arXiv:1204.2593v1 [math.AG] 12 Apr 2012
CHING-JUI LAI
Abstract. Let (X, ∆) be an n-dimensional ǫ-klt log Q-Fano pair. We give an upper bound for the volume Vol(−(KX+ ∆)) = (−(KX+ ∆))n when n = 2 or n = 3 and X is
Q-factorial of ρ(X) = 1. This bound is essentially sharp for n = 2. Existence of an upper bound for anticanonical volumes is related the Borisov-Alexeev-Borisov Conjecture which asserts boundedness of the set of ǫ-klt logQ-Fano varieties of a given dimension n.
Throughout this article, we work over field of complex numbersC. We recall the definition of singularities of pairs and log Q-Fano pairs.
Definition 0.1. A pair (X, ∆) consists of a normal projective variety X and a boundary ∆, i.e., a Q-divisor ∆ with coefficients in [0, 1], such that KX+ ∆ is Q-Cartier. Let π : Y → X
be a log resolution of (X, ∆), the discrepancy a(E, X, ∆) of a divisor E on Y with respect to the pair (X, ∆) is defined by a(E, X, ∆) = multE(KY − π∗(KX + ∆)). We say that (X, ∆)
has only terminal (resp. canonical) singularities if a(E, X, ∆) > 0 (resp. ≥ 0) for any π-exceptional divisor E on Y . We say that (X, ∆) is klt (resp. ǫ-klt for some 0 < ǫ < 1) if a(E, X, ∆) > −1 (resp. > −1 + ǫ) for any divisor E on Y . Note that smaller ǫ corresponds to worse singularities.
A pair (X, ∆) is (weak) log Q-Fano if the Q-Cartier divisor −(KX + ∆) is ample (resp.
nef and big).
For a klt pair (X, ∆) with κ(KX+∆) = −∞, according to the log minimal model program,
there exists a birational map φ : X 99K Y and a morphism Y → Z such that for ∆′ = φ ∗∆,
the pair (Yz, ∆′z) is log Q-Fano with ρ(Yz) = 1 for general z ∈ Z. In particular, log Q-Fano
pairs are the building blocks for pairs with negative Kodaira dimension. It is also expected that the set of mildly singular Q-Fano varieties is bounded.
Definition 0.2. We say that a collection of varieties {Xλ}λ∈Λ is bounded if there exists
h : X → S a morphism of finite type of Noetherian schemes such that for each Xλ, Xλ ∼= Xs
for some s ∈ S.
For example, the set of all the n-dimensional smooth Fano manifolds is bounded by [17]. Boundedness is also known for terminalQ-Fano Q-factorial threefolds of Picard number one by [12] and for canonical Q-Fano threefolds by [18]. However, if one considers the set of all kltQ-Fano varieties with Picard number one of a given dimension, [21] and [23] have shown that birational boundedness fails. The problem is that the category of klt singularities is too big to be bounded since, for example, it contains finite quotients of arbitrarily large order. To get boundedness, one restricts to a smaller class of singularities, known as ǫ-klt singularities. Precisely we have the following conjecture due to A. Borisov, L. Borisov, and V. Alexeev, which is still open in dimension three and higher.
Borisov-Alexeev-Borisov Conjecture. Fix 0 < ǫ < 1, an integer n > 0, and consider the set of all n-dimensional ǫ-klt log Q-Fano pairs (X, ∆). The set of underlying varieties {X} is bounded.
A. Borisov and L. Borisov establish the B-A-B Conjecture for toric varieties in [9]. V. Alexeev establishes the two dimensional B-A-B Conjecture in [2] with a simplified argument given in [3]. Our original motivation for studying the B-A-B Conjecture is that it is related to the conjectural termination of flips in the minimal model program. According to [8], the log minimal model program, the a.c.c.1 for minimal log discrepancies, and the B-A-B Conjecture in dimension ≤ d implies termination of log flips in dimension ≤ d+1 for effective pairs.
The following questions concerning log Q-Fano pairs (X, ∆) are relevant to the B-A-B Conjecture:
(i) The Cartier index of KX + ∆ of an n-dimensional ǫ-klt log Q-Fano pair (X, ∆) is
bounded from above by a fixed integer r(n, ǫ) depending only on n = dim X and ǫ; (ii) The volume Vol(−(KX+ ∆)) = (−(KX+ ∆))n of an n-dimensional ǫ-klt logQ-Fano
pair (X, ∆) is bounded from above by a fixed integer M(n, ǫ) depending only on n = dim X and ǫ;
(iii) (Batyrev Conjecture) For given positive integers n and r, consider the set of all n-dimensional klt log Q-Fano pairs (X, ∆) with r(KX + ∆) a Cartier divisor. The
set of underlying varieties {X} is bounded.
It is clear that the B-A-B Conjecture follows from (i) and (iii). Note that recently C. Hacon, J. McKernan, and C. Xu have announced a proof of the Batyrev Conjecture (iii). In general
it is very hard to establish (i). Ambro in [5] has proved (i) for toric singularities when the boundaries have standard coefficients {1 − 1
ℓ|ℓ ∈Z≥1} ∪ {1}. A necessary condition for (i)
to hold is that we need to restrict the coefficients of boundaries to be in a fixed d.c.c. set. A counterexample for the general statement is given by the set of pairs (P1, 1
N{pt}) for N ≥ 1.
For the convenience of the reader, we include a well-known argument (to the experts) establishing the B-A-B Conjecture via condition (i) and (ii) in the cases ∆ = 0 or ρ(X) = 1. Proposition 0.3. Suppose that ∆ = 0 or ρ(X) = 1, then the B-A-B Conjecture holds if both (i) and (ii) above are true.
Proof. Suppose that ∆ = 0 and let X be any ǫ-klt Q-Fano variety of dimension n. The following statements together imply the B-A-B conjecture in this case:
(1) The divisor N(−KX) is a very ample line bundle for a fixed N depending only on n
and ǫ;
(2) The set of Hilbert polynomials F = {P (t) = χ(OX(−NKX)⊗t)} associated to all
n-dimensional ǫ-kltQ-Fano varieties is finite.
Indeed, statements (1) and (2) imply that the set of n-dimensional ǫ-klt Q-Fano varieties is contained in a finite union of Hilbert schemes `
P (t)∈FHP (t), where each HP (t)is Noetherian.
From (i), there is an upper bound r(n, ǫ) of the Cartier index of KX depending only on
n and ǫ. It follows that rKX is a line bundle for r = r(n, ǫ). By [13], | − mrKX| is base
point free for any m > 0 divisible by a constant N1(n) > 0 depending only on n = dim X.
Since | − mrKX| is ample and base point free for m > 0 sufficiently divisible, it defines
a finite morphism. By [14, Theorem 5.9], the map induced by | − lrKX| is birational for
any l > 0 divisible by a constant N2(n) > 0 depending only on n = dim X. Since a finite
birational morphism of normal varieties is an isomorphism, it follows that there exists an
1An a.c.c. (respectively d.c.c.) set is a set of real numbers satisfying the ascending (descending) chain condition, i.e., it contains no infinite strictly increasing (decreasing) sequences.
effective embedding by |M(−rKX)| for some fixed M > 0 depending only on n = dim X.
Take N = Mr, we have (1).
By [16], the coefficients of the Hilbert polynomial P (t) = h0(O
X(tH)) of a polarized
variety (X, H) with H an ample line bundle can be bounded by the intersection numbers |Hn| and |Hn−1.K
X|. Since by (i) there exists an integer r = r(n, ǫ) > 0 depending only on
n = dim X and ǫ such that −rKX is an ample line bundle, set H = −rKX and apply (ii). It
follows that there are only finitely many Hilbert polynomials for the set of anti-canonically polarized ǫ-klt Fano varieties {(X, −rKX)}.
If ρ(X) = 1, then −(KX + ∆) being ample implies that −KX is also ample. It is clear
that X is also ǫ-klt and hence boundedness follows from the same proof as above. An effective upper bound in (ii) is obtained for smooth Fano n-folds in [17] and for canonicalQ-Fano threefolds in [18]. In this paper, we obtain an effective answer to question (ii) in dimension two, i.e., for log del Pezzo surfaces.
Theorem A. (Theorem 4.3) Let (X, ∆) be an ǫ-klt weak log del Pezzo surface. The volume Vol(−(KX + ∆)) = (KX + ∆)2 satisfies
(KX + ∆)2 ≤ max{64,
8 ǫ + 4}.
Moreover, this upper bound is in a sharp form: There exists a sequence of ǫ-klt del Pezzo surfaces whose volume grows linearly with respect to 1/ǫ.
Let (X, ∆) be an ǫ-klt weak log del Pezzo surface and Xmin be the minimal resolution
of (X, ∆). Alexeev and Mori have shown in [3, Theorem 1.8] that ρ(Xmin) ≤ 128/ǫ5. Also
from [3, Lemma 1.2] (or see the proof of Theorem 4.3), an exceptional curve E on Xmin over
X has degree 1 ≤ −E2 ≤ 2/ǫ. When ∆ = 0, since the Cartier index of K
X is bounded from
above by the determinant of the intersection matrix (Ei.Ej) of the exceptional curves Ei’s
on Xmin over X, it follows that the Cartier index bound r(2, ǫ) in the statement (i) satisfies
(♦) r(2, ǫ) ≤ 2(2/ǫ)128/ǫ5.
An upper bound of (KX + ∆)2 is implicitly mentioned in [2] but not clearly written down.
It is also not clear if the upper bound (♦) is optimal. In view of Theorem A, this seems unlikely.
As a second result, we also obtain an upper bound of the volumes for ǫ-klt Q-factorial log Q-Fano threefolds of Picard number one. Recall that a variety X is Q-factorial if each Weil divisor is Q-Cartier.
Theorem B. (Theorem 5.16) Let (X, ∆) be an ǫ-klt Q-factorial log Q-Fano threefold of ρ(X) = 1. The degree −K3 X satisfies −K3 X ≤ ( 24M(2, ǫ)R(2, ǫ) ǫ + 12) 3,
where R(2, ǫ) is an upper bound of the Cartier index of KS for S any ǫ/2-klt log del Pezzo
surface of ρ(S) = 1 and M(2, ǫ) is an upper bound of the volume Vol(−KS) = KS2 for S
any ǫ/2-klt log del Pezzo surface of ρ(S) = 1. Note that M(2, ǫ) ≤ max{64, 16/ǫ + 4} from Theorem A and R(2, ǫ) ≤ 2(4/ǫ)128·25
/ǫ5
from (♦).
For a Q-factorial ǫ-klt log Q-Fano pair (X, ∆) of ρ(X) = 1, since −(KX + ∆)3 ≤ −KX3
Vol(−(KX + ∆)) = −(KX + ∆)3. However, it is not expected that the bound in Theorem
B is sharp or in a sharp form.
Note that Q-factoriality is a technical assumption. However, this condition is natural in the sense that starting from a smooth variety, each variety constructed by a step of the minimal model program remains Q-factorial. In dimension two, normal surfaces with rational singularities, e.g., klt singularities, are always Q-factorial.
Instead of using deformation theory of rational curves as in [18], the Riemann-Roch formula as in [12], or the sandwich argument of [2], we aim to create isolated non-klt centers by the method developed in [22]. The point is that deformation theory for rational curves on klt varieties is much harder and so far no effective Riemann-Roch formula is known for klt threefolds.
The rest of this paper is organized as follows: In Section 1, we study non-klt centers. In Section 2, we illustrate the general method in [22] for obtaining an upper bound of anticanonical volumes in Theorem A and B. In Section 3, we review the theory of families of non-klt centers in [22]. In Section 4, we study weak log del Pezzo surfaces and prove Theorem A. In Section 5, we prove Theorem B.
Acknowledgment. The author is grateful to Professor Christopher Hacon, Professor James McKernan, and Professor Chenyang Xu for many useful discussions and suggestions.
1. Non-klt Centers
For the theory of the singularities in the minimal model program, we refer to [19].
Definition 1.1. Let (X, ∆) be a pair. A subvariety V ⊆ X is called a non-klt center if it is the image of a divisor of discrepancy at most −1. A non-klt place is a valuation corresponding to a divisor of discrepancy at most −1. The non-klt locus Nklt(X, ∆) ⊆ X is the union of the non-klt centers. If there is a unique non-klt place lying over the generic point of a non-klt center V , then we say that V is exceptional. If (X, ∆) is not klt along the generic point of a non-klt center V , then we say that V is pure.
The non-klt places/centers here are the log canonical (lc) places/centers in [22].
A standard way of creating a non-klt center on an n-dimensional variety X is to find a very singular divisor: Fix p ∈ X a smooth point, if ∆ is a Q-Cartier divisor on X with multp∆ ≥ n,
then p ∈ Nklt(X, ∆). Indeed, consider the blow up π : Y = BlpX → X and let E be the unique
exceptional divisor with π(E) = p, then the discrepancy
a(E, X, ∆) = multE(KY − π∗(KX + ∆)) = (n − 1) − multE(π∗(∆)) ≤ −1,
as n − 1 = multE(KY − π∗KX) and multE(π∗∆) = multp∆ ≥ n.
We can find singular divisors by the following lemma.
Lemma 1.2. Let X be an n-dimensional complete complex variety and D be a divisor with hi(X, O(mD)) = O(mn−1) for all i > 0, e.g., D is big and nef. Fix a positive rational num-ber α with 0 < αn < Dn. For m ≫ 0 and any x ∈ X
sm, there exists a divisor Ex ∈ |mD| with
multx(Ex) ≥ m · α.
Proof. This is [20, Proposition 1.1.31].
We will apply Lemma 1.2 to the case where (X, ∆) is an n-dimensional log Q-Fano pair: Write (−(KX + ∆))n > (ωn)n for some rational number ω > 0, then as hi(X, O(−m(KX + ∆))) = 0
for m > 0 sufficiently divisible by the Kawamata-Viehweg vanishing theorem, we can find for each p ∈ Xsm a Q-divisor ∆p such that ∆p ∼Q −(KX + ∆)/ω and multp(∆p) ≥ n. In particular,
The non-klt centers satisfy the following Connectedness Lemma of Koll´ar and Shokurov, which is simply a formal consequence of the Kawamata-Viehweg vanishing theorem and is the most important ingredient in this paper.
Lemma 1.3. Let (X, ∆) be a log pair. Let f : X → Z be a projective morphism with connected fibers such that the image of every component of ∆ with negative coefficient is of codimension at least two in Z. If −(KX+ ∆) is big and nef over Z, then the intersection of Nklt(X, ∆) with each
fiber Xz= f−1(z) is connected.
Proof. For simplicity, we assume that Z = Spec(C) is a point and (X, ∆) is log smooth, i.e., X is smooth and ∆ has simple normal crossing support. Then the identity map idX : X → X is a log
resolution of (X, ∆) and Nklt(X, ∆) = x∆y. Consider the exact sequence
· · · → H0(X, OX) → H0(X, Ox∆y) → H1(X, OX(−x∆y)) → · · · .
Since −x∆y = KX+ {∆} − (KX+ ∆) and (X, {∆}) is klt, we have H1(X, OX(−x∆y)) = 0 by the
Kawamata-Viehweg vanishing theorem as −(KX + ∆) is nef and big. Since H0(X, OX) ∼=C, we
see that Nklt(X, ∆) = x∆y is connected.
For the general case, see [10, Theorem 17.4].
Here is an example showing that −(KX+∆) being nef and big is necessary in the Connectedness
Lemma 1.3.
Example 1.4. Let X be P1×P1 and denote by F (resp. G) the fiber of the first (resp. second)
projection to P1. Consider ∆
1 = F1+ F2 the sum of two distinct fibers of the first projection to
P1 and ∆
2 = F + G the sum of two fibers with respect to the two different projections to P1.
Then Nklt(X, ∆1) = F1+ F2 is not connected while Nklt(X, ∆2) = F + G is connected. Note that
−(KX + ∆1) is nef but not big while −(KX + ∆2) is nef and big.
Later on, we will produce not only non-klt centers but isolated non-klt centers. The following theorem is the main technique which allows us to cut down the dimension of non-klt centers. Theorem 1.5. ([14, Theorem 6.8.1]) Let (X, ∆) be klt, projective and x ∈ X a closed point. Let D be an effective Q-Cartier Q-divisor on X such that (X, ∆ + D) is log canonical in a neighborhood of x. Assume that Nklt(X, ∆ + D) = Z ∪ Z′ where Z is irreducible, x ∈ Z, and x /∈ Z′. Set
k = dim Z. If H is an ample Q-divisor on X such that (Hk.Z) > kk, then there is an effective
Q-divisor B ≡ H and rational numbers 1 ≫ δ > 0 and 0 < c < 1 such that (1) (X, ∆ + (1 − δ)D + cB) is non-klt in a neighborhood of x, and
(2) Nklt(X, ∆ + (1 − δ)D + cB) = W ∪ W′ where W is irreducible, x ∈ W , x /∈ W′ and
dim W < dim Z.
2. A Guiding Example
The idea in [22] for obtaining an upper bound for the anticanonical volumes is to create isolated non-klt centers and then use the Connectedness Lemma 1.3: For simplicity, we assume that ∆ = 0. Write (−KX)n> (ωn)nfor a positive rational number ω. For each p ∈ Xsm, we can find an effective
Q-divisor ∆p∼Q −KX/ω such that multp∆p ≥ n and hence p ∈ Nklt(X, ∆p). The observation is
that if ω ≫ 0, then for general p ∈ X, p ∈ Nklt(X, ∆p) can not be an isolated point. Indeed, if this
is not true, then for two general points p, q ∈ X, the set Nklt(X, ∆p+ ∆q) would contain {p, q}
as isolated non-klt centers. But the divisor KX + ∆p+ ∆q ∼Q (1 − ω2)(−KX) is nef and big for
ω > 2. By the Connectedness Lemma 1.3, Nklt(X, ∆p+ ∆q) must be connected; a contradiction.
Therefore, for general p ∈ X the minimal non-klt center Vp ⊆ Nklt(X, ∆p) passing through p
is typically positive dimensional. We would like to show that the restricted volume Vol(−KX|Vp)
of non-klt centers by Theorem 1.5. After doing this finitely many times, we get isolated non-klt centers and we are done.
In general, it is hard to find a lower bound of the restricted volume Vol(−KX|Vp) on the minimal
non-klt center Vp. We illustrate McKernan’s method by studying families of non-klt centers to
obtain a lower bound of the restricted volumes on the non-klt center of an ǫ-klt logQ-Fano variety via the following guiding example, cf. [22].
Example 2.1. Let X be the projective cone over a rational normal curve of degree d ≥ 2 with the unique singular point O ∈ X. The blow up π : Y = BlOX → X is a resolution of X where Y is a
P1-bundle f : Y →P1 over P1: X Y ⊇ P1 ∋ Ft ∼=P1 t. π f
It is easy to show that
(a) KY = π∗KX + (−1 + 2/d)E, where E is the unique exceptional divisor and hence X is
ǫ-klt for ǫ = 1/d;
(b) X isQ-factorial of Picard number one and −KX ∼Q(d + 2)l is an ampleQ-Cartier divisor,
where l is the class of a ruling of X. Hence X is an ǫ-klt del Pezzo surface;
(c) Vol(−KX) = d + 4 + 4/d is a linear function of d = 1/ǫ and provides the required example
in Theorem A.
Let p ∈ X be a general point. Then p is not the vertex O and the unique ruling lp passing
through p is the non-klt center of the log pair (X, lp), i.e., lp = Nklt(X, lp). Moreover, the proper
transform Fp of lp on Y is a fiber of theP1-bundle f : Y →P1. In this case, theP1-bundle structure
of Y is a covering family of non-klt centers of X since the map π : Y → X is dominant.
For p, q ∈ X two general points, let lp and lq be the rulings passing through p and q respectively.
Consider the pair KY + (1 − 2/d)E = π∗KX. By the Connectedness Lemma 1.3, the non-klt locus
Nklt(KY + (1 − 2/d)E + π∗(lp+ lq)) containing Fp∪ Fq is connected as
−(KY + (1 − 2/d)E + π∗(lp+ lq)) = −π∗(KX+ lp+ lq) ≡ dπ∗l
is nef and big. In fact, the fibers Fp and Fq are connected in Nklt(KY + (1 − 2/d)E + π∗(lp+ lq))
by E as
Fp∪ Fq⊆ Nklt(KY + (1 − 2/d)E + π∗(lp+ lq)) ⊆ π−1(Nklt(KX + lp+ lq)) = Fp∪ Fq∪ E,
where the second inclusion follows from the definition of non-klt centers. In particular, multE(π∗(lp+ lq)) ≥
2 d = 2ǫ.
By symmetry, π∗lp must contribute multiplicity at least 1/d = ǫ to the component E (and in fact
is exactly 1/d in this case), i.e.,
(2.1) π∗lp ≥ ǫE.
Note that
(2.2) lp ∼Q −KX
pd · Vol(−KX)
By intersecting both sides of (2.1) with a general fiber F of f : Y → P1, we get for the ruling
l = π∗(F ),
(2.3) 1
pd · Vol(−KX)
degl(−KX) = π∗lp.F ≥ ǫE.F.
Since F is a general fiber meeting the horizontal divisor E at a smooth point, E.F ≥ 1. (In this case E.F = 1.) Combining all of these, we obtain a lower bound of the restricted volume degl(−KX),
degl(−KX) ≥ ǫpd · Vol(−KX).
Note that since in this case degl(−KX) = −KX.l = −KY.π∗l ≤ 2, it follows that Vol(−KX) =
K2
X ≤ 4d = 4/ǫ.
In summary, the method of getting an upper bound of the anticanonical volumes is to obtain a lower bound of the restricted volume Vol(−(KX + ∆)|Vp) on the non-klt centers Vp, which can be
outlined in the following steps:
• Suppose that Vol(−(KX + ∆)) = (−(KX + ∆))n > (ωn)n for a positive rational number
ω. We will show that ω > 0 can not be arbitrarily large. • For general p ∈ X, choose
∆p∼Q −(KXω+ ∆),
so that p ∈ Nklt(X, ∆ + ∆p). Let Vp ⊆ Nklt(X, ∆ + ∆p) be the minimal non-klt center
containing p.
• Construct covering families of non-klt centers by “lining up” (part of the) non-klt centers {Vp}, see Section 3. This is the generalization of the P1-bundle structure in the Example
2.1 and is called a covering families of tigers in [22].
• Use the Connectedness Lemma 1.3 to obtain a lower bound of the restricted volume Vol(−(KX + ∆)|Vp)) = (−(KX + ∆)|Vp))
dim Vp,
on the non-klt center Vp in terms of ω and ǫ. This is the most technical part.
• If ω ≫ 0, then we cut down the dimension of non-klt centers by Theorem 1.5. After finitely many steps, we get isolated non-klt centers and hence a contradiction to the Connectedness Lemma 1.3.
The difficulty of this argument arises in dimension three in many places. First of all, the non-klt centers can be of dimension one or two and we have to deal with them case by case. When we have one dimensional covering families of tigers, it is subtle to detect the contribution of the ǫ-klt condition from some horizontal subvariety, which is analogous to the exceptional curve E in Example 2.1. This is done by applying a differentiation argument to construct a better behaved covering family of tigers, see 5.3. In case we have two dimensional non-klt centers, complications arise for computing intersection numbers as the total space Y of a covering family of tigers is in general not Q-factorial. This can be fixed by replacing Y with a suitable birational model. To finish the proof, we also need to run a relative minimal model on the covering family of tigers and study the geometry of all possible outcomes.
3. Covering Families of Tigers The main reference for this section is [22].
Definition 3.1. ([22, Definition 3.1]) Let (X, ∆) be a log pair with X projective and D aQ-Cartier divisor. We say that pairs of the form (∆t, Vt) form a covering family of tigers of dimension
(1) there is a projective morphism f : Y → B of normal projective varieties such that the general fiber of f over t ∈ B is Vt;
(2) there is a morphism of B to the Hilbert scheme of X such that B is the normalization of its image and f is obtained by taking the normalization of the universal family;
(3) if π : Y → X is the natural morphism, then π(Vt) is a minimal pure non-klt center of
KX+ ∆ + ∆t;
(4) π is generically finite and dominant; (5) ∆t∼QD/ω, where ∆t is effective;
(6) the dimension of Vt is k.
Note that by definition k ≤ dim X − 1 and π|Vt : Vt → π(Vt) is finite and birational. The
covering family of tigers is illustrated in the following diagram:
X Y B ⊇ ∋ Vt t. π f
We will sometimes also refer to Vt as the minimal non-klt center of (X, ∆ + ∆t).
For (X, ∆) a logQ-Fano variety, we will always assume that D = −λ(KX+ ∆) for some λ > 0.
In particular, D is assumed to be big and semi-ample.
The existence of a covering family of tigers is achieved by constructing non-klt centers at general points of X and then fitting a sub-collection of them into a fiber space. In order to fit the non-klt centers into a family, we use exceptional non-klt centers so that we patch up the unique non-klt place associated to each of them. The following lemma allows us to create exceptional non-klt centers.
Lemma 3.2. Let (X, ∆) be a log pair and let D be a big and semi-ample Q-Cartier divisor. Write Dn > (ωn)n for some positive rational number ω. In order to find an upper bound of ω and hence an upper bound of Vol(D) = Dn, for every p ∈ X
sm we may assume that there is a divisor
∆p ∼Q D/ω such that the unique minimal non-klt center Vp ⊆ Nklt(X, ∆ + ∆p) containing p is
exceptional.
Proof. By Lemma 1.2, for any p ∈ Xsm we can find an effective divisor ∆′p ∼Q Dω such that
multp∆′p ≥ n and hence p ∈ Nklt(X, ∆ + ∆′p).
Fix p ∈ Xsm, pick 0 < δp ≤ 1 the unique rational number such that (X, ∆ + δp∆′p) is log
canonical but not klt at p. By [4, Proposition 3.2, Lemma 3.4], we can find an effective divisor Mp ∼QD and some rational number a > 0 such that for any rational number 0 < µ < 1, the pair
(X, (1 − µ)(∆ + δp∆′p) + µ∆ + µaMp) has a unique minimal non-klt center Vp passing through p
which is exceptional. If we write
∆p:= (1 − µ)δp∆′p+ µaMp∼Q 1 ω′ p D, then ωp′ = ω (1 − µ)δp+ µaω ,
and (1 − µ)δp + µaω < 1 + 1/n for any n ≥ 1 if we pick 0 < µ ≪ 1 sufficiently small. Hence
ωp′ > ω/(1 + 1/n). Since D is semi-ample, by adding a small multiple of D to ∆p we have
∆p ∼Q D/ωn for ωn = ω/(1 + 2/n), and (X, ∆ + ∆p) has a unique minimal non-klt center Vp
passing through p which is exceptional. If there exists an upper bound of ωn independent of n,
The following proposition is the construction of the covering family of tigers, see [22, Lemma 3.2] or [24, Lemma 3.2].
Proposition 3.3. Let (X, ∆) and ∆p be the same as in Lemma 3.2. Then there exists a covering
family of tigers π : Y → X of weight ω with Vp ⊆ Nklt(X, ∆ + ∆p) the unique minimal non-klt
center passing through p.
Proof. Choose m > 0 an integer such that mD/ω is integral and Cartier and let B be the Zariski closure of points {m∆p|p ∈ Xsm} ∈ |mD/ω|. Replace B by an irreducible component which
contains an uncountable subset Q of B such that the set {p ∈ X|∆p ∈ Q} is dense in X. This is
possible since the ∆p’s cover X. Let H ⊆ X × |mD/ω| be the universal family of divisors defined
by the incidence relation and HB → B the restriction to B. Take a log resolution of HB⊆ X × B
over the generic point of B and extend it over an open subset U of B. By assumption the log resolution over the generic point of B has a unique exceptional divisor of discrepancy −1, since this is true over Q ⊆ B. Let Y be the image of this unique exceptional divisor in X × B with the natural projection map π : Y → X. By construction π : Y → X dominates X.
Possibly taking a finite cover of B and passing to an open subset of B, we may assume that any fiber Vt of f : Y → B over t ∈ B is a non-klt center of KX + ∆ + ∆t. Possibly passing to an
open subset of B, we may assume that f : Y → B is flat and B maps into the Hilbert scheme. Replace B by the normalization of the closure of its image in the Hilbert scheme and Y by the normalization of the pullback of the universal family. After possibly cutting by hyperplanes in B, we may assume that π is generically finite and dominant. The resulting family is the required
covering family of tigers.
In fact, the original construction of covering families of tigers is carried out in a more general setting. For a topological space X, we say that a subset P is countably dense if P is not contained in the union of countable many closed subsets of X.
Corollary 3.4. Let (X, ∆) be a log pair and let D be a bigQ-Cartier divisor. Let ω be a positive rational number. Let P be a countably dense subset of X. If for every point p ∈ P we may find a pair (∆p, Vp) such that Vp is a pure non-klt center of KX + ∆ + ∆p, where ∆p∼Q D/ωp for some
ωp > ω, then we may find a covering family of tigers of weight ω together with a countably dense
subset Q of P such that for all q ∈ Q, Vq is a fiber of π.
Proof. See [22, Lemma 3.2] or [24, Lemma 3.2].
As noted in Example 2.1, we can assume that the covering families of tigers under our consid-eration are always positive dimensional.
Lemma 3.5. Let (X, ∆) be a projective klt pair and D = −(KX + ∆) be a big and nef
Q-Cartier divisor. A covering family of tigers (∆t, Vt) of weight ω > 2 is positive dimensional, i.e.,
k = dim Vt> 0.
Proof. This is [22, Lemma 3.4] and we include the proof for the convenience of the reader. Suppose that there exists a zero dimensional covering family of tigers of weight ω > 2. For p1and p2general,
there are divisors ∆1 and ∆2 with ∆i∼Q D/ω such that pi is an isolated non-klt center of KX +
∆ + ∆i. As p1and p2are general, it follows that ∆2 does not contain p1and Nklt(X, ∆ + ∆1+ ∆2)
contains p1 and p2 as disconnected non-klt centers. But −(KX + ∆ + ∆1+ ∆2) ∼ (1 −2ω)D is nef
and big if ω > 2. This contradicts Lemma 1.3.
Recall that we want to cut down the dimension of non-klt centers via Theorem 1.5. To do so, we study the associated covering families of tigers and obtain a lower bound of restricted volumes on the non-klt centers. If the new non-klt centers after cutting down the dimension are still positive dimensional, then we have to create new covering families of tigers associated to these new non-klt centers and repeat the process. The following proposition enables us to create covering families of tigers of new non-klt centers after cutting down the dimension.
Proposition 3.6. Let (X, ∆) be a log pair and let D be a Q-Cartier divisor of the form A + E where A is ample and E is effective. Let (∆t, Vt) be a covering family of tigers of weight ω and
dimension k. Let At be A|Vt. If there is an open subset U ⊆ B such that for all t ∈ U we may
find a covering family of tigers (Γt,s, Wt,s) on Vt of weight ω′ with respect to At, then for (X, ∆)
we can find a covering family of tigers (Γs, Ws) of dimension less than k and weight
ω′′= 1
1/ω + 1/ω′ =
ωω′ ω + ω′.
Proof. This is [22, Lemma 5.3].
We will apply Proposition 3.6 with the ample divisor D = −(KX + ∆). In the process of
obtaining lower bound of the restricted volume on the non-klt centers, if we have one-dimensional non-klt centers, then we can control the restricted volume of D, cf. [22, Lemma 5.3].
Corollary 3.7. Let (X, ∆) be a log pair and let D be an ample divisor. Let (∆t, Vt) be a covering
family of tigers of weight ω > 2 and dimension one. Then deg(D|Vt) ≤ 2ω/(ω − 2).
Proof. Suppose that deg(D|Vt) > 2ω/(ω − 2). By Lemma 3.2 and Corollary 3.4, we may find a
covering family (Γt,s, Ws,t) of tigers of weight ω′ > 2ω/(ω − 2) and dimension zero on Vt. By
Proposition 3.6, there exists a covering family of tigers of dimension zero and weight ω′′= ωω
′
ω + ω′ > 2,
for X. This contradicts Lemma 3.5.
4. Log Del Pezzo Surfaces
Let (X, ∆) be an ǫ-klt weak log del Pezzo surface. The minimal resolution π : Y → X of (X, ∆) is the unique proper birational morphism such that Y is a smooth projective surface and KY + ∆Y = π∗(KX + ∆) for some effective Q-divisor ∆Y on Y . Note that minimal resolutions
always exist for two-dimensional log pairs. It is easy to see that (Y, ∆Y) is also an ǫ-klt weak log
del Pezzo surface with volume
Vol(Y, ∆Y) = (KY + ∆Y)2 = (KX + ∆X)2 = Vol(X, ∆X).
Replacing (X, ∆) by its minimal resolution, we can assume that X is smooth.
Write (KX+ ∆)2> (2ω)2. For a general point p ∈ X, let ∆p ∼Q−(KX + ∆)/ω be an effective
Q-divisor constructed from Lemma 1.2 such that p ∈ Nklt(X, ∆ + ∆p). Assume that ω > 2. By
Lemma 3.5, the unique minimal non-klt center Fp of (X, ∆ + ∆p) containing p is one dimensional.
Note that for general p ∈ X, Fp ≤ ∆p.
Lemma 4.1. For a very general point p ∈ X, the numerical class F := Fp on X is well-defined
and F is nef.
Proof. The effective integral one cycles Fp satisfy Fp ≤ ∆p ∼Q −(KX + ∆)/ω and hence form a
bounded set in the Mori cone of curves. As C is uncountable, for p ∈ X a very general point the numerical class F := Fp is well-defined. Since {Fp} moves, the class F is nef.
The following lemma shows that if we assume the weight ω is large, then the non-klt centers {Fp} on X already possess a nearly fiber bundle structure analogous to a covering family of tigers.
Lemma 4.2. Assume that ω > 3, then F2 = 0, i.e. F
p ∩ Fq = ∅ for p, q ∈ X two very general
Proof. Assume that Fp∩ Fq6= ∅ for p, q ∈ X two very general points. We can assume that p /∈ ∆q
as p ∈ X is very general. Since by Lemma 4.2 the curve class F = Fpis nef, for H = −(KX+ ∆)/ω
we have
1 ≤ Fp.Fq = Fp.F ≤ ∆p.F = deg(H|Fp),
where the first inequality is true since X is smooth. Since H is big and nef, we can cut down the dimension of the non-klt centers by Theorem 1.52.
To be precise, pick 0 < δ1 ≤ 1 such that the pair (X, ∆ + δ1∆p) is log canonical but not klt
at p. If (X, ∆ + δ1∆p) = {p}, then this contradicts the Connected Lemma 1.3 as p /∈ ∆q and
the non-klt locus Nklt(X, ∆ + δ1∆p + ∆q) containing p and Fq is disconnected, while the divisor
−(KX + ∆ + δ1∆p+ ∆q) is nef and big. Hence we may assume that Nklt(X, ∆ + δ1∆p) is one
dimensional in a neighborhood of p. In particular, Fp ⊆ Nklt(X, ∆ + δ1∆p) is the minimal non-klt
center containing p. By Theorem 1.5, there exists rational numbers 0 < δ ≪ 1, 0 < c < 1, and an effective Q-divisor Bp ≡ H such that Nklt(X, ∆ + (1 − δ)δ1∆p+ cBp) = {p} in a neighborhood of
p. It follows that the set of non-klt centers Nklt(X, ∆ + (1 − δ)δ1∆p+ cBp+ ∆q) containing p and
Fq is disconnected but the divisor −(KX+ ∆ + (1 − δ)δ1∆p+ cBp+ ∆q) is nef and big as ω > 3.
This again contradicts the Connected Lemma 1.3.
Theorem 4.3. Let (X, ∆) be an ǫ-klt weak log del Pezzo surface. Then the anticanonical volume Vol((−KX + ∆)) = (KX + ∆)2 satisfies
(KX + ∆)2 ≤ max{64,
8 ǫ + 4}.
Proof. Replacing (X, ∆) by its minimal resolution, we may assume that X is smooth. Write (KX + ∆)2 > (2ω)2. For each general point p ∈ X, by Lemma 1.2, there exists an effective
Q-divisor ∆p∼Q−(KX+ ∆)/ω such that p ∈ Nklt(X, ∆ + ∆p). From Lemma 3.5, we may assume
that ω > 2 and the unique minimal non-klt center Fp ⊆ Nklt(X, ∆ + ∆p) containing p is one
dimensional. Note that Fp ≤ ∆p for general p ∈ X. By Lemma 4.1 and 4.2, we may assume that
ω > 3 and for very general p ∈ X the numerical class F of Fp is well-defined and nef with F2 = 0.
For two very general points p, q ∈ X, ∆p.∆q > 0 and hence Fp = Supp(Fp) $ Supp(∆p):
Otherwise ∆q ≡ ∆p ≤ N Fp for some N > 0 and 0 < ∆p.∆q ≤ N2Fp2 = N2F2 = 0, a
contradic-tion. By the Connectedness Lemma 1.3, Nklt(X, ∆ + ∆p + ∆q) ⊇ Fp∪ Fq is connected. Denote
Ep = Supp(∆p) − Fp 6= 0. By Lemma 4.2, Fp∩ Fq = ∅ and hence Ep must contain a connected
curve E ≤ Ep such that Fp.E 6= 0, Fq.E 6= 0, and the set Nklt(X, ∆ + ∆p+ ∆q) ⊇ Fp∪ Fq∪ E.
Furthermore, we can assume that E is irreducible since E.Fq 6= 0 as Fq ≡ Fp for q ∈ X a very
general point.
Suppose that E2 ≥ 0 and hence E is nef. Since Nklt(X, ∆ + ∆p+ ∆q) ⊇ Fp∪ Fq∪ E, we have
∆ + ∆p+ ∆q ≥ E and (∆ + ∆p+ ∆q− E).E ≥ 0. For H = −(KX + ∆)/ω, we see that
2 ≥ 2 − 2ga(E) ≥ − (KX + E).E − (∆ + ∆p+ ∆q− E).E
= − (KX + ∆ + ∆p+ ∆q).E
=(ω − 2)H.E.
Write ∆p = ∆′p+ αE where ∆′p∧ E = 0, ∆′p ≥ Fp, and α > 0, we have
H.E = ∆p.E = (∆′p+ αE).E ≥ Fp.E ≥ 1.
The last inequality follows from the fact that X is smooth and Fp.E > 0. Combine the two
inequalities above, we obtain ω ≤ 4.
2By adding a small multiple of −(K
X+ ∆), we may assume that the inequality deg(H|Fq) ≥ 1 is strict
Hence we may assume that E2 < 0, and thus
−2 ≤ 2ga(E) − 2 = (KX+ E).E
= (KX + ∆).E + (1 − ǫ − aE)E2− ∆′.E + ǫE2 ≤ ǫE2,
where ∆ = ∆′+ aEE with ∆′∧ E = 0 and aE ∈ [0, 1 − ǫ) by the ǫ-klt condition. This implies
that 1 ≤ −E2 ≤ 2/ǫ, where the first inequality follows from the fact that E2 ∈Z as X is smooth. Since F2 = 0 for F the numerical class of Fp where p ∈ X is very general, by Nakai’s criterion the
divisor Hs= F + sE with 0 < s ≤ 1/(−E2) is nef and big. By the Hodge index theorem (see [11,
V 1.1.9(a)]), we get the inequality
(4.1) (KX + ∆)2≤
(−(KX + ∆).Hs)2
H2 s
. From ∆.F ≥ 0 and F2 = 0, we have that
(4.2) − (KX + ∆).F ≤ −(KX+ F ).F ≤ 2.
Also for ∆ = ∆′+ aEE with ∆′∧ E = 0 and aE ∈ [0, 1 − ǫ), we have that
−(KX+ ∆).E = − KX.E − ∆′.E − aEE2
≤E2+ 2 − aEE2 = (aE − 1)(−E2) + 2 ≤ 2 − ǫ(−E2).
(4.3)
Put s = 1/(−E2), all together we get (KX + ∆)2≤ (−(KX+ ∆).(F + sE))2 H2 s ≤ (2 + s(2 − ǫ(−E 2)))2 2sE.F + s2E2 ≤ (−E2)(2 − ǫ + 2 −E2) 2 = (−E2)(2 − ǫ)2+ 4(2 − ǫ) + 4 −E2 ≤ 2 ǫ(2 − ǫ) 2+ 4(2 − ǫ) + 4 = 8 ǫ + 4 − 2ǫ
where the first inequality is (4.1), the second inequality follows from (4.2), (4.3), and F2 = 0, the third inequality is given by ignoring the term sE.F ≥ 0, and the last inequality uses 1 ≤ −E2≤ 2/ǫ.
Remark 4.4. Note that by applying Corollary 3.7 one can only obtain an upper bound of order 1/ǫ2. Hence Theorem 4.3 is a non-trivial result.
5. Log Fano Threefolds of Picard Number One
Let (X, ∆) be an ǫ-klt Q-factorial log Q-Fano threefold of Picard number ρ(X) = 1. Note that by hypothesis X is ǫ-klt and −KX is ample with −KX3 ≥ Vol(−(KX + ∆)) = −(KX + ∆)3.
Hence it is sufficient to assume that X is an ǫ-klt Q-factorial Q-Fano threefold of Picard number ρ(X) = 1 and to find an upper bound of Vol(−KX) = −KX3. We will obtain an upper bound
of the anticanonical volumes by studying covering families of tigers. The weight of any covering families of tigers in our study will always be the weight with respect to −KX.
Let X be an ǫ-klt Q-factorial Q-Fano threefold of Picard number ρ(X) = 1 and write the anticanonical volume Vol(−KX) = −KX3 > (3ω)3 for some positive rational number ω. Denote
that for each p ∈ U there exists an effective divisor ∆p ∼Q D/ω with multp∆p ≥ 6. We pick
divisors ∆p’s in the following systematic way so that we can control their multiplicities uniformly.
5.1. Construction. Let ∆U ⊆ U × U be the diagonal and IZ be the ideal sheaf of the subvariety
Z = ∆U ⊆ X × U . For each p ∈ U , by the existence of Q-divisor ∆p ∼Q D/ω with multp∆p ≥ 6,
there exists mp > 0 such that Lmp = mpD/ω is Cartier and H0(X, Lmp ⊗ I
⊗6mp
p ) 6= 0. In
particular, we can write U = ∪Um where m > 0 runs through all sufficiently divisible integers such
that Lm = mD/ω is Cartier and Um = {p ∈ U |H0(X, Lm ⊗ Ip⊗6m) 6= 0}. Moreover, each Um is
locally closed in X by [11, III, Theorem 12.8] and X = ∪Um. Since the base fieldC is uncountable,
X can not be a countable union of locally closed subsets. Thus there exists some m > 0 such that Um is dense in X.
Fix an m > 0 such that Lm = mD/ω is Cartier and Um = {p ∈ U |H0(X, Lm⊗ Ip⊗6m) 6= 0}
is dense in X. Denote prX : X × U → X and prU : X × U → U the projection maps. Since prU : X × U → U is flat, by [11, III,Theorem 12.11], after restricting to a smaller open affine subset of U , we can assume that the map
(prU)∗(pr∗XLm⊗ IZ⊗6m) ⊗C(p) → H0(X, Lm⊗ Ip⊗6m),
is an isomorphism for each p ∈ U where Ip is the ideal sheaf of p ∈ U . Since Um is dense in U ,
the sheaf (prU)∗(pr∗XLm⊗ IZ⊗6m) 6= 0 on U and hence H0(X ⊗ U, pr∗XL ⊗ IZ⊗6m) 6= 0 as U is affine.
Let s ∈ H0(X ⊗ U, pr∗
XL ⊗ IZ⊗6m) be a nonzero section with F = div(s) the corresponding divisor
on X × U . For each p ∈ U , denote Fp = F ∩ (X × {p}) the associated divisor on X ∼= X × {p}.
Since multZ(F ) ≥ 6m, by Lemma 5.1 below, the Q-divisor ∆p = Fp/m ∼Q D/ω on X satisfies
multp∆p ≥ 6 for general p ∈ U .
Lemma 5.1. ([20, Lemma 5.2.11]) Let g : M → T be a morphism of smooth varieties, and suppose that Z ⊆ M is an irreducible subvariety dominating T :
Z
T.
M g
Let F ⊆ M be an effective divisor. For a general point t ∈ T and an irreducible component Zt′ ⊆ Zt, multZ′
t(Mt, Ft) = multZ(M, F ), where multZ(M, F ) is the multiplicity of the divisor F
on M along a general point of the irreducible subvariety Z ⊆ M and similarly for multZ′
t(Mt, Ft).
For a given collection ofQ-divisors {∆p = Fp/m ∼QD/ω|p ∈ U general} associated to a nonzero
section in H0(X ⊗ U, pr∗XL ⊗ IZ⊗6m) as above, by Lemma 3.2, we can modify the ∆p’s so that the
unique non-klt centers Vp ⊆ Nklt(X, ∆p) passing through p are exceptional. By Lemma 3.3 (or in
general Corollary 3.4), we can construct covering families of tigers from these divisors.
In order to obtain an upper bound of ω, which is sufficient for bounding the anticanonical volumes, we will pick up a “well-behaved” nonzero section s ∈ H0(X ⊗ U, p∗L ⊗ IZ⊗6m) and study the corresponding covering families of tigers.
5.2. Cases. By Section 5.1, there exists an open affine subset U ⊆ X and an integer m > 0 such that H0(X ⊗ U, pr∗
XL ⊗ IZ⊗6m) 6= 0. Let s ∈ H0(X × U, pr∗XL × IZ⊗6m) be a nonzero section with
divisor F = div(s) on X × U and {∆p = Fp/m ∼Q D/ω|p ∈ U } be the associated collection of
Q-divisors. We consider two cases:
(1) (Small multiplicity) For each irreducible component W of Supp(F ) passing through Z, multW(F ) ≤ 3m, i.e., for general p ∈ U we have multW(∆p) ≤ 3 for any irreducible
“well-behaved” covering family of tigers of dimension one. We will derive an upper bound of ω by studying this covering family of tigers. See Section 5.3.
(2) (Big multiplicity) There exists an irreducible component W of Supp(F ) passing through Z with multiplicity multW(F ) > 3m, i.e., for general p ∈ U we have multW(∆p) > 3
for some irreducible component W of Supp(∆p) passing through p. We will construct a
covering family of tigers of dimension two and derive an upper bound of ω by studying the geometry of this covering family of tigers. See Section 5.4.
To pick a “well-behaved” nonzero section in H0(X ⊗U, pr∗XL⊗IZ⊗6m), we will apply the following proposition.
Proposition 5.2. ([20, Proposition 5.2.13]) Let X and U be smooth irreducible varieties, with U affine, and suppose that Z ⊆ W ⊆ X × U are irreducible subvarieties such that W dominates X. Fix a line bundle L on X, and suppose we are given a divisor F ∈ |pr∗
X(L)| on X × U . Write
l = multZ(F ) and k = multW(F ). After differentiating in the parameter directions, there exists a
divisor F′∈ |pr∗X(L)| on X × U with the property that multZ(F′) ≥ l − k, and W * Supp(F′).
5.3. Small multiplicity. Let X be an ǫ-klt Q-Fano threefold of Picard number one and write Vol(−KX) = −KX3 > (3ω)3 for some positive rational number ω. Denote D = −2KX, we have
D3 > (6ω)3. By Section 5.1, there is an integer m > 0 such that L = mD/ω is Cartier and an
open affine subset U ⊆ X such that H0(X × U, pr∗XL ⊗ IZ⊗6m) 6= 0. We fix a nonzero section s ∈ H0(X × U, pr∗XL ⊗ IZ⊗6m) with F = div(s) on X × U .
Proposition 5.3. With the set up above. Assume that ω > 4. If we are in the case where all the irreducible components W of Supp(F ) passing through Z satisfy multW(F ) ≤ 3m, then ω < 8/ǫ+4.
In particular, there is an upper bound for the volume Vol(−KX) = −KX3 ≤ (
24 ǫ + 12)
3.
Proof. Let M be the maximum of multW(F ) among all the irreducible components W of Supp(F )
passing through Z. Then M ≤ 3m by the hypothesis. For a fixed irreducible component W of Supp(F ) passing through Z, we can apply Proposition 5.2 to F . We obtain a divisor F′ ∈
|pr∗X(L) ⊗ IZ⊗6m−M| with the property that
multZ(F′) ≥ (6m − M ) ≥ 3m, and W * Supp(F′).
Since there are only finitely many irreducible components of Supp(F ) passing through Z, by taking a generic differentiation, it follows that for a general divisor F′′ ∈ |pr∗
X(L) ⊗ IZ⊗6m−M| we have
W * Supp(F′′) for any irreducible component W of Supp(F ) passing through Z. In particular,
the base locus Bs(|pr∗XL ⊗ IZ⊗6m−M|) contains no codimension one components in a neighborhood of Z.
Let G be a general divisor in |pr∗
XL ⊗ IZ⊗6m−M| and ∆p = Gp/m for p ∈ U general the
corre-spondingQ-divisors on X. It follows that p ∈ Nklt(KX+∆p) as multp∆p ≥ 3. The minimal non-klt
center Vp ⊆ Nklt(KX + ∆p) passing through p must be positive dimensional by Lemma 3.5 as the
weight of ∆p is ω/2 > 2. Note that we may replace |pr∗XL ⊗ IZ⊗6m−M| by |pr∗XL⊗k⊗ I
⊗k(6m−M )
Z |
for any k ≥ 1 and hence we may assume that m ≫ 0. In particular, we have 0 ≤ multW∆p ≪ 1
for W any irreducible component of Supp(∆p), and Vp can only be one-dimensional.
Let π : Y → X and f : Y → B be a one dimensional covering family of tigers of weight ω′ ≥ ω/2
constructed from the ∆p’s above by Lemma 3.2 and Lemma 3.3. By abuse of notation, we still
denote ∆p’s the divisors associated to this covering family of tigers.
Choose p, q ∈ U ⊆ X general. By Lemma 1.3, the non-klt locus Nklt(π∗(KX+∆p+∆q)) ⊇ Vp∪Vq
on Y is connected and it contains a one dimensional cycle Cp,q connecting Vp and Vq. Since Y
is normal, an irreducible component C of Cp,q intersecting Vq satisfies C ∩ Ysm 6= ∅ for p, q ∈ X
Suppose that Σ ⊆ Supp(π∗(∆
p)) is an irreducible component containing C. If f (Σ) = f (C) is
a curve, then Vp ⊆ Σ = f−1(f (C)) as the general fiber of f : Y → B is irreducible. Moreover, we
can assume that Σ is not π-exceptional as there are only finitely many π-exceptional divisors and we choose p ∈ X, and hence Vp, general. Note that there can only be one such Σ once we fix p ∈ X
and C. In particular, Σ ⊆ Supp(π−1∗ (∆p)) is an irreducible component containing Vp, and we can
write π∗(∆
p) = ∆′+ λΣ with ∆′∧ Σ = 0. Moreover, λ ≤ 1/m, where m ≫ 0 by our choice of ∆p
with 0 ≤ multW∆p ≪ 1 for W any irreducible component of Supp(∆p). Also, multCΣ = 1 since Σ
is smooth along C as f (C) passes through a general point of B and Y is smooth in codimension one.
Choose a general point b′∈ f (C), we have that Yb′ is a general fiber of f : Y → B and
2 ω 2 − 2 ≥ 2 ω(−KX.Vt) = π ∗(∆ p).Yb′ = (∆′+ λΣ).Yb > ǫ 2 − 1 m,
where the first inequality follows from Corollary 3.7. The second inequality follows from Σ.Yb≥ 0
and multC∆′ = multC(π∗(∆p)) − λmultCΣ. Since m ≫ 0, we get ω ≤ 8/ǫ + 4.
Remark 5.4. In the proof of Proposition 5.3, the difficulty arises because in general the one cycle C might be contained in Supp(π−1∗ (∆p)). In this case, one can not see the contribution of the
ǫ-klt condition from the intersection number π∗∆
p.Yb for Yb a general fiber over f (C) ⊆ B as
Yb ⊆ Supp(π∗−1(∆p)), cf., Example 2.1. The differentiation argument eliminates the contribution
of irreducible components of Supp(π∗−1(∆p)) along Yb.
5.4. Big multiplicity. Again, let X be an ǫ-klt Q-factorial Q-Fano threefold of Picard num-ber one. Write Vol(−KX) = −KX3 > (3ω)3 for some positive rational number ω and denote
D = −2KX. As before, by Section 5.1, there is an integer m > 0 such that L = mD/ω is Cartier
and an open affine subset U ⊆ X such that H0(X × U, pr∗
XL ⊗ IZ⊗6m) 6= 0. We fix a nonzero
section s ∈ H0(X × U, pr∗XL ⊗ IZ⊗6m) with F = div(s) on X × U . We now consider the case where there exists an irreducible component W of Supp(F ) passing through Z with multiplicity multW(F ) > 3m.
Lemma 5.5. If there exists an irreducible component W of Supp(F ) passing through Z with multiplicity multW(F ) > 3m, then there exists a covering family of tigers of dimension two and
weight ω′ ≥ ω/2.
Proof. Fix W to be one of these irreducible components of Supp(F ). We have the inclusions Z ⊆ W ⊆ X × U with the projection map W → U . Cutting down by hyperplanes on U and restricting to a smaller open subset of U , we may assume that W → U factors through a Hilbert scheme of X and W → X is generically finite. Replace U by the normalization of the closure of its image in the Hilbert scheme and W by the normalization of universal family. We obtain maps π : Y → X and f : Y → B. Note that a general fiber Yb is two dimensional. We claim that the
pairs (∆b = π∗(Yb), Vb = Yb) is a two dimensional covering of tigers of weight ω′ ≥ ω/2.
Since X is Q-factorial and ρ(X) = 1, the integral divisor ∆b = π∗(Yb) for any p ∈ B on X is
Q-linear equivalent to a multiple of −KX. Since W ≤ F , we have π∗(Yb) ≤ Fb for general b ∈ B. In
particular, π∗(Yb) ∼Q−KX/ω′ for some ω′≥ ω/2. Since any two general divisors π∗(Ybi), i = 1, 2,
on X areQ-linear equivalent as the base field is uncountable, and it is clear that Vt= π(Yb) is the
minimal non-klt center of Nklt(X, ∆b), and the lemma follows.
Let π : Y → X with f : Y → B be a covering family of tigers of dimension two and weight ω′ ≥ ω/2 given by Lemma 5.5. We first deal with case where π : Y → X is not birational.
Proposition 5.6. Suppose that the two dimensional covering family of tigers π : Y → X with f : Y → B of weight ω′ ≥ ω/2 is not birational and assume that ω > 12, then ω ≤ 24/ǫ + 12. In
particular, there is an upper bound of volume
Vol(−KX) = −KX3 ≤ (
72 ǫ + 36)
3.
Proof. Let d ≥ 2 be the degree of π : Y → X. Fix an open subset U ⊆ X such that for a gen-eral point p ∈ U there are d divisors ∆ti
p, for some t1, ..., td ∈ B, with π(Yti) ⊆ Nklt(X, ∆
ti
p)
the unique minimal non-klt center passing through p. Consider the collection of Q-divisors {∆′
p= 6d
Pd
i=1∆tpi|p ∈ U }, then multp∆′p ≥ 6, multW′∆′p = d6 ≤ 3 for W′ ⊆ Supp(∆′p) any
ir-reducible component, and ∆′p∼Q −KX
dω′/6.
By the same construction as in Section 5.1, possibly after shrinking U to a smaller open affine subset, there exists an integer m > 0 such that H0(X × U, pr∗XL ⊗ IZ⊗6m) 6= 0 where L = 6m(−KX)/dω′ is Cartier. Let t ∈ H0(X × U, pr∗XL ⊗ IZ⊗6m) be a general nonzero
sec-tion and G = div(t) be the associated divisor on X × U . Note that multZ(G) ≥ 6m and
multW(G) ≤ 6m/d ≤ 3m for any irreducible component W of Supp(G) passing through Z. Indeed,
we know that for general p ∈ U there is the divisor ∆′p with multp∆′p≥ 6 and multW′∆′p= 6
d ≤ 3
for any irreducible component W′ ⊆ Supp(∆′p). Since t is a general section, tp = t|X×{p} is also
a general section for general p ∈ U . Using Lemma 5.1 to compute the multiplicity, we obtain multW(G) = multWp(Gp) ≤ m · multW′∆
′
p ≤ 3m, where Gp = div(tp) and Wp is any irreducible
component of Supp(Gp).
By a differentiation argument and the same construction as in Proposition 5.3, there is a covering family of tigers (∆t, Vt) of dimension one and weight ω′′ ≥ dω′/6 ≥ dω/12, which satisfies the
property that the base locus Bs(|pr∗XL ⊗ IZ⊗6m−M|) contains no codimension one components in a neighborhood of Z, where M is the maximum of multW(G) amongst all the irreducible components
W of Supp(G) passing through Z. Hence by Corollary 3.7, we get 2 ω′′− 2 ≥ 1 ω′′(−KX.Vt) = π ∗∆ p.Yb ≥ ǫ 2. In particular, 4 ǫ + 2 ≥ ω ′′≥ dω 12 ≥ ω 6, and ω ≤ 24/ǫ + 12.
Assumption. From now on, we assume that π : Y → X with f : Y → B is a birational covering family of tigers of dimension two and weight ω′≥ ω/2. Write K
Y + Γ − R = π∗KX where Γ and
R are effective divisors on Y with no common components.
Lemma 5.7. There is a π-exceptional divisor E on Y dominating B. In particular, π : Y → X is not small.
Proof. Suppose that there is no π-exceptional divisors dominating B. Let AB be a sufficiently
ample divisor on B and AY = f∗AB the pull-back. Since ρ(X) = 1, the divisor AX = π∗AY on X
is ample and π∗AX = AY + G for some effective π-exceptional divisor G. By assumption f (G) ⊆ B
has codimension one and hence AY + G ≤ f∗H for some divisor H on B. This is a contradiction
since then AY + G is not big but π∗AX is.
The following lemma is crucial for computing the restricted volume. The key point is that it allows us to control the negative part of the subadjunction −KX|Vt. Note that the proof fails in
higher dimensions, cf. [22, Lemma 6.2].
Lemma 5.8. Let E be a π-exceptional divisor dominating B. For general points p, q ∈ X we have that E ⊆ Nklt(KY + Γ − R + π∗(∆p + ∆q)). In particular, denote H = π∗(−KX). For any
π-exceptional divisor E dominating B we have 2
ω′H ∼Qπ ∗(∆
Proof. Since the construction of covering families of tigers is done via the Hilbert scheme, π is finite on the general fibers Vt of f : Y → B. Recall that π(Vt) ⊆ X is the minimal non-klt center
of (X, ∆p(t)) for some ∆p(t) passing through a general point p(t) ∈ X. We denote ∆p(t) by ∆t for
simplicity.
Let E be a π-exceptional divisor dominating B. Since E ∩Vbis one dimensional for general b ∈ B
and π|Vb is finite, dim π(E) > 0 as π(E) ⊇ π(E ∩ Vb). Since E is irreducible and π-exceptional,
π(E) is an irreducible curve. Fix t1, t2 ∈ B two general points. Pick a general point x ∈ π(E) and
consider its preimage on Vti. Since π is finite on the general fiber Vt, π
−1(x) ∩ V
ti can only be a
discrete finite set. Choose xi∈ π−1(x) ∩ Vti over x for i = 1, 2. Apply the Connectedness Lemma
1.3 to the pair (Y, Γ − R + π∗(∆t1 + ∆t2)) over X. There is a (possibly reducible) curve contained
in π−1(x) ∩ Nklt(Y, Γ − R + π∗(∆
t1 + ∆t2)) connecting x1 and x2. The component of this curve
containing x1 can not lie on Vt1 as the map π is finite on Vt1. As x ∈ π(E) is general, this curve
deforms into a dimension two subset of E by moving x ∈ π(E). Since E is irreducible, the closure of this two dimensional subset coincides with E and hence E ⊆ Nklt(KY + Γ − R + π∗(∆t1+ ∆t2)).
In particular, multE(KY + Γ − R + π∗(∆t1+ ∆t2)) ≥ 1. If E * Supp(Γ), then π
∗(∆
p+ ∆q) ≥ E.
If E ⊆ Supp(Γ), then π∗(∆p+ ∆q) ≥ ǫE since Γ ∈ [0, 1 − ǫ) as X is ǫ-klt.
To study the geometry of the covering family f : Y → B, we would like to run a relative minimal model program of (Y, Γ) over B. However, Y is normal but possibly not Q-factorial. To get a Q-factorial model of (Y, Γ), we adopt Hacon’s dlt models, cf. [15, Theorem 3.1]. In fact, since the volume bound will be obtained by doing a computation on a general fiber Yb, it suffices to modify
Y over an open subset U ⊆ B.
Lemma 5.9. After restricting to an open subset U ⊆ B and replacing Y by a suitable birational model, we can assume that Y is Q-factorial and (Y, Γ) is ǫ/2-klt. Moreover, we can assume for E any π-exceptional divisor dominating U and p, q ∈ X general, we have that
(5.1) 2 ω′H ∼Q π ∗(∆ p+ ∆q) ≥ ǫ 2E. Proof. Fix p, q ∈ X general and consider the pair
(♯) KY + Γ − Rd+ π∗(∆p+ ∆q) − Re∼Q π∗(KX + ∆p+ ∆q)
where R = Rd+ Re with (−)d the sum of components dominating B and (−)e the sum of
compo-nents mapping to points in B. Restricting Y to YU = f−1(U ) for a suitable nonempty open set
U ⊆ B, we may assume that Re= 0 and (♯) becomes
KY + Γ − Rd+ π∗(∆p+ ∆q) ∼Qπ∗(KX+ ∆p+ ∆q).
We abuse the notation: Y is understood to be YU if not specified.
Denote Γp,q= Γ − Rd+ π∗(∆p+ ∆q). Note that Γp,q≥ 0 by Lemma 5.8. Let φ : W → Y be a
log resolution of (Y, Γp,q) and write
KW + φ−1∗ Γp,q+ Q ∼Q φ∗(KY + Γp,q) + P,
where Q, P ≥ 0 are φ-exceptional divisors with Q ∧ P = 0. We aim to modify W by running a relative minimal model program over Y with scaling of an ample divisor so that it contracts Q<1−ǫ/2+ P , where (P
iaiQi)<α :=
P
ai<αaiQi. Note that we define (−)
α≤·<β and (−)≥α in the
same way.
Consider F =P
iFi, where the sum runs over all the φ-exceptional divisors with log discrepancy
in (ǫ/2, 1] with respect to (Y, Γp,q), then
Since (Y, Γ − R) is ǫ-klt, the divisor Γ on Y as well as φ−1
∗ Γ on W has coefficients in [0, 1 − ǫ). For
rational numbers 0 < ǫ < ǫ′ < 1 and 0 < δ, δ′ ≪ 1, we have the following ǫ/2-klt pair KW + φ−1∗ Γ + Q<1−ǫ/2+ δ′Q1−ǫ/2≤·<1+ (1 − ǫ′)(Q≥1)red+ δF ∼Q φ∗(KY + Γp,q) − (φ−1∗ Γp,q− φ−1∗ Γ) − (1 − δ′)Q1−ǫ/2≤·<1− (Q≥1− (1 − ǫ′)(Q≥1)red) + P + δF where (P jbjGj)red:= P
bj6=0Gj. We denote the above pair by (W, Ξ) where
Ξ = φ−1∗ Γ + Q<1−ǫ/2+ δ′Q1−ǫ/2≤·<1+ (1 − ǫ′)(Q≥1)red+ δF.
By [7], a relative minimal model program with scaling of an ample divisor of the pair (W, Ξ) over Y terminates with a birational model ψ : W 99K W′ over Y with φ′ : W′ → Y the induced
map. We obtain the following diagram,
X Y YU B U W′ W π f φ ψ φ′ π′
where π′ : W′ → X is the induced map.
Write KW′+ ΓW′ − RW′ ∼Q π′∗KX where π′ = φ′◦ π. Note that ΓW′ ∈ [0, 1 − ǫ) by the ǫ-klt
condition and ΓW′− (φ′)−1∗ Γ ≥ 0 is φ′-exceptional. It follows by the construction that ΓW′ ≤ ψ∗Ξ.
In particular, (W′, ΓW′) is ǫ/2-klt as the pair (W, Ξ) is ǫ/2-klt and the minimal model program
does not make singularities worse. On W′, the divisor
G = ψ∗(−(φ−1∗ Γp,q− φ−1∗ Γ) − (1 − δ′)Q1−ǫ/2≤·<1− (Q≥1− (1 − ǫ′)(Q≥1)red) + P + δF )
is φ′-nef with φ′∗G ≤ 0 since Γp,q ≥ Γ. By [19, Negativity Lemma 3.39], we have that G ≤ 0. Since
F + P is φ-exceptional and (F + P ) ∧ Q≥1−ǫ/2= 0, it follows that ψ
∗(P + δF ) = 0. In particular,
all the φ′-exceptional divisors on W′ have log discrepancies less than or equal to ǫ/2 with respect
to (Y, Γp,q).
We now show that for any π′-exceptional divisor E′ on W′ dominating U , E′ satisfies the
inequality 2 ω′H ′ ∼ Q π′∗(∆p+ ∆q) ≥ ǫ 2E ′,
where H′ = π′∗(−KX). This easy to see. If E = φ′∗(E′) 6= 0 on YU, then by Lemma 5.8,
E ⊆ Nklt(KY + Γ − R + π∗(∆p+ ∆q)) and hence E′ ⊆ Nklt(KW′ + ΓW′ − RW′+ π′∗(∆p+ ∆q)).
The inequality then follows from the same argument as in Lemma 5.8. If φ′
∗E′ = 0, then by
construction multE′(KW′+ ΓW′− RW′+ π′∗(∆p+ ∆q)) ≥ 1 − ǫ/2. Suppose that E′ ⊆ Supp(RW′),
then 2 ω′H ′ ∼ Qπ′∗(∆p+ ∆q) ≥ E′≥ ǫ 2E ′. If E′ ⊆ Supp(Γ W′), then as ΓW′ ∈ [0, 1 − ǫ) we get 2 ω′H ′∼ Q π′∗(∆p+ ∆q) ≥ ((1 − ǫ 2) − (1 − ǫ))E ′ = ǫ 2E ′.
Remark 5.10. Write Γ = π−1
∗ ∆ + Γd+ Γeand R = Rd+ Re, where (−)d is the sum of components
dominating B and (−)e is the sum of components mapping to points in B. From the proof of
Lemma 5.9, we deduce the following two inequalities :
(5.2) 2 ω′H ∼Qπ ∗(∆ p+ ∆q) ≥ Rd and 2 ω′H ∼Q π ∗(∆ p+ ∆q) ≥ ǫ 2Γd.
Now let π : Y → X with f : Y → U be the modified birational covering family of tigers of dimension two and weight ω′ ≥ ω/2 given by Lemma 5.9, where Y is now Q-factorial. Write KY + Γ − R ∼Qπ∗KX, where Γ, R ≥ 0 are π-exceptional and Γ ∧ R = 0. The pair (Y, Γ) is ǫ/2-klt
with Γ ∈ [0, 1 − ǫ/2) and note that H = π∗(−KX) is semi-ample and big on Y .
Recall that for a projective morphism φ : Z → U , a divisor D on Z is pseudo-effective (PSEF) over U if the restriction of D to the generic fiber is pseudo-effective.
Lemma 5.11. Assume that ω′ > 2 and consider the pseudo-effective threshold of K
Y + Γ over U
with respect to H
τ := inf{t > 0|KY + Γ + tH is PSEF over B}.
Then 1 ≥ τ ≥ 1 − ω2′ > 0.
Proof. Since KY + Γ + H ∼QR ≥ 0, the first inequality is clear. Restricting to a general fiber Yu
of Y over U , we have (KY + Γ + τ H)|Yu =(R − (1 − τ )H)|Yu =(Rd− 2 ω′H)|Yu− (1 − τ − 2 ω′)H|Yu
which can not be PSEF if ω′ > 2 and τ < 1 − 2
ω′ since the first term is non-positive by (5.2) and
the second term is negative.
Now we run a relative minimal model program with scaling for the covering family of tigers f : Y → U . Since (Y, Γ) is ǫ/2-klt and H is semiample and big, we may assume that (Y, Γ + τ′H) remains ǫ/2-klt for any rational number 0 < τ′ < τ . By [7], a relative minimal model program of
(KY + Γ + τ′H) with scaling of H over U terminates with a relative Mori fiber space Y′ → T over
U with dim Y′ > dim T ≥ dim U . Denote the induced maps by g : Y 99K Y′, ψ : Y′ → T , and φ : Y′→ U . We obtain the following diagram,
X Y Y′ U T. B ⊇ π f g φ ψ
For a general fiber Y′
t of ψ : Y′ → T , by construction, the Picard number ρ(Yt′) = 1 and the divisor
−(KY′ + Γ′d)|Y′ t ∼Q(H ′− R d)|Y′ t on Y ′ t is ample.
Lemma 5.12. There exists a divisor E′ on Y′ which is exceptional over X and dominates T .
Proof. Recall that there is a natural map T → U → B. We can extend ψ : Y′ → T to ψ : Y′ → T
over B where (−) stands for a projective compactification of (−). Take a common resolution p : W → X and q : W → Y′ and let A
T be a sufficiently ample divisor on T . Let AY′ = ψ
∗
AT, AW = q∗AY′, and AX = p∗AW. Then p∗AX = AW+ E = q∗AY′+ E = q∗ψ
∗
AT+ E for an effective divisor E on W which is exceptional over X. Since ρ(X) = 1, it follows by the same argument as in Lemma 3.2 that one of the irreducible components of E maps to a divisor E′ on Y′. By the
same argument as in Lemma 3.2 again, one of the irreducible components of the nonzero divisor
Proposition 5.13. If dim T = 2, then ω′ ≤ 8/ǫ + 2.
Proof. By Lemma 5.12, there exists a divisor E′ on Y′ which is exceptional over X and dominates T . Note that Y′ is normal and hence ψ(Sing(Y′)) is a proper subset of T . In particular, a general fiber Y′
t of ψ : Y′ → T is a smooth projective curve and hence E′.Yt′ ≥ 1. Since the divisor
−(KY′+ Γ′d)|Y′
t ∼Q(H
′− R d)|Y′
t is ample, a general fiber Y
′
t is a smooth rational curveP1. From
(5.1), we know that 2 ω′H ′− ǫ 2E ′ ∼ Qeffective. Also from (5.2), −(KY′ + Γ′).Yt′= (H′− R′).Yt′ =(1 − 2 ω′)H ′.Y′ t + ( 2 ω′H − R ′).Y′ t ≥(1 − 2 ω′)H ′.Y′ t. It follows that 2 ω′ ≥ 1 ω′(−(KY′+ Γ ′).Y′ t) ≥ 1 ω′(1 − 2 ω′)H ′.Y′ t ≥ (1 − 2 ω′) ǫ 4E ′.Y′ t ≥ (1 − 2 ω′) ǫ 4
where the first inequality follows by the adjunction formula on P1. Hence ω′ ≤ 8
ǫ + 2.
Proposition 5.14. If dim T = 1, then
ω′ ≤ 4M (2, ǫ)R(2, ǫ)
ǫ + 2
where R(2, ǫ) is an upper bound of the Cartier index of KS for S any ǫ/2-klt log del Pezzo surface
of ρ(S) = 1 and M (2, ǫ) is an upper bound of the volume Vol(−KS) = KS2 for S any ǫ/2-klt log
del Pezzo surface of ρ(S) = 1.
Proof. Since f : Y → U has connected fibers, T ∼= U . Since −(KY′ + Γ′d)|Y′
u ∼Q (H
′− R d)|Y′
u is
ample and ρ(Yu′) = 1 for a general point u ∈ U , we see that −KY′
u∼Q(H
′+ Γ′
d− Rd)|Y′ u
is ample. By Lemma 5.12, let E′ be a divisor on Y′ exceptional over X which dominates U , then
−KY′ u ≡ (H ′+ Γ′ d− Rd)|Y′ u ≥ (1 − 2 ω′)H|Yu′ ≥ (1 − 2 ω′) · ω′ǫ 4 E ′ u
where the second inequality follows by dropping Γ′
d and applying (5.2) while the last one from
(5.1). By intersecting with the ample divisor −KY′
u, this implies that
(−KY′ u) 2 ≥ (ω′− 2)ǫ 4E ′ u.(−KY′ u).
Now (Yu′, Γ′u) is an ǫ/2-klt log del-Pezzo surfaces of Picard number one. Hence Yu′ is an ǫ/2-klt del-Pezzo surface of Picard number ρ(Y′
u) = 1. By Theorem 4.3, (−KY′
u)
2 is bounded above by a
positive number M (2, ǫ) satisfying
M (2, ǫ) ≤ max{64,16 ǫ + 4}. Also, by (♦) the Cartier index of KY′
u has an upper bounded
R(2, ǫ) ≤ r(2,ǫ
2) ≤ 2(4/ǫ)
128·25
/ǫ5
It follows that M (2, ǫ) ≥ (−KY′ u) 2 ≥ 1 R(2, ǫ)(ω ′− 2)ǫ 4E ′ u.(Ample Cartier) ≥ 1 R(2, ǫ)(ω ′− 2)ǫ 4 and hence we get an upper bound
ω′ ≤ 4M (2, ǫ)R(2, ǫ)
ǫ + 2.
Remark 5.15. It has been shown in [6] that a klt log del Pezzo surface has at most four isolated singularities. Also surface klt singularities are classified by Alexeev in [1]. Hence we expect that it is possible to obtain a better upper bound for R(2, ǫ) and M (2, ǫ) in Proposition 5.14.
Theorem 5.16. Let (X, ∆) be an ǫ-klt log Q-Fano threefold of ρ(X) = 1. Then the degree −K3 X
satisfies
−KX3 ≤ (24M (2, ǫ)R(2, ǫ)
ǫ + 12)
3
where R(2, ǫ) is an upper bound of the Cartier index of KS for S any ǫ/2-klt log del Pezzo surface
of ρ(S) = 1 and M (2, ǫ) is an upper bound of the volume Vol(S) = KS2 for S any ǫ/2-klt log del Pezzo surface of ρ(S) = 1. Note that we have M (2, ǫ) ≤ max{64, 16/ǫ + 4} from Theorem 4.3 and R(2, ǫ) ≤ 2(4/ǫ)128·25/ǫ5 from (♦).
Proof. Recall that ω′ ≥ ω/2. The theorem then follows from Propositions 5.3, 5.13 and 5.14. The following example shows that the cone construction analogous to Example 2.1 only provides ǫ-klt Fano threefolds with volumes of order 1/ǫ2.
Example 5.17. (Projective cone of projective spaces) For n ≥ 1 and d ≥ 2, let Pn ֒→ PN be
the embedding by |O(d)| and X be the associated projective cone. The projective variety X is normal Q-factorial of Picard number one with unique singularity at the vertex O. Also, X admits a resolution π : Y = BlOX → X with exceptional divisor E ∼= Pn of normal bundle
OE(E) ∼= OPn(−d). The variety Y is the projective bundle µ : Y ∼= PPn(OPn⊕ OPn(−d)) → Pn
with tautological bundle OY(1) ∼= OY(E). We have:
• OE(E) ∼= OPn(−d) and hence En+1 = (−d)n;
• KY = π∗KX+(−1+n+1d )E and hence X is always klt. Also, X is terminal (resp. canonical)
if and only if n + 1 > d ≥ 2 (resp. n + 1 ≥ d ≥ 2);
• KY = µ∗(KPn + det(E)) ⊗ OY(−rk(E)) ≡ −(n + 1 + d)F − 2E where the vector bundle
E = OPn⊕ OPn(−d) and F = µ∗On P(1); • Fn+1 = 0 and Fn+1−k.Ek = (−d)k−1 for 1 ≤ k ≤ n + 1; • KYn+1= KXn+1+ (−1 +n+1d )n+1En+1 and KYn+1 =−1 d n+1 X k=1 n + 1 − k k (−1 +n + 1 d ) n+1−k(2d)k =−1 d ((d − n − 1) n+1− (−(d + n + 1)n+1));
• In summary, −KX is ample with
(−KX)n+1 =
(d + n + 1)n+1
d .
If n = 2, then we have an ǫ-klt Fano threefold of Picard number one with ǫ = 1/d. The volume Vol(X) = (−KX)3 is of order 1/ǫ2.
