ℚ-Fano Threefolds of Large Fano Index, I

Yuri Prokhorov^{∗}

Received: January 6, 2009 Revised: February 8, 2010

Communicated by Thomas Peternell

Abstract. We study ℚ-Fano threefolds of large Fano index. In particular, we prove that the maximum possible Fano index is attained only by the weighted projective space ℙ(3, 4, 5, 7).

2010 Mathematics Subject Classification: 14J45, 14J30, 14E30. Keywords and Phrases: ℚ-Fano variety, Sarkisov link, linear system

1. Introduction

The Fano index of a smooth Fano variety X is the maximal integer q(X) that divides the anti-canonical class in the Picard group Pic(X) [IP99]. It is wellknown [KO73] that q(X) ≤ dim X + 1. Moreover, q(X) = dim X + 1 if and only if X is a projective space and q(X) = dim X if and only if X is a quadric hypersurface. In this paper we consider generalizations of Fano index for the case of singular Fanos admitting terminal singularities. A normal projective variety X is called Fano if some positive multiple −nK_{X} of its anti-canonical Weil divisor is Cartier and ample. Such X is called a ℚ-Fano variety if it has only terminal ℚ-factorial singularities and its Picard number is one. This class of Fano varieties is important because they appear naturally in the Minimal Model Program. For a singular Fano variety X the Fano index can be defined in different ways. For example, we can define

qW(X) := max{q | −K_{X} ∼ qA, A is a Weil ℚ-Cartier divisor},
qℚ(X) := max{q | −K_{X} ∼_{ℚ} qA, A is a Weil ℚ-Cartier divisor}.

If X has at worst log terminal singularities, then the Picard group Pic(X) and Weil divisor class group Cl(X) are finitely generated and Pic(X) is torsion free (see e.g. [IP99, §2.1]). Moreover, the numerical equivalence of ℚ-Cartier divisors coincides with ℚ-linear one. This implies, in particular, that the Fano indices qW(X) and qℚ(X) defined above are positive integers. If X is smooth, these numbers coincide with the Fano index q(X) defined above. Note also that qℚ(X) = qW(X) if the group Cl(X) is torsion free.

∗The author was partially supported by the Russian Foundation for Basic Research (grants No 06-01-72017-MNTI a, 08-01-00395-a) and Leading Scientific Schools (grants No NSh-1983.2008.1, NSh-1987.2008.1)

Theorem 1.1 ([Suz04]). Let X be a ℚ-Fano threefold. Then qW(X) ∈ {1, …, 11, 13, 17, 19}. All these values, except possibly for qW(X) = 10, occur. Moreover, if qW(X) = 19, then the types of non-Gorenstein points and Hilbert series of X coincide with that of ℙ(3, 4, 5, 7).

It can be easily shown (see proof of Proposition 3.6) that the index qℚ(X) takes values in the same set {1, …, 11, 13, 17, 19}. Thus one can expect that ℙ(3, 4, 5, 7) is the only example of ℚ-Fano threefolds with qℚ(X) = 19. In general, we expect that Fano varieties with extremal properties (maximal degree, maximal Fano index, etc.) are quasihomogeneous with respect to an action of some connected algebraic group. This is supported, for example, by the following facts:

Theorem 1.2 ([Pro05], [Pro07]). (i) Let X be a ℚ-Fano threefold. Assume that X is not Gorenstein. Then −K_{X}^{3} ≤ 125/2 and the equality holds if and only if X is isomorphic to the weighted projective space ℙ(1^{3}, 2).

(ii) Let X be a Fano threefold with canonical Gorenstein singularities. Then −K_{X}^{3} ≤ 72 and the equality holds if and only if X is isomorphic to ℙ(1^{3}, 3) or ℙ(1^{2}, 6, 4).

The following proposition is well-known (see, e.g., [BB92]). It is an easy exercise for experts in toric geometry.

Proposition 1.3. Let X be a toric ℚ-Fano 3-fold. Then X is isomorphic to either ℙ^{3}, ℙ^{3}/μ_{5}(1, 2, 3, 4), or one of the following weighted projective spaces:

ℙ(1^{3}, 2), ℙ(1^{2}, 2, 3), ℙ(1, 2, 3, 5), ℙ(1, 3, 4, 5), ℙ(2, 3, 5, 7), ℙ(3, 4, 5, 7).

We characterize the weighted projective spaces above in terms of Fano index. The following is the main result of this paper.

Theorem 1.4. Let X be a ℚ-Fano threefold. Then qℚ(X) ∈ {1, …, 11, 13, 17, 19}.

(i) If qℚ(X) = 19, then X ≃ ℙ(3, 4, 5, 7). (ii) If qℚ(X) = 17, then X ≃ ℙ(2, 3, 5, 7). (iii) If qℚ(X) = 13 and dim | − K_{X}| > 5, then X ≃ ℙ(1, 3, 4, 5). (iv) If qℚ(X) = 11 and dim | − K_{X}| > 10, then X ≃ ℙ(1, 2, 3, 5).

(v) qℚ(X) ≠ 10.

(vi) If qℚ(X) ≥ 7 and there are two effective Weil divisors A ≠ A_{1} such

that −K_{X} ∼_{ℚ} qℚ(X)A ∼_{ℚ} qℚ(X)A_{1}, then X ≃ ℙ(1^{2}, 2, 3). (vii) If qW(X) = 5 and dim | − \frac{1}{5}K_{X}| > 1, then X ≃ ℙ(1^{3}, 2).

Note that in cases (iii) and (iv) assumptions about | − K_{X}| are really needed. Indeed, there are examples of non-toric ℚ-Fano threefolds with qℚ(X) = 13 and 11.

ℚ-Fano Threefolds of Large Fano Index, I 845

Example 1.5 ([BS07], see also Proposition 3.6). Let X = X_{d} ⊂ ℙ(a_{1}, …, a_{5}) be a general hypersurface of degree d. Assume that X is a ℚ-Fano with qℚ(X) ≥ 10 and such that 𝒪_{ℙ}(1)|_{X} is a primitive element of Cl(X), then X is one of the following:

(i) X = X_{12} ⊂ ℙ(1, 4, 5, 6, 7), qℚ(X) = 11, dim | − K_{X}| = 10;

(ii) X = X_{10} ⊂ ℙ(2, 3, 4, 5, 7), qℚ(X) = 11, dim | − K_{X}| = 8;
(iii) X ≃ X_{12} ⊂ ℙ(3, 4, 5, 6, 7), qℚ(X) = 13, dim | − K_{X}| = 5.

In the proof we follow the use some techniques developed in our previous paper [Pro07]. By Proposition 1.3 it is sufficient to show that our ℚ-Fano X is toric. First, as in [Suz04], we apply the orbifold Riemann-Roch formula to find all the possibilities for the numerical invariants of X. In all cases there is some special element S ∈ |−K_{X}| having four irreducible components. This S should be a toric boundary, if X is toric. Further, we use birational transformations like Fano-Iskovskikh “double projection” [IP99] (see [Ale94] for the ℚ-Fano version). Typically the resulting variety is a Fano-Mori fiber space having “simpler” structure. In particular, its Fano index is large if this variety is a ℚ-Fano. By using properties of our “double projection” we can show that the pair (X, S) is log canonical (LC). Then, in principle, the assertion follows by Shokurov’s toric conjecture [McK01]. We prefer to propose an alternative, more explicit proof. In fact, the image of X under “double projection” is a toric variety and the inverse map preserves the toric structure. In the last section we describe Sarkisov links between toric ℚ-Fanos that start with blow ups of singular points.

Acknowledgements. The work was conceived during the author’s stay at the University of Warwick in the spring of 2008. The author is grateful to Professor M. Reid for invitation, hospitality and fruitful discussions. Part of the work was done at Max-Planck-Institut für Mathematik, Bonn in August 2008. Finally, the author would like to thank the referee for careful reading the manuscript and constructive suggestions.

2. Preliminaries, the orbifold Riemann-Roch formula and its applications

Notation. Throughout this paper, we work over the complex number field ℂ. We employ the following standard notation: ∼ denotes linear equivalence; ∼_{ℚ} denotes ℚ-linear equivalence. Let E be a rank one discrete valuation of the function field ℂ(X) and let D be a ℚ-Cartier divisor on X. a(E, D) denotes the discrepancy of E with respect to a boundary D. Let f: X̃ → X be a birational morphism such that E appears as a prime divisor on X̃. Then ord_{E}(D) denotes the coefficient of E in f^{∗}D.

2.1. The orbifold Riemann-Roch formula [Rei87]. Let X be a threefold with terminal singularities and let D be a Weil ℚ-Cartier divisor on X. Let B = {(r_{P} , b_{P})} be the basket of singular points of X [Mor85a], [Rei87]. Here each pair (r_{P} , b_{P}) correspond to a point P ∈ B of type \frac{1}{r_{P}} (1, −1, b_{P}). For brevity, describing a basket we will list just indices of singularities, i.e., we will write B = {r_{P} } instead of B = {(r_{P} , b_{P})}. In the above situation, the Riemann-Roch formula has the following form

(2.2) χ(D) = \frac{1}{12}D · (D − K_{X}) · (2D − K_{X})+

+ \frac{1}{12}D · c_{2} + ∑_{P ∈B} c_{P} (D) + χ(𝒪_{X}),

where

c_{P} (D) = −i_{P} \frac{r_{P}^{2} − 1}{12r_{P}} + ∑_{j=1}^{i_{P} −1} \frac{\overline{b_{P} j}(r_{P} − \overline{b_{P} j})}{2r_{P}}.

Clearly, computing c_{P} (D), we always may assume that 1 ≤ b_{P} ≤ r_{P} /2.

2.3. Now let X be a Fano threefold with terminal singularities, let q := qℚ(X), and let A be an ample Weil ℚ-Cartier divisor on X such that −K_{X} ∼_{ℚ} qA. By (2.2) we have

(2.4)  χ(tA) = χ(𝒪_{X}) + \frac{t(q + t)(q + 2t)}{12}A^{3} + \frac{tA · c_{2}}{12} + ∑_{P ∈B} c_{P} (tA),

where χ(𝒪_{X}) = 1 and

c_{P} (tA) = −i_{P,t} \frac{r_{P}^{2} − 1}{12r_{P}} + ∑_{j=1}^{i_{P,t}−1} \frac{\overline{b_{P} j}(r_{P} − \overline{b_{P} j})}{2r_{P}}.

If q > 2, then χ(−A) = 0. Using this equality we obtain (see [Suz04])

(2.5)  A^{3} = \frac{12}{(q − 1)(q − 2)} (1 − \frac{A · c_{2}}{12} + ∑_{P ∈B} c_{P} (−A)) .

In the above notation, applying (2.2), Serre duality and Kawamata-Viehweg vanishing to D = K_{X}, we get the following important equality (see, e.g., [Rei87]):

(2.6)  24 = −K_{X} · c_{2}(X) + ∑_{P ∈B} (r_{P} − \frac{1}{r_{P}} ) .

Theorem 2.7 ([Kaw92a], [KMMT00]). In the above notation,

(2.8)  −K_{X} · c_{2}(X) ≥ 0,  ∑_{P ∈B} (r_{P} − \frac{1}{r_{P}} ) ≤ 24.

Proposition 2.9. Let X be a Fano threefold with terminal singularities and let Ξ be an n-torsion element in the Weil divisor class group. Let B^{Ξ} be the collection of points P ∈ B where Ξ is not Cartier. Then

(2.10)  2 = ∑_{P ∈B^{Ξ}} \frac{\overline{b_{P} i_{Ξ,P}} (r_{P} − \overline{b_{P} i_{Ξ,P}} )}{2r_{P}}.

Documenta Mathematica 15 (2010) 843–872

where i_{Ξ,P} is taken so that Ξ ∼ i_{Ξ,P} K_{X} near P ∈ B and is the residue mod r_{P}. Assume furthermore that n is prime. Then

(i) n ∈ {2, 3, 5, 7}.

(ii) If n = 7, then B^{Ξ} = (7, 7, 7).^{†}
(iii) If n = 5, then B^{Ξ} = (5, 5, 5, 5), (10, 5, 5), or (10, 10).
(iv) If n = 3, then ∑_{P ∈B^{Ξ}} r_{P} = 18.
(v) If n = 2, then ∑_{P ∈B^{Ξ}} r_{P} = 16.

Proof. By Kawamata-Viehweg vanishing theorem, Riemann-Roch (2.2), and Serre duality we have χ(𝒪_{X}) = 1,

\begin{aligned}0 & = χ(Ξ) & = 1 + ∑_{P} c_{P} (Ξ), \\ 0 & = χ(K_{X} + Ξ) & = 1 + \frac{1}{12}K_{X} · c_{2}(X) + ∑_{P ∈B} c_{P} (K_{X} + Ξ). \end{aligned}

Subtracting we get

0 = −\frac{1}{12}K_{X} · c_{2}(X) + ∑_{P ∈B}(c_{P} (Ξ) − c_{P} (K_{X} + Ξ)).

Since ni_{Ξ,P} ≡ 0 mod r_{P},

0 = −\frac{1}{12}K_{X} · c_{2}(X) + \frac{1}{12} ∑_{P ∈B} (r_{P} − \frac{1}{r_{P}} ) − ∑_{P ∈B} \frac{\overline{b_{P} i_{Ξ,P}} (r_{P} − \overline{b_{P} i_{Ξ,P}} )}{2r_{P}}.

This proves (2.10). Now assume that n is prime. If P ∈ B^{Ξ}, then n | r_{P}. Write r_{P} = nr′_{P}. Since r_{P} | ni_{P}, i_{P} = r′_{P} i′_{P}, where n ∤ i′_{P}. Let \overline{( )_{n}} be the residue mod n. Then

2 = ∑_{P ∈B^{Ξ}} \frac{\overline{b_{P} i′_{Ξ,P} r′} (nr′_{P} − \overline{b_{P} i′_{Ξ,P} r′_{P}} )}{2nr′_{P}} = \frac{r′_{P} \overline{(b_{P} i′_{Ξ,P})_{n}} (n − \overline{(b_{P} i′_{Ξ,P})_{n}})}{2n}.

Therefore,

4n^{2} = ∑_{P ∈B^{Ξ}} r_{P} \overline{(b_{P} i′_{Ξ,P})_{n}} (n − \overline{(b_{P} i′_{Ξ,P})_{n}}) .

Denote ξ_{P} := \overline{(b_{P} i′_{Ξ,P})_{n}}. Then 0 < ξ_{P} < n, gcd(n, ξ_{P}) = 1, and

4n = ∑_{P ∈B^{Ξ}} r′_{P} ξ_{P} (n − ξ_{P}) ≥ \frac{n^{2}}{4} ∑_{P ∈B^{Ξ}} r′_{P} , 16 ≥ n ∑_{P ∈B^{Ξ}} r′_{P} .

If n ≥ 11, then ∑ r′_{P} = 1, n | r′_{P}, and r_{P} ≥ n^{2} ≥ 121, a contradiction. Therefore, n ≤ 7. Consider the case n = 7. Then ξ_{P} (n − ξ_{P}) = 6, 10, or 12. The only solution is B^{Ξ} = (7, 7, 7). The case n = 5 is considered similarly. If n = 3, then ξ_{P} (n − ξ_{P}) = 3 and ∑ r_{P} = 3 ∑ r′_{P} = 18. Similarly, if n = 2, then ξ_{P} (n − ξ_{P}) = 1 and ∑ r_{P} = 2 ∑ r′_{P} = 16. This finishes the proof. □

†More delicate computations show that this case does not occur. (We do not need this.)

3. Computations with Riemann-Roch on ℚ-Fano threefolds of large Fano index

Lemma 3.1 (see [Suz04]). Let X be a Fano threefold with terminal singularities with q := qW(X), let A := −\frac{1}{q}K_{X}, and let r be the Gorenstein index of X. Then

(i) r and q are coprime; (ii) rA^{3} is an integer.

Lemma 3.2. Let X be a Fano threefold with terminal singularities. (i) If −K_{X} ∼ qL for some Weil divisor L, then q divides qW(X). (ii) If −K_{X} ∼_{ℚ} qL for some Weil divisor L, then q divides qℚ(X). (iii) qW(X) divides qℚ(X). (iv) Let q := qℚ(X) and let K_{X} + qA ∼_{ℚ} 0. If the order of K_{X} + qA in the

group Cl(X) is prime to q, then qW(X) = qℚ(X).

Proof. To prove (i) write −K_{X} ∼ qW(X)A and let d = gcd(qW(X), q). Then d = u qW(X) + vq for some u, v ∈ ℤ. Hence, dA = u qW(X)A + vqA ∼ quL + qvA = q(uL + vA). Since A is a primitive element of Cl(X), q = d and q | qW(X). (ii) can proved similarly and (iii) is a consequence of (ii). To show (iv) assume that Ξ := K_{X} + qA is of order n. By our condition qu + nv = 1, where u, v ∈ ℤ. Put A′ := A − uΞ. Then qA′ = qA − quΞ = qA − Ξ ∼ −K_{X}. Hence, q = qW(X) by (i) and (iii). □

Lemma 3.3. Let X be a Fano threefold with terminal singularities. (i) qℚ(X) ∈ {1, …, 11, 13, 17, 19}.

(ii) If qℚ(X) ≥ 5, then −K_{X}^{3} ≤ 125/2.

Proof. Denote q := qℚ(X) and write, as usual, −K_{X} ∼_{ℚ} qA. Thus n(K_{X} + qA) ∼ 0 for some positive integer n. The element K_{X} + qA defines a cyclic étale in codimension one cover π: X′ → X so that X′ is a Fano threefold with terminal singularities and K_{X′} + qA′ ∼ 0, where A′ := π^{∗}A. Let σ: X′′ → X′ be a ℚ-factorialization. (If X′ is ℚ-factorial, we take X′′ = X′). Run K-MMP on X′′: ψ: X′′ −−→ X̄. At the end we get a Mori-Fano fiber space X̄ → Z. Let A′′ := σ^{−1}(A′) and Ā := ψ_{∗}A′′. Then −KX̄ ∼ qĀ. If dim Z > 0, then for a general fiber F of X̄/Z, we have −K_{F} ∼ qĀ|_{F}. This is impossible because q > 3. Thus dim Z = 0 and X̄ is a ℚ-Fano.

(i) By Lemma 3.2 the number q divides qW(X̄). On the other hand, by Theorem 1.1 we have qW(X̄) ∈ {1, …, 11, 13, 17, 19}. This proves (i). To show (ii) we note that −K3 ; X̄ ≥ −K_{X′′}^{3} = −K_{X′}^{3} ≥ −K_{X′′}^{3}. Here the first inequality holds because for Fanos (with at worst log terminal singularities) the number −\frac{1}{6}K^{3} is nothing but the leading term in the asymptotic Riemann-Roch and dim | − tK_{X′′}| ≤ dim | − tKX̄|. Now the assertion of (ii) follows from Theorem 1.2. □

From Lemmas 3.2 and 3.3 we have Corollary 3.4. Let X be a Fano threefold with terminal singularities.

(i) If −K_{X} ∼ qL for some Weil divisor L and q ≥ 5, then q = qW(X). (ii) If −K_{X} ∼_{ℚ} qL for some Weil divisor L and q ≥ 5, then q = qℚ(X).

Lemma 3.5 (cf. [Suz04]). Let X be a Fano threefold with terminal singularities and let q := qW(X). Assume that qW(X) ≥ 8. Then one of the following holds:

q = 8, B = (3^{2}, 5), (3^{2}, 5, 9), (3, 5, 11), (3, 7), (3, 9), (5, 7), (7, 11),

(7, 13), (11),
q = 9, B = (2, 4, 5), (2^{3}, 5, 7), (2, 5, 13),
q = 10, B = (7, 11),
q = 11, B = (2, 3, 5), (2, 5, 7), (2^{2}, 3, 4, 7),
q = 13, B = (3, 4, 5), (2, 3^{2}, 5, 7),
q = 17, B = (2, 3, 5, 7),
q = 19, B = (3, 4, 5, 7).

In all cases the group Cl(X) is torsion free.

Proof. We use a computer program written in PARI [PARI] ^{‡}. Below is the description of our algorithm. Step 1. By Theorem 2.7 we have ∑_{P ∈B}(1 − 1/r_{P}) ≤ 24. Hence there is only a finite (but very huge) number of possibilities for the basket B = {[r_{P} , b_{P}]}. In each case we know −K_{X} · c_{2}(X) from (2.6). Let r := lcm({r_{P} }) be the Gorenstein index of X. Step 2. By Lemma 3.3 qℚ(X) ∈ {8, …, 11, 13, 17, 19}. Moreover, the condition gcd(q, r) = 1 (see Lemma 3.1) eliminates some possibilities. Step 3. In each case we compute A^{3} and −K_{X}^{3} = q^{3}A^{3} by formula (2.5). Here, for D = −A, the number i_{P} is uniquely determined by qi_{P} ≡ b_{P} mod r_{P} and 0 ≤ i_{P} < r_{P}. Further, we check the condition rA^{3} ∈ ℤ (Lemma 3.1) and the inequality −K_{X}^{3} ≤ 125/2 (Lemma 3.3). Step 4. Finally, by the Kawamata-Viehweg vanishing theorem we have χ(tA) = h^{0}(tA) for −q < t. We compute χ(tA) by using (2.4) and check conditions χ(tA) = 0 for −q < t < 0 and χ(tA) ≥ 0 for t > 0. At the end we get our list. To prove the last assertion assume that Cl(X) contains an n-torsion element Ξ. Clearly, we also may assume that n is prime. By Proposition 2.9 we have ∑_{n|r_{i}} r_{i} ≥ 16. Moreover, ∑_{n|r_{i}} r_{i} ≥ 18 if n = 3. This does not hold in any case of our list. □

Proposition 3.6. Let X be a ℚ-Fano threefold with qℚ(X) ≥ 9. Let q := qℚ(X) and let −K_{X} ∼_{ℚ} qA. Then the group Cl(X) is torsion free, qW(X) = qℚ(X), and one of the following holds:

‡The PARI code is available at http://mech.math.msu.su/department/algebra/staff/ prokhorov/q-fano.

dim |kA|^{q  B  A^{3} |A| |2A| |3A| |4A| |5A| |6A| |7A| | − K|}
9  (2, 4, 5)  \frac{1}{20} 0  1  2  4  6  8  11  19
9 (2, 2, 2, 5, 7) \frac{1}{70} −1 0  0  1  1  2  3  5
10  (7, 11)  \frac{2}{77} −1 0  1  1  3  4  6  13
11 (2, 3, 5)  \frac{1}{30} 0  1  2  3  5  7  9  23
11 (2, 5, 7)  \frac{1}{70} 0  0  0  1  2  3  4  10
11 (2, 2, 3, 4, 7) \frac{1}{84} −1 0  0  1  1  2  3  8
13 (3, 4, 5)  \frac{1}{60} 0  0  1  2  3  4  5  19
13 (2, 3, 3, 5, 7) \frac{1}{210} −1 −1 0  0  0  1  1  5
17 (2, 3, 5, 7) \frac{1}{210} −1 0  0  0  1  1  2  12
19 (3, 4, 5, 7) \frac{1}{420} −1 −1 0  0  0  0  1  8

Proof. First we claim that qW(X) = qℚ(X). Assume the converse. Then, as in the proof of Lemma 3.3, the class of K_{X} + qA is a non-trivial n-torsion element in Cl(X) defining a global cover π: X′ → X. We have K_{X′} + qA′ ∼ 0, where A′ = π^{∗}A. Hence X′ is such as in Lemma 3.5 and by Corollary 3.5 we have Cl(X′) ≃ ℤ · A′ and qW(X′) = qℚ(X′) ≥ q. The Galois group μ_{n} acts naturally on X′. Consider, for example, the case q = 11 and B_{X′} = (2, 3, 5) (all other cases are similar). Then X′ has three cyclic quotient singularities whose indices are 2, 3, and 5. These points must be μ_{n}-invariant. Hence the variety X has cyclic quotient singularities of indices 2n, 3n, and 5n. By Lemma 3.2 we have gcd(q, n) ≠ 1. In particular, n ≥ 11. This contradicts (2.8). Therefore, qW(X) = qℚ(X) and so X is such as in Lemma 3.5. Now we have to exclude only the case q = 9, B = (2, 5, 13). But in this case by (2.6) and (2.5) we have A^{3} = 9/130 and −K_{X} · c_{2} = 621/130. On the other hand, by Kawamata-Bogomolov’s bounds [Kaw92a] we have 2673/130 = (4q^{2} −3q)A^{3} ≤ 4K_{X} ·c_{2} = 1242/65 [Suz04, Proposition 2.2]. The contradiction shows that this case is impossible. Finally, the values of A^{3} and dimensions of |kA| are computed by using (2.5) and (2.4). □

Corollary 3.7. Let X be a ℚ-Fano threefold satisfying assumptions of (i)-(v) of Theorem 1.4. Then X has only cyclic quotient singularities.

Proof. Indeed, in these cases the indices of points in the basket B are distinct numbers and moreover B contains no pairs of points of indices 2 and 4. Then the assertion follows by [Mor85a], or [Rei87] □

Corollary 3.8. Let X be a ℚ-Fano threefold with qℚ(X) ≥ 9. Then dim |A| ≤ 0.

Computer computations similar to that in Lemma 3.5 allow us to prove the following.

Lemma 3.9. Let X be a Fano threefold with terminal singularities, let q := qW(X), and let A := −\frac{1}{q}K_{X}.

(i) If q ≥ 5 and dim |A| > 1, then q = 5, B = (2), and A^{3} = 1/2. (ii) If q ≥ 7 and dim |A| > 0, then q = 7, B = (2, 3), A^{3} = 1/6.

3.10. Proof of (vi) and (vii) of Theorem 1.4. (vii) Apply Lemma 3.9. Then the result is well-known: in fact, 2A is Cartier and by Riemann-Roch dim |2A| = 6 = dim X + 3. Hence X is a variety of Δ-genus zero [Fuj75], i.e., a variety of minimal degree. Then X ≃ ℙ(1^{3}, 2). (vi) Put q := qℚ(X), Ξ := K_{X} + qA, and Ξ_{1} := A − A_{1}. By our assumption nΞ ∼ nΞ_{1} ∼ 0 for some integer n. If either Ξ ≁ 0 or Ξ_{1} ≁ 0, then elements Ξ and Ξ′ define an étale in codimension one finite cover π: X′ → X such that K_{X′} + qA′ ∼ 0 and A′ ∼ A′_{1}, where A′ := π^{∗}A and A′_{1} := π^{∗}A_{1}. If Ξ ∼ Ξ_{1} ∼ 0, we put X′ = X. In both cases, the following inequalities hold: qW(X′) ≥ 7 and dim |A′| ≥ 1. By Lemma 3.9 we have B(X′) = (2, 3) and qℚ(X′) = qW(X′) = 7. Note that the Gorenstein index of X′ is strictly less than qW(X′). In this case, X′ ≃ ℙ(1^{2}, 2, 3) according to [San96]. ^{§} Now it is sufficient to show that π is an isomorphism. Assume the converse. By our construction, there is an action of a cyclic group μ_{p} ⊂ Gal(X′/X), p is prime, such that π is decomposed as π: X′ → X′/μ_{p} → X. Here X′/μ_{p} is a ℚ-Fano threefold and there is a torsion element of Cl(X′/μ_{p}) which is not Cartier exactly at points where X′ → X′/μ_{p} is not étale. There are exactly four such points and two of them are points of indices 2 and 3. Thus the basket of X′/μ_{p} consists of points of indices p, p, 2p, and 3p. This contradicts Proposition 2.9.

Lemma 3.11. Let X be a ℚ-Fano threefold with q := qℚ(X). If there are three effective different Weil divisors A, A_{1}, A_{2} such that −K_{X} ∼_{ℚ} qA ∼_{ℚ} qA_{1} ∼_{ℚ} qA_{2} and A ≁ A_{1}, then q ≤ 5.

Proof. Assume that q ≥ 6. As in 3.10 consider a cover π: X′ → X. Thus on X′ we have A′ ∼ A′_{1} ∼ A′_{2} and −K_{X′} ∼ qA′. Moreover, dim |A′| = 1 according to Lemma 3.9. In this case, the action of Gal(X′/X) on the pencil |A′| is trivial (because there are three invariant members A′, A′_{1}, and A′_{2}). But then A ∼ A_{1} ∼ A_{2}, a contradiction. □

4. Birational construction

4.1. Let X be a ℚ-Fano threefold and let A be the ample Weil divisor that generates the group Cl(X)/∼_{ℚ}. Thus we have −K_{X} ∼_{ℚ} qA. Let ℳ be a mobile linear system without fixed components and let c := ct(X, ℳ) be the canonical threshold of (X, ℳ). So the pair (X, cℳ) is canonical but not terminal. Assume that −(K_{X} + cℳ) is ample.

Recall that, for any point P ∈ X, the class of K_{X} is a generator of the local Weil divisor class group Cl(X, P).

§The result also can be easily proved by using birational transformations similar to that in §4.

Lemma 4.2. Let P ∈ X be a point of index r > 1. Assume that ℳ ∼ −mK_{X} near P, where 0 < m < r. Then c ≤ 1/m.

Proof. According to [Kaw92b] there is an exceptional divisor Γ over P of discrepancy a(Γ) = 1/r. Let φ: Y → X be a resolution. Clearly, Γ is a prime divisor on Y. Write

K_{Y} = φ^{∗}K_{X} + \frac{1}{r}Γ + ∑ δ_{i}Γ_{i}, ℳ_{Y} = φ^{∗}ℳ − ord_{Γ}(ℳ)Γ − ord_{Γ_{i}}(ℳ)Γ_{i},

where ℳ_{Y} is the birational transform of ℳ and Γ_{i} are other φ-exceptional divisors. Then

K_{Y} + cℳ_{Y} = φ^{∗}(K_{X} + cℳ) + (1/r − c ord_{Γ}(ℳ))Γ + …

and so 1/r − c ord_{Γ}(ℳ) ≥ 0. On the other hand, ord_{Γ}(ℳ) ≡ m/r mod ℤ (because mK_{X} + ℳ ∼ 0 near P). Hence, ord_{Γ}(ℳ) ≥ m/r and c ≤ 1/m. □

4.3. In the construction below we follow [Ale94]. Let f: X̃ → X be a K +cℳ - crepant blowup such that X̃ has only terminal ℚ-factorial singularities:

(4.4)  KX̃ + cℳ̃ = f^{∗}(K_{X} + cℳ).

As in [Ale94], we run K + cℳ -MMP on X̃. We get the following diagram (Sarkisov link of type I or II)

(4.5)  \xymatrix{ & X̃ \ar@{-->}[r] \ar[dl]_{f} & X̄ \ar[dr]^{g} \\ X &  &  & X̂}

where the varieties X̃ and X̄ have only ℚ-factorial terminal singularities, ρ(X̃) = ρ(X̄) = 2, f is a Mori extremal divisorial contraction, X̃ −−→ X̄ is a sequence of log flips, and g is a Mori extremal contraction (either divisorial or fiber type). Thus one of the following possibilities holds:

a) dim X̂ = 1 and g is a ℚ-del Pezzo fibration;

b) dim X̂ = 2 and g is a ℚ-conic bundle; or

c) dim X̂ = 3, g is a divisorial contraction, and X̂ is a ℚ-Fano threefold.

In this case, denote q̂:= qℚ(X̂). Let E be the f-exceptional divisor. In all what follows, for a divisor D on X, let D̃ and D̄ denote strict birational transforms of D on X̃ and X̄, respectively. If g is birational, we put D̂ := g_{∗}D̄.

Claim 4.6 ([Ale94]). If the map g of (4.5) is birational, then Ē is not an exceptional divisor. If g is of fiber type, then Ē is not composed of fibers.

Proof. Assume the converse. If g is birational, this implies that the map g ∘ χ ∘ f^{−1} : X −−→ X̂ is an isomorphism in codimension 1. Since both X and X̂ are Fano threefolds, this implies that g ∘ χ ∘ f^{−1} is in fact an isomorphism. On the other hand, the number of K + cℳ -crepant divisors on X̂ is less than that on X, a contradiction. If dim X̂ ≤ 2, then Ē is a pull-back of an ample Weil divisor on X̂. But then nĒ is a movable divisor for some n > 0. This contradicts exceptionality of E. □

4.7. Notation. If |kA| ≠ ∅, let S_{k} ∈ |kA| be a general member. Write

(4.8)  \begin{aligned}KX̃ & = & f^{∗}K_{X} + αE, \\ S̃_{k} & = & f^{∗}S_{k} − β_{k}E, \\ ℳ̃ & = & f^{∗}ℳ − β_{0}E. \end{aligned}

Then

(4.9)  c = α/β_{0}.

Remark 4.10. If α < 1, then a(E, |−K_{X}|) < 1. On the other hand, 0 = K_{X} + |−K_{X}| is Cartier. Hence, a(E, |−K_{X}|) ≤ 0 and KX̃ + f_{∗}^{−1}|−K_{X}| is linearly equivalent to a non-positive multiple of E. Therefore, f_{∗}^{−1}|−K_{X}| ⊂ |−KX̃| and so

dim|−KX̄| = dim|−KX̃| ≥ dim|−K_{X}|.

In our situation X has only cyclic quotient singularities (see Corollary 3.7). So, the following result is very important.

Theorem 4.11 ([Kaw96]). Let (Y ∋ P) be a terminal cyclic quotient singularity of type \frac{1}{r}(1, a, r − a), let f: Ỹ → Y be a Mori divisorial contraction, and let E be the exceptional divisor. Then f(E) = P, f is the weighted blowup with weights (1, a, r − a) and the discrepancy of E is a(E) = 1/r.

We call this f the Kawamata blowup of P.

4.12. Notation. Assume that g is birational. Let F̄ be the g-exceptional divisor and let F̃ and F be its proper transforms on X̃ and X, respectively. Let n be the maximal integer dividing the class of F̄ in Cl(X̄). Let Θ be an ample Weil divisor on X̂ that generates Cl(X̂)/∼_{ℚ}. Write

Ŝ_{k} ∼_{ℚ} s_{k}Θ and Ê ∼_{ℚ} eΘ,

where s_{k}, e ∈ ℤ, s_{k} ≥ 0, e ≥ 1. Note that s_{k} = 0 if and only if S̄_{k} is contracted by g.

Lemma 4.13. In the above notation assume that the group Cl(X) is torsion free. Write F ∼ dA, where d ∈ ℤ, d ≥ 1. Then Cl(X̂) ≃ ℤ ⊕ ℤ_{n} and d = ne.

Proof. Write F̄ ∼ nḠ, where Ḡ is an integral Weil divisor. Then Ē ∼ eΘ̄ + kḠ for some k ∈ ℤ and Cl(X̂) ≃ Cl(X̄)/F̄ℤ ≃ ℤ ⊕ ℤ_{n}. We have

ℤ_{d} ≃ Cl(X)/⟨F ⟩ ≃ Cl(X̄)/⟨Ē, F̄⟩ ≃ ℤ ⊕ ℤ/⟨eΘ̄ + kḠ, nḠ⟩.

Since the last group is of order ne, we have d = ne. □

From now until the end of this section we consider the case where X̂ is a surface.

Lemma 4.14. Assume that X̂ is a surface. Then X̂ is a del Pezzo surface with Du Val singularities of type A_{n}. The linear system |−KX̂| is base point free. If moreover the group Cl(X) is torsion free, then so is Cl(X̂) and there are only the following possibilities:

(i) K2 ; X̂ = 9, X̂ ≃ ℙ^{2};

(ii) K2 ; X̂ = 8, X̂ ≃ ℙ(1^{2}, 2);

(iii) K2 ; X̂ = 6, X̂ ≃ ℙ(1, 2, 3);

(iv) K2 ; X̂ = 5, X̂ has a unique singular point, point of type A_{4}.

Proof. By the main result of [MP08b] the surface X̂ has only Du Val singularities of type A_{n}. Since ρ(X̂) = 1 and X̂ is uniruled, −KX̂ is ample. Further, since both X̄ and X̂ have only isolated singularities and Pic(X̄/X̂) ≃ ℤ, there is a well-defined injective map g^{∗}: Cl(X̂) → Cl(X̄). Hence the group Cl(X̂) is torsion free whenever Cl(X) is. The remaining part follows from the classification of del Pezzo surfaces with Du Val singularities (see, e.g., [MZ88]). □

Lemma 4.15. Let φ: Y → Z be a ℚ-conic bundle (we assume that Y is ℚfactorial and ρ(Y/Z) = 1). Suppose that there are two prime divisors D_{1} and D_{2} such that φ(D_{i}) = Z, the log divisor K_{Y} + D_{1} + D_{2} is φ-linearly trivial and canonical. Suppose furthermore that Z is singular and let o ∈ Z be a singular point. Then o ∈ Z is of type A_{r−1} for some r ≥ 2 and there is a Sarkisov link

\xymatrix{ & Ỹ \ar[dl]_{σ} \ar@{-->}[rr]^{χ} &  & Ȳ \ar[dr]^{φ̄} \\ Y \ar[drr]^{φ} &  &  &  & Z̄ \ar[dll]_{δ} \\  &  & Z}

where σ is the Kawamata blowup of a cyclic quotient singularity \frac{1}{r}(1, a, r − a) over o, χ is a sequence of flips, φ̄ is a ℚ-conic bundle with ρ(Ȳ /Z̄) = 1, and δ is a crepant contraction of an irreducible curve to o. Moreover, if D̄_{i} is the proper transform of D_{i} on Ȳ, then the divisor KȲ + D̄_{1} + D̄_{2} is linearly trivial over Z and canonical.

Proof. Regard Y/Z as an algebraic germ over o. Since D_{i} are generically sections, the fibration φ has no discriminant curve. By [MP08c] the central fiber C := φ^{−1}(o)_{red} is irreducible and by the main result of [MP08b] Y/Z is toroidal, that is, it is analytically isomorphic to a toric contraction:

Y ≃ (ℂ^{2} × ℙ^{1})/μ_{r}(a, r − a, 1)

for some r, a ∈ ℤ with r ≥ 2 and gcd(a, r) = 1. Here the map Y → Z is the projection to Z ≃ ℂ^{2}/μ_{r}(a, r − a). In particular, Y has exactly two singular points and these points are cyclic quotients of types \frac{1}{r}(1, a, r − a) and \frac{1}{r}(−1, a, r − a). Since the pair (Y, D_{1} + D_{2}) is canonical, D_{1} ∩ D_{2} = ∅. On the other hand, the divisor D_{1} + D_{2} ∼ −K_{Y} must contain all points on indices > 1. Hence Sing(Y) = (D_{1} + D_{2}) ∩ C. Further, the divisors D_{i} are quotients of two disjointed sections of ℂ^{2} × ℙ^{1} → ℂ^{2} by μ_{r}. Therefore, D_{i} · C = 1/r. Now consider the Kawamata blowup σ : Ỹ → Y of C ∩ D_{1}. Let E be the exceptional divisor and let D̃_{i} be the proper transform of D_{i}. Since KỸ = σ^{∗}K_{Y} + \frac{1}{r}E and the pair (Y, D_{1} + D_{2}) is canonical, we have

KỸ + D̃_{1} + D̃_{2} = σ^{∗}(K_{Y} + D_{1} + D_{2}).

It is easy to check locally that the proper transform C̃ of the central fiber C does not meet D̃_{1}. Moreover, C̃ ∩ E is a smooth point of Ỹ and E. Thus we have D̃_{1} · C̃ = 0, E · C̃ = 1, and D̃_{2} · C̃ = D_{2} · C = 1/r. Hence, KỸ · C̃ = −1/r. Since the set-theoretical fiber over o in Ỹ coincides with E ∪ C̃, the divisor −KỸ is ample over Z and C̃ generates a (flipping) extremal ray R. Run the MMP over Z in this direction, i.e., starting with R. Assume that we end up with a divisorial contraction φ̄: Ȳ → Z̄. Then φ̄ must contract the proper transform Ē of E. Here Z̄/Z is a Mori conic bundle and the map Y −−→ Z̄ is an isomorphism in codimension one, so it is an isomorphism. Moreover, Z̄/Z has a section, the proper transforms of D_{i}. Hence the fibration Z̄/Z is toroidal over o. Consider Shokurov’s difficulty [Sho85]

d(W) := #{exceptional divisors of discrepancy < 1}.

Then d(Y) = d(Z̄) = 2(r − 1). On the other hand,

d(Z̄) − 1 ≤ d(Ȳ) < d(Ỹ) = r − 1 + a − 1 + r − a − 1 = 2r − 3

(because the map Ỹ −−→ Ȳ is not an isomorphism). The contradiction shows that our MMP ends up with a ℚ-conic bundle. Clearly, the divisor KȲ +D̄_{1}+D̄_{2} is linearly trivial and canonical. By [MP08b] the surface Z̄ has at worst Du Val singularities of type A. Hence the morphism δ is crepant [Mor85b]. □

Corollary 4.16. In the above notation assume that Ȳ is a toric variety. Then so is Y.

Corollary 4.17. Notation as in Lemma 4.15. Assume that the base surface Z is toric. Then so is Y.

Proof. Induction by the number e of crepant divisors of Z. If e = 0, then Y is smooth and Y ≃ ℙ(ℰ), where ℰ is a decomposable rank-2 vector bundle on Z. □

Proposition 4.18. In notation of 4.3, let X̂ be a surface. Let Γ ∈ |−KX̂| and let G := g^{−1}(Γ). Suppose that there are two prime divisors D_{1} and D_{2} such that g(D_{i}) = X̂ and KX̂ + D_{1} + D_{2} + G ∼ 0. Then the pair (X̄, D_{1} + D_{2}) is canonical. If furthermore the surface X̂ is toric, then so are X̄ and X.

Proof. Clearly, we may replace Γ with a general member of | − KX̂|. Note that G is an elliptic ruled surface and K_{G} + D_{1}|_{G} + D_{2}|_{G} ∼ 0. Hence divisors D_{1}|_{G} and D_{2}|_{G} are disjointed sections. This shows that D_{1} ∩ D_{2} is either empty or consists of fibers. Assume that D_{1} ∩ D_{2} ≠ ∅. We can take Γ so that G ∩ D_{1} ∩ D_{2} = ∅. By adjunction −K_{D_{1}} ∼ Ḡ|_{D_{1}} + D_{2}|_{D_{1}}. Since D_{1} is a rational surface (birational to X̂), the divisor Ḡ|_{D_{1}} +D_{2}|_{D_{1}} must be connected, a contradiction. Thus, D_{1} ∩ D_{2} = ∅. Therefore both divisors D_{1} and D_{2} contain no fibers and so D_{1} ≃ D_{2} ≃ X̂. Then the pair (X̄, D_{1} + D_{2}) is PLT by the Inversion of Adjunction. Since KX̄ + D_{1} + D_{2} is Cartier, this pair must be canonical. The second assertion follows by Corollary 4.17. □

5. Case qℚ(X) = 10

Consider the case qℚ(X) = 10. We assume that a ℚ-Fano threefold with qℚ(X) = 10 exists and get a contradiction applying Construction (4.5). By Proposition 3.6 the group Cl(X) is torsion free and B = (7, 11). Recall also that

(5.1)  |A| = ∅, dim |2A| = 0, and dim |3A| = 1.

For r = 7 and 11, let P_{r} be a (unique) point of index r. In notation of §4, take ℳ := |3A|. By (5.1) there exist a (unique) irreducible divisor S_{2} ∈ | − 2K_{X}| and ℳ is a pencil without fixed components. Let S_{3} ∈ ℳ = |3A| be a general member. Apply Construction (4.5). Notations of 4.3 and 4.7 will be used freely. Near P_{11} we have A ∼ −10K_{X}, so ℳ ∼ −8K_{X}. By Lemma 4.2 we get c ≤ 1/8. In particular, the pair (X, ℳ) is not canonical. For some a_{1}, a_{2} ∈ ℤ we can write

\begin{aligned}KX̃ + 5S̃_{2} & = & f^{∗}(K_{X} + 5S_{2}) − a_{1}E & ∼ & −a_{1}E, \\ KX̃ + 2S̃_{2} + 2S̃_{3} & = & f^{∗}(K_{X} + 2S_{2} + 2S_{3}) − a_{2}E & ∼ & −a_{2}E. \end{aligned}

Therefore,

(5.2)  \begin{aligned}KX̄ + 5S̄_{2} + a_{1}Ē & ∼ & 0, \\ KX̄ + 2S̄_{2} + 2S̄_{3} + a_{2}Ē & ∼ & 0, \end{aligned}

where dim |S_{2}| = 0 and dim |S_{3}| = 1. Using (4.8) we obtain

(5.3)  \begin{aligned}5β_{2} & = & a_{1} + α, \\ 2β_{2} + 2β_{3} & = & a_{2} + α. \end{aligned}

Since S_{3} ∈ ℳ is a general member, by (4.9) we have c = α/β_{3} ≤ 1/8, so

(5.4)  8α ≤ β_{3} and a_{2} ≥ 15α + 2β_{2}.

5.5. First we consider the case where f(E) is either a curve or a Gorenstein point on X. Then α and β_{k} are non-negative integers. In particular, a_{2} ≥ 15. From (5.2) and (5.4) we obtain that g is birational. Indeed, otherwise restricting the second relation of (5.2) to a general fiber V we get that −K_{V} is divisible by some number a′ ≥ a_{2} ≥ 15. This is impossible because V is either ℙ^{1} or a smooth del Pezzo surface.

Thus g is birational and X̂ is a ℚ-Fano. Again from (5.2) and (5.4) in notation of 4.12 we get

−KX̂ ∼ 2Ŝ_{2} + 2Ŝ_{3} + a_{2}Ê ∼_{ℚ} (2s_{2} + 2s_{3} + a_{2}e)Θ,

where s_{2}, s_{3} ≥ 0, e ≥ 1, and a_{2} ≥ 15. This immediately gives us qℚ(X̂) ≥ 15 and e = 1, that is, Ê ∼_{ℚ} Θ. By Proposition 3.6 the group Cl(X̂) is torsion free. In particular, Ê ∼ Θ and |Θ| ≠ ∅. On the other hand, again by Proposition 3.6 we have |Θ| = ∅, a contradiction.

5.6. Therefore f(E) is a non-Gorenstein point P_{r} of index r = 7 or 11. By Theorem 4.11 f is Kawamata blowup and α = 1/r. Near P_{r} we can write A ∼ −l_{r}K_{X}, where l_{r} ∈ ℤ and 10l_{r} ≡ 1 mod r. Then S_{k} + kl_{r}K_{X} is Cartier near P_{r}. Therefore, β_{k} ≡ kl_{r}α mod ℤ and so β_{k} = kl_{r}/r + m_{k}, where m_{k} = m_{k,r} ∈ ℤ. Explicitly, we have the following values of α, β_{k}, and a_{k}:

\begin{matrix}r & α & β_{2} & β_{3} & a_{1} & a_{2} \\ 7 & \frac{1}{7} & \frac{3}{7} + m_{2} & \frac{1}{7} + m_{3} & 2 + 5m_{2} & 1 + 2m_{2} + 2m_{3} \\ 11 & \frac{1}{11} & \frac{9}{11} + m_{2} & \frac{8}{11} + m_{3} & 4 + 5m_{2} & 3 + 2m_{2} + 2m_{3} \end{matrix}

Claim 5.7. If r = 7, then m_{3} ≥ 1.

Proof. Follows from c = α/β_{3} ≤ 1/8. □

If g is not birational, then, as above, restricting relations (5.2) to a general fiber V we get

−K_{V} ∼ 5S̄_{2}|_{V} + a_{1}Ē|_{V} ∼ 2S̄_{2}|_{V} + 2S̄_{3}|_{V} + a_{2}Ē|_{V} ,

where E|_{V} ≠ 0 and S_{2}|_{V}, S_{3}|_{V}, and E|_{V} are proportional to −K_{V} (because ρ(X̄/X̂) = 1). Since V is either ℙ^{1}, or a smooth del Pezzo surface, S_{2}|_{V} = 0 and a_{i} ≤ 3. So, r = 7. By the above claim and computations in the table we have a_{2} = 3, m_{1} = 1, and m_{2} = 0. Hence, a_{1} = 2. But then

−K_{V} ∼ 2Ē|_{V} ∼ 2S̄_{3}|_{V} + 3Ē|_{V} ,

a contradiction. Thus g is birational. Below we will use notation of 4.12. Since S̄_{3} is moveable, s_{3} ≥ 1. Put

u := s_{2} + em_{2}, v := s_{3} + em_{3}.

5.8. Case: r = 11. Since Cl(X)/∼_{ℚ} ≃ ℤ·Θ, pushing down (5.2) to X̂ we obtain the following relations

(5.9)  \begin{aligned}q̂ & = & 5s_{2} + (4 + 5m_{2})e & = & 5u + 4e, \\ q̂ & = & 2s_{2} + 2s_{3} + (3 + 2m_{2} + 2m_{3})e & = & 2u + 2v + 3e. \end{aligned}

Assume that u = 0. Then q̂ = 4e. The only solution of (5.9) with q̂ allowed by Proposition 3.6 is the following: q̂ = 8, v = 1, e = 2. Hence, s_{2} = 0 and s_{3} = 1. In particular, dim |Θ| ≥ dim |S_{3}| = 1. On the other hand, by Lemma 4.13 the group Cl(X̂) is torsion free and by Lemma 3.9 the divisor Θ is not moveable, a contradiction. Therefore, u ≥ 1. By the first relation in (5.9) q̂ ≥ 9. Hence the group Cl(X̂) is torsion free (Proposition 3.6). Then by Lemma 4.13 we have F ∼ eA. Since |A| = ∅, e ≥ 2. Again by (5.9) q̂ ≥ 13 and e is odd. Thus, e = 3, u = 1, and q̂ = 17. Further, s_{3} + em_{3} = v = 3 and s_{3} = 3 (because S̄_{3} is moveable). By Proposition 3.6 we have 1 = dim |S_{3}| ≤ dim |3Θ| = 0, a contradiction.

5.10. Case: r = 7. Recall that m_{3} ≥ 1 by Claim 5.7. As in 5.8 write

(5.11)  \begin{aligned}q̂ & = & 5s_{2} + (2 + 5m_{2})e & = & 5u + 2e, \\ q̂ & = & 2s_{2} + 2s_{3} + (1 + 2m_{2} + 2m_{3})e & = & 2u + 2v + e. \end{aligned}

Hence, v = s_{3} + em_{3} ≥ 1 + e. If u = 0, then q̂ = 2e = 2v + e, e = 2v, and q̂ = 4v ≥ 4(1 + e) = 4(1 + 2v), a contradiction. If u = 2, then q̂ is even ≥ 12. Again we have a contradiction. Assume that u ≥ 3. Using the first relation in (5.11) and Proposition 3.6 we get successively

u = 3, q̂ ≥ 17, |Θ| = ∅, e ≥ 2, q̂ ≥ 19, |2Θ| = ∅, e ≥ 3,

and so q̂ ≥ 21, a contradiction. Therefore, u = 1. Then q̂ = 5 + 2e = 2 + 2v + e and 2v = 3 + e = 2v ≥ 2 + 2e. So, e = 1, v = 2, q̂ = 7. Since m_{3} ≥ 1, s_{3} = v−em_{3} = 1. Hence, Ŝ_{3} ∼_{ℚ} Θ. Since dim |Ŝ_{3}| ≥ 1, by (vi) of Theorem 1.4 we have X̂ ≃ ℙ(1^{2}, 2, 3). In particular, the group Cl(X̂) is torsion free. By Lemma 4.13 the divisor F generates the group Cl(X). This contradicts |A| = ∅. The last contradiction finishes the proof of (v) of Theorem 1.4.

6. Case qℚ(X) = 11 and dim | − K_{X}| ≥ 11

In this section we consider the case qℚ(X) = 11 and dim | − K_{X}| ≥ 11. By Proposition 3.6 the group Cl(X) is torsion free and B = (2, 3, 5). Recall that

dim |A| = 0, dim |2A| = 1, and dim |3A| = 2.

It is easy to see that, for m = 1, 2, and 3, general members S_{m} ∈ | − mK_{X}| are irreducible. For r = 2, 3, 5, let P_{r} be a (unique) point of index r. In notation of §4, take ℳ := |2A|. By the above, ℳ is a pencil without fixed components. Apply Construction (4.5). Near P_{5} we have A ∼ −K_{X} and ℳ ∼ −2K_{X}. By Lemma 4.2 we get c ≤ 1/2. In particular, the pair (X, ℳ) is not canonical.

Proposition 6.1. In the above notation, f is the Kawamata blowup of P_{5} and X̂ is a del Pezzo surface with Du Val singularities with K2 ; X̂ = 5 or 6. Moreover, for k = 1, 2 and 3, the image C_{k} := g(S̄_{k}) is a curve on X̂ with −KX̂ · C_{k} = k.

Proof. Similar to (5.2)-(5.3) we have for some a_{1}, a_{2}, a_{3} ∈ ℤ:

(6.2)  \begin{aligned}KX̄ + 11S̄_{1} + a_{1}Ē & ∼ & 0, \\ KX̄ + S̄_{1} + 5S̄_{2} + a_{2}Ē & ∼ & 0, \\ KX̄ + 2S̄_{1} + 3S̄_{3} + a_{3}Ē & ∼ & 0, \end{aligned}

(6.3)  \begin{aligned}11β_{1} & = & a_{1} + α, \\ β_{1} + 5β_{2} & = & a_{2} + α, \\ 2β_{1} + 3β_{3} & = & a_{3} + α. \end{aligned}

Since S_{2} ∈ ℳ is a general member, by (4.9) we have c = α/β_{2} ≤ 1/2, so 2α ≤ β_{2} and a_{2} ≥ 9α + β_{1}. Since 2S_{1} ∼ S_{2}, we have 2β_{1} ≥ β_{2}. Thus β_{1} ≥ α and a_{1}, a_{2} ≥ 10α. First we consider the case where f(E) is either a curve or a Gorenstein point on X. Then α and β_{k} are integers, so a_{1}, a_{2} ≥ 10. Restricting (6.2) to a general fiber of g we obtain that g is birational. Moreover, in notation of 4.12 we have q̂ ≥ 15, the group Cl(X̂) is torsion free, and Ê ∼ Θ. In particular, |Θ| ≠ ∅. This contradicts Proposition 3.6.

6.4. Therefore P := f(E) is a non-Gorenstein point of index r = 2, 3 or 5. As in 5.6 we have the following values of β_{k} and a_{k}:

\begin{matrix}r & β_{1} & β_{2} & β_{3} & a_{1} & a_{2} & a_{3} \\ 2 & \frac{1}{2} + m_{1} & m_{2} & \frac{1}{2} + m_{3} & 5 + 11m_{1} & m_{1} + 5m_{2} & 2 + 2m_{1} + 3m_{3} \\ 3 & \frac{2}{3} + m_{1} & \frac{1}{3} + m_{2} & m_{3} & 7 + 11m_{1} & 1 + m_{1} + 5m_{2} & 1 + 2m_{1} + 3m_{3} \\ 5 & \frac{1}{5} + m_{1} & \frac{2}{5} + m_{2} & \frac{3}{5} + m_{3} & 2 + 11m_{1} & 2 + m_{1} + 5m_{2} & 2 + 2m_{1} + 3m_{3} \end{matrix}

Claim 6.5. If r = 2 or 3, then m_{2} ≥ 1.

Proof. Follows from 1/2 ≥ c = α/β_{2} = 1/rβ_{2}. □

Assume that g is birational. By Proposition 3.6 and Remark 4.10 we have dim | − KX̂| ≥ | − K_{X}| = 23. So, in notation of 4.12, q̂ ≤ 11. If S̄_{1} is not contracted, then by the first relation in (6.2) we have q̂ ≥ 11 + a_{1} ≥ 13, a contradiction. Therefore the divisor S̄_{1} is contracted. By Lemma 4.13 the group Cl(X̂) is torsion free and Ê ∼ Θ. Hence, q̂ = a_{1} ≤ 7, m_{1} = 0, and r ≠ 5. But then m_{2} ≥ 1 (see Claim 6.5) and a_{2} ≥ 5. This contradicts the second relation in (6.2). Therefore g is of fiber type. Restricting (6.2) to a general fiber we get a_{i} ≤ 3. Thus, r = 5 and a_{1} = a_{2} = a_{3} = 2. Moreover, divisors S̄_{1}, S̄_{2}, and S̄_{3} are g-vertical. Since S̄_{3} is irreducible and dim |S̄_{3}| = 2, the variety X̂ cannot be a curve. Therefore X̂ is a surface and the images g(S̄_{1}), g(S̄_{2}), and g(S̄_{3}) are curves. Since dim |S̄_{1}| = 0, we have dim |g(S̄_{1})| = 0. Hence, K2 ; X̂ ≤ 6 and g(S̄_{1}) is a line on X̂. By Lemma 4.14 there are only two possibilities: X̂ ≃ ℙ(1, 2, 3) and X̂ is an A_{4}-del Pezzo surface. □

6.6. Consider the case where X̂ is an A_{4}-del Pezzo surface. Assume that S̄_{6} is g-vertical. By Riemann-Roch for Weil divisors on surfaces with Du Val singularities [Rei87] we have dim |S̄_{6}| = dim |g(S̄_{6})| = 6. On the other hand, dim |S̄_{6}| = dim |S_{6}| = 7, a contradiction. Thus g(S̄_{5}) = X̂. Since K_{X} + S_{5} + S_{6} ∼ 0,

KX̄ + S̄_{5} + S̄_{6} + Ē ∼ 0.

Therefore S̄_{6} and Ē are sections of g. By Proposition 4.18 the pair (X̄, S̄_{6} + Ē) is canonical. Now since S̄_{5} is nef, the map X̄ −−→ X̃ is a composition of steps of the KX̄ + S̄_{6} + Ē-MMP. Hence the pair (X̃, S̃_{6} + E) is also canonical. In particular, S̃_{6} ∩ E = ∅ and so P_{5} = f(E) ∉ S_{6}, a contradiction.

6.7. Now consider the case X̂ ≃ ℙ(1, 2, 3). As above, if g(S̄_{5}) is a curve, then dim |g(S̄_{5})| = 5 and g(S̄_{5}) ∼ 5g(S̄_{1}). On the other hand, g(S̄_{5}) ∼ −\frac{5}{6}KX̂. But then dim |g(S̄_{5})| = 4, a contradiction. Therefore, g(S̄_{5}) = X̂. Similar to (6.2) we have KX̄ + 2S̄_{5} + S̄_{1} + a_{4}Ē ∼ 0. This shows that a_{4} = 0 and S̄_{5} is a section of g. Thus we can write KX̄ + S̄_{5} + G + Ē ∼ 0, where G is a g-trivial Weil divisor, i.e., G = g^{∗}Γ for some Weil divisor Γ. Pushing down this equality to X we get G ∼ 6S̄_{1}, i.e., Γ ∈ | − KX̂|. By Proposition 4.18 varieties X̄ and X are toric. This proves (iv) of Theorem 1.4.

7. Case qℚ(X) = 13 and dim | − K_{X}| ≥ 6

In this section we consider the case qℚ(X) = 13 and dim | − K_{X}| ≥ 6. By Proposition 3.6 B = (3, 4, 5). Recall that

dim |A| = dim |2A| = 0, dim |3A| = 1, dim |4A| = 2, and dim |5A| = 3.

Therefore, for m = 1, 3, 4, and 5, general members S_{m} ∈ | − mK_{X}| are irreducible. For r = 3, 4, 5, let P_{r} be a (unique) point of index r. In notation of §4, take ℳ := |4A|. Since 1 = dim |3A| > dim ℳ = 2, the linear system ℳ has no fixed components. Apply Construction (4.5). Near P_{5} we have A ∼ −2K_{X} and ℳ ∼ −3K_{X}. By Lemma 4.2 we get c ≤ 1/3. In particular, the pair (X, ℳ) is not canonical.

Proposition 7.1. In the above notation, f is the Kawamata blowup of P_{5}, g is birational, it contracts S̄_{1}, and X̂ ≃ ℙ(1^{3}, 2). Moreover, in notation of 4.12 we have Ŝ_{3} ∼ Ŝ_{4} ∼ Ê ∼ Θ and Ŝ_{5} ∼ 2Θ.

Proof. Similar to (5.2)-(5.3) we have for some a_{1}, a_{2}, a_{3} ∈ ℤ:

(7.2)  \begin{aligned}KX̄ + 13S̄_{1} + a_{1}Ē & ∼ & 0, \\ KX̄ + S̄_{1} + 4S̄_{3} + a_{2}Ē & ∼ & 0, \\ KX̄ + S̄_{1} + 3S̄_{4} + a_{3}Ē & ∼ & 0, \end{aligned}

(7.3)  \begin{aligned}13β_{1} & = & a_{1} + α, \\ β_{1} + 4β_{3} & = & a_{2} + α, \\ β_{1} + 3β_{4} & = & a_{3} + α. \end{aligned}

Since S_{4} ∈ ℳ is a general member, by (4.9) we have c = α/β_{4} ≤ 1/3, 3α ≤ β_{4} and a_{3} ≥ 8α + β_{1}. Since 4S_{1} ∼ S_{4}, we have 4β_{1} ≥ β_{4}. Thus β_{1} ≥ α and a_{1} ≥ 12α. First we consider the case where f(E) is either a curve or a Gorenstein point on X. Then α and β_{k} are integers. In particular, a_{1} ≥ 12. From the first relation in (7.2) we obtain that g is birational. Moreover, in notation of 4.12 we have q̂ ≥ 13 and Ê ∼ Θ. In particular, |Θ| ≠ ∅. By Proposition 3.6 we have q̂ = 13, a_{1} = 13, S̄_{1} is contracted, and α = 1. This contradicts (7.3). Therefore P := f(E) is a non-Gorenstein point of index r = 3, 4 or 5. By Theorem 4.11 α = 1/r. Similar to 5.6 we have (here m_{k} ∈ ℤ_{≥0})

\begin{matrix}r & β_{1} & β_{3} & β_{4} & β_{5} & a_{1} & a_{2} & a_{3} \\ 3 & \frac{1}{3} + m_{1} & m_{3} & \frac{1}{3} + m_{4} & \frac{2}{3} + m_{5} & 4 + 13m_{1} & m_{1} + 4m_{3} & 1 + m_{1} + 3m_{4} \\ 4 & \frac{1}{4} + m_{1} & \frac{3}{4} + m_{3} & m_{4} & \frac{1}{4} + m_{5} & 3 + 13m_{1} & 3 + m_{1} + 4m_{3} & m_{1} + 3m_{4} \\ 5 & \frac{2}{5} + m_{1} & \frac{1}{5} + m_{3} & \frac{3}{5} + m_{4} & m_{5} & 5 + 13m_{1} & 1 + m_{1} + 4m_{3} & 2 + m_{1} + 3m_{4} \end{matrix}

Claim 7.4. If r = 3 or 4, then m_{4} ≥ 1.

Proof. Follows from 1/3 ≥ c = α/β_{4} = 1/rβ_{4}. □

If g is not birational, then a_{1} = 3, r = 4, m_{4} ≥ 1, and a_{3} ≥ 3. In this case, a_{2} = a_{3} = 3, g is a generically ℙ^{2}-bundle, and divisors S̄_{1}, S̄_{3}, S̄_{4} are g-vertical. Since dim |S̄_{4}| > 1 and the divisor S̄_{4} is irreducible, we have a contradiction. Therefore g is birational. Below we will use notation of 4.12. By Proposition 3.6 we have dim | − KX̂| ≥ | − K_{X}| = 19 and q̂ ≤ 13. From the first relation in (7.2) we see that S̄_{1} is contracted. By Lemma 4.13 the group Cl(X̂) is torsion free and Ê ∼ Θ. Moreover, m_{1} = 0 (because 13m_{1} < a_{1}e = q̂ ≤ 13). Thus q̂ = a_{1} = 4, 3, and 5 in cases r = 3, 4, and 5, respectively. In cases r = 3 and 4 we have q̂ ≥ 3 + a_{3} ≥ 6, a contradiction. Therefore, r = 5, q̂ = 5, and s_{3} = s_{4} = 1. Since dim |Θ| ≥ 1, by (vi) of Theorem 1.4 we have X̂ ≃ ℙ(1^{3}, 2). Since dim |S_{5}| = 3 and dim |Θ| = 2, s_{5} ≥ 2. Similar to (7.2)-(7.3) we have KX̄ + S̄_{3} + 2S̄_{5} + a_{4}Ē ∼ 0, 2s_{5} + a_{4} = 4, and a_{4} = β_{3} +2β_{5} −α = m_{3} +2m_{5}. Thus, s_{5} = 2 and a_{4} = β_{5} = 0, i.e., P_{5} ∉ S_{5}. □

Lemma 7.5. (i) S_{1} ∩ S_{3} is a reduced irreducible curve.

(ii) S_{1} ∩ S_{3} ∩ S_{4} = {P_{5}}.

Proof. (i) Recall that A^{3} = 1/60 by Proposition 3.6. Write S_{1} ∩ S_{3} = C + Γ, where C is a reduced irreducible curve passing through P_{5} and Γ is an effective 1-cycle. Suppose, Γ ≠ 0. Then 1/4 = S_{1} · S_{3} · S_{5} > S_{5} · C. Since P_{5} ∉ S_{5}, C ⊄ S_{5} and S_{5} · C ≥ 1/4, a contradiction. Hence, S_{1} ∩ S_{3} = C. (ii) Assume that S_{1} ∩ S_{3} ∩ S_{4} ∋ P ≠ P_{5}. Since 1/5 = S_{1} · S_{3} · S_{4} = S_{4} · C and P, P_{5} ∈ S_{4} ∩ C, we have C ⊂ S_{4}. If there is a component C′ ≠ C of S_{1} ∩ S_{4} not contained in S_{5}, then, as above, 1/3 = S_{1} · S_{4} · S_{5} ≥ S_{5} · C + S_{5} · C′ ≥ 1/2, a contradiction. Thus we can write S_{1} ∩ S_{4} = C + Γ, where Γ is an effective 1-cycle with Supp Γ ⊂ S_{5}. In particular, P_{5} ∉ Γ. The divisor 12A is Cartier at P_{3} and P_{4}. We get

\frac{1}{5} = 12A^{3} = 12A · S_{1} · (S_{4} − S_{3}) = 12A · Γ ∈ ℤ,

a contradiction. □

Lemma 7.6. Let X be a ℚ-Fano threefold and D = D_{1} + ⋯ + D_{4} be a divisor on X, where D_{i} are irreducible components. Let P ∈ X be a cyclic quotient singularity of index r. Assume that K_{X} +D∼_{ℚ}0, P ∉ D_{4}, D_{1}∩D_{2}∩D_{3} = {P }, and D_{1} · D_{2} · D_{3} = 1/r. Then the pair (X, D) is LC.

Proof. Let π: (X^{♯}, P^{♯}) → (X, P) be the index-one cover. For k = 1, 2, 3, let D_{k}^{♯} be the preimage of D_{k} and let D^{♯} := D_{1}^{♯} + D_{2}^{♯} + D_{3}^{♯}. By our assumptions D_{1}^{♯} ∩ D_{2}^{♯} ∩ D_{3}^{♯} = {P^{♯}}. Since D_{1} · D_{2} · D_{3} = 1/r, locally near P^{♯} we have D_{1}^{♯} · D_{2}^{♯} · D_{3}^{♯} = 1. Hence D^{♯} is a simple normal crossing divisor (near P^{♯}). In particular, (X^{♯}, D^{♯}) is LC near P^{♯} and so is (X, D) near P. Thus the pair (X, D) is LC in some neighborhood U ∋ P. Since D_{1}∩D_{2}∩D_{3} = {P }, P is a center of LC singularities for (X, D). Let H be a general hyperplane section through P. Write λD_{4} ∼_{ℚ} H, where λ > 0. If (X, D) is not LC in X \U, then the locus of log canonical singularities of the pair (X, D+ϵH −(λϵ+δ)D_{4}) is not connected for 0 < δ ≪ ϵ ≪ 1. This contradicts Connectedness Lemma [Sho92], [Kol92]. Therefore the pair (X, D) is LC. □

7.7. Proof of (iii) of Theorem 1.4. By Lemma 7.6 the pair (X, S_{1} + S_{3} + S_{4} +S_{5}) is LC. Since K_{X} +S_{1} +S_{3} +S_{4} +S_{5} ∼ 0, it is easy to see that a(E, S_{1} + S_{3}+S_{4}+S_{5}) = −1. Thus KX̃+S̃_{1}+S̃_{3}+S̃_{4}+S̃_{5} = f^{∗}(K_{X}+S_{1}+S_{3}+S_{4}+S_{5}) ∼ 0. Therefore the pairs (X̄, S̄_{1} + S̄_{3} + S̄_{4} + S̄_{5} + Ē) and (X̂, Ŝ_{3} + Ŝ_{4} + Ŝ_{5} + Ê) are also LC. It follows from Proposition 7.1 and its proof that X̂ ≃ ℙ(1^{3}, 2), Ê ∼ Ŝ_{3} ∼ Ŝ_{4} ∼ Θ, and Ŝ_{5} ∼ 2Θ. We claim that Ŝ_{3} + Ŝ_{4} + Ŝ_{5} + Ê is a toric boundary (for a suitable choice of coordinates in ℙ(1^{3}, 2)). Let (x_{1} : x′_{1} : x′′_{1} : x_{2}) be homogeneous coordinates in ℙ(1^{3}, 2). Clearly, we may assume that Ê = {x_{1} = 0}, Ŝ_{3} = {x′_{1} = 0}, and Ŝ_{4} = {αx_{1} + α′x′_{1} + α′′x′′_{1} = 0} for some constants α, α′, α′′. Since (X̂, Ŝ_{3}+Ŝ_{4}+Ê) is LC, α′′ ≠ 0 and after a coordinate change we may assume that Ŝ_{4} = {x′′_{1} = 0}. Further, the surface Ŝ_{5} is given by the equation βx_{2} +ψ(x_{1}, x′_{1}, x′′_{1}) = 0, where β is a constant and ψ is a quadratic form. If β = 0, then Ŝ_{3} ∩ Ŝ_{4} ∩ Ê ∩ Ŝ_{5} ≠ ∅ and the pair (X̂, Ŝ_{3} + Ŝ_{4} + Ŝ_{5} + Ê) cannot be LC. Thus β ≠ 0 and after a coordinate change we may assume that Ŝ_{5} = {x_{2} = 0}. Therefore Ŝ_{3} + Ŝ_{4} + Ŝ_{5} + Ê is a toric boundary. Then by Lemma 7.8 below the varieties X̄, X̃, and X are toric. This proves (iii) of Theorem 1.4.

Lemma 7.8 (see, e.g., [McK01, 3.4]). Let V be a toric variety and let Δ be the toric (reduced) boundary. Then every valuation ν with discrepancy −1 with respect to K_{V} + Δ is toric, that is, there is a birational toric morphism Ṽ → V such that ν corresponds to an exceptional divisor.

8. Case qℚ(X) = 17

Consider the case qℚ(X) = 17. By Proposition 3.6 B = (2, 3, 5, 7) and |A| = ∅, dim |2A| = dim |3A| = dim |4A| = 0, dim |5A| = dim |6A| = 1, dim |7A| = 2. Therefore, for m = 2, 3, 5, and 7 general members S_{m} ∈ | − mK_{X}| are irreducible. For r = 2, 3, 5, 7, let P_{r} be a (unique) point of index r. In notation of §4, take ℳ := |5A| and apply Construction (4.5). Near P_{7} we have A ∼ −5K_{X} and ℳ ∼ −4K_{X}. By Lemma 4.2 we get c ≤ 1/4. In particular, the pair (X, ℳ) is not canonical.

Proposition 8.1. In the above notation, f is the Kawamata blowup of P_{7}, g is birational, it contracts S̄_{2}, and X̂ ≃ ℙ(1^{2}, 2, 3). Moreover, in notation of 4.12 we have Ŝ_{3} ∼ Ŝ_{5} ∼ Θ, Ê ∼ 2Θ, and Ŝ_{7} ∼ 3Θ.

Proof. Similar to (5.2)-(5.3) we have for some a_{1}, a_{2}, a_{3} ∈ ℤ:

(8.2)  \begin{aligned}KX̄ + 7S̄_{2} + S̄_{3} + a_{1}Ē & ∼ & 0, \\ KX̄ + S̄_{2} + 5S̄_{3} + a_{2}Ē & ∼ & 0, \\ KX̄ + S̄_{2} + 3S̄_{5} + a_{3}Ē & ∼ & 0, \end{aligned}

(8.3)  \begin{aligned}7β_{2} + β_{3} & = & a_{1} + α, \\ β_{2} + 5β_{3} & = & a_{2} + α, \\ β_{2} + 3β_{5} & = & a_{3} + α. \end{aligned}

Since S_{5} ∈ ℳ is a general member, by (4.9) we have c = α/β_{5} ≤ 1/4, so 4α ≤ β_{5} and a_{3} ≥ 11α + β_{2}. Since S_{2} + S_{3} ∼ S_{5}, we have β_{2} + β_{3} ≥ β_{5} ≥ 4α. Hence, a_{1} ≥ 6β_{2} + 3α and a_{2} ≥ 4β_{3} + 3α. First we consider the case where f(E) is either a curve or a Gorenstein point on X. Then α and β_{k} are integers. In particular, a_{3} ≥ 11 and by the third relation in (8.2) we obtain that g is birational. Moreover, in notation of 4.12 we have q̂ ≥ 11. In particular, the group Cl(X̂) is torsion free and so Ê ≥ 2Θ. Hence, q̂ ≥ 2a_{3} ≥ 22, a contradiction. Therefore P := f(E) is a non-Gorenstein point of index r = 2, 3, 5 or 7. Similar to 5.6 we have α = 1/r and

\begin{matrix}r & β_{2} & β_{3} & β_{5} & β_{7} & a_{1} & a_{2} & a_{3} \\ 2 & m_{2} & \frac{1}{2} + m_{3} & \frac{1}{2} + m_{5} & \frac{1}{2} + m_{7} & 7m_{2} + m_{3} & 2 + m_{2} + 5m_{3} & 1 + m_{2} + 3m_{5} \\ 3 & \frac{1}{3} + m_{2} & m_{3} & \frac{1}{3} + m_{5} & \frac{2}{3} + m_{7} & 2 + 7m_{2} + m_{3} & m_{2} + 5m_{3} & 1 + m_{2} + 3m_{5} \\ 5 & \frac{1}{5} + m_{2} & \frac{4}{5} + m_{3} & m_{5} & \frac{1}{5} + m_{7} & 2 + 7m_{2} + m_{3} & 4 + m_{2} + 5m_{3} & m_{2} + 3m_{5} \\ 7 & \frac{3}{7} + m_{2} & \frac{1}{7} + m_{3} & \frac{4}{7} + m_{5} & m_{7} & 3 + 7m_{2} + m_{3} & 1 + m_{2} + 5m_{3} & 2 + m_{2} + 3m_{5} \end{matrix}

Claim 8.4. (i) If r = 2, then m_{5} ≥ 2 and m_{2} + m_{3} ≥ 2.

(ii) If r = 3, then m_{5} ≥ 1 and m_{2} + m_{3} ≥ 1.

(iii) If r = 5, then m_{5} ≥ 1.

Proof. Note that 1/4 ≥ c = α/β_{5} = 1/rβ_{5} and rβ_{5} ≥ 4. This gives us inequalities for m_{5}. The inequalities for m_{2} + m_{3} follows from β_{2} + β_{3} ≥ β_{5}. □

From this we have min(a_{1}, a_{2}, a_{3}) ≥ 3. Moreover, the equality min(a_{1}, a_{2}, a_{3}) = 3 holds only if r = 7. Therefore the contraction g can be of fiber type only if a_{1} = 3, r = 7, m_{2} = m_{3} = 0, min(a_{1}, a_{2}, a_{3}) = 3, r = 7, m_{2} = m_{3} = m_{5} = 0, a_{3} = 2, and a_{2} = 1. Then g is a del Pezzo fibration of degree 9 and by the first relation in (8.2) divisors Ŝ_{2} and Ŝ_{3} are g-vertical. But then a_{2} = 3, a contradiction. From now on we assume that g is birational. Thus we use notation of 4.12 as usual. Since S̄_{5} is moveable, it is not contracted. Therefore, s_{5} ≥ 1. By (8.2) we have

\begin{aligned}q̂ & = & 7s_{2} + s_{3} + a_{1}e, \\ q̂ & = & s_{2} + 5s_{3} + a_{2}e, \\ q̂ & = & s_{2} + 3s_{5} + a_{3}e. \end{aligned}

Put

u := s_{2} + em_{2},  v := s_{3} + em_{3},  w := s_{5} + em_{5}.

8.5. Case: r = 2. Then a_{3} ≥ 7 and q̂ ≥ 3s_{5} + a_{3} ≥ 10. Hence the group Cl(X̂) is torsion free. So, e ≥ 2 and q̂ ≥ 3s_{5} + 2a_{3} ≥ 17. In this case, |Θ| = ∅. Therefore, s_{5} ≥ 2 and q̂ ≥ 3s_{5} + 2a_{3} ≥ 20, a contradiction.

8.6. Case: r = 3. Then

\begin{aligned}q̂ & = & 7s_{2} + s_{3} + (2 + 7m_{2} + m_{3})e & = & 7u + v + 2e, \\ q̂ & = & s_{2} + 5s_{3} + (m_{2} + 5m_{3})e & = & u + 5v, \\ q̂ & = & s_{2} + 3s_{5} + (1 + m_{2} + 3m_{5})e & = & u + 3w + e. \end{aligned}

Assume that u > 0. Then q̂ ≥ 9. Hence the group Cl(X̂) is torsion free and e ≥ 2. Since dim |S_{5}| = 1 and dim |Θ| ≤ 0, we have s_{5} ≥ 2. Since m_{5} ≥ 1 (see Claim 8.4), we have w ≥ 4 and q̂ > 13. In this case, s_{5} ≥ 5, a contradiction. Therefore, u = 0, m_{2} = 0, s_{3} ≠ 0, m_{3} ≥ 1, and v ≥ 2. So, q̂ = 5v ≥ 10. Then we get a contradiction by (v) of Theorem 1.4.

8.7. Case: r = 5. Then

\begin{aligned}q̂ & = & 7s_{2} + s_{3} + (2 + 7m_{2} + m_{3})e & = & 7u + v + 2e, \\ q̂ & = & s_{2} + 5s_{3} + (4 + m_{2} + 5m_{3})e & = & u + 5v + 4e, \\ q̂ & = & s_{2} + 3s_{5} + (m_{2} + 3m_{5})e & = & u + 3w. \end{aligned}

From the first two relations we have 3u = 2v + e and 1 ≤ u ≤ 2. Further, q̂ − 4u = 3(v + e), so q̂ ≡ u mod 3. If u = 2, then e is even and q̂ = 14 + v + 2e ≥ 18. So, q̂ = 19, a contradiction. Thus u = 1, 3 = 2v +e, and q̂ = 7+v +2e ≥ 9. By (v) of Theorem 1.4 q̂ is odd. Hence, v is even, e = 3, v = 0, q̂ = 13. In this case, s_{5} + 3m_{5} = w = 4. By Claim 8.4 m_{5} = s_{5} = 1. Note that the group Cl(X̂) is torsion free and s_{2} = 1. Thus dim |Θ| > 0. This contradicts Proposition 3.6.

8.8. Case: r = 7. Then

\begin{aligned}q̂ & = & 7s_{2} + s_{3} + (3 + 7m_{2} + m_{3})e & = & 7u + v + 3e, \\ q̂ & = & s_{2} + 5s_{3} + (1 + m_{2} + 5m_{3})e & = & u + 5v + e, \\ q̂ & = & s_{2} + 3s_{5} + (2 + m_{2} + 3m_{5})e & = & u + 3w + 2e. \end{aligned}

Assume that u > 0. Then q̂ ≥ 10, the group Cl(X̂) is torsion free and so e ≥ 2, q̂ ≥ 13, u = 1. From the first two relations we get q̂ + 2 = 7v. Hence, v = 3, q̂ = 19, e = 3, and s_{2} = 0. This contradicts the equality 1 = u = s_{2} + em_{2}.

Therefore, u = 0 and s_{2} = m_{2} = 0. From the first two relations we get q̂ = 7v. Thus, q̂ = 7, v = 1, e = 2, w = 1, m_{3} = m_{5} = 0, and s_{3} = s_{5} = 1. By Lemma 4.13 the group Cl(X̂) is torsion free and so dim |Θ| ≥ dim |S̄_{5}| > 0. From (vi) of Theorem 1.4 we have X̂ ≃ ℙ(1^{2}, 2, 3). In particular, dim |Θ| = 1. Further, similar to (8.2) we have

\begin{matrix}KX̄ + S̄_{3} + 2S̄_{7} + a_{4}Ē ∼ 0, \\ β_{3} + 2β_{7} = a_{4} + α. \end{matrix}

This gives us a_{4} = 2β_{7} and s_{7} + a_{4} = 3. Since dim |S_{7}| = 2, s_{7} > 1, s_{7} = 3, Ŝ_{7} ∼ 3Θ, a_{4} = 0, and β_{7} = 0, i.e., P_{7} ∉ S_{7}.

□

Lemma 8.9. (i) S_{2} ∩ S_{3} is a reduced irreducible curve.

(ii) S_{2} ∩ S_{3} ∩ S_{5} = {P_{7}}.

Proof. (i) Similar to the proof of (i) of Lemma 7.5. (ii) Put C := S_{3} ∩ S_{4}. Assume that S_{2} ∩ S_{3} ∩ S_{5} ∋ P ≠ P_{7}. Since 1/7 = S_{2} ·S_{3} ·S_{5} = S_{5} ·C and P, P_{7} ∈ S_{5} ∩C, we have C ⊂ S_{5}. If there is a component C′ ≠ C of S_{2} ∩ S_{5} not contained in S_{7}, then, as above, 7/15 = S_{2} · S_{7} · S_{7} ≥ S_{7} · C + S_{7} · C′ ≥ 2/5, a contradiction. Thus we can write S_{2} ∩ S_{5} = C + Γ, where Γ is an effective 1-cycle with Supp Γ ⊂ S_{7}. In particular, P_{7} ∉ Γ. The divisor 30A is Cartier at P_{2}, P_{3}, and P_{5}. We get

\frac{120}{210} = 120A^{3} = 30A · S_{2} · (S_{5} − S_{3}) = 30A · Γ ∈ ℤ,

a contradiction. □

Now the proof of (ii) of Theorem 1.4 can be finished similar to 7.7: the pair (X̂, Ŝ_{3} + Ŝ_{5} + Ê + Ŝ_{7}) is LC and the corresponding discrepancy of S̄_{2} is equal to −1.

9. Case qℚ(X) = 19

Consider the case qℚ(X) = 19. By Proposition 3.6 B = (3, 4, 5, 7) and |A| = ∅, |2A| = ∅, dim |3A| = dim |4A| = dim |5A| = dim |6A| = 0, dim |7A| = 1. Therefore, for m = 3, 4, 5, and 7 general members S_{m} ∈ | − mK_{X}| are irreducible. For r = 3, 4, 5, 7, let P_{r} be a (unique) point of index r. In notation of §4, take ℳ := |7A| = |S_{7}| and apply Construction (4.5). Near P_{5} we have A ∼ −4K_{X} and ℳ ∼ −3K_{X}. By Lemma 4.2 we get c ≤ 1/3. In particular, the pair (X, ℳ) is not canonical.

Proposition 9.1. In the above notation, f is the Kawamata blowup of P_{5}, g is birational, it contracts S̄_{3}, and X̂ ≃ ℙ(1^{2}, 2, 3). Moreover, in notation of 4.12 we have Ŝ_{4} ∼ Ŝ_{7} ∼ Θ, Ê ∼ 3Θ, and Ŝ_{5} ∼ 2Θ.

Proof. Similar to (5.2)-(5.3) we have for some a_{1}, a_{2}, a_{3}, a_{4} ∈ ℤ:

(9.2)  \begin{aligned}KX̄ + 5S̄_{3} + S̄_{4} + a_{1}Ē & ∼ & 0, \\ KX̄ + S̄_{3} + 4S̄_{4} + a_{2}Ē & ∼ & 0, \\ KX̄ + S̄_{4} + 3S̄_{5} + a_{3}Ē & ∼ & 0, \\ KX̄ + S̄_{5} + 2S̄_{7} + a_{4}Ē & ∼ & 0, \end{aligned}

(9.3)  \begin{aligned}5β_{3} + β_{4} & = & a_{1} + α, \\ β_{3} + 4β_{4} & = & a_{2} + α, \\ β_{4} + 3β_{5} & = & a_{3} + α, \\ β_{5} + 2β_{7} & = & a_{4} + α. \end{aligned}

Remark 9.4. Since S_{7} ∈ ℳ is a general member, by (4.9) we have c = α/β_{7} ≤ 1/3, so 3α ≤ β_{7} and a_{4} ≥ 5α + β_{5}. Further, S_{3} + S_{4} ∼ S_{7}. Thus, β_{3} + β_{4} ≥ β_{7} ≥ 3α, a_{1} ≥ 4β_{3} + 2α, and a_{2} ≥ 3β_{4} + 2α.

Assume that X̂ is a surface. Then X̂ is such as in Lemma 4.14. From the first and second relations in (9.2) we obtain that S_{3} and S_{4} are g-vertical. Since dim |S̄_{k}| = 0, dim |g(S̄_{k})| = 0, k = 3, 4. Hence, K2 ; X̂ ≤ 6 and the curves g(S̄_{k}) are in fact lines on X̂. In particular, g(S̄_{3}) ∼ g(S̄_{4}). This implies S̄_{3} ∼ S̄_{4} and S_{3} ∼ S_{4}, a contradiction. Now assume that X̂ is a curve and let G be a general fiber of g. Clearly, divisors S̄_{3} and S̄_{4} are g-vertical. If the divisor S̄_{5} is also g-vertical, then k_{3}S̄_{3} ∼ k_{4}S̄_{4} ∼ k_{5}S̄_{5} ∼ G, where the k_{i} are the multiplicities of corresponding fibers. Considering proper transforms on X we get 3k_{3} = 4k_{4} = 5k_{5} and so k_{3} = 20k, k_{4} = 14k, k_{5} = 12k for some k ∈ ℤ. This contradicts the main result of [MP08a]. Therefore the divisor S̄_{5} is g-horizontal. In this case, the degree of the general fiber is 9. As above we have k_{3}S̄_{3} ∼ k_{4}S̄_{4} ∼ G, 3k_{3} = 4k_{4}. So, k_{3} = 4k, k_{4} = 3k, k ∈ ℤ. Again by [MP08a] g has no fibers of multiplicity divisible by 4. From now on we assume that g is birational. Then in notation of 4.12,

(9.5)  q̂ = 5s_{3} + s_{4} + a_{1}e = s_{3} + 4s_{4} + a_{2}e = s_{4} + 3s_{5} + a_{3}e.

Consider the case where f(E) is either a curve or a Gorenstein point on X. Then α and β_{k} are integers. By Remark 9.4

a_{1} + a_{2} = 5(β_{3} + β_{4}) + β_{3} − 2α ≥ 13α ≥ 13.

On the other hand, from (9.5) we obtain 2q̂ ≥ 6s_{3} + 5s_{4} + 13 ≥ 18. So, q̂ ≥ 9 (both S̄_{3} and S̄_{4} cannot be contracted). In this case, the group Cl(X̂) is torsion free and by Lemma 4.13 we have Ê ≥ 3Θ. Since a_{4} ≥ 5, we have Ê ∼ 3Θ, q̂ ≥ 15, and S̄_{3} is contracted. In this situation, |Θ| = ∅, so s_{5}, s_{7} ≥ 2. This contradicts the fourth relation in (9.2).

Therefore P := f(E) is a non-Gorenstein point of index r = 3, 4, 5 or 7. Similar to 5.6 we have α = 1/r and

\begin{matrix}r & β_{3} & β_{4} & β_{5} & β_{7} & a_{1} & a_{2} & a_{3} \\ 3 & m_{3} & \frac{1}{3} + m_{4} & \frac{2}{3} + m_{5} & \frac{1}{3} + m_{7} & 5m_{3} + m_{4} & 1 + m_{3} + 4m_{4} & 2 + m_{4} + 3m_{5} \\ 4 & \frac{1}{4} + m_{3} & m_{4} & \frac{3}{4} + m_{5} & \frac{1}{4} + m_{7} & 1 + 5m_{3} + m_{4} & m_{3} + 4m_{4} & 2 + m_{4} + 3m_{5} \\ 5 & \frac{2}{5} + m_{3} & \frac{1}{5} + m_{4} & m_{5} & \frac{3}{5} + m_{7} & 2 + 5m_{3} + m_{4} & 1 + m_{3} + 4m_{4} & m_{4} + 3m_{5} \\ 7 & \frac{2}{7} + m_{3} & \frac{5}{7} + m_{4} & \frac{1}{7} + m_{5} & m_{7} & 2 + 5m_{3} + m_{4} & 3 + m_{3} + 4m_{4} & 1 + m_{4} + 3m_{5} \end{matrix}

Claim 9.6. (i) If r = 3 or 4, then m_{7} ≥ 1 and m_{3} + m_{4} ≥ 1. (ii) If r = 7, then m_{7} ≥ 1.

Proof. To get inequalities for m_{7} we use 1/3 ≥ c = α/β_{7} = 1/rβ_{7}, rβ_{7} ≥ 3. The inequalities for m_{3} + m_{4} follows from β_{3} + β_{4} ≥ β_{7}. □

Thus, in all cases a_{1}, a_{2} ≥ 1. Put

u := s_{3} + em_{3},  v := s_{4} + em_{4},  w := s_{5} + em_{5}.

9.7. Case: r = 3. Then u + v > e(m_{3} + m_{4}) ≥ e by Claim 9.6. Further,

\begin{aligned}q̂ & = & 5s_{3} + s_{4} + (5m_{3} + m_{4})e & = & 5u + v, \\ q̂ & = & s_{3} + 4s_{4} + (1 + m_{3} + 4m_{4})e & = & u + 4v + e, \\ q̂ & = & s_{4} + 3s_{5} + (2 + m_{4} + 3m_{5})e & = & v + 3w + 2e. \end{aligned}

If u = 0, then v = q̂ = e + 4v, a contradiction. Assume that u ≥ 2. Then q̂ ≥ 10, u ≤ 3, the group Cl(X̂) is torsion free and by Lemma 4.13 we have e ≥ 3. If u = 2, then v ≥ 2, q̂ ≥ 13, v = q̂ − 10, and e ≤ q̂ − 2 − 4v ≤ 2, a contradiction. If u = 3, then v = 2, e = 6, q̂ = 17, and m_{3} = m_{4} = 0. This contradicts Claim 9.6. Therefore, u = 1. Then v = q̂ − 5, 19 = e + 3q̂, and q̂ ≤ 6. We get only one solution: q̂ = 6, u = v = w = e = 1. Recall that m_{3} + m_{4} ≥ 1 by Claim 9.6. Hence either s_{3} = 0 and Ŝ_{4} ∼_{ℚ} Ŝ_{5} ∼_{ℚ} Ê ∼_{ℚ} Θ or s_{4} = 0 and Ŝ_{3} ∼_{ℚ} Ŝ_{5} ∼_{ℚ} Ê ∼_{ℚ} Θ. In both cases Ŝ_{5} ≁ Ê (otherwise S̄_{5} ∼ Ē + lF̄ for some l ∈ ℤ and so S_{5} ∼ lF, a contradiction). Then we get a contradiction by Lemma 3.11.

9.8. Case: r = 4. As in the previous case, u + v > e and

\begin{aligned}q̂ & = & 5s_{3} + s_{4} + (1 + 5m_{3} + m_{4})e & = & 5u + v + e, \\ q̂ & = & s_{3} + 4s_{4} + (m_{3} + 4m_{4})e & = & u + 4v. \end{aligned}

If u is even, then so is q̂. Hence, q̂ ≤ 10. From the first relation we have u = 0, q̂ = 4v, and e = 3v. This contradicts u + v > e. Therefore u is odd. Assume that u = 1. Then q̂ = 5 + v + e = 1 + 4v and e = 3v − 4. Since u + v > e, there is only one possibility: v = e = 2, q̂ = 9. Then the group Cl(X̂) is torsion free. By Lemma 4.13 we have F ∈ |2A| ≠ ∅, a contradiction. Finally, assume u ≥ 3. Then u = 3 and q̂ = 15 + v + e = 3 + 4v ≥ 16. Thus, q̂ = 19, v = 4, and e = 0, a contradiction.

9.9. Case: r = 7. Then

\begin{aligned}q̂ & = & 5s_{3} + s_{4} + (2 + 5m_{3} + m_{4})e & = & 5u + v + 2e, \\ q̂ & = & s_{3} + 4s_{4} + (3 + m_{3} + 4m_{4})e & = & u + 4v + 3e, \\ q̂ & = & s_{4} + 3s_{5} + (1 + m_{4} + 3m_{5})e & = & v + 3w + e. \end{aligned}

In this case, u = (3v + e)/4 > 0. Assume that u ≥ 2. Then q̂ ≥ 13 and the group Cl(X̂) is torsion free. By Lemma 4.13 we have e ≥ 3. Further, u = 2, and q̂ ≥ 17. We get m_{3} = 0, s_{3} = 2, e ≥ 4, q̂ = 19, e = 4, and v = 1. This contradicts the last relation. Therefore, u = 1. Then 3v + e = 4. Assume that e = 4. Then v = 0, q̂ = 13, w = 3, s_{4} = 0, s_{3} = 1, and m_{4} = m_{3} = 0. Since dim |Θ| = dim |2Θ| = 0, we have s_{5} ≥ 3. Recall that m_{7} ≥ 1 by Claim 9.6. Hence, β_{7} ≥ 1 and a_{4} = 2β_{7} ≥ 2. This contradicts the fourth relation in (9.2). Therefore, e < 4. In this case, e = 1, v = 1, and q̂ = 8. Then Ê ∼_{ℚ} Θ and either Ŝ_{3} ∼_{ℚ} Θ or Ŝ_{4} ∼_{ℚ} Θ (because u = v = 1). This contradicts (vi) of Theorem 1.4.

9.10. Case: r = 5. From (9.2) we obtain

(9.11)  \begin{aligned}q̂ & = & 5s_{3} + s_{4} + (2 + 5m_{3} + m_{4})e & = & 5u + v + 2e, \\ q̂ & = & s_{3} + 4s_{4} + (1 + m_{3} + 4m_{4})e & = & u + 4v + e, \\ q̂ & = & s_{4} + 3s_{5} + (m_{4} + 3m_{5})e & = & v + 3w. \end{aligned}

Then e = 3v − 4u. If u ≥ 2, then e = 3v − 4u ≤ 3v − 6, and so v ≥ 3. Hence, q̂ ≥ 15 and the group Cl(X̂) is torsion free. By Lemma 4.13 we have e ≥ 3. So q̂ = 19, e = 3, s_{3} = 0, and 2 = u = em_{3} ≥ 3, a contradiction. Assume that u = 1, then e = 3v − 4 and v ≥ 2. Further, q̂ = 7v − 3 = v + 3w ≤ 19. We get q̂ = 11 and e = 2. This contradicts Lemma 4.13. Therefore, u = 0. Then e = 3v, q̂ = 7v = 7, v = 1, e = 3, and w = 2. By Lemma 4.13 the group Cl(X̂) is torsion free. Thus s_{3} = 0, i.e., S̄_{3} is contracted, s_{4} = 1, s_{5} = 2, and m_{5} = β_{5} = 0. This means, in particular, that P_{5} ∉ S_{5}. From the fourth relation in (9.2) we get a_{4} = 1 and s_{7} = 1. In particular, dim |Θ| > 0 and X̂ ≃ ℙ(1^{2}, 2, 3) by (vi) of Theorem 1.4.

□

Lemma 9.12. (i) S_{3} ∩ S_{4} is a reduced irreducible curve.

(ii) S_{3} ∩ S_{4} ∩ S_{7} = {P_{5}}.

Proof. (i) Similar to the proof of (i) of Lemma 7.5. (ii) Put C := S_{3} ∩ S_{4}. Assume that S_{3} ∩ S_{4} ∩ S_{7} ∋ P ≠ P_{5}. Since 1/5 = S_{3} ·S_{4} ·S_{7} = S_{7} ·C and P, P_{5} ∈ S_{7} ∩C, we have C ⊂ S_{7}. If there is a component C′ ≠ C of S_{3} ∩ S_{7} not contained in S_{5}, then, as above, 1/4 = S_{3} · S_{7} · S_{5} ≥ S_{5} · C + S_{5} · C′ ≥ 2/7, a contradiction. Thus we can write S_{3} ∩ S_{7} = C + Γ, where Γ is an effective 1-cycle with Supp Γ ⊂ S_{5}. In particular, P_{5} ∉ S_{5}. The divisor 84A is Cartier at P_{3}, P_{4}, and P_{7}. We get

\frac{9}{5} = 84A · S_{3} · (S_{7} − S_{4}) = 84A · Γ ∈ ℤ,

a contradiction. □

Now the proof of (i) of Theorem 1.4 can be finished similar to 7.7.

10. Toric Sarkisov links

Proposition 10.1. Let X be a toric ℚ-Fano threefold and let P ∈ X be a cyclic quotient singularity of index r. Let f: X̃ → X be the Kawamata blowup of P ∈ X. Then a general member of | − K_{X}| is a normal surface having at worst Du Val singularities. The linear system | − K_{X}| has only isolated base points. In particular, −KX̃ is nef and big. The map f: X̃ → X can be completed by a toric Sarkisov link (cf. (4.5)).

Proof. This can be shown by explicit computations in all cases of Proposition 1.3. Consider, for example, the case X = ℙ(3, 4, 5, 7). Let x_{3}, x_{4}, x_{5}, x_{7} be quasi-homogeneous coordinates in ℙ(3, 4, 5, 7). A section S ∈ | − K_{X}| is given by a quasi-homogeneous polynomial of degree 19. By taking this polynomial as a general linear combination of x_{3}^{5}x_{4}, x_{3}^{3}x_{5}^{2}, x_{3}^{4}x_{7}, x_{4}x_{5}^{3}, x_{4}^{3}x_{7}, x_{5}x_{7}^{2} we see that the base locus of | − K_{X}| is the union of four coordinate points and the surface S has only quotient singularities. Since K_{S} is Cartier, the singularities of S are Du Val. Further, we can write KX̃ + S̃ = f^{∗}(K_{X} + S) ∼ 0, where S̃ is the proper transform of S. Hence, S̃ ∈ | − KS̃| and the linear system | − KX̃| has only isolated base points outside of f^{−1}(P). In particular, −KX̃ is nef. It is easy to check that −K3 ; X̃ > 0, i.e., −KX̃ is big. Recall that ρ(X̃) = 2. So, the Mori cone NE(X̃) has exactly two extremal rays, say R_{1} and R_{2}. Let R_{1} is generated by f-exceptional curves. If −KX̃ is ample, we run the MMP starting from R_{2}. Otherwise we make a flop in R_{2} and run the MMP. Clearly, we obtain Sarkisov link (4.5). □

Explicitly, for weighted projective spaces from Proposition 1.3, we have the

following diagram of Sarkisov links. Here an arrow X_{1} ⟶^{\frac{1}{r}} X_{2} indicates that there is a Sarkisov link described above that starts from the Kawamata blowup of a cyclic quotient singularity of index r > 1 on X_{1} and the target variety is X_{2}.

[figure: ℙ^{1}-bundle; ℙ(1, 2, 3, 5); \frac{1}{3}; \frac{1}{2}; \frac{1}{5}; ℚ-conic ; bundle; ℙ(1^{3}, 2); \frac{1}{2}; ℙ(3, 4, 5, 7); \frac{1}{5}; \frac{1}{7}; \frac{1}{4}; \frac{1}{3}; ℙ(1^{2}, 2, 3); \frac{1}{3}; \frac{1}{2}; ℙ^{3}; ℙ(1, 3, 4, 5); \frac{1}{5}; \frac{1}{4}; \frac{1}{3}; ℙ(2, 3, 5, 7); \frac{1}{5}; \frac{1}{3}; \frac{1}{2}; \frac{1}{7}; del Pezzo fibration]

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Yuri Prokhorov Department of Algebra Faculty of Mathematics Moscow State University Moscow 117234 Russia prokhoro@gmail.com