Abstract and Keywords
This chapter proves a generalization of Shokurov's finite generation conjecture to pairs (X,B) that are not necessarily log canonical. This is an example of a large program of work with the aim of extending Mori theory to non log canonical pairs.
In this chapter we prove finite generation of Shokurov algebras in dimension one and two, in the presence of singularities worse than klt, see Theorem 9.3.1.
In the preliminary, we briefly recall the codimension one adjunction formula for log pairs, which is used elsewhere in this volume. We also recall the notion of discriminant of a log pair with respect to a fibration, which is used in the proof of Theorem 9.3.1. The discriminant plays an important role in higher-codimensional adjunction [Kaw97, Kaw98], see also Chapter 8 of this volume.
We consider algebraic varieties defined over an algebraically closed field of characteristic zero. A log pair (X , B) is a normal variety X endowed with a ℚ-Weil divisor B, such that K + B is ℚ-Cartier. Note that B may have negative coefficients. The locus where (X , B) has Kawamata log terminal singularities is an open subset of X, whose complement, the non-klt locus, is denoted by nklt(X , B). A fibration is a proper surjective morphism of normal varieties f : X → Y such that OY = f∗O X . We assume the reader is familiar with Shokurov's teminology of b-divisors.
9.2.1 Codimension one adjunction
The adjunction formula (K + S)|S = KS , where S is a non-singular divisor in a non-singular variety X, is a useful tool in the study of algebraic varieties. Its singular version was introduced by Reid, Kawamata and Shokurov. Below, we follow Shokurov [Sho93b].
Let (X , B) be a log pair, let W be a prime divisor on X such that mult W(B) = 1 and let ν: Wν → W be the normalisation.
Let μ: Y → X be a log resolution, write μ ∗(K + B) = KY + BY . Note that BY is a well-defined ℚ-divisor, since we use the same top rational form in the definitions of
(p. 164 ) both K and KY . We denote by E the proper transform of W on Y. Since E is non-singular, the induced morphism E → W factors through the normalisation Wν :
We can write BY = E + B′, where B′ is a ℚ-divisor that does not contain E in its support. Define BE = B′|E . By the classical adjunction formula mentioned above, we have
μ ∗(K + B) |E = (KY + E + B′) |E = KE + BE .
Since the above diagram commutes, we obtain KE + BE = f ∗ ν ∗(K + B). Since f is birational, we infer K E + BE = f ∗(KW ν + BW ν), where
BWν = f*(BE ).
The ℚ-Weil divisor BW ν is called the different of (X , B) on Wν . It is well defined, the above construction being independent of the choice of the log resolution.
Proposition 9.2.1 The following properties hold:
(1) (Wν , BW ν) is a log pair and the following adjunction formula holds
(K + B)| W ν = KW ν + BW ν.
(2) BW ν is effective if B is effective in a neighbourhood of W.
(3) If (X , B) has log canonical singularities near W, then (Wν , BW ν) has log canonical singularities.
(4) Let D be a ℚ-Cartier divisor on X such that mult W (D) = 0. Then (X,B + D) is a log pair, mult W (B + D) = 0 and
(B + D) W ν = BW ν + D |Wν. .
9.2.2 The discriminant of a log pair
Let f : X → Y be a fibration and let (X , B) be a log pair structure on X such that (X , B) has log canonical singularities over the generic point of Y. The singularities of the log pair (X , B) over the codimension one points of Y define a ℚ-Weil divisor BY on Y, called the discriminant of (X , B) on Y.
Let P ⊂ Y be a prime divisor. Since Y is normal, there exists an open set U ⊂ Y such that U ∩ P ≠ ∅ and P |U is a Cartier divisor. Let aP be the largest real number t such that the log pair (f −1(U), B |f −1 (U) + tf ∗(P|U)) has log canonical singularities over the generic point of P. It is clear that aP = 1 for all but finitely many prime divisors P of Y. The discriminant is defined by the following formula
(p. 165 ) Let μ: Y′ → Y be a birational contraction. Letting X′ be the normalisation of the graph of the rational map μ−1οf : X ⤏ Y′, we obtain a birational contraction μX and a fibre space f′ making the following diagram commute.
Let (X′, BX′ ) be the induced log-crepant log pair structure on X′, that is, KX′ + BX′ . Then (X′, BX′ ) has log canonical singularities over the generic point of Y′, and the discriminant of (X′, BX′ ) on Y′ is a well-defined ℝ-Weil divisor BY′ . We have
BY = μ *(BY′ ).
Thus, the set of ℚ-Weil divisors B = (BY′ ) Y′ is a ℚ-b-divisor of Y, called the discriminant ℚ-b-divisor induced by (X, B) via f. By construction, B depends only on the discrepancy ℚ-b-divisor A(X , B).
Lemma 9.2.2 Let f : X → Y be a fibration and let (X , B) be a log pair having Kawamata log terminal singularities over the generic point of Y. Let B be the induced discriminant ℚ-b-divisor of Y. Then
O Y (⌈−B⌉) ⊆ f * OX (⌈A(X, B)⌉).
Proof Clearly this is a local statement on the base. Thus it is enough to show the above inclusion at the level of global sections. Fix a non-zero rational function a ∈ k(Y)× such that . We claim that
Let E be a prime b-divisor of X. If then mult E (f * a) = 0 and mult E (⌈A(X, B)⌉) ≥ 0 by assumption. If there exists a commutative diagram
with the following properties:
(i) μ and μX are birational contractions, μX : (X′, BX′ ) → (X , B) is log crepant, and f′ is a fibre space.
(ii) is a prime divisor P on Y′.
(iii) There exists an open set U in Y′ such that P ∩ U ≠ ∅, P|U is non-singular, f′ −1(U) is non-singular and contains a simple normal crossings divisor ∑l Q l on f′ −1(U) such that B X′|f′ −1(U) = ∑l b l Q l and f′ *(P |U ) = ∑l m l Q l.
(iv) There exists l 0 such that
The multiplicity bP = mult P (B) is computed as follows
By assumption, mult P (a) + ⌈−bP ⌉ ≥ 0, so that mult P (a) + 1−bP > 0. The above formula implies that mult Ql0 (f * a) + 1−bl0 > 0, which means that the b-divisor has non-negative multiplicity at E. □
Proposition 9.2.3 Let f : X → Y be a fibre space and let (X, B) be a log pair having Kawamata log terminal singularities over the generic point of Y. Let B be the induced discriminant ℝ-b-divisor of Y. Let π : Y → S be a proper morphism. Let (D i ) i≥1 be a sequence of ℝ-b-Cartier ℝ-b-divisors of Y such that the sequence (f * D i ) i≥1 is asymptotically A(X, B)-saturated, relative to S. Then (D i ) i≥1 is asymptotically saturated with respect to −B, relative to S.
Proof We claim that the following inclusion
O Y (⌈−B + D⌉) ⊆ f * OX (⌈A(X, B) + f * D⌉)
holds for every ℝ-b-Cartier ℝ-b-divisor D of Y. Indeed, we may replace f : (X , B) → Y birationally, so that D = D̅, where D is an ℝ-Cartier ℝ-divisor on Y. Then (X , B − f * D) is a log pair having Kawamata log terminal singularities over the generic point of Y, with discriminant ℝ-b-divisor B − D, and A(X , B – f * D) = A(X , B) − f * D. The claim follows from Lemma 9.2.2 applied to f : (X , B − f * D) → Y.
Let ν = π οf. By assumption, there exists a positive integer I such that the following inclusion holds for every I|i, j:
ν * OX (⌈A(X , B) + jf * D i ⌉) ⊆ ν * OX (jf * D j ).
We have ν * OX (jf * D j ) = π * OY (j D j ), and from above we obtain
π * OY (⌈−B + j D i ȉ) ⊆ ν * OX (⌈A(X, B) + jf * D i ⌉).
Therefore, π * OY (⌈−B + j D i ⌉) ⊆ π * OY (j D j ). □
9.3 Non-klt finite generation
In this section we show that Shokurov algebras in dimension one and two are finite generated, in the presence of singularities worse than klt [Sho03, Conjecture 5.26, Example 4.41, Corollary 6.42]. Compared to the original statement, Theorem 9.3.1 contains two simplifications: we no longer assume that the boundary is effective, or that the algebra is ample on the non-klt locus.
(p. 167 ) Theorem 9.3.1 Let (X , B) be a log pair, let π : X → S be a proper surjective morphism, and let (D i ) i≥1 be a sequence of ℚ-b-divisors of X such that
(i) i D i is mobile/S, for every i.
(ii) D i ≤ D j for i|j.
(iii) The limit lim i→∞ D i = D is an ℝ-b-divisor of X.
Assume, moreover, that the following properties hold:
(1) −(K + B) is π-nef and π-big.
(2) D • is asymptotically A(X , B)-saturated over S; equivalently, there exists a positive integer I such that for every I|i, j, the following inclusion holds:
π * OX (⌈A(X, B) + j D i ⌉) ⊆ π * OX (j D j ).
(3) There exists an open neighbourhood U ⊆ X of nklt(X , B) such that D i|U = D |U for every i ≥ 1
If dim(X) ≤ 2, then D i = D for i sufficiently large and divisible.
Remark 9.3.2 Assumption (1) is redundant if dim(X) = 1.
Proof We may assume that S is affine. Since D 1 is mobile/S, there exists a rational function a ∈ k(X)× such that By (ii), we obtain for every i ≥ 1. The two sequences (D i )i and satisfy the same properties with respect to (X, B), and their stabilisation is equivalent. Therefore, we assume from now on that D i ≥ 0 for every i. After a truncation, we may also assume that I = 1 in (2).
(I) Assume dim(X) = dim(S) = 1. The problem is local, so we may assume that X = S is the germ of a non-singular curve at a point P. We have B = b · P, D i = d i · P and D = d · P. We have di ≤ d and lim i→∞ di = d. If b ≤ 1, then P ∈ nklt (X, B), hence D i = D for every i, by (3). Assume now that b < 1. Asymptotic saturation is equivalent to
⌈−b + jdi ⌉ ≤ jdj , ∀ i, j.
Letting i converge to infinity, we obtain
⌈−b + jd⌉ ≤ jdj , ∀ j.
In particular, ⌈−b + jd⌉ ≤ jd, which is equivalent to
If d ∉ ℚ, the Diophantine Approximation implies 1 ≤ b, contradicting our assumption. Therefore, d is rational. Let I′ be a positive integer such that I′d ∈ ℤ. We infer from the above that j(d − dj ) ≤ ⌊b⌋ ≤ 0 for I′|j. Therefore, b ≥ 0 and di = d for I′|i.
(II) Assume dim(X) = 1, dim(S) = 0. Thus, each D i is an effective ℚ-divisor Di of the non-singular proper curve X. Let D = lim i→∞ Di . If D = 0, then Di = 0 for every i. Otherwise, D is an ample ℝ-divisor. In particular, there exists a positive integer
(p. 168 ) I′ such that deg(I′ D − K − B)> 1. Asymptotic saturation means that for every i, j, the following inclusion holds
H 0(X, ⌈−B + jDi ⌉) ⊆ H 0(X, jDj ).
For fixed j, the divisor ⌈−B + jDi ⌉ coincides with ⌈−B + jD⌉ for some sufficiently large integer i. Therefore, for every j we have
H 0(X, ⌈−B + jD⌉) ⊆ H 0(X , jDj ).
We have ⌈−B + jD⌉ = K + ⌈jD − K − B⌉ and deg ⌈jD − K − B⌉ ≥ 2 for every I′ |j. Therefore, the linear system |⌈−B + jD⌉| is base-point free for I′|j. In particular, asymptotic saturation becomes
⌈−B + jD⌉ ≤ jDj , ∀ I′ |j.
A pointwise argument as in (I) implies that Di = D for i sufficiently large and divisible.
(III) Assume dim(X) = 2 and D i is big/S for some i; passing to a truncation, we may assume that D i is big/S for every i.
First, if dim(S) = 0, we may also assume that −(K + B).(i D i)X > 1 for every i. Indeed, each ℚ-b-divisor D i is b-big, and −(K + B) is nef and big. Therefore, −(K + B).D i,X > 0. Since D i , X ≤ D X , we obtain
−(K + B) · D X > 0.
In particular, lim i→∞ −(K + B) · (i D i ) X = +∞, so we may assume after a truncation that −(K + B) · (i D i ) X > 1 for everyi.
The log pair (X, B) has Kawamata log terminal singularities on the open set V = X \nklt(X , B). Therefore, there exist only finitely many geometric valuations E of k(X) such that cX (E) ∩ V ≠ ∅ and mult E A(X , B) ≤ 0. We consider a log crepant resolution μ: (Y , BY ) → (X , B) with the following properties:
(a) Y is non-singular and BY and D i , Y , for every i, are supported by a simple normal crossings divisor on Y.
(c) Every valuation E of X, for which cX (E) ∩ V ≠∅ and mult E A(X , B) ≤ 0, has acentre of codimension one on Y.
Let M = i D i for some i. If dim(S) = 0, then
−(KY + BY ) · M Y =−(K + B) · (i D i ) X > 1.
Proposition 9.3.3 applies for M and (Y, BY ), hence the linear system |M| Y is base-point free∕S on μ −1(V). Equivalently, the restriction of the ℚ-b-divisor to μ −1(V) is zero. On the other hand,D i|U = D 1|U by assumption, hence the support of the ℚ-divisor D i,Y − D 1,Y does not intersect μ −1(U). We infer by (b) that the restriction of the ℚ-b-divisor O Y( D i ) to μ −1(U) is zero. The open sets μ −1(U), μ −1(V) cover Y, hence O Y ( D i ) = 0. This holds for every i, hence we obtain
(p. 169 ) In particular, D Y is a π-nef ℝ-divisor and D Y −(KY + BY ) is π-nef and big. Furthermore, μ −1(U) is an open neighbourhood of nklt(Y, BY ) and D i |μ−1(U) = D|μ −1(U) for every i. We infer by [Amb05b], Theorem 3.3 that D i = D for i sufficiently large and divisible. Note that [Amb05b], Theorem 3.3 is stated only for characteristic sequences of functional algebras, but its proof is valid in our setting.
(IV) Assume dim(X) = 2 and, for all i, D i is not big∕S. After a truncation, there exists a rational map with connected fibres f : X ⤏ Y, defined over S, such that dim(Y) = 1 and there exist effective ℚ-divisors Di on Y such that for every i ≥ 1. It is clear that the sequence (D i ) i≥1 and its limit D = lim i→∞ Di satisfy the properties(i)–(iii) in the statement of the theorem.
If μ: X′ → X is a resolution of singularities and μ*(K + B) = KX′ + BX′ is the induced crepant log pair structure on X′, then the sequence (D i ) i satisfies the same properties (1)–(3) with respect to (X′, BX′) and μ −1(U). Therefore, we may assume that f is a morphism.
The sequence (D i ) i is constant in a neighbourhood of f (nklt(X, B)). If f (nklt(X, B)) = Y, we are done. Otherwise, f (nklt(X , B)) ≠ Y, that is, (X , B) has Kawamata log terminal singularities over the generic point of Y. In this case, the discriminant BY of (X , B) on Y is well defined. Since (D i ) i is A(X , B)-asymptotically saturated, we infer by Proposition 9.2.3 that the sequence (Di ) i is asymptotically saturated with respect to A(Y, BY )=−BY .
It is clear that Y, BY ) is a log pair structure on Y. Furthermore, the inclusion nklt(Y, BY ) ⊂ f (nklt(X , B)) implies that Di = D over a neighbourhood of nklt(Y , BY ). Therefore, the sequence (Di ) i and (Y , BY ) → S satisfy the hypothesis of the theorem, possibly except for property (1). We have not used the assumption (1) in the proof of(I) and (II), hence Di = D for i sufficiently large and divisible. Therefore, D i = D for i sufficiently large and divisible. □
Proposition 9.3.3 Let (X , B) be a 2-dimensional log pair and let π : X → S be a proper surjective morphism such that −(K + B) is π-nef and π-big. Let M be a mobile/S and b-big/S b-divisor of X such that
π * OX (⌈A(X , B) + M⌉) ⊆ π * OX (M).
If dim(S) = 0, assume, moreover, that −(K + B) · M X > 1. Then the relative base locus of the linear system |M| X is included in the set of points of X where the log pair (X , B) does not have terminal singularities.
Proof We may assume that S is affine. Let μ: Y → X be a resolution of singularities such that and Supp(BY ) is a simple normal crossings divisor, where μ *(K + B) = KY + BY . The general member C ∈ |M|Y is non-singular curve, intersecting Supp(BY) transversely. By saturation, we have
H0 (Y,⌈−BY ⌉ + C) ⊆ H0 (Y,C)
(1) The restriction map H 0(Y, ⌈−BY + C⌉)→ H 0(C, ⌈−BY + C⌉|C ) is surjectve. Indeed, the cokernel is included in
H 1(Y, ⌈−BY⌉) = H 1(Y, KY + ⌈−μ *(K + B)⌉).(p. 170 )
Since −μ *(K + B) is μ-nef and μ-big, we infer by Kawamata–Viehweg vanishing that H 1(Y, ⌈−BY ⌉) = 0, hence the claim holds.
(2) The linear system |⌈−BY + C⌉|C| is base-point free. Indeed, assume that dim (S) > 0. Then C an affine curve, hence OC (⌈−BY + C⌉|C) is generated by global sections. If dim(S) = 0, then C is a non-singular projective curve and the following identity holds by adjunction:
⌈−BY + C⌉|C = KC + ⌈−μ *(K + B)|C ⌉.
By assumption, we have deg(⌈−μ *(K + B)|C ⌉) ≥ 2. The claim follows now from a standard argument.
(3) BY is effective in a neighbourhood of C. Indeed, it follows from (1) and (2) that the sheaf OY (⌈−BY + C⌉) is generated by global sections in a neighbourhood of C. Therefore, the above saturation implies that ⌈−BY + C⌉ ≤ C in an open neighbourhood of C, which is equivalent to BY ≥ 0 near C.
It is clear that(3) implies that |M| X is base-point free at the terminal points of (X , B). □