proof of (i) is the following: the two vertical maps exist and are surjective.The maps ρ and ρ, restricting a horizontal one-cycle to the special fiber, also exist and make the outer diagram commute.We show that the diagonal map γ, induced by lifting closed points to horizontal curves which intersect the special fiber transversally, makes the lower triangle of the above diagram commute.Then we show that γ factorises through rational equivalence which gives the map γ.The map γ is surjective and inverse to ρ and therefore an isomorphism.For j ≥ 1 the situation is slightly more subtle since already the bottom map in Diagram ( 2) is surjective but not injective.This approach to studying relative zerocycles was used by Kollár in [31,Sec. 18] assuming, like we do, that X k is separably rationally connected, and by Kerz, Esnault and Wittenberg in [28,Sec. 4 and 5] working with finite coefficients prime to the residue characteristic but without the assumption of X k being separably rationally connected.
Proof.Consider an element We need to show that T ≡ 0 ∈ CH d (X).In order to do so, we make several reduction steps.
Step 1.We may assume that all the Z i intersect X k in the same point x and that s = 2. Indeed, let Z ′ be a lift of x intersecting X k transversally.Then we may assume that T is of the form where Z and Z ′ are integral and n is the intersection multiplicity of Z with X k , since if n i is the intersection multiplicity of Step 2. In this step we reduce to the case that n = 1, i.e.Z also intersects X k transversally.Let O Z be the ring of integers in K(Z).Note that O Z is a henselian discrete valuation ring.We base change X along the map Spec(O Z ) → S and denote the base change of X by restrict to the same point in X O Z , i.e. their difference is in the kernel of ρ.Then, if we have that , where p : X O Z → X is the projection. 2tep 3. Let f : P 1 k → X k be the rational curve with image the point x.Taking the diagonal map we may now show that Z OZ = (Z OZ , (1 : 0)) and [31,Thm. 15] (up to passing to a product with P m for some m ∈ N), there is a genus zero comb over C * ⊂ X × P 1 k over k, with handle C, which is smooth at z 1 and z 2 and such that By [1,Thm 51.5], there is a smooth S-scheme U and a family of stable genus zero curves containing Z OZ ∪ Z ′ OZ and a point u ∈ U (k) corresponding to C * (in the notation of loc.cit., P = Z OZ ∪ Z ′ OZ ).Indeed, since X k is smooth and C * a local complete intersection, there is an exact sequence of sheaves on This gives an exact sequence Using the Mayer-Vietoris sequence for closed covers, one can show that . The case j ≥ 1 is proved similarly, the important part of the case d = 1 being treated in Proposition 3.5.For the reduction to the case d = 1 we proceed as above with the following modifications.In Step 1 we start with an additional α ∈ K M j (k(Z)) supported on Z.We embed Z OZ and Z ′ OZ as Z OZ = (Z OZ , (1 : 0), (1 : 0)) and Z ′ OZ = (Z ′ OZ , (1 : 0), (1 : 0)) into X × P 2 OZ .In particular they intersect the special fiber in the same point, say p.As in Step 3 of the proof of Theorem 4.2 we can find relative curves of genus zero C S and C ′ S containing Z OZ and Z ′ OZ respectively with the property that C S and C ′ S intersect transversally in p.Let Z ′′ OZ the intersection of the two relative curves at p. Then we apply Proposition 3.5 to Proof of Theorem 4.1.In the case of (i) it remains to prove that γ factorises through γ.
. As in the proof of [44,Lem. 7.2], we can find a relative curve Z ⊂ X (that is of dimension 2 and flat over S) containing x which is regular at x and such that Z ∩ X k contains {x} with multiplicity 1.Let Z 0 be the special fiber of Z and denote by ∪ i∈I Z (i) 0 ∪ {x} the union of the pairwise different irreducible components of Z 0 .Here the irreducible components different from {x} are indexed by I. Let z be the generic point of Z. Now as in the proof of [36,Lem. 2.1], we can find a lift α ∈ k(z) × of α 0 which specialises to α 0 in k(x) × and to 1 in K(Z which implies the above factorisation. In the case of (ii) we can use the assumption on the special fiber to show that CH d+j (X, j) is generated by Milnor K-groups supported on just one transversal curve lifting rational point . By Proposition 4.2 it is sufficient to show that there is an . By making a base change along the morphism Z 1 → S we may assume that K(Z 1 ) with the property that i div(f i ) = p − q.We lift each C i and f i exactly as in (i) and denote these lifts by C i and f i .Then, by Proposition 4. (i) There is a zigzag of isomorphisms (ii) Assume that CH d (X k ) ∼ = Z.Then for all j ≥ 0 the pullback map induces an isomorphism Proof.(i) The fact that the composition is zero and that the second map is an isomorphism by Theorem 4.1(i) implies that the first map in the localisation sequence There is a commutative diagram with exact rows The first row is the localisation exact sequence for higher Chow groups and the second row is exact by assumption of the Gersten conjecture.The first and third vertical maps are isomorphisms by Theorem 4.1(ii).By a diagram chase this implies that the map g * induces the given isomorphism.

Higher zero-cycles
Let A be a henselian discrete valuation ring with quotient field K and residue field k.Let X be a smooth and projective scheme over A with generic fiber X K and special fiber X k .Let d be the relative dimension of X over Spec(A).Generalising Theorem 4.1(ii) in a different direction, one may ask the following question: Question 5.1.Assume that the special fiber X k of X is separably rationally connected.Is the restriction map In this section we study the case i = 1, in which we can make use of the isomorphisms of Proposition 2.2.Note further that if the Gersten conjecture holds for the Milnor K-sheaf K M d,X , then these groups are isomorphic to We recall how the restriction map is defined on Rost's Chow groups with coefficients in Milnor K-theory.Let π be some fixed local parameter of A. We define the restriction map to be the composition The map res π depends on the choice of π but the induced map on homology is independent of the choice (for more details see [36,Sec. 2]).In the following we assume that we have fixed π and drop it from the notation.Theorem 5.2.Assume that the special fiber X k of X is separably rationally connected.Then the map is surjective.
Proof.We closely follow the proof of [36,Prop. 2.2].The diagram to consider is the following: Here Z i (X) := C i (X, −i) are just the cycles of dimension i.The restriction map in the lowest degree res : Z 1 (X) → Z 0 (X k ) agrees with the restriction map on cycles defined by Fulton in [17,Rem. 2.3].We want to show that the middle horizontal map induces a surjection on homology.First note that the map res : C 2 (X, −1) → C 1 (X k , 0) is surjective by [36,Lem. 2.1].Let x ∈ Ker[res : Z 1 (X) → Z 0 (X k )].We show that there is a ξ ∈ Ker[res : C 2 (X, −1) → C 1 (X k , 0)] with ∂(ξ) = x.By a diagram chase this then implies the theorem.In particular we do not need the upper line of the above diagram.
We first treat the case of relative dimension d = 1 assuming that X is as in Proposition 3.1(ii).In this case the above diagram becomes the following the diagram We consider the following short exact sequence of sheaves: where M * X;X k (resp.O * X;X k ) denotes the sheaf of invertible meromorphic functions (resp.invertible regular functions) relative to Spec(A) and congruent to 1 in the generic points of X k , i.e. in each O X,µ , and Div(X, X k ) is the sheaf associated to M * X;X k /O * X;X k .In other words, Div(X, X k )(U ) is the set of relative Cartier divisors on U ⊂ X which restrict to zero in Z 0 (X k ).For the concept of relative meromorphic functions and divisors see [20,Sec. 20,21.15].We want to show that Div(X, X k )(X)/M * X;X k (X) = 0.The short exact sequence (3) implies that Div(X, X k )(X)/M * X;X k (X) injects into Pic(X, X k ).But the latter group fits into the exact sequence in which the first map is surjective, since it is possible to lift units, which can be shown using the Stein factorization.Furthermore, the map on the right is an isomorphism by Proposition 3.1.Therefore Pic(X, X k ) = 0. Let d > 1.As noted in the beginning of the proof, it suffices to show that for x ∈ Ker[res : Remark 5.3.Let j : X K → X denote the inclusion of the generic fiber.Again, the group CH d (X, 1) links the groups CH d (X K , 1) and CH d (X k , 1): The importance of the group CH d (X, 1) comes from the fact that it is related to the torsion in CH d (X K ) (see [36,Sec. 3]).
Corollary 5.4.Let the notation be as in Theorem 5.2.Assume that the Gersten conjecture holds for the Milnor K-sheaf K M d,X .Then which, by the remarks at the beginning of the section, is isomorphic to The statement therefore follows from Theorem 4.1 and Theorem 5.2.
6 Conjectures in the non-smooth case and the case of rational surfaces In this section we study the case of smooth rationally connected varieties over local fields which do not have good reduction.More precisely, we would like to understand how the Chow group of 1-cycles of a regular model of such a variety over the ring of integers relates to the Chow group of zero cycles of the generic fiber and to a cohomological version of the Chow group of zero cycles of the special fiber.One motivation is to use finiteness results about the latter to deduce the following conjecture: Conjecture 6.1 (Kollár-Szabó).Let X K be a d-dimensional, smooth, projective, separably rationally connected variety defined over a local field.Then CH d (X K ) 0 := Ker[deg : CH 0 (X K ) → Z] is finite.
Conjecture 6.1 is known to be true if X K has good reduction.This is due to Kollár and Szabó [33,Thm. 5].They even show that CH d (X K ) 0 = 0.This also follows from the main theorem of [31] or our Corollary 4.3, combined with the following theorem: In general, it cannot be expected that CH d (X K ) 0 = 0.In [46, Thm.1.1], Saito and Sato calculate this group for certain cubic surfaces to be isomorphic to Z/3 ⊕ Z/3.Nevertheless, Conjecture 6.1 holds in dimension two: Theorem 6.3.(Colliot-Thélène [9, Thm.A]) Let X K be a smooth projective rational surface over a local field K. Then CH 2 (X K ) 0 is finite.
From now on let A be a henselian discrete valuation ring with local parameter π, residue field k and quotient field K. Let X be a regular scheme flat and projective over A with separably rationally connected generic fiber X K .Assume that the special fiber X k (in the following also denoted by X 1 ) of X is a simple normal crossing divisor.We denote the respective inclusions by j : X K → X and i : X k → X and set X n = X × A A/π n .Let d be the relative dimension of X over Spec(A).As mentioned above, we intend to study Conjecture 6.1 for CH d (X K ) using different (cycle) class maps out of CH d (X).We begin with a few remarks on these maps.Assuming the Gersten conjecture for Milnor K-theory, there is an isomorphism CH d (X)/p r ∼ = H d (X, K M d,X /p r ) which, by composition, allows us to define a restriction map This is the Zariski side of the story.If K is a p-adic local field, the right p-adic étale motivic theory can be defined by Sato's p-adic étale Tate twists, which are objects T r (n) ∈ D b (X ét , Z/p r ) fitting into a distinguished triangle of the form (see [48]).The Galois symbol map 4 The map i * K M d,X /p r → i * H d (T r (d) X ) factorises through K M d,Xn /p r for n large enough.Via a norm argument one first reduces this statement to the case in which the residue field is large and therefore the improved Milnor K-theory coincides with ordinary Milnor K-theory.This case can be deduced from the short exact sequence 0 → 1 + π n O X → O × X → O × Xn → 0, which induces the unit filtration on Milnor K-theory, noticing that in the Nisnevich topology the group (1 + π n O X )/p r = 0 for n large enough by Hensel's lemma [14,II,Lem. 2].Indeed, the kernel of the restriction map i * K M d,X → K M d,Xn is locally generated by elements of the form {1 + π n a, f 1 , . . ., f d−1 } with f 1 , . . ., f d−1 units [26, Lemma 1.3.1].In [37] we showed that the resulting map lim ) is an isomorphism in the smooth case.In sum, there are commutative diagrams The bottom map on the left is an isomorphism by étale proper base change.The map cl X is surjective by [46].Precomposing the vertical maps with maps from p r -torsion subgroups, we get a commutative diagram (4) Remark 6.4.(i) In [28,Conj. 10.1], Kerz, Esnault and Wittenberg conjecture that, assuming the Gersten conjecture for Milnor K-theory for the isomorphism on the left, the map is an isomorphism for any regular flat projective scheme over A. In [37] we showed that this map is surjective in the smooth case even with integral coefficients.
(ii) We would like to study the map res d | tor using the map res d /p r .For this the map σ needs to be injective.Then we can deduce the injectivity of res d | tor from the conjectural injectivity of res d /p r .If CH d (X)[p r ] is of finite exponent, then σ is injective for r >> 0.
(iii) If p = ch(k), then by Corollary B.12 the above diagram becomes the following diagram: For the notation and definition of the cdh-cohomology group on the right see Section B.3.In this case the map res d /ℓ r is known to be an isomorphism by the main result of [28] assuming k is finite or separably closed.
Saito and Sato prove the following important theorem concerning the composition cl X •σ = cl X,p−tors,r .Theorem 6.5.([47, Cor.4.3],[45, Thm.1.5]) Let A be a henselian discrete valuation ring with perfect residue field k and quotient field K. Let X be a regular scheme flat and projective over A with generic fiber X K .Let d be the relative dimension of X over A. Assume that H 2 (X K , O XK ) = 0. Let ch(k) = p, ch(K) = 0 and assume that the special fiber is a simple normal crossing divisor.Then the map cl X,p−tors,r : CH 2 (X) p−tors → H 4 (X, T r (2)) is injective for sufficiently large r.Remark 6.6.The condition H 2 (X K , O XK ) = 0 is satisfied if X K is a smooth projective separably rationally connected variety of dimension 2. In this case, by [30, Ch.IV, Cor.3.8], H 0 (X K , Ω m XK ) = 0 for all m > 0 and by Serre duality, H Proposition 6.7.Let K be a local field with residue field k and ring of integers A. Let X be a regular flat projective scheme over A. Assume that the generic fiber X K of X is rationally connected of dimension 2, i.e. a rational surface, then the group CH 2 (X) is finitely generated and the group CH 2 (X) tors is finite.
Proof.The group CH 2 (X) fits into the exact sequence CH 1 (X k ) → CH 2 (X) → CH 2 (X K ) → 0. By Theorem 6.3, CH 2 (X K ) is the direct sum of Z and a finite group.The group CH 1 (X k ) is finitely generated since the Picard group of each component of X k is finitely generated since k is finite.Therefore CH 2 (X) tors is finite.
Theorem 6.8.Let the notation be as in Proposition 6.7 and ch(k) = p and ch(K) = 0. We make the following additional assumptions: (i) The reduced special fiber X k is a simple normal crossing divisor.
(ii) The Gersten conjecture holds for the Milnor K-sheaf K M 2,X and the induced map Then the map is an isomorphism.
Proof.By Theorem 6.5 and the commutativity of Diagram (4) the map res 2 is injective on p-torsion.By Proposition 6.7 and and the commutativity of Diagram (5) the map res 2 is injective on ℓ-torsion.
Combined we get an injection Since by Proposition 6.7 the group CH 2 (X) is finitely generated, the theorem now follows combining the injectivity on the torsion subgroup with assumption (ii) and the fact that the map res 2 /ℓ r is an isomorphism for all ℓ = p by the main theorem of [28].Conjecture 6.9.Let A be a henselian discrete valuation ring with residue field k and quotient field K. Let X be a regular scheme flat and projective over A with separably rationally connected generic fiber X K .Let d be the relative dimension of X over Spec(A).Assume the Gersten conjecture for Milnor K-theory.Then the restriction map ) is an isomorphism.
Remark 6.10.Conjecture 6.9 should be viewed as an analogue of the conjecture of Kerz, Esnault and Wittenberg described in Remark 6.4(ii) and as a generalisation of Theorem 6.8.If CH d (X K ) is p-torsion free, then we expect that lim Indeed, s is always an isomorphism for fields by [50] and [41].The lower horizontal sequence is exact by Proposition A.1(ii) and therefore s is always injective.By the above remarks the upper horizontal sequence is exact for r ≤ 2. For r = 3 the motivic-to-K-theory spectral sequence implies that CH 2 (k, 3) ∼ = K 3 (k) is finite by Quillen's calculation of the K-groups of finite fields.The result then follows from the fact that s is an isomorphism with finite coefficients.

B Cohomological Chow groups of zero-cycles
Let X be a scheme of finite type over a field k.The G-theory G n of X, defined using the abelian category of coherent sheaves on X, is related to Bloch's higher Chow groups via a version of the Grothendieck-Riemann-Roch theorem for singular schemes; this theorem is due to Baum-Fulton-MacPherson [2] and Bloch [7] and says that there is a sequence of isomorphisms This is considered to be the homological side of the story since Bloch's higher Chow groups form a Borel-Moore homology theory.On the cohomological side there are two versions of K-theory.Algebraic K-theory, defined by Quillen using the category of locally free sheaves on X, and KH-theory, a homotopy invariant version of algebraic K-theory, defined by Weibel.Algebraic K-theory should be related to a cohomological version of Bloch's higher Chow groups.In the case of zero-cycles Levine and Weibel [34] have defined a version of the Chow group of zero-cycles on X in terms of the smooth points and certain good curves on X, which is related to K 0 (X) (for a precise definition see Definition B.8).
Finally, KH-theory is related to cdh-motivic cohomology.If A is a ring and ℓ ∈ A × , then there is an isomorphism K n (A, Z/ℓZ) ∼ = KH n (A, Z/ℓZ) (see [51,Prop. 1.6]).This indicates that with finite coefficients cohomological Chow groups should be closely related to cdh-motivic cohomology groups.
In Section B.1 and B.2 we define cohomological versions of the higher zero-cycles CH d+j (X, j) which we have encountered in Section 2. This extends the definition of Levine and Weibel for zero-cycles and is related to higher K-theory.In Section B.3 we relate these groups to cdh-motivic cohomology.For more results of this nature we refer the reader to [6].

B.1 The case of curves
In this and the next subsection we assume for simplicity that all residue fields are large in order to be able to work with ordinary Milnor K-theory.In the following let k be a field and X be an equidiensional quasi-projective k-scheme.Let X sm denote the disjoint union of the smooth loci of the d-dimensional irreducible components of X.Let X sing denote the singular locus of X.A smooth closed point of X will mean a closed point lying in X sm .Let Y X be a closed subset not containing any d-dimensional component of X and such that X sing ⊆ Y .Let Z 0 (X, Y ) be the free abelian group on closed points of X \ Y .We shall sometimes write Z 0 (X, X sing ) as Z 0 (X).More generally, we make the following definition:

Proposition 4 . 2 .
Let the notation be as in Theorem 4.1.Then the diagram
For the reduction to the case d = 1 we now proceed as in the proof of Theorem 4.2 with the following modifications.We embed Z OZ and Z ′ OZ as Z OZ = (Z OZ , (1 : 0), (1 : 0)) andZ ′ OZ = (Z ′ OZ , (1 : 0), (1 : 0)) into X × P 2 OZ .In particular they intersect the special fiber in the same point, say p.As in Step 3 of the proof of Theorem 4.2 we can find relative curves of genus zero C S and C ′ S containing Z OZ and Z ′ OZ respectively with the property that C S and C ′ S intersect transversally in p.Let Z ′′ OZ the intersection of the two relative curves at p. Then we apply the case d = 1 proved above to Z OZ − Z ′′ OZ and Z ′ OZ − Z ′′ OZ in C S and C ′ S respectively.

Theorem 6 . 2 .
Thm. 1]) If k is a finite field and X k a d-dimensional, smooth, projective, separably rationally connected variety over k, then CH d (X k ) 0 = 0.

Definition B. 1 .
Z 0 (X, X sing , j) := x∈Z0(X,Xsing)K M j (x).Let C be a reduced scheme of pure dimension 1 over k and {η 1 ,• • • , η r } be the set of generic points of C. Let O C,Z denote the semi-local ring of C at S = Z ∪ {η 1 , • • • , η r }.Let k(C)denote the ring of total quotients of C and write O × C,Z for the group of units in O C,Z .Notice that O C,Z coincides with k(C) if |Z| = ∅.Since C is reduced, it is Cohen-Macaulay and therefore O × C,Z is the subgroup of group of units in the ring of total quotients k(C) × consisting of those f which are regular and invertible in the local rings O C,x for every x ∈ Z. Definition B.2.We letK M j (O C,Z ) := (O × C,Z ) ⊗j / < a 1 ⊗ • • • ⊗ a j | a i + a i ′ = 1 for some 1 ≤ i < i ′ ≤ j > .
[39,h is the composition of basic maps defined in Definition 2.3.α:X⇒Y is called a correspondence.The next lemma shows that a correspondence is compatible with the boundary maps d X and d Y .The lemma is formulated in[43, Prop.4.6]for schemes over fields but the proofs transfer almost word by word to our situation.Indeed, either the the steps of the proof reduce to equal characteristic or to statements which are known for arbitrary discrete valuation rings.For more details we refer to[39, Sec.2].
Proposition 2.15.([18, Prop.7.4.4])Let C be a smooth projective curve over k.For a closed point P , let ∂ M P . The proposition now follows from Proposition 3.1 applied to the irreductible component C ′ S of C S which contains P .Indeed, since C ′ S is smooth around x 1 and x 2 , [ Z OZ ] − [ Z ′ OZ ] restricts to zero in Pic((C ′ S ) k ) and therefore, by Proposition 3.1, also [ Z OZ 3Let s be the closed point of S. By Hensel's lemma there is a section σ : S → U such that σ(s) = u.σ(S) corresponds to a relative curve C S , which contains P , with special fiber C *