S T Éphane

Julien Sebag
email: julien.sebag@univ-rennes1.fr

Stéphane Lamy
email: slamy@math.univ-toulouse.fr

BIRATIONAL SELF-MAPS AND PIECEWISE ALGEBRAIC GEOMETRY

Let X be a smooth projective complex variety, of dimension 3, whose Hodge numbers h 3,0 (X), h 1,0 (X) both vanish. Let f : X X be a birational map that induces an isomorphism U ∼ = V on (dense) open subvarieties U, V of X. Then we show that the complex varieties (X \ U ) red , (X \ V ) red are piecewise isomorphic.

Introduction

In [START_REF] Gromov | Endomorphisms of symbolic algebraic varieties[END_REF], Misha Gromov asks the following question: Question 1. Let X, Y be algebraic varieties that admit embeddings to a third one, say X ֒→ Z and Y ֒→ Z, such that the complements Z \ X and Z \ Y are biregular isomorphic. How far are X and Y from being birationally equivalent?

Let k be an arbitrary field. Let Z[Var k ] be the free abelian group generated by the isomorphism classes of k-schemes of finite type. One defines the Grothendieck ring of varieties over k, denoted by K 0 (Var k ), as the quotient of Z[Var k ] by the following "scissor" relations: whenever X is a k-scheme of finite type, and F is a closed subscheme of X, we impose that

[X] = [F ] + [X \ F ].
The multiplication in K 0 (Var k ) is defined by

[X] • [X ′ ] := [X × k X ′ ].
for any pair of k-schemes of finite type X, X ′ .

Independently of [START_REF] Gromov | Endomorphisms of symbolic algebraic varieties[END_REF] 1 , Michael Larsen and Valery Lunts ask, in [11, Question 1.2], the following question in order to understand the "geometric nature" of the Grothendieck ring of varieties: Question 2. Let X, Y be reduced separated schemes of finite type over k = C such that [X] = [Y ] in K 0 (Var k ). Is it true that X, Y are piecewise isomorphic?

Recall that two k-schemes of finite type X, Y are called piecewise isomorphic, if one can find a finite set I, a partition (V i ) i∈I of X in (locally closed) subschemes of X, and a partition (W i ) i∈I of Y in (locally closed) subschemes of Y , such that, for any i, i ∈ I, there exists an isomorphism of k-schemes f i : (V i ) red → (W i ) red . Note that this property only depends on the reduced scheme structures of X and Y . Up to now, this question is open in general (see [START_REF] Liu | The Grothendieck ring of varieties and piecewise isomorphisms[END_REF], [START_REF] Sebag | Variations on a question of Larsen and Lunts[END_REF] for positive elements of answer in this direction, or [START_REF] Hruchovski | Integration in valued fields[END_REF]Theorem 13.1] that uses a different point of view).

The fact that, without any condition on k, a positive answer to Question 2 implies a precise answer to Question 1 is easy to see. We adopt the notations of Question 1. Because of the relations lying in K 0 (Var k ), we deduce that [Z \ X] = [Z \ Y ], and that [X] = [Y ] in K 0 (Var k ). If Question 2 has a positive answer, we conclude that X, Y are piecewise isomorphic. Because of the definition of a piecewise isomorphism, this remark proves in particular the existence of a bijection σ between the set of the irreducible components of maximal dimension of respectively X and Y , such that (X i ) red and (σ(X i )) red are birationally equivalent, for any irreducible component X i of X.

Besides, note that Question 2 holds true in particular for any pair X, Y of separated k-schemes of finite type of dimension at most 1, if k is supposed to be algebraically closed of characteristic 0 (see [START_REF] Liu | The Grothendieck ring of varieties and piecewise isomorphisms[END_REF]Proposition 6]).

In the present article, we investigate the following conjectural reformulation of Question 1 based on Question 2 (see also [START_REF] Liu | The Grothendieck ring of varieties and piecewise isomorphisms[END_REF]Remark 13], or [START_REF] Sebag | Variations on a question of Larsen and Lunts[END_REF]).

Conjecture 1. Let k be an algebraically closed field of characteristic zero. Let X be a separated scheme of finite type over k. Let f : X X be a birational map that induces an isomorphism of k-schemes U ∼ = V on dense open subsets U, V of X. Then the k-schemes (X \ U ) red , (X \ V ) red are piecewise isomorphic.

Note that, by a classical argument of birational geometry, if X is a smooth projective scheme over k, of arbitrary dimension d, having no rational curves, then such a map f is, in fact, an isomorphism of k-schemes. Thus Conjecture 1 holds true in that case, but remains open, and particularly relevant, for separated k-schemes of finite type having an infinite number of rational curves. (The case where the separated k-scheme of finite type X has only finitely many rational curves can be solved positively for example by [START_REF] Liu | The Grothendieck ring of varieties and piecewise isomorphisms[END_REF]Theorem 5].) Precisely, we prove the following theorem, which is the main result of this article (see Theorem 24, and Remark 25 for a slightly more general statement).

Main Theorem. Let X be a smooth irreducible projective C-variety, of dimension 3, whose Hodge numbers h 3,0 (X), h 1,0 (X) both vanish. Let f : X X be a birational map that induces a C-isomorphism U ∼ = V on dense open subvarieties U, V of X. Then the C-varieties (X \ U ) red , (X \ V ) red are piecewise isomorphic.

The proof of such a statement is based on the following ingredients.

a) The use of the Weak Factorization Theorem for birational maps (see [START_REF] Abramovich | Torification and factorization of birational maps[END_REF]) and its combinatoric; b) The use of intermediate Jacobians to control the weak factorizations of f (see [START_REF] Clemens | The intermediate Jacobian of the cubic threefold[END_REF]); c) The positive answer to Larsen-Lunts' question in the case of curves (see [START_REF] Liu | The Grothendieck ring of varieties and piecewise isomorphisms[END_REF]Proposition 6]). This article is mostly self-contained, for the convenience of the reader. Sections 3 and 4 give an overview of some tools of birational, piecewise algebraic and complex geometry that we use in the proof of Theorem 24. Besides, we establish in these sections some technical lemmas, sometimes not explicitly described in the literature, that we need, in a crucial way, in the proof of Theorem 24 (e.g., see Lemma 10, Corollary 12, Propositions 15, 21, Corollary 23). Then Section 5 is devoted to our main theorem and its proof (see Theorem 24). In Sections 6 and 7 we give respectively some explicit examples and some further comments.

The authors would thank Qing Liu and François Loeser for their comments on a first version of this article.

Conventions

Let k be a field. A curve (resp. surface) is a separated k-scheme of finite type purely of dimension 1 (resp. 2). More generally, a k-variety is a separated k-scheme of finite type. If x is a point in a k-variety, we define its dimension as the transcendence degree over k of the residue field κ(x) of X at x. If X, Y are two piecewise isomorphic k-varieties, the datum of the family of the k-isomorphisms (f i ) i∈I (given, by definition, between the reduced locally closed strata of X, Y ) is called a piecewise isomorphism.

If k is an algebraically closed field, we say that a k-variety is an abelian variety if X is a smooth irreducible projective abelian group scheme over k.

If X is a C-variety, we denote by X an := X(C) the analytic space associated to X.

In K 0 (Var k ), we denote by L the class of the affine line, and by [X] the class of the k-scheme of finite type X. Let A be a ring. A ring morphism K 0 (Var k ) → A is called a motivic measure.

Birational and piecewise algebraic geometry

In this section, we review some tools of birational and piecewise algebraic geometry that we use in the proof of Theorem 24. We also establish some useful preliminary lemmas.

3.1. Birational maps and the Weak Factorization Theorem. In this paragraph, we review some well-known results on birational maps such as the Weak Factorization Theorem established in [1, Theorem 0.1.1] and we assume that k is an algebraically closed field of characteristic zero. Let X, Y be two k-varieties. A rational map f : X Y that induces an isomorphism of k-schemes U ∼ = V from a dense open subvariety of X onto a dense open subvariety of Y is called a birational map. In particular, there exists a unique maximal open subvariety Dom(f ) of X, such that f | Dom(f ) : Dom(f ) → Y is a morphism of k-schemes. If x ∈ X, we say that a birational map f : X Y is a local isomorphism (for the Zariski topology) at x if there exists an open subvariety U of Dom(f ) that contains x such that the restric-

tion of f | Dom(f ) to U is an open immersion of k-schemes. Note that, if f : X Y is a local isomorphism at x, then x ∈ Dom(f ).
Remark also that the set of points of X, at which a birational map f : X Y is not a local isomorphism, is a closed subset of X, that we call exceptional set of f . We denote it by Exc(f ).

The following lemmas are recalled for convenience for the reader. Lemma 4 should be kept in mind for the proof of Theorem 24.

Lemma 2. Let X, Y be two integral k-varieties. Let f : X Y be a birational map. Let x ∈ Dom(f ) be a point. Let us set y := f (x). The following assertions are equivalent. a) The map f is a local isomorphism at x; b) The morphism f | Dom(f ) induces an isomorphism of k-algebras O Y,y → O X,x ; c) The morphism f | Dom(f ) is étale at x. d ) The morphism f | Dom(f ) induces an isomorphism of k-algebras O Y,y → O X,x . Proof. Since f | Dom(f ) is dominant, the induced morphism of local k-algebras f x : O Y,y → O X,x is injective.
Assume that f | Dom(f ) is étale at x. Thus, the local morphism f x is flat, hence faithfully flat. Since f is birational, we deduce that f x is an isomorphism. It shows that point c) implies point b). Besides, f | Dom(f ) then induces an isomorphism of k-algebras κ(y) → κ(x). So point c) is equivalent to point d).

Remark 3. Let n ∈ N, n ≥ 1. To emphasize Lemma 2, consider the birational C-morphism f : A n C → A n C defined by the datum of (f i ) 1≤i≤n ∈ (C[x 1 , . . . , x n ]) n . Then we deduce from the Formal Inverse Function Theorem that the set of the C-points of A n C at which f is not a local isomorphism is defined as the C-points of the hypersurface of A n C defined by the determinant of the Jacobian matrix of f , i.e., defined by the equation det((∂f i /∂x j ) 1≤i,j≤n ) = 0.

Lemma 4. Let f : X Y be a birational map between two integral k-varieties. Assume that Y is normal. Let U 0 ⊂ Dom(f ) be the open set of points of X at which f is a local isomorphism. Let us set V 0 := f (U 0 ). Then a) The morphism of k-schemes f |U 0 : U 0 → Y is an open immersion. So the map f induces an isomorphism of k-schemes U 0 → V 0 . b) Assume that X, Y are proper and smooth over k. Each irreducible component of codimension 1 in (X \ U 0 ) red is ruled.
Proof. a) From the definition of U 0 and from Lemma 2, we deduce that the restriction of the morphism f | Dom(f ) to U 0 is separated, birational and locally quasi-finite. From Grothendieck's form of Zariski's Main Theorem applied to the morphism f |U 0 , we conclude that f |U 0 is an open immersion. So it induces an isomorphism of k-schemes from U 0 to V 0 . b) By eliminating the indeterminacies of f (e.g., see [1, Lemma 1.3.1]), we find a composite of blow-ups π : Z → X, with smooth centers disjoint from U 0 , and a birational morphism g : Z → Y such that g = f • π. Let η be the generic point of an irreducible component X η of maximal dimension in (X \ U 0 ) red . Note that η ∈ Dom(f ). The strict transform by π of X η (considered with its reduced scheme structure) is a divisor Z η of Z, birationally equivalent to X η . But, because of the choice of U 0 and the dimension of η, the subvariety Z η necessarily is an irreducible component of the exceptional locus of the morphism g. By Abhyankar's Theorem on exceptional sets of birational morphisms, we deduce that Z η is ruled, i.e., birationally equivalent to P 1 C × C S where S is an integral k-variety of dimension dim(X) -2. So X η is ruled.

Remark 5. Note that Lemma 4 in particular shows that the set of points of X at which f is a local isomorphism is the largest open subset U of X such that f |U is an isomorphism of k-schemes.

The Weak Factorization Theorem was a great step in the understanding of the geometry of birational maps. We recall below the statement of this theorem.

Let X, Y be smooth irreducible proper (resp. projective) k-varieties. Let f : X Y be a birational map that induces an isomorphism of k-schemes U → V on a dense open subvariety U of X onto a dense open subvariety V of Y . We say that f admits a weak factorization if there exist birational morphisms b i : Z i → X i , c i : Z i → X i+1 , for i, 1 ≤ i ≤ n between smooth irreducible proper (resp. projective) k-varieties such that: f : X Y be a birational map. The elimination of the indeterminacies of f produces a weak factorization of f . Indeed, in that case, one knows that there exist composites of blow-ups of points b : Z → X, c : Z → Y making the following diagram commute (see [START_REF] Liu | Algebraic geometry and arithmetic curves[END_REF]Theorem 9.2.7]):

Z 1 b1 }} } } } } } } c1 2 2 f f f f f f f f Z n bn } } | | | | | | | | cn 4 4 i i i i i i i i X 1 X 2 . . . X n X n+1 a) X := X 1 , Y := X n+1 ; b) For any i, 1 ≤ i ≤ n,
Z b ~~~~~~~c 1 1 d d d d d d d X f G G Y
The existence of such a strong factorization in any dimension is still a conjecture.

The following remark should be kept in mind for the proof of Theorem 24.

Remark 8. Let b ij be a blow-up that appears in the expression of b i . Then two cases occur. a) First, the strict transform of its exceptional divisor might be equal for some j ′ to the exceptional divisor of a blow-up c i ′ j ′ appearing in the expression of c i ′ . Note that, in that case, the strict transform is then contracted by c i ′ j ′ . b) On the contrary, it can happen that no blow-up appearing in any c i contracts the strict transform of its exceptional divisor. In that case, we can consider that strict transform in Y . Then it is equal to a divisor of Y contained in Y \ V . Note that if V is chosen as the maximal open subset on which f -1 is well-defined and injective, then all the irreducible components, of dimension dim(Y ) -1, in Y \ V are realized this way, and in particular are ruled. (This last assertion follows from Lemma 4, or from the Weak Factorization Theorem).

Cohomology theories and examples of motivic measures.

In this paragraph, we assume that k is a subfield of the field of complex numbers, except when stated otherwise. Let X be a smooth proper k-variety. We denote by H * dR (X) the algebraic de Rham cohomology of X. Then

dim k H n dR (X) = i+j=n dim k H i (X, Ω j X/k )
(Hodge's decomposition theorem) for any integer n. Let us denote by H p,q (X) the C-vector space H q (X, Ω p X/k ) ⊗ k C, for any pair p, q, 0 ≤ p, q ≤ dim(X). The integer h p,q (X) := dim C H p,q (X) is called the (p, q)-Hodge number of X. We denote by H * (X) := H * (X an , Z) the integral singular cohomology of X an . By Grothendieck's results ( [START_REF] Grothendieck | On the de Rham cohomology of algebraic varieties[END_REF], Theorem 1 ′ ), we know that the canonical map

H * dR (X) ⊗ k C ≃ H * (X) ⊗ Z C is an isomorphism. For i, 1 ≤ i ≤ 2dim(X), we denote by b i (X) := dim Q H i (X) ⊗ Z Q = dim C H i (X) ⊗ Z C the i-th Betti number of X.
By Deligne's mixed Hodge theory, the assignment to a C-variety X of its Hodge polynomial X → H X (u, v) defines a unique ring morphism

K 0 (V ar C ) → Z[u, v]
(see [START_REF] Srinivas | The Hodge characteristic[END_REF]Theorem 1.1] or [5, §6.1]). Recall that, if X is a smooth proper C-variety, its Hodge polynomial is defined as

HD X (u, v) := p,q≥0 h p,q (X)u p v q .
We call Poincaré polynomial of a C-variety X the polynomial P X ∈ Z[u], defined by P X (u) := HD X (u, u). This assignment defines a unique ring morphism

K 0 (V ar C ) → Z[u]. Recall that P X (u) = 2 dim X n=0 dim C H n dR (X)u n = 2 dim X n=0
b n (X)u n if X is a smooth proper C-variety, by Hodge Decomposition Theorem.

One defines the Euler characteristic with compact support to be the ring morphism

χ c : K 0 (V ar C ) → Z defined by χ c ([X]) = P X (-1), for any C-variety X. If X is a smooth proper C-variety, then χ c ([X]) := 2 dim X n=0 (-1) n b n (X).
This definition corresponds to the usual topological Euler characteristic associated to X an .

Example 9 (see [10, Proposition 2.3(a)]). Let X, Y be two connected k-curves. Then X, Y are piecewise isomorphic if and only if X red , Y red are birationally equivalent with χ c (X) = χ c (Y ).
These constructions have the following useful consequence. By the scissor relations in K 0 (Var k ) and by the assumption, we have

[X red ] = [X] = [Y ] = [Y red ].
Since the statement only depends on the reduced scheme structures of X and Y , we can assume that X, Y are reduced C-varieties. Let X 1 , . . . , X m (resp. Y 1 , . . . , Y n ) be the irreducible components of X (resp. Y ). By Nagata's Compactification Theorem and Hironaka's Theorem on resolution of singularities, we can find smooth irreducible proper k-varieties X1 , . . . , Xn (resp. Ȳ1 , . . . , Ȳn ) such that, for any i,

1 ≤ i ≤ m (resp. j, 1 ≤ j ≤ n), Xi is birationally equivalent to X i (resp. Ȳj is birationally equivalent to Y j ), and smooth irreducible proper k-varieties C 1 , . . . , C s , D 1 , . . . , D t that transform the equality of classes [X] = [Y ] in K 0 (Var k ) in the following one: m i=1 [ Xi ] + s α=1 ε α [C α ] = n j=1 [ Ȳj ] + t β=1 ε β [D β ],
with ε α , ε β ∈ {1, -1}. By computing the degrees and the leading coefficients of the Poincaré polynomials associated to the left and right-hand terms, we obtain the conclusion of the statement. Indeed, by Serre Duality, one knows that:

h dim(Z),dim(Z) (Z) = b 2dim(Z) (Z) = 1,
for any smooth irreducible proper k-variety Z.

We can also prove the following lemma.

Lemma 11. Let k be an algebraically closed field of characteristic zero. Let X, Y be piecewise isomorphic smooth irreducible proper k-varieties. Let f : X Y be a birational map. Let

Z 1 b1 }} } } } } } } c1 0 0 a a a a a a a a Z n bn Ð Ð Ð Ð Ð Ð Ð Ð Ð Ð cn 2
L n i=1 si j=1 [C ij ][P αj -2 k ] = L n i=1 ti j=1 [D ij ][P βj -2 k ]. Proof. If b 1 = id X , or c n = id Y , eliminate X = X 1 or Y = X n+1 from the diagram.
Let V be a smooth irreducible proper k-variety. Let W be the blow-up of V , with smooth irreducible center C of codimension α in V . Then, by the blow-up formula in K 0 (Var k ), we deduce that:

[W ] -[C][P α-1 k ] = [V ] -[C],
or equivalently, using the identity

P γ k = γ i=0 L i , [W ] -L[C][P α-2 k ] = [V ]
(see, for example, [START_REF] Liu | The Grothendieck ring of varieties and piecewise isomorphisms[END_REF]Example 1]). Applying this formula to each blow-up appearing in the b i , and to each blow-up appearing in the c j , we deduce the result by adding all the obtained relations.

Corollary 12. Let X, Y be piecewise isomorphic smooth irreducible proper Cvarieties. Let f : X Y be a birational map. Let Proof. The proof follows from computing the leading coefficients of Poincaré polynomials in the formula of Lemma 11.

Z 1 b1 }} } } } } } } c1 0 0 a a a a a a a a Z n bn Ð Ð Ð Ð Ð Ð Ð Ð Ð Ð
Remark 13. a) With the notations of Lemma 11, assume that the dimension of X (and so that of Y by Lemma 10) is equal to 3. By looking at the proof of Lemma 11, we see that (1) can be refined in:

(2)

L n i=1 m i (L + 1) + ℓ [C ij ℓ ] = L n i=1 n i (L + 1) + ℓ ′ [D ij ℓ ′ ]
where m i (resp. n i ) is the number of blow-ups appearing in the expression of b i (resp. c i ), with centers a rational curve (i.e., P 1 k ) or a point, and where the C ij ℓ , D ij ℓ ′ are smooth irreducible projective k-curves, whose genus g is at least 1. If we compute Poincaré polynomials of the left and right-hand terms, then we deduce, by comparing the coefficients of the monomial of degree 3, that:

n i=1 ℓ g(C ij ℓ ) = n i=1 ℓ ′ g(D ij ℓ ′ ),
where g( * ) is the genus of the k-curve * . In Proposition 21, we will see how to improve this result by introducing intermediate Jacobians. b) One can prove Corollary 12 directly and independently from Lemma 11 by computing Picard numbers, which increase by one under blow-up of an irreducible center. c) Corollary 12 can be formulated when k is any algebraically closed field of characteristic zero.

3.3. Larsen-Lunts' motivic measure and some consequences. In this paragraph we assume that k is an algebraically closed field of characteristic zero. In [START_REF] Larsen | Motivic measures and stable birational geometry[END_REF], M. Larsen and V. Lunts have constructed a motivic measure that is a very important piece in the understanding of the Grothendieck ring of varieties and its structure. Precisely we have:

Theorem 14 (Larsen-Lunts). Let Z[sb] be the ring generated by the classes of smooth irreducible proper k-varieties for the stably birational equivalence relation. There exists a unique surjective ring morphism

SB : K 0 (Var k ) → Z[sb],
that assigns to the class in K 0 (Var k ) of a smooth irreducible proper k-variety X its stably birational class in Z[sb]. Besides, its kernel is the ideal of K 0 (Var k ) generated by L.

It is important to note that the description of the image of the morphism SB in terms of stably birational classes of k-varieties only holds for classes of smooth irreducible proper k-varieties, which form a set of generators of the ring K 0 (Var k ). The image of a k-variety which is not irreducible proper and smooth is defined as a linear combination of stably birational equivalence classes of smooth irreducible proper k-varieties. In particular, the image by SB of such a variety can be "noneffective". To illustrate this point, consider the following example: It is easy to show that, if Y is a rational k-variety of dimension 1, then [13, Proposition 7] implies the following formula

SB([Y ]) = χ c ([Y ]) -c Y ,
where c Y is the number of irreducible components of dimension 1 of Y . See also [START_REF] Ivorra | Géométrie algébrique par morceaux, K-équivalence et motifs[END_REF]Remark 1.6]. (Note that the canonical homomorphism Z → K 0 (Var k ) is injective.)

The realization morphism has the following useful consequence.

Proposition 15 ([13, Proposition 6]). Let X, Y be k-varieties such that [X] = [Y ] in K 0 (Var k ). Assume that the dimension of X is at most 1. Then X, Y are piecewise isomorphic.
Proof. The proof below follows the arguments of the proof of [START_REF] Liu | The Grothendieck ring of varieties and piecewise isomorphisms[END_REF]Proposition 6]. Since the statement only depends on the reduced scheme structures of X and Y , we can suppose that X and Y are reduced. Assume first that the dimension of X is one. By Lemma 10 the dimension of Y is equal to 1 too. Let X1 , . . . , Xn (resp. Ȳ1 , . . . , Ȳn ) be smooth irreducible projective models of the irreducible components X 1 , . . . , X n (resp. Y 1 , . . . , Y n ) of X (resp. of Y ). So, the relation [X] = [Y ] can be transformed, in K 0 (Var k ), in the following one:

n i=1 [ Xi ] = n i=1 [ Ȳi ] + m,
with m ∈ Z. By applying the morphism SB, we conclude that m = 0, and, up to renumbering, that SB( Xi ) = SB( Ȳi ). Thus Xi , Ȳi are stably birational, for any i,

1 ≤ i ≤ n. Let us fix i, 1 ≤ i ≤ n. From a birational map Xi × k P s k Ȳi × k P s
k , we deduce, if Xi is rational (i.e., k-isomorphic to P 1 k , because of the assumption on k), the existence of a dominant morphism of k-schemes P 1 k → Ȳi . Thus, by Lüroth Problem, we conclude that Ȳi is k-isomorphic to P 1 k , and that X i , Y i are birationally equivalent. If X i is not rational, the symmetry of the argument proves that Y i is not rational. By [START_REF] Liu | The Grothendieck ring of varieties and piecewise isomorphisms[END_REF]Theorem 2], there exists an isomorphism of k-schemes Xi → Ȳi . So, X i , Y i are again birationally equivalent. These remarks imply the existence of dense open subvarieties U of X and V of Y , and of a k-isomorphism

f 1 : U → V . The complements of U in X and of V in Y are disjoint sums of points. A (non canonical) k-isomorphism f 0 : (X \ U ) red → (Y \ V ) red
exists if and only if the cardinals of the underlying sets of (X \ U ) red and (Y \ V ) red are equal. But this is the case thanks to the equality Card((

X \ U ) red ) = χ c ([(X \ U ) red ]) = χ c ([(Y \ V ) red ]) = Card((X \ U ) red ).
If the dimension of X is zero, then X, Y are disjoint sums of points. We conclude in that case directly by computing Euler characteristics.

Remark 16. Assume that L is not a zero divisor in K 0 (Var k ). Then it follows from the formula of Lemma 11 that

n i=1 si j=1 [C ij ][P αj -2 k ] = n i=1 ti j=1 [D ij ][P βj-2 k ].
If all the C ij , D ij are smooth irreducible projective k-curves or points, it follows from Proposition 15 that there exists a bijection σ from the set of the C ij to the set of the D ij such that, for any C ij , the curves C ij , σ(C ij ) are k-isomorphic. See §7 for a discussion around cancellation problems.

4. Jacobians 4.1. Principally polarized abelian varieties. Recall that a polarization on a complex torus V /Λ is a positive definite hermitian form H on V such that the imaginary part ImH takes integral values on Λ × Λ. Equivalently a polarization is given by an ample line bundle L on X; the first class Chern c 1 (L) then corresponds to ImH. A polarization is called principal if the Hermitian form is unimodular, or equivalently if H 0 (X, L) = C. The pair (X, H), or equivalently (X, L), is called a principally polarized abelian variety. In this situation a divisor associated with a section of L is called a theta divisor and is denoted Θ. The datum of Θ is another way to specify a polarization on X. Note that such a divisor Θ is always reduced.

A morphism f : (X 1 , H 1 ) → (X 2 , H 2 ) between principally polarized abelian varieties is defined as a morphism of C-varieties such that H 1 = f * H 2 . In particular such a morphism is always injective.

For any line bundle L on a principally polarized abelian variety (X, Θ) we can define a morphism X → P ic 0 (X), x → τ *

x L ⊗ L -1 , where τ x is the translation by x. The kernel of this morphism is a disjoint union of subtori of X; we denote by K(L) 0 the irreducible component of the kernel that contains the identity, and by π : X → X/K(L) 0 the quotient map. Then if H 0 (X, L) = 0, there exists an ample line bundle Θ on X/K(L) 0 such that π * Θ = L (see [START_REF] Debarre | of Cours Spécialisés[END_REF]Theorem VI.5

.1]).

There is a natural notion of product for principally polarized abelian varieties (see [START_REF] Debarre | of Cours Spécialisés[END_REF]VI.6

.1]): The product (X 1 , Θ 1 ) × (X 2 , Θ 2 ) is defined as (X 1 × X 2 , Θ) where Θ = p * 1 Θ 1 + p * 2 Θ 2
, and p 1 , p 2 are the first and second projections on X 1 × X 2 . We say that a principally polarized abelian variety (X, Θ) is indecomposable if it cannot be written as such a (nontrivial) product. Then any principally polarized abelian variety (X, Θ) admits an essentially unique factorization into indecomposable factors, which is directly related to the irreducible components of Θ (see [4, VI.9] or [3, Lemma 3.20]): Theorem 17. Let (X, Θ) be a principally polarized abelian variety, and let

D 1 , • • • , D s be the irreducible components of Θ. For each k = 1, . . . , s, denote X k = X/K(D k ) 0 , π k : X → X k the projection and Θ k a divisor on X k such that π * k Θ k = D k . Then (X, Θ) = (X 1 , Θ 1 ) × • • • × (X s , Θ s )
is a factorization into indecomposable principally polarized abelian varieties. Furthermore, if (X, Θ) = k=1,...,s ′ (Y k , Θ ′ k ) is another such factorization, then s = s ′ and up to a change of index we have (X k , Θ k ) isomorphic to (Y k , Θ ′ k ) for all k. 4.2. Intermediate Jacobians. In this paragraph, we quickly overview the theory of intermediate Jacobians, following [START_REF] Clemens | The intermediate Jacobian of the cubic threefold[END_REF] and [START_REF] Voisin | Hodge theory and complex algebraic geometry. I[END_REF].

Let X be a smooth irreducible projective C-variety. For n ∈ N such that 2n-1 ≤ dim X, consider the C-vector space

V n = H n-1,n (X) × H n-2,n+1 (X) × • • • × H 0,2n-1 (X).
Then the nth intermediate Jacobian of X is defined as the complex torus:

J n (X) = V n /H 2n-1 (X, Z),
where H 2n-1 (X, Z) is identified to its image via the group morphism ι : H 2n-1 (X, Z) → V n , which is obtained by composing the canonical morphism H 2n-1 (X, Z) → H 2n-1 (X, C) and the projection H 2n-1 (X, C) → V n . In general J n (X) has no reason to define an abelian variety.

First we specialize to the case where dim X = 3 and n = 2. Consider the hermitian form defined on V 2 by

H(α, β) = i X α ∧ β.
Recall that if ω ∈ H 1,1 (X) is the Kähler form induced by the ambient Fubini-Study metric, the Lefschetz operator L : H k (X) → H k+2 is defined by L(α) = ω ∧ α, and the primitive subspace H p,q prim (X) is defined as the kernel of the restriction of L 4-p-q to H p,q (X). One can show that V 2 admits a decomposition

V 2 = H 1,2
prim (X) × LH 0,1 (X) × H 0,3 (X), and that H is positive definite on H 1,2 prim (X) and negative definite on LH 0,1 (X) × H 0,3 (X). If we assume H 0,1 (X) = H 0,3 (X) = 0, we obtain a principal polarization on J(X) = J 2 (X) (see [START_REF] Voisin | Hodge theory and complex algebraic geometry. I[END_REF]Theorem 6.32] or [3, Lemma 3.4]). Remark that threefolds which are complete intersections gives examples of varieties satisfying H 0,1 (X) = H 0,3 (X) = 0.

If instead we specialize to the case where dim X = 1 and n = 1, we recover the usual Jacobian J(X) = J 1 (X) of a curve X. In this case a canonical principal polarization comes from the hermitian metric defined on H 0,1 (X) by

H(α, β) = -i X α ∧ β.
Recall the classical result of Torelli which says that we can recover a curve from its principally polarized Jacobian (see [6, p. 359]): Theorem 18. Let X, X ′ be two smooth irreducible projective C-curves. If their Jacobians (J(X), H) and (J(X ′ ), H ′ ) are isomorphic as principally polarized varieties, then X is isomorphic to X ′ .

Note also that for a curve X, a theta divisor on J(X) is irreducible, because it is the image (up to translation) of the (g -1)-symmetric product of X under a natural morphism X (g-1) → J(X) (see [6, p. 338]). In particular J(X) is indecomposable by Theorem 17.

Blow-ups.

For our purposes it is important to understand how the intermediate Jacobian of a threefold is affected under the blow-up of a smooth center. First, at the level of the cohomology, we have the following fact (this is a particular case of [START_REF] Voisin | Hodge theory and complex algebraic geometry. I[END_REF]Theorem 7.31]): Proposition 19. Let X be a smooth irreducible projective C-variety of dimension 3, and let Z → X be the blow-up of a smooth irreducible center C.

(1) If C is a point, then we have canonical isomorphisms of C-vector spaces

H 1,1 (Z) ∼ = H 1,1 (X) ⊕ C;
H p,q (Z) ∼ = H p,q (X) for (p, q) = (1, 1).

(2) If C is a curve, then we have canonical isomorphisms of C-vector spaces

H 1,1 (Z) ∼ = H 1,1 (X) ⊕ C; H 2,1 (Z) ∼ = H 2,1 (X) ⊕ H 1,0 (C); H 1,2 (Z) ∼ = H 1,2 (X) ⊕ H 0,1 (C);
H p,q (Z) ∼ = H p,q (X) for other (p, q).

Note in particular that the condition H 0,3 (X) = H 0,1 (X) = 0 is preserved under such blow-ups. Now at the level of the Jacobians we have (see [START_REF] Clemens | The intermediate Jacobian of the cubic threefold[END_REF]Lemma 3.11]): Proposition 20. Let X be a smooth irreducible projective C-variety of dimension 3, with H 0,3 (X) = H 0,1 (X) = 0, and let Z → X be the blow-up of a smooth irreducible center C.

(1) If C is a point or a rational curve, then J(Z) ∼ = J(X).

(2) If C is a non rational curve, then J(Z) ∼ = J(X) × J(C). These are isomorphisms of principally polarized abelian varieties, where J(X), J(Z) and J(C) are equipped with their canonical polarizations defined above.

4.4.

Consequences. For our main result, Theorem 24, we are mainly interested in varieties of dimension 3 over C. In this context, we have the following result.

Proposition 21. Let X be a smooth irreducible projective C-variety of dimension 3 whose Hodge numbers h 3,0 (X), h 1,0 (X) both vanish. Let f : X X be a birational map. Let

Z 1 b1 ~~~~~~~~c 1 2 2 f f f f f f f f Z n bn } } | | | | | | | | cn 2 2
e e e e e e e e X X 2 . . . X n X be a weak factorization of f . For each 1 ≤ i ≤ n, let C bi (resp. C ci ) be the (finite) set of all the blow-ups π C , with center a smooth irreducible non rational C-curve C, appearing in the expression of the birational morphisms b i (resp. c i ). Then there exists an isomorphism of principally polarized abelian varieties over C

J(X) ×   πC ∈∪ n i=1 C b i J(C)   ≃ J(X) ×   π C ′ ∈∪ n i=1 Cc i J(C ′ )   .
In particular, we have the following properties: Proof. Let us fix i, 1 ≤ i ≤ n, and consider the diagram of blow-ups with irreducible smooth centers

Z i bi }} } } } } } } ci 4 4 h h h h h h h h X i X i+1
coming from the weak factorization of f . Recall that the morphisms b i and c i are finite sequences of blow-ups along irreducible smooth centers. By Proposition 20, we know that only blow-ups along non rational curves affect the intermediate Jacobians, and precisely we obtain a canonical isomorphism of principally polarized abelian varieties over C, for each i,

1 ≤ i ≤ n: J(X i ) ×   πC ∈C b i J(C)   ∼ = J(Z i ) ∼ = J(X i+1 ) ×   π C ′ ∈Cc i J(C ′ )   .
By taking the product of respectively the left and right-hand sides of these n isomorphisms, and canceling the factors J(X 2 ), . . . , J(X n ) which appear once on both side (this makes use of Theorem 17), we obtain the expected isomorphism of principally polarized abelian varieties over C

(3)

J(X) × πC ∈C b J(C) ≃ J(X) ×   π C ′ ∈Cc J(C ′ )   .
Then assertion a) follows from (3) and Theorem 17; and assertion b) follows from a) and Torelli's Theorem 18.

Corollary 22. Let X be a smooth irreducible projective C-variety of dimension 3 whose Hodge numbers h 3,0 (X), h 1,0 (X) both vanish. Let f : X X be a birational map. Let Corollary 23. Let X be a smooth irreducible projective C-variety of dimension 3 whose Hodge numbers h 3,0 (X), h 1,0 (X) both vanish. Let f : X X be a birational map. Let Proof. The result follows from Proposition 21, from the unicity of the factorization into indecomposable factors (Theorem 17) and from Torelli's Theorem 18.

Z 1 b1 ~~~~~~~~c 1 2 2 f f f f f f f f Z n bn } } | | | | | | | |
Z 1 b1 ~~~~~~~~c 1 2 2 f f f f f f f f Z n bn } } | | | | | | | | cn 2
5. The main theorem 5.1. Preliminary results and remarks. We assume that k is algebraically closed of characteristic zero. Let X be a k-variety, of dimension d ≥ 0, and let f : X X be a map that induces an isomorphism U ∼ = V on dense open subvarieties U, V of X.

Components of maximal dimension.

By Lemma 10, we see that the k-varieties (X \ U ) red and (X \ V ) red have automatically the same dimension, and the same number of irreducible components of maximal dimension.

If U is not the complement of the exceptional set of f , the variety (X \U ) red could contain non-uniruled irreducible components of maximal dimension (see Lemma 4). But then, the set of non-uniruled irreducible components of maximal dimension of (X \U ) red is automatically in bijection with that of of (Y \U ) red , by [START_REF] Liu | The Grothendieck ring of varieties and piecewise isomorphisms[END_REF]Theorem 2]. In particular, if U 0 is the complement of the exceptional set of f , and V 0 := f (U 0 ), then the set of all the irreducible components of maximal dimension in U 0 \ U is in bijection with that of V 0 \ V .

State of the art.

We gather here what was known about Conjecture 1 before this work. a) Let X be a k-variety of dimension d ≥ 0. Assume that X contains only finitely many rational curves. It follows from [START_REF] Liu | The Grothendieck ring of varieties and piecewise isomorphisms[END_REF]Theorem 5] that Conjecture 1 holds true. (See also [9, §13].) b) Assume that the dimension of (X \ U ) red is at most 1, then Conjecture 1 holds true for X in that case by Proposition 15 (see also [START_REF] Liu | The Grothendieck ring of varieties and piecewise isomorphisms[END_REF]Proposition 6]). This remark implies in particular that Conjecture 1 holds true for any kvariety X of dimension 2. c) Assume that X is projective, of dimension at least 3, that the projective kvarieties (X \ U ) red , (X \ V ) red are smooth surfaces over k. Then Conjecture 1 holds true for X in that case by [START_REF] Liu | The Grothendieck ring of varieties and piecewise isomorphisms[END_REF]Theorem 4]. d) Assume that the k-varieties (X \ U ) red , (X \ V ) red have dimension 2, and that their irreducible components of maximal dimension are rational, then Conjecture 1 holds true for X in that case by [START_REF] Sebag | Variations on a question of Larsen and Lunts[END_REF]. One can prove directly this statement by Lemma 10 and Proposition 15. e) Let X be a smooth irreducible proper k-variety with a birational map f : X X which admits a weak factorization obtained by blowing-up smooth irreducible projective rational k-surfaces, smooth irreducible projective rational k-curves or points. Then, if the dimension of (X\U ) red is at most 2, the k-varieties (X\U ) red and (X \ V ) red are piecewise isomorphic. Indeed, the assumptions implies that each irreducible component of dimension 2 of (X \U ) red or (X \V ) red is rational. We conclude by Lemma 10 and Proposition 15.

Proof of Main Theorem.

In this paragraph, we prove the following theorem which is the main result of this article.

Theorem 24. Let X be a smooth irreducible projective C-variety of dimension 3, whose Hodge numbers h 1,0 (X), h 3,0 (X) both vanish. Let f : X X be a birational map that induces an isomorphism U ∼ = V on dense open subvarieties U, V of X. Then the C-varieties (X \ U ) red , (X \ V ) red of X are piecewise isomorphic.

Proof. First we observe that it is sufficient to prove that the irreducible components of dimension 2 of (X \ U ) red and (X \ V ) red are mutually birationally equivalent. Indeed, if there exist an open dense subscheme U 1 of (X \ U ) red , an open dense subscheme V 1 of (X \ V ) red , and an isomorphism

f 2 : U 1 → V 1 of C-varieties, then Proposition 15 produces a piecewise isomorphism (f 0 , f 1 ) from ((X \ U ) \ U 1 ) red to ((X \ V ) \ V 1 ) red . The datum of (f 0 , f 1 , f 2 )
is a piecewise isomorphism between (X \ U ) red and (X \ V ) red . From now, we will only prove that there exists a bijection σ from the set of all the irreducible components of dimension 2 of (X \ U ) red to those of (X \ V ) red such that, for any such a component F of (X \ U ) red , the C-varieties F and σ(F ) are birationally equivalent. Now the proof proceeds from a precise analysis of the weak factorization of f , via Proposition 21. First, by §5.1.1 we can assume without loss in generality that U is the largest open subset of X on which the restriction of f is an isomorphism. In particular all components of dimension 2 in (X \ U ) red and (X \ V ) red are ruled.

Let

Z 1 b1 ~~~~~~~~c 1 2 2 f f f f f f f f Z n bn } } | | | | | | | | cn 2 2
e e e e e e e e X X 2 . . . X n X be a weak factorization of f . Let η be the generic point of an irreducible component of dimension 2 of (X \ U ) red . Thanks to our hypothesis, we know that there exists a smooth irreducible projective curve C η such that the residue field κ(η) of X at η is isomorphic to κ(C η )(T ), the quotient field of the ring of polynomials in T with coefficients in the functions field of C η . We also know that such a point must be contracted by a morphism c i , because of our assumption on U and the dimension of η. If we replace c i by b i , the generic points η ′ of the irreducible components of dimension 2 of (X \ V ) red verify the same properties.

So, it is sufficient to prove that for any smooth irreducible projective curve C of genus g(C) ≥ 1, the number of generic points η in (X \ U ) red with residue field isomorphic to κ(C)(T ) is equal to the number of generic points η ′ in (X \V ) red with κ(η ′ ) ≃ κ(C)(T ). Indeed, by Lemma 10, we conclude that the number of rational irreducible components in (X \ U ) red will coincide with that in (X \ V ) red .

Let C be a smooth irreducible projective curve C of genus g(C) ≥ 1. By Corollary 23, we know that in the weak factorization of f , the number of blow-ups with center C appearing in the b i is equal to those appearing in the c i .

We define now a numbering on the set of the blow-ups, with center C, that appear in the b i as follows. In b 1 , consider the first blow-up of this kind. If the strict transform of its exceptional divisor is the exceptional divisor of a blow-up appearing in a c j , we denote it by b(1), and by c(1) the corresponding blow-up in c j . If not, consider the second blow-up in b 1 and so on. When we have tested all the blow-ups of this kind in b 1 , we apply the same procedure to b 2 and so on. Then, if it exists, we have found the first numbered blow-up b(1), with center C. It determines ipso facto c(1) that is also a blow-up with center C. At this step, we search for the second numbered blow-up b(2) such that the strict transform of its exceptional divisor is k-isomorphic to the exceptional divisor of a blow-up appearing in a c j ′ , distinct from the blow-up c(1) by construction, and so on.

At the end, we have numbered all the blow-ups with center C appearing in the b i , by b(ℓ), 1 ≤ ℓ ≤ s, except a finite number of them which are those whose strict transform are the irreducible components of (X \ V ) red birationally equivalent to

C × C P 1
C . Then we can split the formula in Proposition 21 as follows (4) J(X)

×   C =C J( C)   ×   1≤ℓ≤s J(C)   × ℓ∈Γ J(C) ≃ J(X) ×   C′ =C J( C′ )   ×   1≤ℓ≤s J(C)   × ℓ ′ ∈Γ ′ J(C)
where Γ is a set in bijection with the (finite) set of irreducible component in (X \ V ) red , birationally equivalent to C × C P 1 C , and where, by symmetry, Γ ′ is a (finite) set in bijection with the set of irreducible component in (X \ U ) red , birationally equivalent to C × C P 1 C . By unicity of the factorization of a principally polarized abelian variety (Theorem 17), we deduce from (4) that Γ is in bijection with Γ ′ . In other words the numbers of irreducible components birationally equivalent to C × C P 1 C in (X \ U ) red and (X \ V ) red are equal.

Remark 25. The arguments of the proof of Theorem 24 also imply the following statement. Let X, Y be smooth projective C-varieties of dimension 3 (possibly reducible), with h 3,0 (X) = h 1,0 (X) = h 3,0 (Y ) = h 1,0 (Y ) = 0, and such that there exists an isomorphism J(X) ∼ = J(Y ) of principally polarized abelian varieties over C. If there exists a birational map f : X Y that induces an isomorphism of C-schemes U → V between dense open subvarieties U of X and V of Y , then the C-varieties (X \ U ) red and (Y \ V ) red are piecewise isomorphic.

Corollary 26. Let X be a smooth irreducible projective C-variety, of dimension 3, whose Hodge numbers h 3,0 (X), h 1,0 (X) both vanish. Let f : X X be a birational map that induces an isomorphism of C-schemes U ∼ = V on dense open subvarieties U, V of X. Let A be a ring, and µ : K 0 (Var k ) → A be a motivic measure. Then

µ([(X \ U ) red ]) = µ([(X \ V ) red ]).
In particular, the projective C-varieties (X \U ) red , (X \V ) red have the same Hodge-Deligne polynomial, the same Poincaré polynomial, the same (topological) Euler characteristic, and SB([(X \ U ) red ]) = SB([(X \ V ) red ]).

Proof. By Theorem 24, [(X\U ) red ] = [(X\V ) red ] in K 0 (V ar C ). So, by the definition of a motivic measure, µ([(X \ U ) red ]) = µ([(X \ V ) red ]). Recall that the Hodge-Deligne realization morphism, the Poincaré realization morphism, the (topological) Euler characteristic, the morphism SB are examples of motivic measure (see §3.2).

Examples

One particularly rich source of examples for Theorem 24 is when we consider X equal to the projective space. We give below three explicit examples in this setting. The first one is in dimension 2, and illustrates the fact that the exceptional loci of a transformation and its inverse might be quite far from being isomorphic, despite the fact that they are piecewise isomorphic.

The two last examples are in dimension 3, and as such are direct illustrations of Theorem 24.

Example 27 (see [START_REF] Alberich-Carramiñana | Geometry of the plane Cremona maps[END_REF]Example 2.1.14]). Consider the plane Cremona map f :

P 2 C P 2
C defined by the datum of the three following homogeneous polynomials, expressed in homogeneous coordinates x, y, z:

F = (x 3 -yz(x + y))(x 2 -yz)(x + y), G = x 2 (x 2 -yz)(x + y) 2 , H = x 3 (x 3 -yz(x + y)).
The determinant of the Jacobian matrix

  ∂ x F ∂ y F ∂ z F ∂ x G ∂ y G ∂ z G ∂ x H ∂ y H ∂ z H   is equal to Jac(f ) = -6x 3 y 2 z 2 (x + y) 3 (x + z) 3 (y -z) 2 .
We deduce the (reduced) equation of the contracted curves:

xyz(x + y)(x + z)(y -z) = 0.
We see that the exceptional locus is equal to the disjoint union of with Jacobian determinant Jac(f -1 ) = 6x 4 y(x + y) 2 (x 2 -yz)(x 3 -y 2 z -xyz)(x 2 y + 2x 3 -y 2 z -xyz).

•
So the exceptional locus of f -1 has six irreducible components described as follows with their reduced structures: three lines, a conic and two nodal cubics. We can write this locus as the disjoint union of • Three points [0 : 0 : 1], [0 : 1 : 0], and [-1 : 1 : 1]; • One curve isomorphic to P 1 C minus one point (line y = 0), two curves isomorphic to P 1 C minus two points (corresponding to the two other lines), and three other curves isomorphic to P 1 C minus three points (corresponding to the conic and the nodal cubics).

We see that we can realize a piecewise isomorphism between the exceptional locus of f and f -1 by choosing an arbitrary additional point on each one of the lines x = 0 and x + y = 0, and two arbitrary additional points on the line y = 0.

Example 28. We give now an example in dimension 3. Consider a general transformation of P 3 C of bidegree [START_REF] Alberich-Carramiñana | Geometry of the plane Cremona maps[END_REF][START_REF] Debarre | of Cours Spécialisés[END_REF], meaning that f has degree 2 and f -1 has degree 4: see [START_REF] Pan | Transformations birationnelles quadratiques de l'espace projectif complexe à trois dimensions[END_REF], where this transformation is denoted by gen [4] . One representant (up to left and right composition by linear automorphisms) is for instance We compute the Jacobian determinants of f and f -1 :

Jac(f ) = xyz(x + y + z); Jac(f -1 ) = x 2 y 2 z 2 (yz + xz + xy) 3 .
The exceptional set of f is equal to the disjoint union of:

• The point p = [0 : 0 : 0 : 1];

• Six curves isomorphic to P 1 C \ {p}; • Four surfaces isomorphic to P 2 C minus three lines with a common point. On the other hand the exceptional set of f -1 is equal to the disjoint union of:

• The point p = [0 : 0 : 0 : 1];

• Three curves isomorphic to P 1 C \ {p}; • Three surfaces isomorphic to P 2 C minus two lines; • A surface isomorphic to a quadratic cone minus three rules, which is also isomorphic to P 2 C minus three lines with a common point. By choosing a general line passing through p in each plane x = 0, y = 0, z = 0, we can recover a stratification of Exc(f -1 ) which realizes a piecewise isomorphism with Exc(f ).

Example 29. The following example is probably the simplest birational self-map g of P 3

C such that (a) Exc(g) and Exc(g -1 ) are not isomorphic; (b) Exc(g) contains an irreducible component which is not a rational surface.

Consider the smooth plane cubic C ⊂ P 2 C with homogeneous equation q 3 (x, y, z) := y 2 z -x 3 + xz 2 = 0. Then f : (x, y, z) xq 3 x 3 , yq 3 x 3 , zq 3

x 3 is a birational transformation of P 3 C (expressed in an affine chart C 3 ), with inverse

f -1 : (x, y, z) x 4 q 3 , yx 3 q 3 , zx 3 q 3 .
Note that we could construct a similar example starting from any irreducible plane curve C of arbitrary degree (see [START_REF] Pan | Une remarque sur la génération du groupe de Cremona[END_REF]). The associated transformation contracts a surface birational to C × P 1 C : We see that transformations contracting only rational surfaces are far from being the rule.

The transformations f and f -1 have same exceptional loci, so we now compose f with the cubo-cubic involution

σ : (x, y, z) 1 x , 1 y , 1 z
to get a more interesting example. We set g = σ •f , so g -1 = f -1 •σ, and explicitly

g : (x, y, z) x 3 xq 3 , x 3 yq 3 , x 3 zq 3 , g -1 : (x, y, z) y 2 z 2 xq 4 , xyz 2 xq 4 , xy 2 z xq 4 ,
where q 4 (x, y, z)

:= x 3 z -y 2 z 2 + x 2 y 2 satisfies q 3 1 x , 1 y , 1 z = q 4 (x, y, z) x 3 y 2 z 2 .
Exc(g) Exc(g We compute the Jacobian determinants: Jac(g) = -5x 7 y 2 z 2 t 2 q 3 ; Jac(g -1 ) = -5x 2 y 4 z 4 t 2 q 4 .

). •1 component ∼ = P 1 C \ 4 points : line (p 1 , p 3 ) minus {p 1 , p 3 , p 4 , p 5 } •1 component ∼ = C \ 4 points : •1 component ∼ = C \ 4 points : q 4 = 0 minus {p 1 , p 2 , p 3 } q 3 = 0 minus {p 2 , p 3 , p 4 , p 5 } Surfaces •2 components ∼ = A 2 C minus 2 lines : •3 components ∼ = A 2 C minus 2 lines : come from x = 0, z = 0. come from x = 0, y = 0, z = 0. •1 component ∼ = A 2 C minus 4 lines : comes from y = 0. •1 component ∼ = (C * ) 2 \ {q 3 = 0} : •1 component ∼ = (C * ) 2 \ {q 4 = 0} : comes from t = 0. comes from t = 0. •Cone over C \ {4 points} •Cone over C ′ \ {3 points}
The exceptional locus of g is the union of the 4 coordinates planes and of the cone over the smooth elliptic curve {t = q 3 (x, y, z) = 0}, isomorphic to C. Similarly, the exceptional locus of g is the union of the 4 coordinates planes and of the cone over the singular quartic curve C ′ = {t = q 4 (x, y, z) = 0}.

We set and we note that in the plane t = 0, the singular locus of the curve xyzq 3 = 0 is equal to the five points {p 1 , . . . , p 5 }, and the singular locus of xyzq 4 = 0 is equal to the three points {p 1 , p 2 , p 3 }.

p
Observe also that there exists a birational morphism π : C → C ′ , since C is isomorphic to the desingularisation of C ′ , and π induces an isomorphism between C ′ \ {p 1 , p 2 , p 3 } and C minus four points. Indeed p 2 is an ordinary double point for C ′ , hence π -1 (p 2 ) is equal to two points, and p 1 , p 3 are respectively a smooth and a cuspidal double point for C ′ hence their preimage by π both consist of one single point.

We have natural stratifications of the exceptional loci, as detailled in Table 1.

Note that by construction the quadratic involution [x : y : z] [yz : xz : xy] realizes an isomorphism between (C * ) 2 \ {q 3 = 0} and (C * ) 2 \ {q 4 = 0}. We can realize a piecewise isomorphism by picking two well-chosen new rules in {y = 0} ⊂ Exc(g -1 ), and at most four well-chosen new rules in both cones over the elliptic curve in Exc(g) and Exc(g -1 ).

Further comments

In this section, we assume for simplicity that k is an algebraically closed field of characteristic zero. As we have shown in the introduction, Conjecture 1 is linked to a stronger problem: Question 2 (see [START_REF] Larsen | Motivic measures and stable birational geometry[END_REF]Question 1.2]).

The strategy of resolution of Question 2 in [START_REF] Liu | The Grothendieck ring of varieties and piecewise isomorphisms[END_REF] is stopped by different problems of cancellation (geometric and motivic). Besides, some preexisting geometric problems of cancellation can be formulated, or interpreted via Question 2, in the Grothendieck ring of varieties.

We list here some problems related to these cancellation questions.

7.1. Zero divisor. Is L a zero divisor in K 0 (Var k ) ? That question corresponds to a "piecewise" cancellation problem. Let a ∈ K 0 (Var k ) such that L • a = 0. One can represent the element a as a formal some of classes of the following type

a = m i=1 [X i ] - n i=1 [Y i ],
where all the X i , Y i are smooth irreducible projective k-varieties. Now, if L is not a zero divisor in K 0 (Var k ) then a = 0. Thus we are studying the validity of the following formula

L • ([⊔ m i=1 X i ]) = L • ([⊔ n i=1 Y i ]) ⇒ [⊔ m i=1 X i ] = [⊔ n i=1 Y i ].
Note that, if L is a regular element of K 0 (Var k ), we deduce formally that Question 2 admits a positive answer for any pair of k-varieties of dimension 2 (see [START_REF] Sebag | Variations on a question of Larsen and Lunts[END_REF]). In that case, Conjecture 1 would be true for any k-varieties of dimension 3. 7.2. Rationality. Is it possible to characterize rationality in K 0 (Var k )? That question admits a positive answer in dimension at most 2, by [13, Proposition 7, Lemma 12]. In particular, is it possible that a stably rational (or unirational, i.e., dominated by P dim(X) k

), non-rational, smooth proper k-variety X, of dimension at least 3, has the same class in K 0 (Var k ) than a rational smooth proper k-variety (of the same dimension)? 7.3. Cancellation. In affine geometry, Zariski's Cancellation Problem can be formulated as follows. If X, Y is a pair of affine k-varieties such that X × k A 1 k , Y × k A 1 k are k-isomorphic, is it true that X, Y are k-isomorphic? One knows that some further assumptions are necessary to hope for a positive answer: indeed there exist counterexamples (e.g., Danielewski's counterexample) formed by a couple X, Y of non-isomorphic affine k-varieties such that

X × k A 1 k , Y × k A 1 k are k-isomorphic.
We can formulate the question in K 0 (Var k ). We are interested in couples of affine k-varieties X, Y such that L[X] = L[Y ]. If L is a regular element of K 0 (Var k ), we conclude that any such a couple verifies [X] = [Y ]. If Question 2 admits a positive answer, then X, Y must be piecewise isomorphic. Up to our knowledge, and up to now, all the counterexamples to Zariski's Cancellation Problem (in affine geometry) do not contradict this conclusion. Moreover, it follows from [13, Theorem 2, Proposition 6], that any counterexample which does not verify that piecewise condition must be of dimension at most 3.

So it seems to us interesting to study the following question (see [START_REF] Sebag | Variations on a question of Larsen and Lunts[END_REF]). Does there exist a couple of integral affine k-varieties X, Y , of dimension d ≥ 3, such that X × k A 1 k , Y × k A 1 k are k-isomorphic, and X, Y not birationally equivalent?

Lemma 10 .

 10 Let k be an arbitrary field. Let us fix an algebraic closure k of k. Let X, Y be k-varieties such that [X] = [Y ] in K 0 (Var k ). Then the dimension of X and Y are equal. Besides, the number of irreducible components of maximal dimension of Xk is equal to that of Yk. Proof. See, for example, [13, Proposition 4, Corollary 5]. Below, we give a proof when k equals C, that is the framework of Theorem 24.

  factorization of f . Then the number of all the (non-empty) blow-ups appearing in the expression of the b i , 1 ≤ i ≤ n, is equal to the number of all (non-empty) the blow-ups appearing in the expression of the c i , 1 ≤ i ≤ n.

  a) The sets C b := ∪ n i=1 C bi and C c := ∪ n i=1 C ci have same cardinality; b) There exists a bijection σ : C b → C c such that, for any π C , π C ∈ C b , the center C of π C is isomorphic over k to the center of σ(π C ).

  factorization of f . Let Cb (resp. Cc ) be the (finite) set of all of the blowups with center a point or a rational curve, appearing in the expression of any of the birational morphism b i (resp. c i ), 1 ≤ i ≤ n. Then there exists a bijection Cb → Cc . Proof. This follows from Proposition 21, which says that C b and C c have the same cardinality, and from Corollary 12, which says that C b ∪ Cb and C c ∪ Cc have the same cardinality.

  Seven points [0 : 0 : 1], [0 :1 : 0], [1 : 0 : 0], [-1 : 1 : 0], [-1 : 1 : 1], [0 : 1 : 1], [-1 : 0 : 1]; • Six curves isomorphic to P 1C minus three points. The inverse of f has the formf -1 : [x : y : z] [-xyz(x + y)(x + z)(y -z) : -x 2 y(x + z)(y -z) 2 : z 2 (y + x) 2 (x 2 -yz)]

f

  : [x : y : z : t] [yz : xz : xy : t(x + y + z)] with inverse f -1 : [x : y : z : t] [(yz + xz + xy)yz : (yz + xz + xy)xz : (yz + xz + xy)xy : xyzt].

  Two reduced k-varieties X, Y are birationally equivalent if there exist a dense open subscheme of X and a dense open subscheme of Y that are isomorphic as k-schemes. We say that two reduced k-varieties X, Y are stably birational if there exist integers m, n ∈ N such that X × k P m

k , Y × k P n k are birationally equivalent. If k is algebraically closed, we say that an integral k-variety X is rational, if it is birationally equivalent to P dim(X) k . More generally, if k is algebraically closed, a reduced k-variety is said rational if its irreducible components are all rational. An integral k-variety X, of dimension d, is said uniruled if there exists a k-variety Y of dimension d -1, and a dominant rational map Y × k P 1 k X. If k is an algebraically closed field of characteristic zero, and if d = 2, a smooth projective k-variety X is uniruled if and only if it is ruled.

  the b i and the c i are composites of blow-ups, with smooth irreducible centers, disjoint from U (note that, in our notation and for convenience in the following computations, c n : Z n → Y or b 1 : Z 1 → X could be the blow-up with empty center, i.e., c n = id Y and b 1 = id X ).

	Theorem 6 ([1, Theorem 0.1.1]). Let X, Y be two smooth irreducible proper k-
	varieties. Let f : X	Y be a birational map. Then f admits a weak factorization.
	Example 7. In dimension 2, a stronger version of Theorem 6 goes back to Os-
	car Zariski. Let X, Y be two smooth irreducible projective k-surfaces, and let

  be a weak factorization of f . If 1 ≤ i ≤ n, let us denote by C ij , for 1 ≤ j ≤ s i , the (non-empty) smooth irreducible center (of codimension α j ) of the blow-ups defining b i , and by D ij , for 1 ≤ j ≤ t i , the (non-empty) smooth irreducible centers (of codimension β

	X	. . .	2 e e e e e e e e Y

j ) of the blow-ups defining c i .Then

[START_REF] Abramovich | Torification and factorization of birational maps[END_REF]

  : lines (p 0 , p i ), i = 1, . . . 5, Lines (p 0 , p i ), i = 1, . . . 3, and(p 1 , p 2 ), (p 2 , p 3 ). and(p 1 , p 2 ), (p 2 , p 3 ), (p 1 , p 3

			-1 )
	Points Curves	•6 points p 0 , . . . , p 5 •7 components ∼ = C	•4 points p 0 , . . . , p 3

* : •6 components ∼ = C *

Table 1 .

 1 Stratification of exceptional loci in Example 29.

In homogeneous coordinates we have

g : [x : y : z : t] [x 2 yzt : x 3 zt : x 3 yt : yzq 3 ],

g -1 : [x : y : z : t] [y 2 z 2 t : xyz 2 t : xy 2 zt : xq 4 ].

Mathematics Subject Classification. 14E07.[START_REF] Abramovich | Torification and factorization of birational maps[END_REF] The existence of a link between Question 1 and the structure of K 0 (Var k ) is already mentioned in [7, 3G ′′′ ]. 1


Danh-Tai Hoang
email: danh-tai.hoang@u-cergy.fr

H T Diep
email: diep@u-cergy.fr

Hexagonal-Close-Packed Lattice: Phase Transition and Spin Transport

Keywords: numbers: 05.70.Fh, 75.10.Hk, 75.40.Mg, 75.47.-m : I

We study the ground state (GS) and the phase transition in a hexagonal-close-packed lattice with both XY and Ising models by using extensive Monte Carlo simulation. We suppose the in-plane interaction J1 and inter-plane interaction J2, both antiferromagnetic. The system is frustrated with two kinds of GS configuration below and above a critical value of η = J1/J2 (ηc). For the Ising case, one has η c = 0.5 which separates in-plane ferromagnetic and antiferromagnetic states, while for the XY case η c = 1/3 separates the collinear and non collinear spin configurations. The phase transition is shown to be of first (second) order for η > (<)η c . The spin resistivity is calculated for the Ising case. It shows a rounded maximum at the magnetic transition in the second-order region, and a discontinuity in the first-order region of η.

INTRODUCTION

Frustrated spin systems have been subject of intensive investigations during the last 30 years [START_REF]Frustrated Spin Systems[END_REF]. These systems are very unstable due to the competition between antagonist interactions or due to a geometrical frustration such as in the antiferromagnetic (AF) triangular lattice. A spin is said frustrated when it cannot find an orientation which "fully" satisfies all the interactions with its neighbors [START_REF] Toulouse | [END_REF]3]. As a consequence of the frustration, the ground state (GS) is very highly degenerate. In the Ising case the GS degeneracy is often infinite as in the triangular lattice, the face-centered cubic (FCC) and hexagonalclose-packed (HCP) lattices, with AF interaction. In the case of vector spins, the GS is non collinear such as the 120-degree configuration in the XY and Heisenberg AF stacked triangular lattice (STL). In two dimensions (2D), several frustrated systems with Ising spin model have been exactly solved [START_REF] Liebmann | Statistical Mechanics of Periodic Frustrated Ising Systems[END_REF]5]. Among the most interesting models one can mention the frustrated generalized Kagome lattice [START_REF] Debauche | [END_REF] and the honeycomb lattice [7] where exotic features such as the existence of several phase transitions, the reentrance and the disorder lines have been exactly found by mapping these systems into vertex models [START_REF] Baxter | Exactly Solved Models in Statistical Physics[END_REF]. In three dimensions (3D), the situation is complicated. The renormalization group (RG) [START_REF] Wilson | [END_REF][START_REF] Zinn-Justin | Quantum Field Theory and Critical Phenomena[END_REF], which provided a good understanding of the nature of the phase transition in non frustrated systems, encounters much of difficulties in application to frustrated systems. Among the most studied subjects during the last 20 years, one can mention the nature of the phase transition in the XY and Heisenberg STL. After a long debate [START_REF]Renormalization Group Approaches to Frustrated Magnets in D=3 by B. Delamotte[END_REF] on whether it is a second-, or a first-order transition or it belongs to a new universality class, the controversy has recently ended with the conclusion of a first-order transition [START_REF] Ngo | [END_REF]13].

In this paper, we are interested in the HCP antiferromagnet with Ising and XY spin models. Our purpose is to study its properties such as the ground state, the phase transition and the spin transport, in the case of anisotropic exchange interactions. The isotropic nearestneighbor (NN) AF interaction has been studied for Ising [14] and XY and Heisenberg spins [15]. These isotropic cases have been shown to undergo a phase transition of first order, and the infinite GS degeneracy is reduced to 6 at low temperatures [14,15]. The effect of anisotropic interaction and anisotropy on the GS in the case of vector spins has been studied [16].

In section II, we show our model and analyze the ground state. Results of Monte Carlo (MC) simulations are shown and discussed in section III. Concluding remarks are given in section V.

II. MODEL AND GROUND-STATE ANALYSIS

We consider the HCP lattice shown in Fig. 1. The stacking direction is z. The Hamiltonian is given by

H = - (i,j) J i,j S i • S j (1)
where S i is the spin at lattice site i and J ij is the AF interaction between nearest-neighbors (NN). We suppose that J ij = J 1 if the NN are on the xy triangular plane, and J ij = J 2 if the NN are on two adjacent planes (see Fig. 1). The GS can be determined by the steepestdescent method. This method is very simple [17] (i) we generate an initial configuration at random (ii) we calculate the local field created at a site by its neighbors using (1) (iii) we align the spin of that site along the calculated local field to minimize its energy (iv) we go to another site and repeat until all sites are visited: we say we make one sweep (v) we do a large number of sweeps per site until a good convergence is reached. However, one can also minimize the interaction energy as shown below to calculate the GS configuration. Since both interactions are AF (negative), for simplicity, we fix J 2 = -1 and vary J 1 . The unit of energy is taken as

|J 2 | and the temperature T is in the unit of |J 2 |/k B where k B is the Boltzmann constant.
Let us recall the GS in the case of isotropic interaction, namely J 1 = J 2 [15]. For the HCP lattice, each spin is shared by eight tetrahedra (four in the upper half-space and four in the lower half-space along the z axis) and a NN bond is shared by two tetrahedra. The GS spin configuration of the system is formed by stacking neighboring tetrahedra. In the GS, one has two pairs of antiparallel spins on each tetrahedron. Their axes form an arbitrary angle α. The degeneracy is thus infinite (see Fig. 2a of Ref. [15]). Of course, the periodic boundary conditions will reduce a number of configurations, but the degeneracy is still infinite. Of these GS's, one particular family of configurations of interest for both XY and Heisenberg cases is when α = 0. The GS is then collinear with two spins up and the other two down. The stacking sequence is then simplest: there are three equivalent configurations since there are three ways to choose the parallel spin pair in the original tetrahedron.

We now examine the case where J 1 = J 2 .

• Ising case:

The steepest descent method with varying J 1 (J 2 = -1 gives two kinds of GS spin configuration: the first consists of xy ferromagnetic planes stacked antiferromagnetically along the z direction, while the second one is the stacking of xy AF planes such that each tetrahedron has two up and two down spins. The transition between the two configurations is determined as follows: one simply writes down the respective energies of a tetrahedron and compares them

E 1 = 3(-J 1 + J 2 ) (2) E 2 = J 1 + J 2 (3)
One sees that

E 1 < E 2 when J 1 > 0.5J 2 , i.e. |J 1 | < 0.5|J 2 |. Thus the first configuration is more stable when |J 1 | < 0.5|J 2 |.
• XY case:

Consider one tetrahedron of the HCP lattice. The Hamiltonian for this cell is given by

H c = -J 1 (S 1 • S 2 + S 2 • S 3 + S 3 • S 1 ) -J 2 (S 1 • S 4 + S 2 • S 4 + S 3 • S 4 ) (4)
Suppose that |S i | = 1, one has

H c = -J 1 [cos α + cos β + cos(α + β)] -J 2 [cos γ + cos(γ -α) + cos(γ -α -β)] (5)
where the angles are defined in Fig. 2. The steepest descent method shows that while β and α have unique values for a given J 1 /J 2 , α is arbitrary, just as in the case of isotropic interaction [15] discussed above. To simplify the formulae, we take α = 0 in the following. The energy of the cell is written as

H c = -J 1 [1 + 2cosβ] -J 2 [2 cos γ + cos(γ -β)] (6)
The critical values of β and γ are determined from the relations

∂H c ∂β = 2J 1 sin β -2J 2 sin(γ -β) = 0 (7) ∂H c ∂γ = 2J 2 sin γ + J 2 sin(γ -β) = 0 (8)
We find the following solutions:

(i) β = 0, γ = 0, (ii)β = 0, γ = π, (iii) β = π, γ = 0, (iv) β = π, γ = π, and (v) cos β = 1 4(J1/J2) 2 -5 4 , cos γ = -1+3(J 1 /J 2 ) 2 4(J1/J2)
. By comparing the energy values at these solutions, we obtain the minimum energy with the last solution: one has where cos β and cos γ are given above.

H c = - 1 + 3(J 1 /J 2 ) 2 2(J 1 /J 2 ) (9)
Because -1 ≤ cos β ≤ 1 and -1 ≤ cos γ ≤ 1, the above solution is valid for

1 3 ≤ J 1 /J 2 ≤ 1.
We plot cos β, cos γ, β and γ in Fig. 3 where we observe that the non collinear GS configuration occurs in the interval 1/3

≤ η = J 1 /J 2 ≤ 1.
.

III. PHASE TRANSITION: RESULTS

We consider a sample size of L×L×L z where L and L z vary from 12 to 36 but L z can be different from L in order to detect the dependence of the GS on J 1 . The exchange interaction |J 2 | = 1 is used as the unit of energy. We use here MC simulations with a histogram technique to detect first-order transition. We equilibrate the system and average physical quantities over several millions of MC steps per spin. The averaged energy U and the heat capacity C V are calculated by

U = H ( 10
)
C V = U 2 -U 2 k B T 2 (11)
where ... indicates the thermal average taken over microscopic states at T . The order parameter M is defined from the sublattice magnetization by

M = K | i∈K S i | ( 12
)
where S i belongs to the sublattice K. Note that there are at least four sublattices which define the ordering of the spins on the tetrahedra if a long-range order is observed. The susceptibility is defined by

χ = M 2 -M 2 k B T ( 13
)
In MC simulations, we work at finite sizes, so for each size we have to determine the "pseudo" transition which corresponds in general to the maximum of the specific heat or of the susceptibility. The maxima of these quantities need not to be at the same temperature. Only at the infinite size, they should coincide. The theory of finite-size scaling [18][19][20] permits to deduce properties of a system at its thermodynamic limit. In order to check the first-order nature of the transition, we used the histogram technique which is very efficient in detecting weak first-order transitions and in calculating critical exponents of second-order transitions. [19,20] The main idea of this technique is to make an energy histogram at a temperature T 0 as close as possible to the transition temperature. Often, one has to try at several temperatures in the transition region. Using this histogram in the formulae of statistical physics for canonical distribution, one obtains energy histograms in a range of temperature around T 0 . In second-order transitions, these histograms are gaussian. They allows us to calculate averages of physical quantities as well as critical exponents using the finite-size scaling. In first-order transitions, the energy histogram shows a double-peak structure.

A. Ising case

In the following, the results in different figures are shown with L = 18 and L z = 8 (16 atomic planes along z). Since the GS changes at η c = 0.5, we show here examples on both sides of this value. Figure 4 shows the energy per spin E, the specific heat per spin C V , the order parameter M and the susceptibility χ, for η = 0.3. The transition is of second order. On the other side, we show in Fig. 5 the energy per spin and the order parameter versus T , for η = 0.85 and 1. We find a strong first-order transition in both cases. The discontinuity of E and M at the transition is very large. We show in Fig. 6 the energy histogram taken at the transition temperature for three values η = 0.3 (black), 0.85 (blue) and 1 (red). As seen, the first case is a gaussian distribution indicating a second-order transition, while the last two cases show a double-peak structure indicating a first-order transition.

We have calculated the critical temperature T C as a function of η. The phase diagram is shown in Fig. 7 where I and II indicate the ordering of the first, and second kinds, respectively. P indicates the paramagnetic phase. Note that the transition line between I and P is a second-order line, while that between II and P is a first-order line.

B. XY case

In the XY case, the change of the GS takes place at η = 1/3. Let us show in Fig. 8 the result for η = 0.3 where the GS is composed of ferromagnetic planes antiferromagnetically stacked in the z direction. The transi- tion is of second order. We show now the result for the non collinear GS region in Fig. 9. The energy and the order parameter show clearly a discontinuity at the transition for η = 0.58 and 1. Using the histogram method described above, we have calculated the histogram shown in Fig. 10 for η = 0.3, 0.58 and 1. For η = 0.3 which is in the collinear region of the GS, the histogram is gaussian, confirming the secondorder transition observed in the data shown above. For η = 0.58 and 1 belonging to the non collinear region, the histogram shows a two-peak structure which confirms the first-order character of the transitions in this region.

The two peaks are very well separated with the dip going down to zero, indicating an energy discontinuity. The distance between the two peaks is the latent heat ∆E.

We show in Fig. 11 the transition temperature versus η where I and II indicate the collinear and non collinear phases, respectively. P denotes the paramagnetic state. The line separating I and P is a second-order transition line, while that separating II and P is the first-order one.

To close this section, we emphasize that all 3D frustrated systems known so far undergo a first-order transition: let us mention the AF STL [START_REF] Ngo | [END_REF]13], the FCC antiferromagnets [21], the simple cubic fully frustrated lattices [22][23][24][25], helimagnets [26] and the HCP lattice studied here.

IV. SPIN TRANSPORT IN THE ISING CASE

We have recently studied the spin transport in magnetically ordered materials by using an efficient MC simulation method [27,28]. The tour-de-force of the method was realized when we studied the semiconducting MnTe where the agreement with experiment is excellent [29]. In ferromagnets, the spin resistivity shows a sharp peak at the magnetic phase transition. The origin of this anomaly comes from the spin-spin correlation [30][31][32]. In antiferromagnets, such a peak is not sharp and often a broad maximum [33]. We have also studied the spin resistivity in two frustrated cases: the FCC Ising case [34] and the J 1 -J 2 simple cubic lattice [35]. The results show that the discontinuous transition in these systems gives rise to a discontinuity of the spin resistivity.

Using the method which has been described in detail elsewhere [27,34], we calculate the current of itinerant spins moving across the crystal under an electric field applied in the x direction. We suppose that each itinerant spin σ i interacts with the lattice spins around it within a sphere of radius D 1 at any point on its trajectory. The interaction Hamiltonian reads

H l = - j I ij σ i • S j ( 14
)
where the sum is performed over all lattice spins in the sphere around the itinerant spin σ i and I ij is the interaction which depends on the distance between σ i and S j . We suppose also the following interaction between an itinerant spin with its neighboring itinerant spins within a sphere of radius D 2

H i = - j K ij σ i • σ j ( 15
)
For simplicity, we suppose

I ij = I 0 exp(-Br ij ) (16) K ij = K 0 exp(-Cr ij ) ( 17
)
where I 0 , B, K 0 and C are constants chosen in such a way that the energy of an itinerant spin is smaller than that of a lattice spin. This choice is made to avoid the influence of itinerant spins on the ordering of the lattice spins. Note that in the almost-free electron model, I 0 K 0 0, and in semiconductors they are larger but still weak with respect to J 1 and J 2 . A discussion on the choice of different parameters has been given for example in Ref. [27]. We suppose here a concentration of one itinerant per two lattice cells. At such a low concentration, the averaged distance between itinerant spins is much larger than the cutoff distance D 2 . In spite of this, due to the attractive nature of their interaction, Eq. ( 17), we have to use a chemical potential term to insure that itinerant spins do not form clusters. This potential is taken of the form D[n( r) -n 0 ] where D is a positive constant, n( r) the concentration of itinerant spins in the sphere of cutoff radius D 2 centered at the position r of the itinerant spin under consideration, and n 0 the averaged concentration. We perform the simulation by taking into account the relaxation time of the lattice spins [18,28,29] which is given by

τ L = A |1 -T /T C | zν ( 18
)
where A is a constant, ν the correlation critical exponent, and z the dynamic exponent. We see that as T tends to T C , τ L diverges. This phenomenon is known as the critical slowing-down. For the Ising spin model, ν = 0.638 (3D Ising universality) and z = 2.02 [36]. We have previously shown that τ L strongly affects the shape of ρ at T C [28]. By choosing A = 1, we fix τ L = 1 at T = 2T C deep inside the paramagnetic phase far above T C . This value is what we expect for thermal fluctuations in the disordered phase. The spin resistivity ρ versus T for the two typical cases η = 0.3 and 1 is shown in Fig. 12 where distances D 1 and D 2 are in unit of the NN distance, energy constants I 0 , K 0 and D are in unit of |J 2 | = 1. We observe here that in the second-order region ρ has a rounded maximum and in the first-order region it undergoes an almost discontinuous jump at the transition.

We have varied the radius D 1 to see its effect on the resistivity ρ at the transition. We observe the same effect which has been seen in some other antiferromagnets [34,35]: at a given T , ρ oscillates slightly with distance. We have found that this oscillation comes from the oscillatory behavior of the difference between the numbers of up and down spins in the sphere as D 1 varies.

V. CONCLUSION

We have studied in this paper some properties of the HCP antiferromagnet with Ising and XY spin models. The in-plane J 1 and inter-plane J 2 interactions are supposed to be different. As a result, the GS spin configuration depends on the ratio η = J 1 /J 2 . We show that there exists a critical value η c where the GS changes. For the Ising case, we find η c = 0.5 below (above) which the spins in the xy planes are ferromagnetic (antiferromagnetic). For the XY case, the GS is collinear below η c = 1/3, and is non collinear above that value. The nature of the transition changes from a second order below η c to a first order above η c for both Ising and XY cases. We have also studied the spin resistivity in the Ising case. We found that the shape of ρ depends on the nature of the transition: in the second-order region, a rounded maximum is observed at the transition while in the first-order region, ρ undergoes a discontinuity at the transition as we have observed in other frustrated cases. These findings may help to understand the transition nature and the spin transport in different compounds with HCP structure.

J 1 J 2 ZFIG. 1 .

 121 FIG.1. HCP lattice. The in-plane and inter-plane interactions are indicated by J 1 and J 2 .

FIG. 2 .FIG. 3 .

 23 FIG. 2. Ground state in the XY case. The tetrahedron is projected on the xy plane. The spins are numbered from 1 to 4. See text for comments.

FIG. 5 .FIG. 6 .

 56 FIG. 5. Ising case (color online): Energy E and order parameter M for η = J1/J2 = 0.85 (blue void circles) and 1 (red triangles). See text for comments.

FIG. 7 .

 7 FIG. 7. Ising case: TC versus η. I, II and P denote the first, second and paramagnetic phases, respectively. See text for comments.

FIG. 9 .

 9 FIG.9. XY case (color online): E and M versus T for η = 0.58 (blue void circles) and 1 (red triangles). See text for comments.

PFIG. 10 .

 10 FIG. 10. XY case (color online): Energy histogram P versus E for η = 0.3 (black circles), 0.58 (blue void circles), 1 (red triangles) at the respective transition temperatures. See text for comments.

FIG. 11 .

 11 FIG. 11. XY case: TC versus η. I, II and P denote the collinear, non collinear and paramagnetic phases, respectively. See text for comments.

FIG. 12 .

 12 FIG. 12. Ising case: Spin resistivity ρ versus T for η = 0.3 (upper) and 1 (lower). L = 18, L z = 8 (16 planes in the z direction), D 1 = D 2 = 1, = 1, I 0 = 2, K 0 = 0.5, B = C = 1, A = 1, D = 0.5. All distances are in unit of the NN distance, energy constants are in unit of |J2| = 1. See text for comments.