THREE-DIMENSIONAL TERMINAL CONTRACTIONS

S. A. KUDRYAVTSEV

Abstract. The analog of Miles Reid’s general elephant conjecture is that every three-dimensional terminal extremal contraction is not exceptional and hence the canonical divisor K has 1-, 2-, 3-, 4- or 6-complement. In this paper the approach to the conjecture is suggested. The main results are the following ones. The structure theorem about the inductive log surface (\overline{S}, Diff\overline{S}(0)) is proved. Shokurov’s conjecture on the characterization of toric varietis is proved for the three-dimensional small contractions. The basic special log flips of terminal type are classified.

Introduction

The problem of the existence of a “good” divisor in the (multiple) anticanonical linear system is a fundamental one in the classification of singularities, contractions and varieties arising in the log minimal model program. In solving this problem the powerful method is a theory of complements. It allows to reduce the question to some log variety of a smaller dimension, called an inductive log variety (for instance, see the papers [23], [18], [15]).

Roughly speaking, an n-complement of a canonical divisor K is an effective ℚ-divisor D^{+} such that n(K + D^{+}) ∼ 0 and K + D^{+} is a log canonical divisor. Other advantages of this method are that it works in any dimension, every n-complement of our local contraction, singularity gives an n-complement of some inductive log variety and every n-complement of the inductive log variety is extended to an ncomplement of our local contraction.

For the three-dimensional terminal contractions the following conjecture takes place. It is the analog of Miles Reid’s general elephant conjecture in the theory of complements. Conjecture 0.1. Let f: X → (Z ∋ P) be a contraction of threedimensional variety X with the following properties:

(1) the variety X has the ℚ-factorial terminal singularities; (2) the divisor −K_{X} is f-ample;

This work was done with the partial support of the Russian Foundation for Basic Research (grant no. 02-01-00441), the Leading Scientific Schools (grant no.

00-15-96085) and INTAS-OPEN (grant no. 2000#269.)

(3) dim Z > 0, Z is a germ near P and ρ(X/Z) = 1. Then this contraction is not exceptional, in particular there exists 1-, 2-, 3-, 4- or 6-complement of a divisor K_{X}.

See chapter 4 for the discussion of this conjecture.

We suggest the very direct and natural approach to prove it. Assume that f is exceptional. Equivalently, the inductive log surface (\overline{S}, Diff\overline{S}(0)) is exceptional (see chapter 3). The exceptional log surfaces belong to the finite number of algebraic families [23]. Therefore it is possible to obtain their complete and detailed classification. Such classification was done in the papers [8], [23] for δ(\overline{S}, Diff\overline{S}(0)) = 1, 2, where

δ(\overline{S}, Diff\overline{S}(0)) = #\{E | E is a divisor (not nessecarily exceptional)

with a discrepancy a(E, Diff\overline{S}(0)) ≤ −\frac{6}{7}\}.

Since δ ≤ 2 [23, Theorem 5.1] it remains to classify the log surfaces with δ = 0. In the case δ = 0 the models are already constructed [9].

Let us remember the construction of the inductive log surface (\overline{S}, Diff\overline{S}(0)) (see chapter 3). At first we take an inductive blow-up (purely log terminal blow-up) of a variety X at some terminal point and obtain the pair (Y, S). After it the log minimal model program is applied and we obtain the pair (\overline{Y} , \overline{S}) required.

Problem. Describe the class of all log del Pezzo surfaces, generic ℙ^{1}- fibrations which can be the exceptional divisors of some purely log terminal blow-ups of three-dimensional terminal singularities.

The class mentioned above is restricted. Thus, it is “difficult” to obtain (after some transformation) the exceptional log surface (\overline{S}, Diff\overline{S}(0)) from “very” non-exceptional log surface (S, Diff_{S}(0)). Unfortunately, this information is not enough to prove conjecture 0.1. Therefore we suggest to study the birational map Y −−→ \overline{Y}. It is the composition of anti-log flips with respect to −(K_{Y} + S). These log flips are special and have the terminal singularities outside S. They are called the special log flips of terminal type (see definition 2.14).

Problem. Describe the three-dimensional special log flips of terminal type.

The advantage of described above appoach is that if it is realized then the classification of terminal extremal contractions (may be not very explicit) will be obtained immediately.

This paper is organized in the following way. In chapter 1 the main definitions and preliminary results are collected. In chapter 2 the method of a study of special flips is given. The idea is to make a reduction to the toric special flips. For this purpose Shokurov’s conjecture on the characterization of toric varieties is proved for the three-dimensional small contractions (see theorem 2.4). Also the basic special log flips of terminal type are classified (see theorem 2.20). In chapter 3 we recall the construction of model. In chapter 4 the theorem about the structure of a transformation (S, Diff_{S}(0)) −−→ (\overline{S}, Diff\overline{S}(0)) is proved (see theorem 4.7).

I am grateful to Professors Yu.G. Prokhorov and V.V. Shokurov for useful discussions.

The work has been completed during my stay at Max-Planck-Institut für Mathematik in 2003. I would like to thank MPIM for the hospitality, support and stimulating atmosphere.

1. Preliminary facts

All varieties are algebraic and are assumed to be defined over ℂ, the complex number field. The main definitions, notations and notions used in the paper are given in [5], [16].

Definition 1.1. Put Φ_{sm} = {1−1/m | m ∈ ℤ_{>0} ∪{∞}}. A coefficient d is called standard if d ∈ Φ_{sm}.

Definition 1.2. For fix n ∈ ℕ put

𝒫_{n} = {a | 0 ≤ a ≤ 1, ⌞(n + 1)a⌟ ≥ na}.

Definition 1.3. Let (X, D) be a contraction of varieties, where D is a subboundary. Then a ℚ-complement of K_{X} +D is a log divisor K_{X} +D′ such that D′ ≥ D, K_{X} + D′ is log canonical and n(K_{X} + D′) ∼ 0 for some n ∈ ℕ.

Definition 1.4. Let X be a normal variety and let D = S + B be a subboundary on X such that B and S have no common components, S is an effective integral divisor and ⌞B⌟ ≤ 0. Then we say that K_{X} + D is n-complementary if there is a ℚ-divisor D^{+} (called an n-complement) such that

(1) n(K_{X} + D^{+}) ∼ 0 (in particular, nD^{+} is an integral divisor);

(2) the divisor K_{X} + D^{+} is log canonical;

(3) nD^{+} ≥ nS + ⌞(n + 1)B⌟.

Definition 1.5. Let (X, D) be a projective log variety of dimension d such that (1) the pair (X, D) is log canonical;

(2) every coefficient of a divisor D is standard;

(3) the divisor −(K_{X} + D) is nef and big;

(4) there exists a ℚ-complement of a divisor K_{X} + D. Then the pair (X, D) is called a log Fano variety.

Put

Compl(X, D) = min{m | the divisor K_{X} + D has m-complement}

and

N_{d} = sup{Compl(X, D) | (X, D) is a log Fano variety of dimension d}.

It was proved that N_{1} = 6 [22, Example 5.2] and N_{2} < ∞ [23]. Hypothetically, N_{d} < ∞ for every d ≥ 1 and N_{2} = 66 (see [8, Corollary 2.4]).

Put

Φ^{d} = Φ_{sm} ∪ [\frac{N_{d−1}}{N_{d−1} + 1}, 1].

Note that a ∈ 𝒫_{n} if a ∈ Φ^{d} and n ≤ N_{d−1} [18, Lemma 1.4].

The invariance of the complements with respect to the log minimal model program and their inductive properties are shown in the following three theorems.

Theorem 1.6. [22, Lemma 5.4] Let f: X → Y be a birational contraction and let D be a subboundary on X. Assume that K_{X} + D is n-complementary for some n ∈ ℕ. Then K_{Y} + f(D) is also ncomplementary.

Theorem 1.7. [23, Lemma 4.4] Let f: X → Z be a birational contraction of varieties and let D be a subboundary on X. Assume that

(1) the divisor K_{X} + D is f-nef;

(2) the coefficient of every non-exceptional component of D meeting Exc f belongs to 𝒫_{n};

(3) the divisor K_{Z} + f(D) is n-complementary. Then the divisor K_{X} + D is also n-complementary.

Theorem 1.8. [16, Proposition 4.4.1] Let f: X → (Z ∋ P) be a contraction and D be a boundary on X. Put S = ⌞D⌟ and B = {D}. Assume that

(1) the divisor K_{X} + D is purely log terminal;

(2) the divisor −(K_{X} + D) is f-nef and f-big;

(3) S ≠ 0 near f^{−1}(P);

(4) every coefficient of D belongs to 𝒫_{n}.

Further, assume that near f^{−1}(P) ∩ S there exists an n-complement K_{S} + Diff_{S}(B)^{+} of K_{S} + Diff_{S}(B). Then near f^{−1}(P) there exists an n-complement K_{X} + S + B^{+} of K_{X} + S + B such that Diff_{S}(B)^{+} = Diff_{S}(B^{+}).

Definition 1.9. Let (X/Z ∋ P, D) be a contraction of varieties, where D is a boundary.

(1) Assume that Z is not a point (local case). Then (X/Z ∋ P, D) is said to be exceptional over P if for any ℚ-complement K_{X} + D′ of K_{X} +D near the fiber over P there exists at most one divisor E such that a(E, D′) = −1.

(2) Assume that Z is a point (global case). Then (X, D) is said to be exceptional if every ℚ-complement K_{X} + D′ of K_{X} + D is kawamata log terminal.

Example 1.10. Let (Z ∋ P) be a normal singularity. Suppose that there is an effective divisor H such that (Z, H) is a log canonical pair and ⌞H⌟ ≠ 0. Then the singularity (Z ∋ P) is not exceptional. Therefore the three dimensional terminal singularity is not exceptional because there is a divisor having only Du Val singularities in the anticanonical linear system | − K_{Z}| [21, §6.4]. Hence every threedimensional divisorial contraction f: X → (Z ∋ P) is not exceptional if the singularity (Z ∋ P) is terminal.

2. Three-dimensional toric flips

2.1. We refer the reader to [12] for the basics of a toric geometry. Let N ≅ ℤ^{n} be a lattice of rank n and M = Hom(N, ℤ) be the dual lattice. For a fan Δ in N the corresponding toric variety is denoted by T_{N}(Δ). For a k-dimensional cone σ ∈ Δ the closure of corresponding orbit is denoted by V (σ). This is a closed subvariety of codimension k in T_{N}(Δ).

Example 2.2. 1) Let

σ = ⟨e_{1}, e_{2}, e_{3}, e_{4}⟩ = ⟨(1, 0, 0), (0, 1, 0), (0, 0, 1), (1, 1, −1)⟩

be a cone in N ≅ ℤ^{3} and Δ′ = {σ, all faces of σ} be a fan. Then T_{N}(Δ′) is a three-dimensional terminal hypersurface singularity x_{1}x_{2} + x_{3}x_{4} = 0 ⊂ (ℂ^{4}, 0). Let

Δ = {⟨e_{1}, e_{2}, e_{3}⟩, ⟨e_{1}, e_{2}, e_{4}⟩, their faces}

and

Δ^{+} = {⟨e_{1}, e_{3}, e_{4}⟩, ⟨e_{2}, e_{3}, e_{4}⟩, their faces}

be the fans. Then T_{N}(Δ) → T_{N}(Δ′) and T_{N}(Δ^{+}) → T_{N}(Δ′) are the small resolutions. The birational map T_{N}(Δ) −−→ T_{N}(Δ^{+}) is the famous Atiyah’s flop.

2) Let f: X → Z be a toric flopping contraction (with respect to K_{X}) of a smooth 3-fold X. Then f is the previous Atiyah’s contraction, in particular ρ(X/Z) = 1 [24, Example 6]. The generalization of this fact is in proposition 2.3.

Proposition 2.3. Let (Z ∋ P) be a three-dimensional toric terminal singularity. Then either (Z ∋ P)≅_{an}(ℂ^{3} ∋ 0)/ℤ_{r}(1, −1, q), (r, q) = 1 or (Z ∋ P)≅_{an}(x_{1}x_{2} + x_{3}x_{4} = 0 ⊂ (ℂ^{4}, 0)).

Proof. We give an non-combinatorical proof. If the singularity (Z ∋ P) is ℚ-factorial then we have the first possibility by terminal lemma (for example, see [12, §1.6]). Assume that the singularity (Z ∋ P) is not ℚfactorial. Let (Z′ ∋ P′) → (Z ∋ P) be a canonical cover. Let Z̃ → Z′ be a ℚ-factorialization. Since the variety Z̃ has the ℚ-factorial toric terminal singularities of index one only then Z̃ is a smooth variety. By [24, Example 6] we obtain that

(Z′ ∋ P′)≅_{an}(x_{1}x_{2} + x_{3}x_{4} = 0 ⊂ (ℂ^{4}, 0)).

Hence (Z′ ∋ P′) = (Z ∋ P) by the classification of terminal singularities [11] (of course, it is clear without classification). □

In the next theorem Shokurov’s conjecture on the characterization of toric varieties is proved for the three-dimensional small contractions.

Theorem 2.4. Let f: (X, D = ∑_{i=1}^{r} D_{i}) → (Z ∋ P) be a small contraction of three-dimensional normal ℚ-gorenstein variety X. Assume that the divisor K_{X} + D is log canonical and the divisor −(K_{X} + D) is f-nef. Then

r ≤ σ(X/Z) + 3,

where σ(X/Z) is the rank of group Weil divisors on X modulo the numerical equivalence over Z. Moreover, the equality holds if and only if the log variety (X/Z ∋ P, D) is analytically isomorphic to a toric pair, in particular K_{X} + D ∼ 0 over Z.

Proof. The strategy of proof is the same one as in the paper [19]. The question is local, so (Z ∋ P) is a sufficiently small analytical neighborhood. The sufficiency is well-known. Let r ≥ σ(X/Z) + 3 and a divisor K_{X} + D′ be a ℚ-complement of K_{X} + D. It exists since we can add to the divisor D the necessary number of general hyperplane sections of a variety X. So, by abundance theorem [5, Theorem 8.4] the ℚ-complement required is constructed. Let D′ = ∑ d_{i}D′_{i}. Then ∑ d_{i} ≥ σ(X/Z)+3. Actually, in lemma 2.9 we will prove that D′ = D.

Proposition 2.5. Let Y be a log canonical variety. Let B be an effective ℚ-Weil divisor not containing any minimal log canonical center of LCS(Y, 0). Then there exists a small contraction (ℚ-factorialization) Ỹ → Y such that KỸ and B̃ are ℚ-Cartier divisors, where B̃ is the proper transform of a divisor B.

Proof. In the proof of this proposition we use the log minimal model program and abundance theorem [5, Theorem 8.4], which are proved in dimension ≤ 3. Let φ: Y′ → Y be a minimal log terminal modification [16, Corollary 3.1.3]. If Y is a kawamata log terminal variety then φ is a required small contraction. Assume that Y is not a kawamata log terminal variety. Write K_{Y′} + ∑ E_{i} = φ^{∗}K_{Y}. By hypothesis of proposition the center of every E_{i} on Y does not lie in Supp B. Run (K_{Y′} + ∑ E_{i} + εB′)–MMP over Y for 0 < ε ≪ 1, where B′ is a proper transform of B. After some log flips we obtain a log variety (\overline{Y} , ∑ \overline{E}_{i} + ε\overline{B}). By abundance theorem the linear system |m(K\overline{Y} +∑ \overline{E}_{i} +ε\overline{B})| is free over Y for some m ≫ 0 and gives a birational contraction q: \overline{Y} → Ỹ. It is obvious that q contracts every \overline{E}_{i}. Moreover, if (K\overline{Y} + ∑ \overline{E}_{i} + ε\overline{B}) · C = 0 for a curve C then \overline{B} · C = (K\overline{Y} + ∑ \overline{E}_{i}) · C = 0. Hence KỸ is a ℚ-Cartier divisor and Ỹ is a required variety. □

Proposition 2.6. [5] Let (Y ∋ P, B = ∑ b_{i}B_{i}) be a log canonical pair such that every component of a boundary B is ℚ-Cartier. Then ∑ b_{i} ≤ dim Y. Moreover,

a) if the equality holds then (Y ∋ P, ⌞B⌟) is a ℚ-factorial toric pair;

b) if (Y ∋ P) is not a kawamata log terminal singularity then ∑ b_{i} ≤ dim Y − 2.

Proof. The first and second statements are followed from [5, Theorems 18.22, 18.23]. Let us prove the last one. The proposition is trivial if dim Y = 2. Let us prove it by induction. We have dim Y ≥ 3, ∑ b_{i} > dim Y − 2 and hence P is not a minimal log canonical center of LCS(Y, 0). By taking cyclic covers we may assume that B_{i} = {f_{i} = 0} are Cartier divisors. Let B = {∑ c_{i}f_{i} = 0} for general c_{i} ∈ ℂ. By the choice of a divisor B we obtain that a(E, B) ≥ a(E, B_{i}) for all i and every E. Therefore the pair (Y, B + ∑ b′_{i}B_{i}) is log canonical, where b′_{i} ≥ 0 and ∑ b′_{i} = ∑ b_{i} − 1. By inversion of adjunction [5, Theorem 17.6] the variety B is log canonical and not kawamata log terminal. By the inductive hypothesis the proposition is proved for (B, ∑ b′_{i}B_{i}|_{B}). So, ∑ b′_{i} ≤ dim Y − 3, a contradiction. □

Continue the proof of our theorem.

Lemma 2.7. The variety X has the kawamata log terminal singularities.

Proof. Replace the variety X with its ℚ-factorialization outside minimal log canonical centers of LCS(X, 0) by proposition 2.5. In particular, D′_{i} is a ℚ-Cartier divisor for every i. By applying some log minimal model program over Z we can contract every curve from Exc f, which does not contain any minimal log canonical center of LCS(X, 0). We obtain X −−→ W and W ≠ Z by proposition 2.6. Therefore it can be assumed that every curve from Exc f contains a minimal log canonical center of LCS(X, 0). By connectedness lemma [5, Theorem 17.4] we obtain that LCS(X, 0) ∩ Exc f is a curve. Also, if there is an exceptional curve C ⊄ LCS(X, 0) then C ∩ D′ = ∅. Thus there exists an irreducible exceptional curve lying in LCS(X, 0) such that at least three components of D′ intersect and do not contain it. Let X −−→ X^{+} be a log flip in this curve and D′^{+} be a proper transform of D′. Then the divisor K_{X^{+}} + D′^{+} is not log canonical along a flipped locus, a contradiction. □

Replace the variety X with its ℚ-factorialization. By the log minimal model program we can assume without loss of generality that Exc f = C is an irreducible curve (ρ(X/Z) = 1).

Definition 2.8. Let B = ∑ b_{i}B_{i} be a ℚ-Weil divisor. Then ||B|| = ∑ b_{i}.

Put

D^{hor} = ∑_{i: D′_{i}·C>0} d_{i}D′_{i} and D^{vert} = ∑_{i: D′_{i}·C≤0} d_{i}D′_{i}.

Actually, corollary 2.10 implies that D′_{i} · C ≠ 0 for all i. Let f^{+} : X^{+} → Z be a log flip of f and Exc f^{+} = C^{+}.

Lemma 2.9. [19, Lemma 2.10] We have ||D^{hor}|| = ||D^{vert}|| = 2. Hence D = D′.

Proof. Since the divisor K_{X} + D is log canonical then ||D^{vert}|| ≤ 2. Since the divisor K_{X^{+}} + D^{+} is log canonical then ||D^{hor}|| ≤ 2. □

Corollary 2.10. We have C ⊄ Supp D^{hor} and C^{+} ⊄ Supp(D^{vert})^{+}.

Let S be an irreducible component of a divisor D^{vert} and F = D −S. The divisorial log contraction (S, Diff_{S}(F)) → (f(S) ∋ P) is toric by [23, Theorem 6.4]. In particular, it is analitically isomorphic to the toric blow-up of a cyclic singularity. This implies that the singularities of X are toric by proposition 2.6. Also we may assume that the divisor −(K_{X} + S) is f-ample by replacing X with X^{+} and S · C < 0 by corollary 2.10. Moreover, by taking the toric blow-ups of X it can be assumed that the surface S is smooth and the variety X is smooth outside C. Consider two possibilities A) and B).

2.11. A) Let X be a smooth variety.

Lemma 2.12. Our theorem is enough to prove for a flopping contraction (with respect to K_{X}) of a smooth variety X.

Proof. Since the divisor −K_{S} is f-ample then f: S → f(S) is the contraction of a (–1) curve C and (K_{X} + S) · C = −1. Let S · C = −m, K_{X} · C = m − 1, where m ≥ 2. Using the natural section of O_{X}(S) we can construct a degree m-cyclic cover φ: X̃ → X ramified along S (cf. [5, Theorem 5.4]). Let C̃ = φ^{−1}(C), Z̃ be the normalization of Z in the function field of the variety X. Let f̃: X̃ → Z̃ be the induced contraction of a curve C̃. By the ramification formula

KX̃ · C̃ = φ^{∗}(K_{X} + \frac{m − 1}{m}S) · C̃ = K_{X} · C + \frac{m − 1}{m}S · C = 0.

□

We obtain that (Z ∋ P)≅_{an}(g = 0 ⊂ (ℂ^{4}, 0)) is an isolated cDV singularity and

(K_{Z} + f(S) + f(F))|_{f(S)} = K_{f(S)} + L_{1} + L_{2},

where L_{k} = D_{i_{k}}|_{f(S)}. Also

(f(S), L_{1} + L_{2})≅_{an}(ℂ^{2} ∋ 0, {x = 0} + {y = 0}).

Hence the pair (Z ∋ P, H + f(S) + D_{i_{1}}) is log canonical, where H is a general hyperplane section (induced from the general hyperplane section of ℂ^{4}) passing through P. Thus (H, f(S)|_{H} + D_{i_{1}}|_{H}) is a log canonical pair, that is, H has a cyclic (Du Val) singularity at the point P. By [1], [3] the singularity of H is 𝔸_{1}. So,

(Z ∋ P)≅_{an}(xy + z^{2} + t^{2l} = 0 ⊂ (ℂ^{4}, 0))

and f(D) = {x = 0} + {y = 0}. Since the pair (Z ∋ P, f(D)) is log canonical then l = 1. Hence f: X → (Z ∋ P) is the Atiyah’s toric contraction (see example 2.2).

2.13. B) Let X be a singular variety along the curve C, that is, X is isomorphic to \frac{1}{m}(1, q) × C in some small analytical neighborhood of C by [14, Lemma 3.9]. We can take the toric blow-ups along C and reduce our problem to case A) (all conditions remain valid and varieties do not have singularities along flipping curves). This completes the proof of our theorem. □

Definition 2.14. Let X be a normal ℚ-factorial n-dimensional variety and S = ∑ S_{i} ⊂ X be a reduced Weil divisor. Let f: (X, S) → (Z ∋ P) be a small contraction with ρ(X/Z) = 1. Then f is called a special flipping contraction if the following conditions are satisfied:

(1) the divisor K_{X} + S is log terminal;

(2) the divisor −(K_{X} + S) is f-ample;

(3) the divisor −S_{i} is f-ample for every i.

In chapter 3 we will consider some elementary transformations. In the terminal category they have a special structure. Therefore the next definition is introduced.

The special flipping contraction is called a special flipping contraction of terminal type if the following conditions are satisfied:

(1) the divisor S is irreducible, that is, (X, S) is a purely log terminal pair;

(2) the variety X^{+} is terminal outside S^{+}, where f^{+} : X^{+} → Z is a flip of f and S^{+} is a proper transform of S.

Remark 2.15. In example 2.2 the toric singularity is not ℚ-factorial. The well-known criterion asserts that the variety T_{N}(Δ) is ℚ-factorial if and only if σ is a simplicial cone for all σ ∈ Δ (for example, see [10, Lemma 14-1-1]).

2.16. Now let us remind the local description of three-dimensional ℚ-factorial toric flips [20] (see also [10, §14]). Let X be a threedimensional ℚ-factorial toric variety and f: X → Z be a toric small contraction of extremal ray (in some log minimal model program). By definition ρ(X/Z) = 1 and hence Exc f = C is an irreducible curve. Let f^{+} : X^{+} → Z be a flip of f. Also Exc f^{+} = C^{+} is an irreducible curve. We have the following local description: X = T_{N}(Δ), X^{+} = T_{N}(Δ^{+}), Z = T_{N}(Δ′), where

Δ = {⟨e_{1}, e_{2}, e_{3}⟩, ⟨e_{1}, e_{2}, e_{4}⟩, their faces},

Δ′ = {⟨e_{1}, e_{2}, e_{3}, e_{4}⟩, their faces}

and

Δ^{+} = {⟨e_{1}, e_{3}, e_{4}⟩, ⟨e_{2}, e_{3}, e_{4}⟩, their faces}.

There is a relation

a_{1}e_{1} + a_{2}e_{2} + a_{3}e_{3} + e_{4} = 0,

where a_{1} < 0, a_{2} < 0, a_{3} > 0. Note that C = V (⟨e_{1}, e_{2}⟩) and C^{+} = V (⟨e_{3}, e_{4}⟩).

Definition 2.17. In examples 2.18, 2.19 we will use the notation of section 2.16 and the next agreements. Let e_{1} = (1, 0, 0), e_{2} = (q, r, 0), e_{3} = (0, 0, 1), where 0 ≤ q < r. Put S = V (⟨e_{2}⟩) on X and S^{+} = V (⟨e_{2}⟩) on X^{+}.

Example 2.18. Let q = 0, r = 1, that is, X is an non-singular variety. Put e_{4} = (1, a, −1), where a ≥ 1. Then f is a special flipping contraction of terminal type and we have the following description.

(1) The variety X^{+} has a singularity of type \frac{1}{a}(−1, 1, 1) outside S^{+} only.

(2) The surface S^{+} is non-singular.

(3) The relation is −e_{1} − ae_{2} + e_{3} + e_{4} = 0 and K_{X} · C = a − 1, S · C = −a.

Example 2.19. Let 0 < q < r, (q, r) = 1. In all cases the variety X has a singularity along the curve C only. Its type is \frac{1}{r}(−q, 1) × C. Let us consider four cases.

I) Let e_{4} = (a, 1, −1), where a = 1, 2. Then f is a special flipping contraction of terminal type and we have the following description.

(1) The variety X^{+} has a singularity of type \frac{1}{ar−q}(−r, −r, 1) on S^{+} only.

(2) The surface S^{+} has a singularity of type \frac{1}{ar−q}(1, 1). It is nonsingular if and only if a = 1, r = q + 1.

(3) The relation is −(a − \frac{q}{r})e_{1} − \frac{1}{r}e_{2} + e_{3} + e_{4} = 0 and K_{X} · C = a − 2 − \frac{q−1}{r}, S · C = −\frac{1}{r}.

II) Let e_{4} = (2a + 1, 2, −1), where either a = 0 and \frac{q}{r} > \frac{1}{2}, or a = 1. Then f is a special flipping contraction of terminal type and we have the following description.

(1) The variety X^{+} has the singularities of types \frac{1}{2}(1, 1, 1) outside S^{+} and \frac{1}{(2a+1)r−2q}(−q + ra, −q + ra, 1) on S^{+}.

(2) The surface S^{+} has a singularity of type \frac{1}{(2a+1)r−2q}(1, 1). It is non-singular if and only if a = 0, r = 2q + 1.

(3) The relation is −(2a+1− \frac{2q}{r})e_{1} − \frac{2}{r}e_{2} +e_{3} +e_{4} = 0 and K_{X} ·C = 2a − 1 − \frac{2q−2}{r}, S · C = −\frac{2}{r}.

III) Let e_{4} = (a + 1, a, −1), where 3 ≤ a < \frac{r}{r−q}. Then f is a special flipping contraction of terminal type and we have the following description.

(1) The variety X^{+} has the singularities of types \frac{1}{a}(−1, 1, 1) outside S^{+} and \frac{1}{a(r−q)+r}(1, r − q, r − q) on S^{+}.

(2) The surface S^{+} has a singularity of type \frac{1}{a(r−q)+r}(1, 1). It is singular always.

(3) The relation is −(a + 1 − \frac{qa}{r})e_{1} − \frac{a}{r}e_{2} + e_{3} + e_{4} = 0 and K_{X} · C = a − 1 − \frac{qa−a}{r}, S · C = −\frac{a}{r}.

IV) Let e_{4} = (1, a, −1), where 3 ≤ a < \frac{r}{q}. Then f is a special flipping contraction of terminal type and we have the following description.

(1) The variety X^{+} has the singularities of types \frac{1}{a}(−1, 1, 1) outside S^{+} and \frac{1}{(r−aq}(−q, −q, 1) on S^{+}.

(2) The surface S^{+} has a singularity of type \frac{1}{r−aq}(1, 1). It is nonsingular if and only if a = \frac{r−1}{q}.

(3) The relation is −(1 − \frac{qa}{r})e_{1} − \frac{a}{r}e_{2} + e_{3} + e_{4} = 0 and K_{X} · C = −1 − \frac{qa−a}{r}, S · C = −\frac{a}{r}.

Theorem 2.20. Let f: (X, S) → (Z ∋ P) be a three-dimensional special flipping contraction of terminal type. Assume that S is an nonsingular surface and X is an non-singular variety outside Exc f. If X is a singular variety then we also assume that Exc f is an irreducible curve. Then f is a toric contraction from examples 2.18 and 2.19.

Proof. If X is an non-singular variety then every irreducible exceptional curve is (−1) curve on a surface S. Therefore Exc f is an irreducible curve always. By the same argument as in lemma 2.12 and section 2.13 it is enough to prove that f is toric for a flopping contraction (with respect to K_{X}) of a smooth variety X. Similarly as in proof of theorem 2.4 (after lemma 2.12) we can show that

(Z ∋ P)≅_{an}(xy + z^{2} − t^{2l} ⊂ (ℂ^{4}, 0)) and f(S) = {x = z − t^{l} = 0}.

Hence the surface S has a Du Val singularity 𝔸_{l−1}. Therefore l = 1 and f: X → (Z ∋ P) is the Atiyah’s toric contraction (see example 2.2).

Put S = V (⟨e_{2}⟩). Consider two possibilities.

Let X be an non-singular variety. Then we may assume that e_{1} = (1, 0, 0), e_{2} = (0, 1, 0), e_{3} = (0, 0, 1) and e_{4} = (a, b, c), where a > 0, b > 0, c < 0. Since T_{N}(⟨e_{1}, e_{2}, e_{4}⟩) is an non-singular variety then c = −1. After the log flip T_{N}(⟨e_{2}, e_{3}, e_{4}⟩) is an non-singular variety by [22, Corollary 3.7]. Hence a = 1. We obtain that these log flips are from example 2.18.

Let X be a singular variety along curve C = Exc f and X is isomorphic to \frac{1}{r}(−q, 1) × C in some small analytical neighborhood of C, where q > 0. Then we may assume that e_{1} = (1, 0, 0), e_{2} = (q, r, 0), e_{3} = (0, 0, 1) and e_{4} = (a, b, c), where c < 0, b > 0 and a > \frac{qb}{r} > 0. Easy calculations show that c = −1 always. By terminal lemma (for example, see [12, §1.6]) the vectors e_{1}, e_{3}, e_{4} coincide in some ℤ-basis of N with (1, 0, 0), (0, 1, 0), (1, p, t) up to permutation, where either p = 0, t = 1, or 0 < p < t, (p, t) = 1. Easy calculations show that e_{4} = (α, 1, −1), e_{4} = (2α + 1, 2, −1) or e_{4} = (1 + kα, α, −1). Taking into account that (K_{X} + S) · C < 0 we obtain that these log flips are from example 2.19. □

Example 2.21. In an non-ℚ-factorial case the toric flips have more complicated structure. Let

Δ = {⟨e_{1}, e_{2}, e_{3}, e_{4}⟩, ⟨e_{3}, e_{4}, e_{5}⟩, their faces},

Δ′ = {⟨e_{1}, e_{2}, e_{3}, e_{4}, e_{5}⟩, their faces}

and

Δ^{+} = {⟨e_{1}, e_{2}, e_{5}⟩, ⟨e_{2}, e_{3}, e_{5}⟩, ⟨e_{1}, e_{4}, e_{5}⟩, their faces},

where e_{1} = (1, 0, 0), e_{2} = (0, 1, 0), e_{3} = (0, −1, 1), e_{4} = (−1, 0, 1), e_{5} = (a, b, c), a > 0, b > 0, c < 0, a+b+c < 0, (a, b, c) = 1. Let X = T_{N}(Δ), X^{+} = T_{N}(Δ^{+}), Z = T_{N}(Δ′), f: X → Z and f^{+} : X^{+} → Z. The reader can check the following statements:

(1) Exc f = C = V (⟨e_{3}, e_{4}⟩) is an irreducible curve;

(2) Exc f^{+} = C_{1}^{+} ∪C_{2}^{+} = V (⟨e_{1}, e_{5}⟩)∪V (⟨e_{2}, e_{5}⟩) is an union of two irreducible curves;

(3) the variety X has (at most) two singularities on C: \frac{1}{−(a+b+c)}(b, a, 1), x_{1}x_{2} + x_{3}x_{4} = 0 ⊂ (ℂ^{4}, 0);

(4) the variety X^{+} has (at most) three singularities: \frac{1}{−c}(−a, −b, 1) at the point C_{1}^{+} ∩ C_{2}^{+}, \frac{1}{a}(−c, −b − c, 1) on C_{1}^{+}, \frac{1}{b}(−a − c, −c, 1) on C_{2}^{+};

(5) the transformation X −−→ X^{+} is an anti-log flip with respect to

K_{X} + D, where D = V (⟨e_{5}⟩). Note also that the singularities on X and X^{+} can be non-isolated. If (a, b, c) = (1, 1, −3) then this log flip takes place in example 4.1.

3. Construction of model

In this section we consider the construction of model for a local contraction [18], [15].

3.1. Let f: (X, D) → (Z ∋ P) be a contraction of d-dimensional variety X with the following properties:

(1) every coefficient of a boundary D belongs to Φ^{d};

(2) the pair (X, D) is kawamata log terminal;

(3) the divisor −(K_{X} + D) is f-nef and f-big;

(4) dim Z > 0, Z is a germ near P;

(5) ρ(X/Z) = 1 and the variety X is ℚ-factorial.

Let us remark that same construction of model takes place without condition (5) (see [18]). In our case we obtain some information about S −−→ \overline{S} (see proposition 3.8 and section 4).

3.2. By Kodaira’s lemma there exists an effective ℚ-divisor D′ such that the pair (X, D + D′) is kawamata log terminal and the divisor −(K_{X} + D + D′) is ample over Z.

Let H = ∑ h_{i}H_{i} be an effective divisor on Z containing the point P, where H_{i} is a ℚ-Cartier divisor for all i. Let c = c(X, D + D′, f^{∗}H) be the log canonical threshold of a divisor f^{∗}H for the pair (X, D + D′). Put D′′ = D′ + cf^{∗}H. Then the divisor K_{X} + D + D′′ is anti-ample over Z, log canonical and not kawamata log terminal. Let us consider two cases A) and B).

3.3. A) Let the divisor K_{X}+D+D′′ be not purely log terminal. Take an inductive blow-up g: Y → X of the pair (X, D + D′′) [16, Proposition 3.1.4]. Write

K_{Y} + S + D_{Y} + D′′_{Y} = g^{∗}(K_{X} + D + D′′),

where D_{Y}, D′′_{Y} are the proper transforms of D and D′′. By the properties of an inductive blow-up the variety X is ℚ-factorial, Exc g = S is an irreducible divisor, ρ(Y/X) = 1, the divisor K_{Y} +S+D_{Y} +(1−ε)D′′_{Y} is purely log terminal and anti-ample over X for any ε > 0.

If the divisor −(K_{Y} + S + D_{Y}) is nef over Z then (Y, S) is a required model. Put \overline{Y} = Y, \overline{S} = S, \overline{D}_{Y} = D_{Y} and q = f ∘ g. It is clear that theorems 3.5 and 3.6 hold in this case. Therefore we assume that the divisor −(K_{Y} + S + D_{Y}) is not nef over Z.

The Mori cone \overline{NE}(Y/Z) is polyhedral and generated by two extremal rays since the divisor K_{Y} + S + D_{Y} + (1 − δ)D′′_{Y} is purely log terminal and anti-ample over Z for 0 < δ ≪ 1 (see [18, Lemma 3.9]). Let the prime divisors F_{1}, F_{2} generate N^{1}(Y/Z). For n ≫ 0 take a general element

F ∈ | − n(K_{Y} + S + D_{Y} + (1 − δ)D′′_{Y}) − F_{1} − F_{2}|

and put

B = (1 − δ)D′′_{Y} + \frac{1}{n}(F + F_{1} + F_{2}).

We obtain K_{Y} + S + D_{Y} + B ≡ 0 over Z. For some 0 < ε ≪ 1 the divisor K_{Y} + S + D_{Y} + (1 + ε)B is purely log terminal. Apply (K_{Y} + S + D_{Y} + (1 + ε)B)–MMP over Z. In the result one of two possibiliites holds [18], [15].

I) After a sequence of (K_{Y} + S + D_{Y} + (1 + ε)B)–flips we obtain the pair (\overline{Y} , \overline{S} + \overline{D}_{Y}) such that the divisor −(K\overline{Y} + \overline{S} + \overline{D}_{Y}) is nef over Z, ρ(\overline{Y} /Z) = 2.

[figure: Z; X; f; Y = Y_{1}; g = g_{1}; X_{1}; r_{1}; −−→; Y_{2}; g_{2}; Z; =; …; =; …; …; r_{n−1}; X_{n−1}; Z; g_{n}; Y_{n} = \overline{Y}; Z; =; q]

Proposition 3.4. [18, Lemma 3.11]. The divisor −(K\overline{S} + Diff\overline{S}(\overline{D}_{Y})) is nef and big over q(\overline{S}).

Proof. It is easily shown that the Mori cone \overline{NE}(Y_{i}/Z) is polyhedral and generared by two extremal rays. Hence, by base point free theorem the linear system |−m(K\overline{Y} +\overline{S} +\overline{D}_{Y})| is free over Z and gives a contraction h: \overline{Y} → W, where m ≫ 0. For every curve C in the fiber of h we have C · \overline{B} = 0. Therefore C · \overline{B}_{i} = 0 for some component \overline{B}_{i} of a divisor \overline{B} by the construction of B. Hence C ⊂ Supp(\overline{B}). □

II) After a sequence of (K_{Y} + S + D_{Y} + (1 + ε)B)–flips we obtain the pair (Y_{n}, S_{n} + (D_{Y})_{n}) and a divisorial contraction r_{n}: Y_{n} → \overline{Y} such that the divisor K_{Y_{n}} + S_{n} + (D_{Y})_{n} is r_{n}-ample. After it the divisor −(K\overline{Y} + \overline{S} + \overline{D}_{Y}) is ample over Z, ρ(\overline{Y} /Z) = 1.

[figure: Z; X; f; Y = Y_{1}; g = g_{1}; X_{1}; r_{1}; −−→; Y_{2}; g_{2}; Z; =; …; =; …; …; r_{n−1}; X_{n−1}; Z; g_{n}; Y_{n}; Y_{n+1} = \overline{Y}; r_{n}; Z; =; q]

Note that the divisor −(K\overline{S} + Diff\overline{S}(\overline{D}_{Y})) is ample over q(\overline{S}).

In both cases I) and II) we have the following properties of the constructed model.

Theorem 3.5. [18, Lemma 3.13] Let there exists an n-complement of K\overline{S} + Diff\overline{S}(\overline{D}_{Y}) near q^{−1}(P). Then there exists an n-complement of K_{X} + D near f^{−1}(P).

Proof. The n-complement of a divisor K\overline{S}+Diff\overline{S}(\overline{D}_{Y}) is extended to an n-complement K\overline{Y} +\overline{S}+\overline{D}_{Y}^{+} of a divisor K\overline{Y} +\overline{S}+\overline{D}_{Y} by theorem 1.8. In the case I) the image of K\overline{Y} +\overline{S} +\overline{D}_{Y}^{+} is an n-complement K_{Y} +S +D_{Y}^{+} of a divisor K_{Y} + S + D_{Y}. Therefore K_{X} + g(D_{Y}^{+}) is an n-complement of a divisor K_{X} + D by theorem 1.6. In the case II) a pull-back of K\overline{Y} + \overline{S} + \overline{D}_{Y}^{+} is an n-complement of a divisor K_{Y_{n}} + S_{n} + (D_{Y})_{n} by theorem 1.7. As above we obtain an n-complement of a divisor K_{X} + D. □

Theorem 3.6. [18, Corollary 3.18] The local contraction f: (X, D) → (Z ∋ P) is exceptional if and only if f(g(S)) = q(\overline{S}) = P and the log variety (\overline{S}, Diff\overline{S}(\overline{D}_{Y})) is exceptional.

Definition 3.7. The images of divisors, curves of Y on Y_{i} are denoted with index i.

Proposition 3.8. 1) The divisor −S_{i} is g_{i}-ample. In particular, Exc g_{i} ⊂ S_{i} [17, Lemma 4.3].

2) The sets Exc(S −−→ \overline{S}) and Exc(\overline{S} −−→ S) are connected.

Proof. Let us prove the first statement at first (it was proved in [17, Lemma 4.3] for the non-small contractions). Let the Mori cone \overline{NE}(Y_{i}/Z) be generated by the rays G_{i} and R_{i}. They gives the contractions g_{i} and r_{i} respectively.

We claim that the divisor −S_{i} is not nef over Z for i = 1, …, n − 1. Assume the converse. Consider three cases.

1) Let Exc(f) ≠ f^{−1}(P). Let l ⊂ g^{−1}(f^{−1}(P′)) be a curve, where P′ ≠ P. It is obvious that l_{i} does not generate an extremal ray. Since 0 = S_{i} · l_{i} = S_{i} · (αR_{i} + βQ_{i}), where α, β ≥ 0 then S_{i} · R_{i} = S_{i} · Q_{i} = 0. Hence S_{i} ≡ 0 over Z, a contradiction.

2) Let Exc(f) = f^{−1}(P) and f be a divisorial contraction to a point P. Then S_{i} = Exc(Y_{i} → Z), a contradiction with the construction.

3) Let Exc(f) = f^{−1}(P) and f be a small contraction. Then S_{i} = Exc(Y_{i} → Z). It is enough to prove that −(K_{Y_{i}} + S_{i} + (D_{Y})_{i}) is nef over Z. Let c′ = c(X, D, f^{∗}H) be a log canonical threshold. Without loss of generality it can be assumed that g is the inductive blow-up of a divisor K_{X} + D + cf^{∗}H. Let B′ be a proper transform of cf^{∗}H on Y. We have B′ + αS ≡ 0 over Z for some α > 0. Hence B′_{i} + αS_{i} ≡ 0 over Z and B′_{i} is nef over Z. Since ρ(X/Z) = 1 there are two possibilities.

Let K_{Y} + S + D_{Y} + B′ ≡ 0 over Z. Then the divisor −(K_{Y_{i}} + S_{i} + (D_{Y})_{i}) = B′_{i} is nef over Z.

Let K_{Y} +S +D_{Y} +B′ ≢ 0 over Z. Then −(K_{Y} +S +D_{Y} +(1−ε)B′) is ample over Z for some 0 < ε ≪ 1. It is clear that (K_{Y} + S + D_{Y} + (1 − ε_{1})B′) · R_{1} = 0 and (K_{Y} + S + D_{Y} + (1 − ε_{1})B′) · Q_{1} < 0 for some ε < ε_{1} < 1. After the log flip Y −−→ Y_{2} we have B′_{2} ·Q_{2} > 0, (K_{Y_{2}} +S_{2} + (D_{Y})_{2}+(1−ε_{1})B′_{2})·Q_{2} = 0 and (K_{Y_{2}} +S_{2}+(D_{Y})_{2}+(1−ε_{1})B′_{2})·R_{2} < 0. In fact, if (K_{Y_{2}} + S_{2} + (D_{Y})_{2} + (1 − ε_{1})B′_{2}) · R_{2} > 0 then the divisor K_{Y_{2}} +S_{2}+(D_{Y})_{2}+(1−ε_{1}+δ)B′_{2} is nef over Z for some 0 < δ ≪ ε_{1}−ε, a contradiction with [5, Lemma 6.3]. By the same argument on every step there is ε_{j−2} < ε_{j−1} < 1 such that (K_{Y_{j}} +S_{j}+(D_{Y})_{j}+(1−ε_{j−1})B′_{j})·Q_{j} = 0 and (K_{Y_{j}} + S_{j} + (D_{Y})_{j} + (1 − ε_{j−1})B′_{j}) · R_{j} < 0. Hence the divisor

−(K_{Y_{i}} +S_{i} +(D_{Y})_{i}) = −(K_{Y_{i}} +S_{i} +(D_{Y})_{i} +(1−ε_{i−1})B′_{i})+(1−ε_{i−1})B′_{i}

is nef over Z. This completes the proof of claim.

It is clear that S · Q_{1} < 0 and S · R_{1} > 0. Hence S_{2} · Q_{2} < 0. Assume our statement is proved on Y_{i−1}. Then S_{i−1} · Q_{i−1} < 0 and hence S_{i−1} · R_{i−1} > 0. After the log flip Y_{i−1} −−→ Y_{i} we have S_{i} · Q_{i} < 0. This completes the proof of first statement.

Let us prove the second one. Let it is true for S −−→ S_{i−1}. Let us prove it for S −−→ S_{i}. Assume the converse. We have three nontrivial factorizations of Y_{i−1} → Z. These are Y_{i−1} → X_{i−2} → Z, Y_{i−1} → X_{i−1} → Z and Y_{i−1} → W → Z, where Y_{i−1} → W is induced by r_{i}. We obtain a contradiction with ρ(Y_{i−1}/Z) = 2 by [5, Lemma 6.1]. □

3.9. B) Let the divisor K_{X} + D + D′′ be purely log terminal. Put S = ⌞D + D′′⌟. By connectedness lemma [5, Theorem 17.4] the divisor S is irreducible and hence normal. Let D = ∑ d_{i}D_{i}. Put

D_{X} = ∑_{i: D_{i}≠S} d_{i}D_{i} and D′′_{X} = D + D′′ − D_{X} − S.

Let the divisor −(K_{X} + S + D_{X}) be nef over Z. Then theorems 3.5 and 3.6 hold, where \overline{Y} = Y, \overline{S} = S, \overline{D}_{Y} = D_{X} and q = f. Let us remark that the local contraction f: (X, D) → (Z ∋ P) can be exceptional only in the cases when f is a divisorial contraction to a point P or dim Z = 1.

Let the divisor −(K_{X} + S + D_{X}) be not nef over Z. By the choice of a divisor H the contraction f is not small. Therefore f is a divisorial contraction and f(S) ≠ 0. Then an n-complement of K_{f(S)} + Diff_{f(S)}(f(D_{X})) near P is extended to an n-complement of K_{Z} + f(S) + f(D_{X}) by theorem 1.8. By theorem 1.7 we obtain an n-complement of K_{X} + S + D_{X} and K_{X} + D. In particular, the local contraction f: (X, D) → (Z ∋ P) is non-exceptional in this case.

4. Three-dimensional contractions

Now we give two examples illustrating the construction of model. In this examples the variety X is not ℚ-factorial. We can also apply our construction of model (without a ℚ-factorialization of X) since the inductive blow-up exists [6, Theorem 1.5]. See the paper [7] for the calculation details. By abuse of notation the coordinates are denoted t, z, x, y for every variety.

Example 4.1. Consider the three-dimensional terminal hypersurface singularity (Z ∋ P) = (t^{2} + z^{2} + x^{2}y + y^{4} = 0 ⊂ (ℂ^{4}, 0)). We have a following diagram.

[figure: (Z ∋ P); (X, E); f; (Y, S); g; (Ỹ, S̃); ψ; (\overline{Y} , \overline{S}); h; φ]

The birational morphism f: X → (Z ∋ P) is induced by the weighted blow-up of ℂ^{4} with weights (1, 1, 1, 1). Therefore

Exc f = E = t^{2} + z^{2} ⊂ ℙ(1, 1, 1, 1) = E^{1} + E^{2},

where E^{1} and E^{2} are the planes intersecting along the line l = E^{1} ∩E^{2}. It is obvious that Sing X = P_{1} ∪ P_{2} ⊂ l and

(X ∋ P_{1})≅_{an}(t^{2} + z^{2} + x^{2}y + y^{2} = 0 ⊂ (ℂ^{4}, 0)),
(X ∋ P_{2})≅_{an}(t^{2} + z^{2} + xy = 0 ⊂ (ℂ^{4}, 0)).

Thus X has the terminal singularities. Let D′ = \frac{1}{2}\{x = 0\}+ \frac{7}{8}\{t = 0\} be a divisor on Z. Then the divisor K_{X} +D + \frac{3}{8}E = f^{∗}(K_{Z} +D′) is log canonical, where D is a proper transform of D′. Moreover, the point P_{1} is the unique log canonical center of K_{X} + D + \frac{3}{8}E. Therefore the inductive blow-up g: Y → X of a pair (X, D + \frac{3}{8}E) at the point P_{1} is given by the weighted blow-up of ℂ^{4} with weights (2, 2, 1, 2). Then

Exc g = S = t^{2} + z^{2} + xy + y^{2} ⊂ ℙ(1, 1, 1, 1) = 𝔽_{0}

and Diff_{S}(0) = \frac{1}{2}C = \frac{1}{2}\{x = 0\}.

The birational morphism φ: \overline{Y} → (Z ∋ P) is induced by the weighted blow-up of ℂ^{4} with weights (4, 4, 3, 2). Therefore

Exc φ = \overline{S} = t^{2} + z^{2} + xy + y^{4} ⊂ ℙ(2, 2, 3, 1)

and Diff\overline{S}(0) = \frac{1}{2} \overline{C} = \frac{1}{2}\{x = 0\}. By Noether’s formula ρ(\overline{S}) = 2.

It is easily shown that the divisors S and \overline{S} have the same valuation of a function field K(Z), since they correspond to same toric blow-up of ℂ^{4}.

We can see that the divisor −(K_{Y} +S) is not nef over Z (for example, (K_{Y} + S) · l = 1). The birational map ψ: Y −−→ Ỹ is the anti-log flip with respect to K_{Y} + S in the curve l. This log flip Y → X_{1} ← Ỹ is toric by criterion 2.4 since K_{Y} + (E_{1})_{Y} + (E_{2})_{Y} + S + D_{1} + D_{2} ∼ 0 over X_{1} and σ(Y/X_{1}) = 2, where D_{i} are the proper transforms of the divisors \{y = t ± \sqrt{−1} · z = 0\} of Z. We can see that this log flip is of example 2.21. After this log flip the divisor −(KỸ + S̃) is not nef over Z again (for example, (KỸ + S̃) · l̃_{i} = \frac{1}{2}).

The birational morphism h: Ỹ → \overline{Y} induces the contraction of the images of E^{1} and E^{2}.

Let us consider the transformation S ← S̃ → \overline{S}.

[figure: S = 𝔽_{0}; l_{2}; C; l_{1}; P; S̃; C̃; l̃_{1}; \frac{1}{3}(1, 1); l̃_{2}; \overline{S}; \overline{C}; \frac{1}{3}(1, 1); 𝔸_{1}; 𝔸_{1}]

In this diagram l_{i} = S ∩ E_{Y}^{i}, where E_{Y}^{i} are the proper transforms of E^{i} on Y. The curves l_{i} are the rulings of 𝔽_{0}. The log flip ψ induces a divisorial extraction at the point P = l_{1} ∩ l_{2}. After it l̃_{i}^{2} = −2 and two exceptional curves appear. They intersect at the singularity \frac{1}{3}(1, 1). The birational morphism h induces the contraction of l̃_{1}, l̃_{2}.

The divisor K\overline{S} + Diff\overline{S}(0) has 1-complement K\overline{S} + \{x = 0\} + \{y = 0} ∼ 0. Thus K_{X} has 1-complement. It is

K_{X} + E + {x = 0}_{X} + {y = 0}_{X} = f^{∗}(K_{Z} + {x = 0} + {y = 0}) ∼ 0.

Example 4.2. Consider the three-dimensional exceptional canonical singularity (Z ∋ P) = (t^{3} + z^{3} + x^{4} + y^{n} = 0 ⊂ (ℂ^{4}, 0)), where n = 5, 7, 11 [7]. In the notations of example 4.1 let the birational morphisms f, φ be induced by the weighted blow-ups of ℂ^{4} with weights (1, 1, 1, 1) and (4n, 4n, 3n, 12) respectively. We have

(\overline{S}, Diff\overline{S}(0)) = (t^{3} + z^{3} + x + y ⊂ ℙ(1, 1, 3, 3), \frac{3}{4}\{x = 0\}+

+\frac{n − 1}{n}\{y = 0\}) = (ℙ(1, 1, 3), \frac{3}{4}\overline{C}_{1} + \frac{n − 1}{n}\overline{C}_{2}),

where \overline{C}_{i} are the general hyperplane sections of a cone.

Also

Exc f = E = t^{3} + z^{3} ⊂ ℙ(1, 1, 1, 1) = E^{1} + E^{2} + E^{3},

where E^{i} are the planes intersecting along the line l = E^{1} ∩ E^{2} ∩ E^{3}. It is obvious that Sing X = P_{1} ⊂ l and

(X ∋ P_{1})≅_{an}(t^{3} + z^{3} + x^{4}y + y^{n−3} = 0 ⊂ (ℂ^{4}, 0)).

Note that K_{X} ≡ 0 over Z (in particular −K_{X} is f-nef and f-big). Thus the divisors K_{Y} + S and KỸ + S̃ are not nef over Z. If n = 5 then X is a terminal variety.

Put D′ = (1 − \frac{n}{12})\{y = 0\} and take an inductive blow-up g at the point P_{1}. It is induced by the weighted blow-up of ℂ^{4} with weights (4n − 12, 4n − 12, 3n − 12, 12). We obtain

(S, Diff_{S}(0)) = (t^{3} + z^{3} + xy + y^{n−3} ⊂ ℙ(n − 3, n − 3, 3n − 12, 3),

\frac{3}{4}C_{1} = \frac{3}{4}\{x = 0\}).

Let us consider the transformation S ← S̃ → \overline{S}.

[figure: 𝔸_{n−4}; l_{3}; 𝔸_{n−4}; C̃_{1}; l_{1}; 𝔸_{n−4}; l_{2}; \frac{1}{3n−12}(1, 1); \frac{1}{3}(1, 1); S; 𝔸_{n−4}; l̃_{3}; 𝔸_{n−4}; C̃_{1}; l̃_{1}; 𝔸_{n−4}; l̃_{2}; \frac{1}{3}(1, 1); S̃; C̃_{2}; \overline{S} = ℙ(1, 1, 3)]

The log flip ψ induces the minimal resolution of a singularity \frac{1}{3n−12}(1, 1). After it a new component C̃_{2} of Diff appears, that is, Ỹ has a singularity along the flipped curve C̃_{2}. This log flip Y → X_{1} ← Ỹ is toric by criterion 2.4 since K_{Y} + (E_{1})_{Y} + (E_{2})_{Y} + S + D ∼ 0 over X_{1} and σ(Y/X_{1}) = 1, where D is the proper transforms of a divisor {y = 0} of Z. This log flip is from example 2.19 I). If n = 5, 7, 11 then (q, r, a) = (2, 5, 1), (5, 7, 2), (1, 11, 2) respectively.

The birational morphism h contracts the curves l̃_{i} = Ẽ^{i} ∩ S̃ to the smooth points of \overline{S}, where Ẽ^{i} are the images of E^{i} on Ỹ.

The divisor K\overline{S} + Diff\overline{S}(0) has 5-, 8-, 12-complement for n = 5, 7, 11 respectively. Hence K_{X} has 5-, 8-, 12-complement for n = 5, 7, 11 respectively.

The next conjecture is the analog of Reid’s general elephant conjecture in the theory of complements.

Conjecture 4.3. Let f: X → (Z ∋ P) be a contraction of threedimensional variety X with the following properties:

(1) the variety X has the ℚ-factorial terminal singularities;

(2) the divisor −K_{X} is f-ample;

(3) dim Z > 0, Z is a germ near P and ρ(X/Z) = 1. Then this contraction is not exceptional, in particular there exists 1-, 2-, 3-, 4- or 6-complement of a divisor K_{X}.

Remark 4.4. If the contraction f: X → (Z ∋ P) is not exceptional then the log surface (\overline{S}, Diff\overline{S}(0)) is not exceptional by theorem 3.6. Hence there exists 1-, 2-, 3-, 4- or 6-complement of a divisor K\overline{S} + Diff\overline{S}(0) by [23, Theorem 2.3] and this n-complement is extended to an n-complement of K_{X} by theorem 3.5.

Remark 4.5. It is worth to note that previous conjecture 4.3 can be generelized by many ways. For instance, we can reject from the assumptions that the variety X is ℚ-factorial and ρ(X/Z) = 1. If the variety X is not ℚ-factorial then the contraction X → (Z ∋ P) is exceptional if and only if the contraction X̃ → (Z ∋ P) is exceptional, where X̃ is a ℚ-factorialization of X. Let us remark that the divisor −KX̃ is nef and big over Z and not ample over Z.

Finally note that we cannot reject from the condition (2) and assume only that the divisor −K_{X} is f-nef and f-big by example 4.2 (n = 5).

Remark 4.6. 1) Conjecture 4.3 was proved in some cases.

A) If f is a divisorial contraction then it is non-exceptional by example 1.10. Besides, for a divisorial contraction it is enough to require that (Z ∋ P) is a terminal singularity only.

B) If f is a divisorial contraction to the point P then the divisor K_{X} has a strong 1-complement D^{+}, that is, K_{X} + D^{+} ∼ 0 is a purely log terminal divisor [2].

C) Conjecture 4.3 was proved if any singularity of X has an index one or two and the relative dimension of f is equal to one [4], [13].

D) Some special case was considered in the paper [17].

2) Reid’s general elephant conjecture under the assumption that the central fiber (over the point P) is irreducible was proved for the birational contractions [4], [2]. If X → (Z ∋ P) is a Mori conic bundle then see the paper [13] for some results about the general elephant conjecture in this case.

Theorem 4.7. Conjecture 4.3 must be proved in the case A) I) for the non-divisorial contractions (see chapter 3) only. Moreover,

\overline{S} = S_{n} → S_{n−1} → … → S_{1} = S

is a birational contraction with the following properties.

(1) The set Exc(\overline{S} → S) is connected.

(2) θ_{i}: S_{i} → S_{i−1} is an non-trivial divisorial contraction for every i = n, …, 2. The divisor −(K_{S_{i}} + Diff_{S_{i}}(0)) is θ_{i}-ample.

(3) If C is an irreducible curve from Exc θ_{i} then there are two possibilities.

(a) Let C ⊄ Sing Y_{i}. Then Diff_{S_{i}}(0) = (θ_{i})_{∗}^{−1}(Diff_{S_{i−1}}(0)) near

a curve C.

(b) Let C ⊂ Sing Y_{i}. Then Diff_{S_{i}}(0) = (θ_{i})_{∗}^{−1}(Diff_{S_{i−1}}(0)) +

\frac{n−1}{n}C near a curve C for some n ≥ 2.

Proof. By example 1.10 and remark 4.6 we can assume that f is an non-divisorial contraction. The log surface (S, Diff_{S}(0)) is not exceptional since it is exceptional divisor of an inductive blow-up (purely log terminal blow-up) of some terminal singularity [14]. Therefore, if the divisor −(K_{Y} + S) is nef over Z then f is not exceptional. Hence we have the cases A) and B) in the construction of model. It is obvious that dim Z = 1 in the case B) and in the case A) II). Thus, f is not exceptional by [17, Proposition 3.1]. Therefore, the case A) I) must be considered only.

It remains to show that S_{j} → S_{j−1} is a birational morphism by proposition 3.8. The key observation is that if we change the divisor H then the construction of model for an new H will coincide with the previous construction. Let H_{i} denotes the image of a divisor H on Y_{i}. If f has a relative dimension one then H_{i} ∩ S_{i} is a general curve of S_{i} and H_{i} does not contain any irreducible curve from Exc g_{i}. Since −H_{j} is r_{j}-nef then any flipping curve of r_{j}: Y_{j} → X_{j} does not lie in S_{j}.

Consider the last case dim Z = 1. Let F be a general fiber of f (it is a smooth del Pezzo surface). Consider the exact sequence

0 → O_{X}(−K_{X} − F) → O_{X}(−K_{X}) → O_{F} (−K_{F}) → 0.

By Kawamata-Viehweg vanishing theorem R^{1}f_{∗}O_{X}(−K_{X} − F) = 0. Hence there is a surjection

H^{0}(X, O_{X}(−K_{X})) → H^{0}(F, O_{F} (−K_{F})) → 0.

Therefore h^{0}(X, O_{X}(−K_{X})) ≥ 2. Let G ∈ |−K_{X}| be a general element. Consider the G instead of f^{∗}H and apply above-mentioned arguments.

□

Remark 4.8. The transformations from items (a) and (b) are illustrated in examples 4.1 and 4.2 respectively. Also such examples can be constructed in the ℚ-factorial case too.

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Department of Algebra, Faculty of Mathematics, Moscow State University, 117234 Moscow, Russia

  E-mail address: kudryav@mech.math.msu.su

Max-Plank-Institut für Mathematik, Vivatsgasse 7, D-53111 Bonn, Germany

  E-mail address: kudryav@mpim-bonn.mpg.de