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Sequences of point blow-ups from a combinatorial point of view

We define equivalence relations for both sequences of point blow-ups of smooth d-dimensional projective varieties and for associated sequential morphisms and we prove that there exists a bijection between the respective equivalence classes of sequences of point blow-ups and the associated sequential morphisms. Moreover we show some diferences when dealing with the more general case of sequences of blow-ups of smooth 3-dimensional projective varieties, where all possible dimensions for the centers are allowed.


Introduction
Sequences of blow-ups of smooth varieties along smooth centers are useful for general algebraic geometric purposes, in particular, for resolution and classification of singularities. We will assume, additionally, that the center of each blow up is transversal to the already created exceptional divisors by the precedent blow ups, i.e. that at every point there are suitable regular systems of parameters such that both the center and the divisor components are locally defined by ideals generated by some of those parameters. * First author was partially supported by PGC2018-096446-B-C21 † Second author was partially supported by PGC2018-095392-BI00 The composition of the successive blow ups of such sequences is a projective (and therefore proper) birational morphisms Z → X, where Z and X are smooth algebraic varieties which are respectively called sky and ground. A morphism obtained in that way will be called sequential morphism. Each sequential morphism has an exceptional divisor which is the reduced divisor of Z whose support consists of those points of Z at which the morphism is not a local isomorphism. By the choice of the successive centers, the exceptional divisor of a sequential morphism has only normal crossing. In fact, in this paper, we restrict to the case of point blow-ups, as it is involved in many geometric contexts, in particular, the study of algebraic curves and surfaces.
Campillo, González-Sprinberg and Lejeune-Jalabert in [1] studied sequences of point blow-ups using several combinatorial objects associated to the sequential morphism such as the d-ary intersection form on the abelian group of divisors with exceptional support, and Campillo and Reguera in [3] also studied other equivalent objects as the weighted dual polyhedron or the weighted tree. These results are based on the proximity of infinitely near points. More details and other applications of proximity can be found on the survey [2]. More recently, Stepanov has studied in [12] another combinatorial object, the dual complex associated to a log-resolution, and has proved that the dual complexes of two log-resolutions of an isolated singularity are homotopy equivalent. Moreover, in [4] de Fernex, Kollár and Xu have proved that in many cases, for instance in the case of isolated singularities, we can do even better by selecting a minimal representative such that it is well defined up-to piecewise linear homeomorphism.
Throughout this paper we revisit some of the results in [1], [3] focusing in the d−ary intersection form. Our approach is geometrical in the sense that it makes an intensive use of intersection theory, looking forward it could become useful for the more general case of varieties of dimension d ≥ 3, where all possible dimensions for the centers are allowed. More precisely, we will focus on birational contractions (see [11]) starting from the final components. Moreover we consider the case of any base field K which is perfect but non necessarily algebraically closed.
For sequences of point blow-ups, a combinatorial equivalence notion can be defined for them, see definition 2.25. For their associated sequential morphism another combinatorial equivalence notion of equivalence can be defined in terms of the intersection forms of their divisor with exceptional support, see definition 2.20.
Naturally, sequences of blow-ups and sequential morphisms which are algebraically isomorphic are equivalent for their corresponding notions.
Our first main result, theorem 3.1, states that two sequences of point blow ups are combinatorially equivalent if and only if their associated sequential morphisms are combinatorially equivalent too. In particular, the equivalence of sequences of blow ups whose composition is a sequential morphism becomes also another well defined equivalence notion for those morphisms. To prove all this, we show a rather strong result, which is interesting for its own (see theorem 5.14).
Namely, in our second main result, theorem 3.2, we show that the intersection form on the sky Z determines those exceptional components which are final, i.e.
divisors obtained by one of the point blow ups of the sequence. Also, we show that the intersection form on the blow down of the final components can be explicitly computed in terms of the intersection form of Z. This allows to recover from the sequential morphism, not only the blow up sequence up to combinatorial equivalence, but to recover, up to algebraic equivalence, the blow up sequence itself.
Finally, we illustrate with some examples that the information contained in the intersection form is minimal in some sense for our classification purpose and as well as highlight some of the main differences with the more general case in dimension d = 3.

Sequences of blow ups and sequential morphisms
Fix a perfect field k and chose an algebraic closure k. Throughout this paper, unless otherwise stated, a variety will mean a reduced projective scheme over a perfect field K, with K an algebraic extension of k, so it is also perfect, such that K ⊂ k, and a point will mean a closed point.
Definition 2.1. A sequence of blow-ups over K is defined as a sequence of blowups at smooth closed subvarieties C i of smooth d−dimensional projective varieties such that for i ∈ {0, 1, ..., s − 1}: i. if we denote by C i+1 to the center of π i+1 , then C i+1 is a smooth subvariety iii. if we denote by E j j the exceptional divisor of π j , and for k > j we denote by E k j the strict transform of E j j in Z k , then C i+1 has simple normal crossings We denote by π the composition π 1 • π 2 • ... • π s−1 • π s . A morphism π : Z → Z 0 which can be expressed, in at least one way, as a composition of blow ups with the conditions in definition 2.1 will be called a sequential morphism over K.
Remark 2.2. Given a sequence of blow-ups π, we denote by π s,i : Remark 2.3. We will refer to X = Z 0 and Z as the ground and the sky of the sequential morphism π respectively.
Definition 2.4. The length m over K of a sequence of blow ups is defined as s i=1 #C i , where #C i denotes the number of irreducible components of C i over K. Notice that it coincides with the number of irreducible components of the exceptional divisor E (over K too). Therefore, the length depends on the sequential morphism π and it can be also called the length of π over K, and it will be denoted by m = lenght K (π). Notice that s ≤ m, and s = m exactly when all the blow up centers are irreducible over K.
Remark 2.6. Moreover we will denote by H β the irreducible components over K of the exceptional divisor E of π, that is we have E = β H β .
Definition 2.7. We denote by E Z,π,K (or simply E Z if there is no confusion) to be the abelian group of divisors of Z of the form m β=1 n β H β where n β are integers, that is, the free group generated by the irreducible components of E over K. The Definition 2.8. Given a sequential morphism π : Z → Z 0 , we consider the d-ary multilinear intersection form defined by intersecting cycles in the sky Z and taking degrees, that is where H i1 · H i2 · H i3 · · · H i d is an intersection class of 0−cycles in the abelian group A 0 (Z), and deg stands for the degree.
Remark 2.9. For the sake of simplicity we will denote H i1 · H i2 · · · H i d = Definition 2.10. Given a sequential morphism π as in definition 2.1, it induces . In this way, the multilinear form of intersection give rise to a multilinear form We say that Φ Z,EZ is the multilinear form associated to π. The permutation group S m acts on the set of multilinear forms Z m × · · · × Z m → Z by interchanging the elements of the standard basis of Z m . It is clear that if we denote by Ψ Z,EZ to the orbit of Φ Z,EZ , then Ψ Z,EZ does not depend on the labeling of the elements of the basis {H i }.
In order to consider different fields K, with k ⊂ K ⊂ k, we define the notion of compatible partition of the exceptional divisor E.
Combinatorial compatibility with a sequential morphism 2.11 will mean compatibility of the d−ary multilinear intersection form. Compatibility with a sequence of point blow-ups 2.12 will mean compatibility of proximity relations and degrees of the residue field extensions.
Also we will define the notion of algebraic compatibility, stronger than combinatorial, where the partition comes, by fiber product, from a sequential morphism 2.14 (resp. a sequence of blow-ups 2.15) defined over a smaller field K, with k ⊂ K ⊂ K.
Definition 2.11. Given a sequential morphism π : Z s → Z 0 as in definition 2.1 and a partition E = ⊔ l i=1 F i , we will say that the partition is combinatorially compatible with π if for each i = 1, .., l, and H j1 , H j2 ∈ F i there exists σ ∈ S m such that .., Z 0 , π) be a sequence of blow ups of length m, and H 1 , ..., H m the irreducible components of the exceptional divisor over K of the associated sequential morphism. For each i, with i = 1, 2, .., m, let r(i) be the integer such that the image of H i at Z r(i) is a component of the center (codimension at least 2) whose blow-up creates H i . If j is different from i, and the image of H j at Z r(i) has codimension 1 and contains the image of H i at Z r(i) , then H i is said to be proximate to j and we denote it by H i → H j . It is clear that one has r(i) > r(j) when H i is proximate to H j .
For sequences of point blow-ups we denote deg(H i ) = [K(P i ) : K], where P i is the point in the center of π r(i) such that the image of H i in Z r(i) is P i . Definition 2.12. Given a sequence of point blow-ups (Z 0 , ..., Z s , π) and a partition E = ⊔ l i=1 F i , we will say that the partition is combinatorially compatible with the sequence (Z 0 , ..., Z s , π) if for each i = 1, .., l and H j1 , Definition 2.14. Given a sequential morphism π : Z s → Z 0 as in definition 2.1 and a partition E = ⊔ l i=1 F i , we will say that the partition is algebraically compatible with the morphism π if there exist a smaller field K ⊂ K with k ⊂ K, there are K−varieties Z 0 and Z and a K−morphism Z such that the exceptional divisor of π, E, has irreducible components H 1 , ..., H l and for each i = 1, ..., l then ∀H ∈ F i β(H) = H i Definition 2.15. Given a sequence of point blow-ups (Z 0 , ..., Z s , π) and a partition E = ⊔ l i=1 F i , we sill say that the partition is algebraically compatible with the sequence (Z 0 , ..., Z s , π) if there exist a smaller field K ⊂ K with k ⊂ K and there .., s ,such that the exceptional divisor of ( Z 0 , ..., Z l , π) has irreducible components H 1 , ..., H l and for each i = 1, ..., l then Remark 2.16. Note that since k is perfect then K ⊂ K is a separable algebraic extension, so K and K are both perfect fields.
A combinatorially (resp. algebraically) marked sequential morphism is denoted is a partition combinatorially (resp. algebraically) compatible with π. The same notation holds for sequences.
Note also that if a partition is algebraically compatible with a sequential morphism (resp. a sequence) then the partition is combinatorially compatible with the sequential morphism (resp. the sequence). Now we define our notions of equivalence (algebraic and combinatorial) for marked sequential morphisms (definitions 2.17 and 2.20).
Definition 2.17. We say that two algebraically marked sequential morphisms and there exist isomorphisms a and b such that the following diagram is commu- Definition 2.19. Given a combinatorially marked sequential morphism (π : Z s → we can also consider the d-ary multilinear intersection form associated to the partition where F Z is the free abelian group generated by {F i } and by an abuse of notation The intersection form is defined by intersecting cycles in the sky Z and taking degrees, that is where F i1 · F i2 · F i3 · · · F i d is a intersection class of 0−cycles in the abelian group A 0 (Z), and deg stands for the degree.
Definition 2.20. Given two combinatorially marked sequential morphisms (π : are equivalent, and we denote it by Moreover, the combinatorially marked sequential morphisms (π : Definition 2.22. Given a variety X we will call a brick blow-up with ground X to a sequential morphism obtained as a composition of point blow-ups with disjoint Definition 2.23. We say that two algebraically marked sequences of point blow Remark 2.24. If two algebraically marked sequences of blow ups, (Z s , ..., alg , are algebraically equivalent, then their associated sequential morphisms are also algebraically equivalent. Therefore, in particular, one Definition 2.25. We say that two combinatorially marked sequences of point blow

Statements of the main results
Throughout this paper we will restrict ourselves to the case of sequences of point blow-ups, that is sequences of blow-up in which all centers are closed points. Notice that in this case, for each exceptional component Our main theorem states the following result.
are equivalent as in definition 2.20 Furthermore, using the necessary condition in remark 2.21, that is π it follows a stronger result which is the core result of this article.
Theorem 3.2. Given two algebraically marked sequential morphisms (π : Z s → ) alg respectively such that they are algebraically equivalent over K as in definition 2.23.

Some useful intersection theory
In this section we will recall some known results on intersection theory which we will use in the proof of our results. We do not presents an exhaustive list, but the aim of this section is to list some results which either are critical in our development, or are very specific in intersection theory, or we did not find any proof in the literature.
Along this section, X and Z will stand for smooth algebraic varieties over K, the considered maps Z → X for proper morphisms, and d for the dimension of X.
for simplicity we will denote also by V to the equivalence class whenever there is not possible confusion.
its image by f , then one defines the push-forward of the cycle associated to V as follows denotes the degree of the extension of function fields . This definition allows also to define, by linearity, the push-forward f * (α) for any cycle α.
The next result known as the projection formula is specially useful and it will play an essential role in some of our intersection computations.
The following proposition is obtained as a particular case of the one above. Then, one has Let P be a closed point of a variety X. Let X be the blow-up of X at P , and let E be the exceptional divisor. We have then a fibre square In order to apply the projection formula (either proposition 4.4 or proposition 4.5) then we need previously the following result is the residue field of O P,X , and e P V is the multiplicity of P on V .
Since D and F meet regularly in X, then (see [8,Proposition 3.6.]): So, we have the following exact sequences of vector bundles on G In addition, the canonical embedding of Y in X is a regular imbedding, with normal bundle where π is the projection from Y to Y , and F is the exceptional divisor on Y of such projection.
Corollary 4.11. [6, Theorem 2.1.] The Chow ring of P n is where ς ∈ A 1 (P n ) is the rational equivalence class of all the hyperplane; more generally, the class of all closed subschemes of codimension k and degree d is dς k Proposition 4.12. [6, Proposition 9.13] If X is a smooth projective variety and F ⊂ E are vector bundles on X of ranks s and r respectively, then

Proof of the main theorems
Definition 5.2. Let π : Z s → Z 0 be a sequential morphism. We say that and irreducible component H of E is final with respect to K if it exists a sequence of blow-ups (Z s , ..., Z 0 , π) associated to π : Z s → Z 0 such that H is final with respect to this sequence. Proof. Set P i ∈ Z i−1 , P j ∈ Z j−1 , to be the points such that H i maps to P i and H j maps to P j . If H β is final with β ∈ {i, j}, then H β ∼ = P d−1 K(P β ) and Let us suppose that H i ∩ H j = ∅. Then either P i is proximate to P j or P j is proximate to P i . In the first case H j ∼ = P In order to prove the necessary condition, we will prove that H i ∩ H j = ∅ ⇒ ∃r = 0, s = 0 such that (H i ) s (H j ) r = 0, it is enough to prove it in the case of a sequence of point blow-ups of length m = 3 since the general result follows by induction.
First let π 1 : Z 1 → Z 0 be the blow-up with center P 1 . Now we blow-up Z 1 with center P 2 such that P 2 ∈ E 1 1 , that is such that P 2 is proximate to P 1 . Thus we have the following diagram Then it follows by propositions 4.3 and 4.12 that By proposition 4.8 Furthermore we have the following commutative diagram where all the morphisms are regular imbeddings Now due to the commutativity of the diagram above and the result of lemma 4.7 then By induction on r and s respectively it follows Finally we can conclude that Note that (H 2 1 ) s (H 2 2 ) r = 0 and furthermore if we denote by ∆ 1,2 to deg(j H 2 1 ,Z2 * ((i H 2 Let π 3 : Z 3 → Z 2 be the blow-up of Z 2 with center P 3 , such that P 3 ∈ H 2 1 ∩ H 2 2 , that is P 3 is proximate to P 1 and to P 2 . Then it follows that by theorem 3.2 and due to the projection formula, propositions 4.4 and 4.5, then Since (H 2 1 ) s (H 2 2 ) r = 0 and furthermore that it is of the form (H 2 1 ) s (H 2 2 ) r = (−1) s−1 ∆ 1,2 , then it must exist r, s with r = 0 and s = 0 such that (H 3 1 ) s (H 3 2 ) r = 0.
For the more general case, let us suppose that {P α1 , P α2 , ..., P α k } are proximate to both P 1 and P 2 . Then by iterating equation (1) so it must exist r, s with r = 0 and s = 0 such that (H α k 1 ) s (H α k 2 ) r = 0.
for every j such that H i ∩H j = ∅ (see lemma 5.5 for a numerical characterization) and for all natural numbers r and s with r + s = d.
Proof. We have the following commutative diagram where we denote by D i,j to the scheme theoretic intersection H i ∩H j and all the morphism are regular imbeddings First let us suppose that H i is final .Then H i ∼ = P d−1 and by proposition 4.9 N Hi/Zn = O Hi (−1), so it follows by proposition 4.3 that (1)).

Moreover as we have seen in proposition 4.3
H i · H j = j Hi,Z * j * Hi,Z (H j ), and D i,j determines a projective subbundle on H i , so by proposition 4.12 Since E is simple normal crossing divisor, then by lemma 4.7: N Hj /Z | Di,j ∼ = N Di,j /Hi .

It follows by propositions 4.3 and 4.12 that
Now due to the commutativity of the diagram above and the result of lemma 4.7 then So we can conclude that and more concretely Now let us suppose that H i is not final. If P α is proximate to P i , then we have the following commutative diagram Among all the index satisfying α → i there must exist an index j such that j → i but that there not exists k with k → i and k → j. Let j be such index. Since H i is not final then by proposition 4.8 its normal bundle satisfies Now, by the projection formula (see proposition 4.4) with n i + α→i n α = d, and then Furthermore, by an analogous reasoning to the case when H i is final then By induction on r and s respectively it follows so it follows that If d is even then If d is odd then Remark 5.7. Note that whereas in [3] the authors show two different numerical characterizations for final divisors, distinguishing when the dimension d is even or odd, our numerical characterization works for both cases.
Proposition 5.8. Given an algebraically marked sequence (Z s , ..., for every j such that F i ∩F j = ∅ and for all natural numbers r and s with r+s = d.
Proof. This is a consequence of proposition 5.6, since if H, H ′ ∈ F i and H = H ) alg algebraically marked sequential morphisms that are algebraically equivalent. If we Now we will define a key tool for our proof of theorems 3.1 and 3.2, that of a regular projective contraction.
Definition 5.11. Let X ′ be a d−dimensional variety, L be an effective Cartier divisor on X ′ and Y an r−dimensional variety with r < d − 1. Then we say that L is contractible to Y within X ′ if there exist a variety X and a proper birational ii. L is a closed subset of X ′ which consists of the points where f is not an isomorphism We call this triple (X ′ , f, X) a contraction of L to Y , or simply a contraction.
We shall say that L is regularly and projectively contractible to Y within X ′ , when moreover iii. X is a non-singular projective variety Before continuing, we need to prove the following technical lemma that is crucial for the uniqueness of the regular projective contractions.
Lemma 5.12. Let X and Y be two affine normal varieties such that X = Spec(A) and Y = Spec(B). Let π : Z → X and π ′ : Proof. By [9, Theorem 3.2.1], since π and π ′ are proper morphisms then π * (O Z ) and π ′ * (O Z ′ ) are a coherent sheaves on X and Y respectively. Since X and Y are both normal, then Proof. Since H i is final there must exist an isomorphism between the two opens sets U i ⊂ Z i and V i ⊂ Z s via π s,i . After shrinking U i if necessary, we may assume An alternative construction of the contraction if K = K.
and there exist two proper morphisms is a simple normal crossing divisor.
So we have proved that there exists a regular projective contraction (Z, f m , X m−1 ) of H i to a point P i ∈ X m−1 .
Following the notations of definition 5.1, let W i = π i (U i ). Then by definition 5.11 is an isomorphism. Now we define g : X m−1 → Z i−1 as follows: g| W i = π s,i−1 | W i and g| Wi = Id Wi . By our construction of X m−1 g is well defined, and by the definition g : X m−1 → Z i−1 is a sequence of point blow-ups.
Hence the composition X m−1 → Z i−1 → Z 0 is a sequence of blow-ups.
Theorem 5.14. Let (π : Z s → Z 0 , ⊔ l i=1 F i ) alg be an algebraically marked sequential morphism. Given the d−ary multilinear intersection form associated to the partition I Z,⊔ l i=1 Fi (see definition 2.19) we can recover all the algebraically marked sequences of point blow-ups that are associated to algebraically marked sequential morphisms in the same algebraic equivalence class of (π : Proof. Since ⊔ l i=1 F i is a partition algebraically compatible with π then ∃ K ⊂ K as in definition 2.14. If H ∈ F i is final then H = β(H) is final for π : Z s → Z 0 .
We will prove this result first by contracting one irreducible component of the exceptional divisor E each time.
Since the set formed by final divisors is not empty, let us suppose that H i is final, then by proposition 5.6 there exists a regular projective contraction ( Z s , f l , X l−1 ) of H i to a point such that X l−1 is the sky of a sequence of point blow-ups with The next step in our proof refers to how to obtain the intersection form in X l−1 associated to the simple normal crossing divisor D X l−1 .
If we denote by H X l−1 ,i to f l ( H i ), then by the projection formula (proposition 4.5) Applying the result of theorem 4.6 then where δ ij ,i = 1 if H i ∩ H ij = ∅ (see numerical characterization in lemma 5.5) and δ ij ,i = 0 otherwise.
Remark 5.15. It follows then that by iterating the above process, that is by contracting a final divisor at each step, we will obtain a sequence of point blow-ups of length l. The algebraically marked sequence obtained depends on the choice of final components. Below we will prove that all the algebraically marked sequential morphisms associated to the sequences constructed in this way are algebraically equivalent. Before proving this, we need the following lemma    In order to prove proposition 5.16 we need the following definition.
Definition 5.19. We say that two sequences of point blow-ups obtained as in remark 5.15, that is through the composition of regular projective contractions from a fixed sky Z and a fixed simple normal crossing divisor E, have the same end if at least the first contraction is common to both. i.e. one has Proof of Proposition 5.16. Let us suppose then that Z is the sky of a sequence of n + 1 point blow ups and that proposition 5.16 is true for sequences of length lower or equal than n. If two sequences obtained as above ρ : have the same end, then it is clear that both are associated to algebraically marked sequential morphism in the same algebraic equivalence class. It is a direct consequence of the fact that by hypothesis the assertion is true for sequences of length lower or equal than n.
are equivalent too as in definition 2.20 First we will prove that if two combinatorially marked sequential morphisms ) comb are combinatorially equivalent then the associated combinatorially marked sequences of points blow-ups are combinatorially equivalent too. To begin with, we need a numerical characterization of proximity. where Z s = X m → X m−1 → · · · → X α → · · · → X 0 = Z 0 is any sequence of contractions obtained as in remark 5.15.
Proof. If P i is proximate to P j then H i i ∩ H i j = ∅ i). Moreover, if P i ∈ Z r , then H i i is final for the sequence π r+1 • π r • · · · • π 1 , for some r ≥ i and we have ii). Conversely, if H Xα,i is final for the sequence π r+1 • π r • · · · • π 1 for some r ≥ i then by proposition 5.13 there exist a regular projective contraction f α : Remark 5.22. The result of the previous lemma also holds for characterizing numerically the proximity between elements of the combinatorially compatible par- Proof of Theorem 3.1. Assume that the combinatorially marked sequential mor-

Furthermore, by equation (2)
so it follows then that there exists τ ∈ S l−1 such that . Furthermore, by iterating the above process, then for α = 1, .., l. So as a consequence of lemma 5.21 any two combinatorially marked sequential morphisms combinatorially equivalent preserve the proximity relations. Moreover, ) so combinatorially equivalent sequential morphism also preserve degrees.
Conversely assume now that the combinatorially marked sequences of point blowups with l = l ′ are combinatorially equivalent. We want to prove that their associated combinatorially marked sequential morphisms (π : Moreover, as a consequence proposition 4.4 Finally, and as a consequence of theorem 4.6 ) alg respectively such that they are algebraically equivalent over K as in definition 2.23.
Proof. If two algebraically marked sequences of point blow-ups are algebraically equivalent, then it follows directly by definition 2.23 that the associated algebraically marked sequential morphisms are algebraically equivalent too. Now we will prove that if two algebraically marked sequential morphism (π : Z s → ) alg are algebraically equivalent, then there exist algebraically marked sequences of point blow-ups associated to them that are algebraically equivalent too. By theorem 5.14 given a certain sky Z associated to an algebraically marked sequential morphism (π : Z s → Z 0 , ⊔ l i=1 F i ) alg , all the algebraically marked sequences of point blow-ups obtained by regular projective contractions are associated to algebraically marked sequential morphisms in the same algebraic equivalence class. Since (π : ) alg are algebraically equivalent, then ∃ K ⊂ K such that there exist an isomorphism b : Z → Z ′ . By applying proposition 5.13 and proposition 5.16 we conclude the result by scalar extension × Spec( K) Spec(K).

Examples and comments
In the following example we will show that the numerical information that appears in the d−ary multilinear intersection form I Z,EZ is minimal in some sense for our classification purpose.
Example 6.1. Consider a d−dimensional smooth algebraic variety X over k, where k is a perfect field not algebraically closed, and d is an even natural number.
Let P = P 1 be a k−rational point on X and let π 1 : Z 1 → X be the blowing up with center P . Take a point P 2 on Z 1 such that the degree of the field extension k ⊂ k(P 2 ) is 1, and a point Q 2 in Z 1 such that [k(Q 2 ) : k] = 2. Now, let π 2 : Z 2 → Z 2 (respectively π ′ 2 : Z ′ 2 → Z 1 ) be the blowing up of P 2 (of Q 2 , respectively) and take a point P 3 in Z 2 such that P 3 is not proximate to P 1 and such that the degree of the field extension k(P 2 ) ⊂ k(P 3 ) is 3 (respectively, a point Q 3 in Z ′ 2 no proximate to P 1 such that [k(Q 3 ) : k(Q 2 )] = 1). Consider now the blow-up π 3 : Z 3 → Z 2 (π ′ 3 : Z ′ 3 → Z ′ 2 ) of Z 2 (Z ′ 2 ) with center P 3 (Q 3 respectively).
The next two examples pretends to highlight some of the differences that appear in the more general case, even when we restrict to the case of varieties of dimension d = 3.
Example 6.2. Let C 1 , C 2 ⊂ P 2 ⊂ P 3 be two smooth plane curves intersecting transversally at a point P , and let π 1 : Z 1 → Z 0 = P 3 be the blow-up of P 3 with center C 1 = C 1 . If we denote by C 2 to the strict transform of C 2 , let π 2 : Z 2 → Z 1 be the blow-up of Z 1 with center C 2 = C 2 Note that although C 2 is not proximate to C 1 in the usual sense, the blow-up with center C 2 modifies E 1 1 by creating an exceptional curve and increasing its Picard number by one. Consequently, E 2 1 is not a projective bundle any more, so it can not be regularly contractible from Z 2 by [11, Theorem 3] and thus it can not be final.
This fact lead us to introduce a new proximity relation between the centers of a sequence of blow-ups Definition 6.3. Let us suppose that C j ∩ E j−1 i = ∅. Then we say that C j is proximate to C i , and we write C j → C i if and only if C j ⊂ E j−1 i . Otherwise we say that C j is t−proximate to C i , and we write C j t − → C i .
The unique final divisor of the sequence above in example 6.2 is E 2 , and following the above notations C 2 is t−proximate to C 1 , and we would denote it by C 2 t − → C 1 .
Example 6.4. Let C ⊂ P 3 be a smooth curve of degree d and genus g, and let π 1 : Z 1 → Z 0 = P 3 be the blow-up of P 3 with center C 1 = C. If P ∈ C is a closed point, and we denote by F to the inverse image π −1 1 (P ), let π 2 : Z 2 → Z 1 be the blow-up of Z 1 with center C 2 = F .