On equivalent conjectures for minimal log discrepancies on smooth threefolds
aa r X i v : . [ m a t h . AG ] M a r ON EQUIVALENT CONJECTURES FOR MINIMAL LOGDISCREPANCIES ON SMOOTH THREEFOLDS
MASAYUKI KAWAKITA
Abstract.
On smooth threefolds, the ACC for minimal log discrepancies is equiv-alent to the boundedness of the log discrepancy of some divisor which computes theminimal log discrepancy. We reduce it to the case when the boundary is the productof a canonical part and the maximal ideal to some power. We prove the reducedassertion when the log canonical threshold of the maximal ideal is either at mostone-half or at least one. Introduction
Let P ∈ X be the germ of a smooth variety and a = Q ej =1 a r j j be an R -ideal on X .We write mld P ( X, a ) for the minimal log discrepancy of the pair ( X, a ) at P . For asubset I of the positive real numbers, we mean by a ∈ I that the exponents r j in a belong to I . ACC stands for the ascending chain condition while DCC stands for thedescending chain condition. This paper discusses the ACC conjecture for minimal logdiscrepancies on smooth threefolds, which was conjectured by Shokurov [4], [27] forarbitrary lc pairs. Conjecture A.
Fix a subset I of the positive real numbers which satisfies the DCC.Then the set { mld P ( X, a ) | P ∈ X a smooth threefold , a an R -ideal , a ∈ I } satisfies the ACC. We approach it with the theory of the generic limit of ideals introduced by de Fernexand Mustat¸˘a [7]. Our earlier work [17] shows the finiteness of the set of mld P ( X, a ) inwhich the germ P ∈ X of a klt variety and the exponents in a are fixed. Instead, if theexponents in a move in an infinite set satisfying the DCC, then we require the stabilityof minimal log discrepancies for generic limits. This stability connects Conjecture Ato the following important conjectures equivalently, as it was indicated essentially byMustat¸˘a and Nakamura [26].The first is the ACC for a -lc thresholds, a generalisation of lc thresholds. Conjecture B.
Fix a non-negative real number a and a subset I of the positive realnumbers which satisfies the DCC. Then the set { t ∈ R ≥ | P ∈ X a smooth threefold , a , b R -ideals , mld P ( X, ab t ) = a, ab ∈ I } satisfies the ACC. The second is a uniform version of the m -adic semi-continuity, which was proposedoriginally by Mustat¸˘a. Conjecture C.
Fix a finite subset I of the positive real numbers. Then there existsa positive integer l depending only on I such that if P ∈ X is the germ of a smooththreefold and if a = Q ej =1 a r j j and b = Q ej =1 b r j j are R -ideals on X satisfying that Partially supported by JSPS Grant-in-Aid for Scientific Research (C) 16K05099. r j ∈ I and a j + m l = b j + m l for any j , where m is the maximal ideal in O X defining P , then mld P ( X, a ) = mld P ( X, b ) . The last is the boundedness of the log discrepancy of some divisor which computesthe minimal log discrepancy, proposed by Nakamura.
Conjecture D.
Fix a finite subset I of the positive real numbers. Then there existsa positive integer l depending only on I such that if P ∈ X is the germ of a smooththreefold and a is an R -ideal on X satisfying that a ∈ I , then there exists a divisor E over X which computes mld P ( X, a ) and satisfies the inequality a E ( X ) ≤ l . The first main result of this paper is to reduce these conjectures to the case whenthe boundary is the product of a canonical part and the maximal ideal to some power.
Theorem 1.1.
Conjectures A , B , C and D are equivalent to Conjecture . Conjecture 1.2.
Let P ∈ X be the germ of a smooth threefold and m be the maximalideal in O X defining P . Fix a positive rational number q and a non-negative rationalnumber s . Then there exists a positive integer l depending only on q and s such thatif a is an ideal on X satisfying that ( X, a q ) is canonical, then there exists a divisor E over X which computes mld P ( X, a q m s ) and satisfies the inequality a E ( X ) ≤ l . Our earlier work derives this boundedness when ( X, a q ) is terminal or s is zero. Theorem 1.3.
Let P ∈ X be the germ of a smooth threefold and m be the maximalideal in O X defining P . (i) Fix a positive rational number q and a non-negative rational number s . Thenthere exists a positive integer l depending only on q and s such that if a is an idealon X satisfying that ( X, a q ) is terminal, then there exists a divisor E over X whichcomputes mld P ( X, a q m s ) and satisfies the inequality a E ( X ) ≤ l . (ii) Fix a positive rational number q . Then there exists a positive integer l dependingonly on q such that if a is an ideal on X satisfying that ( X, a q ) is canonical, then thereexists a divisor E over X which computes mld P ( X, a q ) and satisfies the inequality a E ( X ) ≤ l . The second main result is to prove Conjecture 1.2 when the lc threshold of themaximal ideal is either at most one-half or at least one.
Theorem 1.4.
Let P ∈ X be the germ of a smooth threefold and m be the maximalideal in O X defining P . Fix a positive rational number q and a non-negative rationalnumber s . (i) There exists a positive integer l depending only on q and s such that if a is anideal on X satisfying that ( X, a q ) is canonical and that mld P ( X, a q m / ) is not positive,then there exists a divisor E over X which computes mld P ( X, a q m s ) and satisfies theinequality a E ( X ) ≤ l . (ii) There exists a positive integer l depending only on q and s such that if a isan ideal on X satisfying that ( X, a q ) is canonical and that ( X, a q m ) is lc, then thereexists a divisor E over X which computes mld P ( X, a q m s ) and satisfies the inequality a E ( X ) ≤ l . Once Theorem 1.4(i) is established, it is relatively simple to obtain Conjecture 1.2when s is close to zero in terms of a scale determined by q . Corollary 1.5.
Conjecture holds when s is at most /n for some integer n greaterthan one such that nq is integral. N EQUIVALENT CONJECTURES ON SMOOTH THREEFOLDS 3
We shall explain the outline of our research. Fix the germ P ∈ X of a smooththreefold and a positive rational number q . For a sequence { a i } i ∈ N of ideals on X , itsgeneric limit a is defined on the spectrum ˆ P ∈ ˆ X of the completion of the local ring O X,P ⊗ k K , where K is an extension of the ground field k . The stability of minimallog discrepancies means the equalitymld ˆ P ( ˆ X, a q ) = mld P ( X, a qi )for infinitely many i , to which any of Conjectures A to D is equivalent (Theorem 4.6).The strategy employed in this paper is to pursue Conjecture D.Our previous work [18] derived the above stability except for the case when ( ˆ X, a q )has the smallest lc centre of dimension one, which implies that mld ˆ P ( ˆ X, a q ) is atmost one. By this result, in order to prove Conjecture D, one has only to considerthose ideals a which have mld P ( X, a q ) less than one. We begin with the ACC for1-lc thresholds [29]. Using it together with the classification of divisorial contractions[13], [20], we construct a birational morphism Y → X with bounded log discrepanciesby which ( X, a ) can be replaced with a pair ( Y, ( a ′ ) q b q ) satisfying that ( Y, ( a ′ ) q ) iscanonical and that b has bounded colength (Theorem 5.1).We study the generic limit a of a sequence of ideals a i on X such that ( X, a qi ) iscanonical. We may assume that ( ˆ X, a q ) has the smallest lc centre ˆ C of dimensionone. Then the mld ˆ P ( ˆ X, a q ) equals one by the canonicity of ( X, a qi ), and so does themld P ( X, a qi ). By our result [19] in dimension two, there exists a divisor ˆ E over ˆ X computing mld η ˆ C ( ˆ X, a q ) = 0 which is obtained at the generic point η ˆ C of ˆ C by aweighted blow-up. On our extra condition that mld ˆ P ( ˆ X, a q ) = 1, we find ˆ E for whichthe weighted blow-up at η ˆ C is extended to the closed point ˆ P (Theorem 6.2).We associate the minimal log discrepancy on ˆ X with that on ˆ E by precise inversionof adjunction (Section 7). The generic limit b of a sequence of ideals of boundedcolength satisfies that b O ˆ C = ˆ m b O ˆ C for some integer b , where ˆ m is the maximal idealin O ˆ X . Then Conjecture D is reduced to the case when b is the maximal ideal m tothe power of b , which completes Theorem 1.1.Suppose that the lc threshold of m with respect to ( X, a q ) is at most one-half. Underour assumptions on the generic limit a involved, we derive a special conclusion thatmld P ( X, a q m s ) = 1 − s , which includes the boundedness stated in Theorem 1.4(i). Asimilar argument is applied to the case of lc threshold at least one, Theorem 1.4(ii).Conjecture 1.2 remains open when mld P ( X, a q ) equals one and mld P ( X, a q m / ) ispositive. In this case, every divisor E over X computing mld P ( X, a q ) satisfies thatord E m equals one. We supply a classification of the centre of E on a certain weightedblow-up of X (Theorem 9.1). 2. Preliminaries
We shall fix the notation and review the basics of singularities in birational geometry.Refer to [22] for details.We work over an algebraically closed field k of characteristic zero. We omit to writethe bases of tensor products over k and of products over Spec k when it is clear. A variety is an integral separated scheme of finite type over Spec k . The dimension of ascheme means the Krull dimension. A variety of dimension one (resp. two) is called a curve (resp. a surface ).The germ is considered at a closed point algebraically. When we work on thespectrum of a noetherian ring, we identify an ideal in the ring with its coherent ideal MASAYUKI KAWAKITA sheaf. For an irreducible closed subset Z of a scheme, we write η Z for the genericpoint of Z .The round-down ⌊ r ⌋ of a real number r is the greatest integer at most r . The natural number starts from zero. Orders . Let X be a noetherian scheme and Z be an irreducible closed subset of X .The order of a coherent ideal sheaf a on X along Z is the maximal ν ∈ N ∪ { + ∞} satisfying that a O X,η Z ⊂ I ν O X,η Z for the ideal sheaf I of Z , and it is denoted byord Z a . If Y → X is a birational morphism from a noetherian normal scheme, thenwe set ord E a = ord E a O Y for a prime divisor E on Y . The ord Z f for a function f in O X stands for ord Z ( f O X ).Suppose that X is normal. For an effective Q -Cartier divisor D on X , we setord Z D = r − ord Z O X ( − rD ) for a positive integer r such that rD is Cartier, whichis independent of the choice of r . The notion of ord Z D is extended to R -Cartier R -divisors by linearity. R -ideals . An R - ideal on a noetherian scheme X is a formal product a = Q j a r j j offinitely many coherent ideal sheaves a j on X with positive real exponents r j . For apositive real number t , the a to the power of t is a t = Q j a tr j j . The cosupport of a is the union of the supports of O X / a j for all j . The order of a along an irreducibleclosed subset Z of X is ord Z a = P j r j ord Z a j . The a is said to be invertible if all a j are invertible. The pull-back of a by a morphism Y → X is a O Y = Q j ( a j O Y ) r j . For asubset I of the positive real numbers, we mean by a ∈ I that all exponents r j belongto I .If a is invertible, then the R -divisor A = P j r j A j for which a j = O X ( − A j ) is calledthe R -divisor defined by a . When we work on the germ P ∈ X , the R -divisor definedby a general member in a means an R -divisor P j r j ( f j ) on X with a general member f j in a j . The a is said to be m -primary if all a j are m -primary, where m is the maximalideal in O X defining P . Convention . It is sometimes convenient to allow an exponent in an R -ideal to bezero. We define a coherent ideal sheaf to the power of zero as the structure sheaf. The minimal log discrepancy . A subtriple ( X, ∆ , a ) consists of a normal variety X ,an R -divisor ∆ on X such that K X + ∆ is R -Cartier, and an R -ideal a on X . The( X, ∆ , a ) is called a triple if ∆ is effective. We omit to write a or ∆ and call ( X, ∆)or ( X, a ) a ( sub ) pair when a = O X or ∆ = 0. The ∆ or a is called the boundary when( X, ∆) or ( X, a ) is a pair.A prime divisor E on a normal variety Y equipped with a birational morphism π : Y → X is called a divisor over X , and the closure of the image π ( E ) is called the centre of E on X and denoted by c X ( E ). We write D X for the set of all divisors over X . Two elements in D X are usually identified when they define the same valuationon the function field of X . The log discrepancy of E with respect to ( X, ∆ , a ) is a E ( X, ∆ , a ) = 1 + ord E K Y/ ( X, ∆) − ord E a , where K Y/ ( X, ∆) = K Y − π ∗ ( K X + ∆).Let Z be a closed subvariety of X . The minimal log discrepancy of ( X, ∆ , a ) at thegeneric point η Z ismld η Z ( X, ∆ , a ) = inf { a E ( X, ∆ , a ) | E ∈ D X , c X ( E ) = Z } , It is either a non-negative real number or minus infinity. We say that E ∈ D X computes mld η Z ( X, ∆ , a ) if c X ( E ) = Z and a E ( X, ∆ , a ) = mld η Z ( X, ∆ , a ) (or is negative when N EQUIVALENT CONJECTURES ON SMOOTH THREEFOLDS 5 mld η Z ( X, ∆ , a ) = −∞ ). It is often enough to study the case when Z is a closed pointby the relation mld η Z ( X, ∆ , a ) = mld P ( X, ∆ , a ) − dim Z for a general closed point P in Z . It is sometimes convenient to use the minimal log discrepancy of ( X, ∆ , a ) in aclosed subset W of X which is defined bymld W ( X, ∆ , a ) = inf { a E ( X, ∆ , a ) | E ∈ D X , c X ( E ) ⊂ W } . Singularities . The subtriple ( X, ∆ , a ) is said to be log canonical ( lc ) (resp. Kawa-mata log terminal ( klt )) if a E ( X, ∆ , a ) ≥ >
0) for all E ∈ D X . It is said tobe purely log terminal ( plt ) (resp. canonical , terminal ) if a E ( X, ∆ , a ) > ≥ >
1) for all E ∈ D X exceptional over X . For a closed point P in X , ( X, ∆ , a ) is lcabout P iff mld P ( X, ∆ , a ) is not minus infinity. When ( X, ∆ , a ) is lc, the lc threshold with respect to ( X, ∆ , a ) of a non-trivial R -ideal b on X is the maximal real number t such that ( X, ∆ , ab t ) is lc.Let Y be a normal variety birational to X . A centre c Y ( E ) on Y of E ∈ D Y suchthat a E ( X, ∆ , a ) ≤ non-klt centre on Y of ( X, ∆ , a ). The union of allnon-klt centres on Y is called the non-klt locus on Y of ( X, ∆ , a ). When we just saya non-klt centre or the non-klt locus, we mean that it is on X .When ( X, ∆ , a ) is lc, a non-klt centre of ( X, ∆ , a ) is often called an lc centre . Whenwe work on the germ of a variety, an lc centre contained in every lc centre is called the smallest lc centre . The smallest lc centre exists and it is normal [10, Theorem 9.1].The index of a normal Q -Gorenstein singularity P ∈ X is the least positive integer r such that rK X is Cartier at P . Birational transformations . A reduced divisor D on a smooth variety X is said tobe simple normal crossing ( snc ) if D is defined at every closed point P in X by theproduct of a part of a regular system of parameters in O X,P . A stratum of D = P i ∈ I D i is an irreducible component of T i ∈ I ′ D i for a subset I ′ of I . For a smooth morphism X → S , the D said to be snc relative to S if every stratum of D is smooth over S .A log resolution of a subtriple ( X, ∆ , a ) is a projective birational morphism from asmooth variety Y to X such that • the exceptional locus is a divisor and a O Y is invertible, • the union of the exceptional locus, the support of the strict transform of ∆, andthe cosupport of a O Y is snc, and • it is isomorphic on the maximal open locus U in X such that U is smooth, a O U is invertible, and the union of the support of ∆ | U and the cosupport of a O U issnc.Let ( X, ∆ , a ) be a subtriple, where a = Q j a r j j , and Y be a normal variety birationalto X . A subtriple ( Y, Γ , b ) is said to be crepant to ( X, ∆ , a ) if a E ( X, ∆ , a ) = a E ( Y, Γ , b )for any divisor E over X and Y . Suppose that Y is smooth and has a birationalmorphism to X whose exceptional locus is a divisor P i E i . The weak transform on Y of a is the R -ideal a Y = Q j ( a jY ) r j defined by a jY = a j O Y ( P i (ord E i a j ) E i ) . Remark that this notion is different from that of the strict transform, the j -th ideal ofwhich is P f ∈ a j f O Y ( P i (ord E i f ) E i ) (see [11, III Definition 5]). The definition of theweak transform a Y is extended to the case when Y is normal as far as P i (ord E i a j ) E i is Cartier for any j . We introduce Definition 2.2.
The pull-back of ( X, ∆ , a ) by Y → X is the subtriple ( Y, ∆ Y , a Y ) inwhich ∆ Y = − K Y/ ( X, ∆) + P ij ( r j ord E i a j ) E i . MASAYUKI KAWAKITA
The pull-back ( Y, ∆ Y , a Y ) is crepant to ( X, ∆ , a ). Weighted blow-ups . Let P ∈ X be the germ of a smooth variety. Let x , . . . , x c bea part of a regular system of parameters in O X,P and w , . . . , w c be positive integers.For w ∈ N , let I w be the ideal in O X generated by all monomials x s · · · x s c c suchthat P ci =1 s i w i ≥ w . The weighted blow-up of X with wt( x , . . . , x c ) = ( w , . . . , w c ) isProj X ( L w ∈ N I w ). Remark . If x ′ , . . . , x ′ c is a part of another regular system of parameters such that x ′ i ∈ I w i \ I w i +1 for any i , then the weighted blow-up of X with wt( x ′ , . . . , x ′ c ) =( w , . . . , w c ) is the same that is obtained by wt( x , . . . , x c ) = ( w , . . . , w c ).Its explicit description is reduced to the case of the affine space by an ´etale mor-phism. Let o ∈ A d be the germ at origin of the affine space with coordinates x , · · · , x d and Y be the weighted blow-up of A d with wt( x , . . . , x d ) = ( w , . . . , w d ). One mayassume that w , . . . , w d have no common divisors. Then Y is covered by the affinecharts U i = A d / Z w i ( w , . . . , w i − , − , w i +1 , . . . , w d ) for 1 ≤ i ≤ d , and the exceptionaldivisor is isomorphic to the weighted projective space P ( w , . . . , w d ) (see [24, 6.38] fordetails).Here the notation A d / Z r ( a , . . . , a d ) stands for the quotient of A d by the cyclicgroup Z r of order r whose generator sends the i -th coordinate x i of A d to ζ a i x i ,where ζ is a primitive r -th root of unity. The x , . . . , x d on this quotient are called orbifold coordinates . An isolated cyclic quotient singularity means that the spectrumof the completion of its local ring coincides with the regular base change of some A d / Z r ( a , . . . , a d ), in which it is said to be of type r ( a , . . . , a d ).In terms of toric geometry following the notation in [12], by setting N = Z d + Z v where v = r ( a , . . . , a d ), the quotient A d / Z r ( a , . . . , a d ) is the toric variety T N (∆)which corresponds to the cone ∆ spanned by the standard basis e , . . . , e d of Z d . For e = r ( w , . . . , w d ) ∈ N ∩ ∆, the weighted blow-up of A d / Z r ( a , . . . , a d ) with respectto wt( x , . . . , x d ) = r ( w , . . . , w d ) is defined by adding the ray generated by e . Adjunction . Let X be a normal variety and S + B be an effective R -divisor on X such that S is reduced and has no common components with the support of B .Suppose that they form a pair ( X, S + B ). Then one has the adjunction ν ∗ ( K X + S + B | S ) = K S ν + B S ν on the normalisation ν : S ν → S of S , in which B S ν is an effective R -divisor called the different on S ν of B (see [23, Chapter 16] or [28, Section 3]). Example . Let X = A / Z r (1 , w ) with orbifold coordinates x , x such that w iscoprime to r . Let S be the curve on X defined by x and P be the origin of X . Then K X + S | S = K S + (1 − r − ) P .The singularity on X is associated with that on S ν by Theorem 2.5 (Inversion of adjunction) . Notation as above. (i) ([23, Theorem 17.6]) (
X, S + B ) is plt about S iff ( S ν , B S ν ) is klt. In this case, S is normal. (ii) ([15]) ( X, S + B ) is lc about S iff ( S ν , B S ν ) is lc. R -varieties . The notions explained above make sense over the ring R of formalpower series over a field of characteristic zero, which has been discussed by de Fernex,Ein and Mustat¸˘a [6], [7]. We mean by an R -variety an integral separated scheme offinite type over Spec R . We consider regular R -varieties instead of smooth R -varieties. N EQUIVALENT CONJECTURES ON SMOOTH THREEFOLDS 7
The canonical divisor K X on a normal R -variety X is defined by the sheaf of specialdifferentials in [6]. Let Y → X be a birational morphism between regular R -varieties.The relative canonical divisor K Y/X is the effective divisor defined by the zeroth Fittingideal of Ω
Y/X [6, Remark A.12]. In particular, K Y/X is independent of the structureof X as an R -variety. Remark . Let P ∈ X be the germ of a normal Q -Gorenstein R -variety and a be an R -ideal on X . Let X ′ be either • the spectrum of the completion of the local ring O X,P , or • X × Spec R Spec R ′ , where R ′ is the completion of R ⊗ K K ′ for a field extension K ′ of K ,which has a regular morphism π : X ′ → X . Then K X ′ = π ∗ K X , by which one has that a E ′ ( X ′ , a O X ′ ) = a E ( X, a ) and mld P ′ ( X ′ , a O X ′ ) = mld P ( X, a ) for any components E ′ of E × X X ′ and P ′ of P × X X ′ . Lemma 2.7.
Let P ∈ X be the germ of an R -variety. Let ˆ X be the spectrum of thecompletion of the local ring O X,P and ˆ P be its closed point. (i) Let m be the maximal ideal in O X defining P and ˆ m be the maximal ideal in O ˆ X . Then the pull-back defines a bijective map from the set of m -primary R -ideals on X to the set of ˆ m -primary R -ideals on ˆ X . (ii) Suppose that X is normal. Then the base change defines a bijective map fromthe set of divisors over X with centre P to the set of divisors over ˆ X with centre ˆ P .Proof. The (i) follows from the isomorphisms O X / m l ≃ O ˆ X / ˆ m l , while (ii) follows fromthe property that blowing-up commutes with flat base changes. q.e.d.By Lemma 2.7, in order to study the minimal log discrepancy at the closed pointof the germ P ∈ X , one may often replace P ∈ X with a germ the completion of thelocal ring of which is isomorphic to that of O X,P .3.
The generic limit of ideals
We recall the generic limit of ideals on a fixed germ. It was introduced by de Fernexand Mustat¸˘a [7] and simplified by Koll´ar [21]. We follow our style of the definition in[18].Let P ∈ X be the germ of a scheme of finite type over k and m be the maximalideal in O X defining P . Let S = { ( a i , . . . , a ie ) } i ∈ N be an infinite sequence of e -tuplesof ideals in O X . For every positive integer l , the ideal ( a ij + m l ) / m l in O X / m l for i ∈ N and 1 ≤ j ≤ e corresponds to a closed point P ij ( l ) in the Hilbert scheme H l parametrising ideals in O X / m l . The H l is a scheme of finite type over k and thereexists a natural rational map H l +1 → H l . Take the Zariski closure Z j ( l ) of the subset { P ij ( l ) } i ∈ N in H l . By finding a locally closed irreducible subset Z l of Z ( l ) × · · · × Z e ( l )inductively, one obtains a family of approximations of S defined below. Definition 3.1. A family F = ( Z l , ( a j ( l )) j , N l , s l , t l ) l ≥ l of approximations of S con-sists of a fixed positive integer l and for every l ≥ l , • a variety Z l , • an ideal sheaf a j ( l ) on X × Z l for every 1 ≤ j ≤ e which is flat over Z l andcontains m l O X × Z l , • an infinite subset N l of N and a map s l : N l → Z l ( k ), where Z l ( k ) denotes theset of the k -points in Z l , and • a dominant morphism t l : Z l +1 → Z l ,such that MASAYUKI KAWAKITA • a j ( l ) O X × Z l +1 = a j ( l + 1) + m l O X × Z l +1 by id X × t l , • a j ( l ) i = a ij + m l for i ∈ N l , where a j ( l ) i = a j ( l ) ⊗ O Zl k is the ideal in O X givenby the closed point s l ( i ) ∈ Z l , • the image of N l by s l is dense in Z l , and • N l +1 is contained in N l and t l ◦ s l +1 = s l | N l +1 .For the above F , let K = lim −→ l K ( Z l ) be the union of the function fields K ( Z l ) of Z l by the inclusions t ∗ l : K ( Z l ) → K ( Z l +1 ). Let ˆ X be the spectrum of the completionof the local ring O X,P ⊗ k K . Let ˆ P be the closed point of ˆ X and ˆ m be the maximalideal in O ˆ X . Definition 3.2.
The generic limit of S with respect to F is the e -tuple ( a , . . . , a e )of ideals in O ˆ X defined by a j = lim ←− l a j ( l ) K , where a j ( l ) K = a j ( l ) ⊗ O Zl K is the ideal in O X ⊗ k K given by the natural K -pointSpec K → Z l . Remark . Let R be the completion of the local ring O X,P . In the literature, thegeneric limit is defined for a sequence of e -tuples of ideals in R . When a ij are idealsin R , the generic limit is defined in the same way just by replacing the condition a j ( l ) i = a ij + m l in Definition 3.1 with a j ( l ) i = ( a ij + m l R ) ∩ O X .By the very definition, one has Lemma 3.4.
Let ( a , b ) be a generic limit of a sequence { ( a i , b i ) } i ∈ N of pairs of idealsin O X . (i) If a i ⊂ b i for any i , then a ⊂ b . (ii) The a + b and ab are the generic limits of { a i + b i } i ∈ N and { a i b i } i ∈ N . The generic limit depends on the choice of F but remains the same after the re-placement of F with a subfamily. Definition 3.5.
A family F ′ = ( Z ′ l , ( a ′ j ( l )) j , N ′ l , s ′ l , t ′ l ) l ≥ l ′ of approximations of S iscalled a subfamily of F if l ′ is at least l and if there exists an open immersion i l : Z ′ l → Z l for every l ≥ l ′ such that • t l ◦ i l +1 = i l ◦ t ′ l , • a j ( l ) O X × Z ′ l = a ′ j ( l ) by id X × i l , and • N ′ l is a subset of N l and i l ◦ s ′ l = s l | N ′ l . Convention . Later we shall often replace F with a subfamily, but we retain thesame notation F = ( Z l , ( a j ( l )) j , N l , s l , t l ) l ≥ l to avoid intricacy.The theory of the generic limit of ideals was developed for the study of the singu-larities on the germ P ∈ X . When X is klt, the singularities on ˆ X are associated withthose on X (see [6]). The existence of log resolutions supplies Lemma 3.7.
Notation as above and assume that X is klt. Then ˆ X is klt, and afterreplacing F with a subfamily but using the same notation, mld ˆ P ( ˆ X, Q ej =1 ( a j + ˆ m l ) r j ) = mld P ( X, Q ej =1 a j ( l ) r j i ) for any positive real numbers r , . . . , r e and for any i ∈ N l and l ≥ l . N EQUIVALENT CONJECTURES ON SMOOTH THREEFOLDS 9
Remark . (i) Let ˆ E be a divisor over ˆ X with centre ˆ P . Then replacing F witha subfamily (but using the same notation as in Convention 3.6), one can descend ˆ E to a divisor E l over X × Z l for any l ≥ l , that is, E l ′ = E l × Z l Z l ′ when l ≤ l ′ , andˆ E = E l × X × Z l ˆ X . Let E i be any connected component of the fibre of E l at s l ( i ) ∈ Z l ,which is independent of l as far as i ∈ N l . Replacing F with a subfamily again, forany i ∈ N l and 1 ≤ j ≤ e , E i is a divisor over X and satisfies thatord ˆ E a j = ord ˆ E ( a j + ˆ m l ) = ord E i ( a ij + m l ) = ord E i a ij < l,a ˆ E ( ˆ X, a ) = a E i ( X, a i ) . (ii) Let ˆ π : ˆ Y → ˆ X be a projective birational morphism isomorphic outside ˆ P . Thenˆ π is descendible as stated in [18, Proposition A.7], that is, after replacing F with asubfamily, there exist projective morphisms π l : Y l → X × Z l such that π l ′ = π l × Z l Z l ′ when l ≤ l ′ and such that ˆ π = π l × X × Z l ˆ X . Remark . The E l is treated in [18] as if it has connected fibres, which should havebeen corrected appropriately.Now we fix positive real numbers r , . . . , r e and consider the R -ideals a i = Q ej =1 a r j ij and a = Q ej =1 a r j j . The a is called the generic limit of the sequence { a i } i ∈ N of R -idealson X with respect to F . The most important achievement at present is the followingtheorem due to de Fernex, Ein and Mustat¸˘a. Indeed, as an application, they provedfirst the ACC for lc thresholds restricted on smooth varieties. Theorem 3.10 ([5], [6]) . Notation as above and assume that X is klt. If ( ˆ X, a ) is lc,then so is ( X, a i ) for any i ∈ N l after replacing F with a subfamily which depends on r , . . . , r e . Theorem 3.10 is a corollary to the effective m -adic semi-continuity of lc thresholds,which was globalised in [18, Theorem 4.11]. We prove its relative version. Theorem 3.11.
Let X be a klt variety and X → T be a morphism to a variety.Suppose that every closed fibre of X → T is klt. Let a = Q j a r j j be an R -ideal on X and Z be an irreducible closed subset of X which dominates T . Suppose that mld η Z ( X, a ) =0 and it is computed by a divisor E over X . Then after replacing X and T with theirdense open subsets, the following hold for any t ∈ T . • The fibre of E at t is non-empty, and its arbitrary connected component E t isa divisor over a component X t of the fibre of X at t . • The centre Z t on X t of E t is smooth. • If an R -ideal b = Q j b r j j on X t satisfies that a j O X t + p j = b j + p j for any j , where p j = { f ∈ O X t | ord E t f > ord E a j } , then ( X t , b ) is lc about Z t and mld η Zt ( X t , b ) = 0 .Proof. Take a log resolution π : Y → X of ( X, a I Z ), where I Z is the ideal sheaf of Z ,such that E is realised as a divisor on Y . We may shrink T so that T and Y → T aresmooth and so that the union F of the exceptional locus of π and the cosupport of a I Z O Y is an snc divisor relative to T . Replace X with an open subset X ′ containing η Z such that Z ′ = Z | X ′ is smooth over T and such that if the restriction S ′ = S | π − ( X ′ ) of a stratum S of F satisfies that S ′ = ∅ and π ( S ′ ) ⊂ Z ′ , then S ′ → Z ′ is smooth.Set n = dim Z − dim T . Then for any t ∈ T and z ∈ Z t ,mld z ( X t , m nz · a O X t ) = 0for the maximal ideal sheaf m z on X t defining z , and it is computed by the divisor G z obtained by the blow-up of Y t = Y × X X t along a component of E t ∩ π − ( z ). This is verified from the local description at each closed point y in π − ( z ). Indeed, let v , . . . , v s be a part of a regular system of parameters in O Y,y such that F is definedat y by Q sl =1 v l . Since every stratum of F mapped into Z is smooth over Z , theyare extended to a part v , . . . , v s , w , . . . , w n of a regular system of parameters in O Y,y such that their images form a part of a regular system of parameters in O Y t ,y and suchthat m z O Y t ,y = ( w , . . . , w n , Q sl =1 v m l l ) O Y t ,y , where m l is the order of I Z along the divisor defined by v l . (Note that the corre-sponding expression in the proof of [18, Theorem 4.11] is incorrect).Since ord G z a j O X t = ord E a j and ord G z f ≥ ord E t f for any f ∈ O X t , by [5, Theorem1.4] we conclude that mld z ( X t , m nz b ) = 0 for the b in the statement. Hence ( X t , b ) islc about Z t , and mld η Zt ( X t , b ) = 0 by a E t ( X t , b ) = 0. q.e.d. Corollary 3.12.
Let X be a klt variety and X → T be a morphism to a variety.Suppose that the fibre X t at every closed point t in T is klt. Let a = Q j a r j j be an R -ideal on X and Z be a closed subset of X such that ( X, a ) is lc about Z . Set Z t = Z × X X t and let I t denote the ideal sheaf of Z t on X t . Then there exists apositive integer l such that after replacing T with its dense open subset, for any t ∈ T if an R -ideal b = Q j b r j j on X t satisfies that a j O X t + I lt = b j + I lt for any j , then ( X t , b ) is lc about Z t .Proof. We shall prove it by noetherian induction on Z . Let Z be an irreduciblecomponent of Z , which may be assumed to dominate T . Let I Z be the ideal sheafof Z and r be the non-negative real number such that mld η Z ( X, a I rZ ) equals zero.Applying Theorem 3.11, after shrinking T there exist open subset X ′ of X containing η Z and a positive integer l such that for any t ∈ T , if an R -ideal b = Q j b r j j on X t satisfies that a j O X ′ t + I l Z O X ′ t = b j O X ′ t + I l Z O X ′ t for any j on X ′ t = X ′ × X X t , then( X ′ t , b O X ′ t ) is lc about Z × X X ′ t . Thus the assertion is reduced to that for the closureof Z \ ( Z ∩ X ′ ), which follows from the hypothesis of induction. q.e.d.4. Singularities on a fixed variety
In this section, we fix the germ P ∈ X of a klt variety and review an approach tothe study of mld P ( X, a ) for R -ideals a which uses the generic limit of ideals on X .Our earlier work shows the discreteness for log discrepancies a E ( X, a ). Theorem 4.1 ([17]) . Let P ∈ X be the germ of a klt variety. Fix a finite subset I ofthe positive real numbers. Then the set { a E ( X, a ) | a an R -ideal , a ∈ I, E ∈ D X , ( X, a ) lc about η c X ( E ) } is discrete in R . We shall explain the equivalence of several important conjectures on a fixed germwith the help of Theorems 3.10 and 4.1.
Conjecture 4.2.
Let P ∈ X be the germ of a klt variety. (i) (ACC for minimal log discrepancies) Fix a subset I of the positive real numberswhich satisfies the DCC. Then the set { mld P ( X, a ) | a an R -ideal , a ∈ I } satisfies the ACC. N EQUIVALENT CONJECTURES ON SMOOTH THREEFOLDS 11 (ii) (ACC for a -lc thresholds) Fix a non-negative real number a and a subset I ofthe positive real numbers which satisfies the DCC. Then the set { t ∈ R ≥ | a , b R -ideals , mld P ( X, ab t ) = a, ab ∈ I } satisfies the ACC. (iii) (uniform m -adic semi-continuity) Fix a finite subset I of the positive real num-bers. Then there exists a positive integer l depending only on X and I such thatif a = Q ej =1 a r j j and b = Q ej =1 b r j j are R -ideals on X satisfying that r j ∈ I and a j + m l = b j + m l for any j , where m is the maximal ideal in O X defining P , then mld P ( X, a ) = mld P ( X, b ) . (iv) (boundedness) Fix a finite subset I of the positive real numbers. Then thereexists a positive integer l depending only on X and I such that if a is an R -ideal on X satisfying that a ∈ I , then there exists a divisor E over X which computes mld P ( X, a ) and satisfies the inequality a E ( X ) ≤ l . (v) (generic limit) Let r , . . . , r e be positive real numbers and { a i = Q ej =1 a r j ij } i ∈ N be a sequence of R -ideals on X . Notation as in Section , so set the generic limit a = Q ej =1 a r j j on ˆ P ∈ ˆ X . Then mld ˆ P ( ˆ X, a ) = mld P ( X, a i ) for any i ∈ N l afterreplacing F with a subfamily but using the same notation.Remark . We provide a few remarks on Conjecture 4.2(v).(i) Lemma 3.7 means the equalitymld ˆ P ( ˆ X, Q ej =1 ( a j + ˆ m l ) r j ) = mld P ( X, Q ej =1 ( a ij + m l ) r j )for any i ∈ N l after replacing F with a subfamily. Take a divisor ˆ E over ˆ X whichcomputes mld ˆ P ( ˆ X, a ) and choose an integer l ≥ l such that ord ˆ E a j ≤ l ord ˆ E ˆ m forany j . Then for l ≥ l , the left-hand side equals mld ˆ P ( ˆ X, a ) while the right-hand sideis at least mld P ( X, a i ). Thus after replacing l with l , one has the inequalitymld ˆ P ( ˆ X, a ) ≥ mld P ( X, a i )for any i ∈ N l . The intrinsic part in Conjecture 4.2(v) is the opposite inequality.(ii) In particular, Conjecture 4.2(v) holds when mld ˆ P ( ˆ X, a ) is not positive by The-orem 3.10. The conjecture also holds when ( ˆ X, a ) is klt [18, Theorem 5.1]. Thus, weknow that Conjecture 4.2(v) holds unless ( ˆ X, a ) is not klt but mld ˆ P ( ˆ X, a ) is positive.We prepare basic lemmata. Lemma 4.4.
Let I and J be subsets of the positive real numbers both of which satisfythe DCC. Then the set { rs | r ∈ I, s ∈ J } satisfies the DCC.Proof. Let { r i s i } i ∈ N be an arbitrary non-increasing sequence where r i ∈ I and s i ∈ J .It is enough to show that r i s i is constant passing to a subsequence. We claim that thereexists a strictly increasing sequence { i j } j ∈ N such that { r i j } j ∈ N is a non-decreasingsequence. Indeed, let i be a number such that r i attains the minimum of the set { r i | i ∈ N } , which exists since this set satisfies the DCC. If one constructed i , . . . , i j ,then take i j +1 as a number such that r i j +1 attains the minimum of the set { r i | i > i j } .By replacing { r i s i } i ∈ N with { r i j s i j } j ∈ N , we may assume that r i is non-decreasing.Applying the same argument to { s i } i ∈ N , we may also assume that s i is non-decreasing.Then the sequence { r i s i } i ∈ N becomes both non-increasing and non-decreasing, so r i s i must be constant. q.e.d. Lemma 4.5.
Let P ∈ X be the germ of a normal Q -Gorenstein variety and a , . . . , a e be R -ideals on X . Let t , . . . , t e be non-negative real numbers such that P ei =1 t i = 1 . (i) mld P ( X, Q ei =1 a t i i ) ≥ P ei =1 t i mld P ( X, a i ) . (ii) If a divisor E over X computes all mld P ( X, a i ) , then mld P ( X, Q ei =1 a t i i ) = P ei =1 t i mld P ( X, a i ) and it is computed by E .Proof. Let F be a divisor over X which computes mld P ( X, Q ei =1 a t i i ). Then,mld P ( X, Q ei =1 a t i i ) = a F ( X, Q ei =1 a t i i ) = e X i =1 t i · a F ( X, a i ) ≥ e X i =1 t i mld P ( X, a i ) , which is (i). On the other hand, if E computes all mld P ( X, a i ), thenmld P ( X, Q ei =1 a t i i ) ≤ a E ( X, Q ei =1 a t i i ) = e X i =1 t i · a E ( X, a i ) = e X i =1 t i mld P ( X, a i ) , which with (i) shows the assertion (ii). q.e.d. Theorem 4.6.
Let P ∈ X be the germ of a klt variety. Then the five statements inConjecture are equivalent.Proof. Step
1. The generic limit of ideals was invented from the insight of the impli-cation from (v) to (i). Mustat¸˘a informed us the proof of this implication and we wroteit in [18, Proposition 4.8]. Note that the proof in [18] works even if X has klt singu-larities. We also note that though the statement in [18] assumes the assertion in (v)for ideals a ij in the completion of the local ring O X,P , its proof uses only the assertionfor ideals in O X which is exactly (v). We derived from (v) in fact the following ACCwhich was formulated by Cascini and McKernan [25].(vi) Fix subsets I of the positive real numbers and J of the non-negative real numbersboth of which satisfy the DCC. Then there exist finite subsets I of I and J of J suchthat if a is an R -ideal on X satisfying that a ∈ I and mld P ( X, a ) ∈ J , then a ∈ I and mld P ( X, a ) ∈ J . The assertion (i) follows from (vi) immediately. We shall derive (ii) from (vi).Let { t i } i ∈ N be a non-decreasing sequence of positive real numbers such that thereexist ideals a i and b i on X satisfying that mld P ( X, a i b t i i ) = a and a i b i ∈ I . It isenough to show that T = { t i | i ∈ N } satisfies the ACC. By Lemma 4.4, the set IT = { rt | r ∈ I, t ∈ T } satisfies the DCC. Applying (vi) to I ∪ IT and { a } , oneobtains a finite subset I of I ∪ IT such that a i b t i i ∈ I for any i . Particularly, T iscontained in the set I − I = { r − s | r ∈ I, s ∈ I } which satisfies the ACC. Step
2. The conjecture (iv) was proposed by Nakamura. His joint work [26] withMustat¸˘a shows the equivalence of (iii), (iv) and (v). They treated the assertion in (v)for ideals in the completion, but their proof works for our (v). They also provided adirect proof of the implication from (iv) to (i) which uses the ACC for lc thresholdson X and Theorem 4.1. We write the argument from (v) to (iv) as Lemma 4.7 sinceit will be used later. Step
3. Hence it is enough to show the implications from (i) to (iv) and from(ii) to (iv). If (iv) were false, then there would exist a strictly increasing sequence { l i } i ∈ N and a sequence { a i } i ∈ N of R -ideals on X such that a i ∈ I and such that everydivisor E i over X computing mld P ( X, a i ) satisfies the inequality a E i ( X ) ≥ l i . Theassertion (iv) for those a whose mld P ( X, a ) is not positive will be proved in Theorem4.8 independently. We assume that mld P ( X, a i ) is positive for any i here.By Theorem 4.1, the set M = { a E ( X, a ) | a ∈ I, E ∈ D X , ( X, a ) lc } N EQUIVALENT CONJECTURES ON SMOOTH THREEFOLDS 13 is discrete in R . In particular, all mld P ( X, a i ) belong to a finite set since they arebounded from above by mld P X . Thus we may assume that mld P ( X, a i ) is constant,say m , which is positive by our assumption. We may assume that a i is non-trivial,then m is less than mld P X . By the discreteness of M , there exists a real number m ′ greater than m such that r M for any real number m < r ≤ m ′ .Let t i be the positive real number such that mld P ( X, a − t i i ) = m ′ , which exists andsatisfies that 0 < t i < m < m ′ < mld P X . Take a divisor E i over X whichcomputes mld P ( X, a − t i i ). Then a E i ( X, a i ) < m ′ , so E i also computes mld P ( X, a i ) = m by the property of m ′ , and thus ord E i a i = a E i ( X ) − m ≥ l i − m . Since t i ord E i a i = a E i ( X, a − t i i ) − a E i ( X, a i ) = m ′ − m , one has the estimate t i ≤ ( m ′ − m ) / ( l i − m ) when l i > m , showing that t i approaches to zero as i increases.This contradicts the ACC for m ′ -lc thresholds in (ii). It is sufficient to verify thatour situation also contradicts the ACC for minimal log discrepancies in (i). By passingto a subsequence, we may assume that t i are less than one-half and form a strictlydecreasing sequence whose limit is zero. Then { − (1 − t i ) t i } i ∈ N is a strictly increasingsequence. We set T = { − (1 − t i ) t i | i ∈ N } , which satisfies the DCC.Note that 1 − (1 − t i ) t i = (1 − t i )(1 − t i )+ t i . Because E i computes both mld P ( X, a − t i i )and mld P ( X, a i ), by Lemma 4.5(ii) one has thatmld P ( X, a − (1 − t i ) t i i ) = (1 − t i ) mld P ( X, a − t i i ) + t i mld P ( X, a i ) = m ′ − t i ( m ′ − m )which is computed by E i . But then mld P ( X, a − (1 − t i ) t i i ) is strictly increasing. Thiscontradicts (i) for IT = { rt | r ∈ I, t ∈ T } since IT satisfies the DCC by Lemma4.4. q.e.d. Lemma 4.7.
Let P ∈ X be the germ of a klt variety. Let r , . . . , r e be positivereal numbers and { a i = Q ej =1 a r j ij } i ∈ N be a sequence of R -ideals on X . Notationas in Section , so set the generic limit a = Q ej =1 a r j j on ˆ P ∈ ˆ X . If mld ˆ P ( ˆ X, a ) =mld P ( X, a i ) for any i ∈ N l , then there exists a positive rational number l such that forinfinitely many indices i , there exists a divisor E i over X which computes mld P ( X, a i ) and satisfies the equality a E i ( X ) = l .Proof. Take a divisor ˆ E over ˆ X which computes mld ˆ P ( ˆ X, a ). As in Remark 3.8(i),replacing F with a subfamily, one can descend ˆ E to a divisor E l over X × Z l for any l ≥ l . For a component E i of the fibre of E l at s l ( i ) ∈ Z l , one may assume that a ˆ E ( ˆ X, a ) = a E i ( X, a i ) for any i ∈ N l . Then E i computes mld P ( X, a i ) and a E i ( X )equals the constant a ˆ E ( ˆ X ). q.e.d. Theorem 4.8.
Let P ∈ X be the germ of a klt variety. Fix a finite subset I of thepositive real numbers. Then there exists a positive integer l depending only on X and I such that if a is an R -ideal on X satisfying that a ∈ I and that mld P ( X, a ) is notpositive, then there exists a divisor E over X which computes mld P ( X, a ) and satisfiesthe inequality a E ( X ) ≤ l .Proof. Let { a i } i ∈ N be an arbitrary sequence of R -ideals on X such that a i ∈ I andsuch that mld P ( X, a i ) is not positive. It is sufficient to show the existence of a positiverational number l such that for infinitely many indices i , there exists a divisor E i over X which computes mld P ( X, a i ) and satisfies the equality a E i ( X ) = l .Write a i = Q e i j =1 a r ij ij so r ij ∈ I . We may assume that every a ij is non-trivial. Let r be the minimum of the elements of I . Then mld P ( X, a i ) ≤ mld P X − P e i j =1 r ij ≤ mld P X − re i . Let e ′ denote the greatest integer such that re ′ ≤ mld P X . If ( X, a i )is lc, then e i ≤ e ′ . If ( X, a i ) is not lc and e i > e ′ , then we may replace a i with a ′ i = Q e ′ +1 j =1 a r ij ij because every divisor computing mld P ( X, a ′ i ) = −∞ also computesmld P ( X, a i ). Hence by passing to a subsequence, we may assume that e i is constant,say e , and that r ij is constant, say r j , for each 1 ≤ j ≤ e . That is, a i = Q ej =1 a r j ij .Following Section 3, we construct a generic limit a = Q ej =1 a r j j of { a i } i ∈ N . We use thenotation in Section 3, so a is an R -ideal on ˆ P ∈ ˆ X . If mld ˆ P ( ˆ X, a ) were positive, thenthere would exist a positive real number t such that ( ˆ X, a ˆ m t ) is lc. By Theorem 3.10,( X, a i m t ) is lc for infinitely many i , which contradicts that mld P ( X, a i ) is not positive.Thus mld ˆ P ( ˆ X, a ) is not positive. Then by Remark 4.3(ii), mld ˆ P ( ˆ X, a ) = mld P ( X, a i )for any i ∈ N l after replacing F with a subfamily, and the existence of l follows fromLemma 4.7. q.e.d.The conjectures hold in dimension two. Theorem 4.9.
Conjecture holds when X is a klt surface.Proof. By Theorem 4.6, it is enough to verify one of the statements. The (iv) is statedin [26, Theorem 1.3]. Alternatively, one may derive (i) from [2, Theorem 3.8], or derive(v) from [16] by replacing X with its minimal resolution. q.e.d.Roughly speaking, our former work [18] asserts a part of the conjectures in dimensionthree in the case when the minimal log discrepancy is greater than one. Theorem 4.10.
Let P ∈ X be the germ of a smooth threefold. Let r , . . . , r e be positivereal numbers and { a i = Q ej =1 a r j ij } i ∈ N be a sequence of R -ideals on X . Notation asin Section , so set the generic limit a = Q ej =1 a r j j on ˆ P ∈ ˆ X . Then the pair ( ˆ X, a ) satisfies one of the following cases. The mld ˆ P ( ˆ X, a ) is not positive.
2. ( ˆ X, a ) is klt.
3. ( ˆ X, a ) is lc and has the smallest lc centre which is normal and of dimensiontwo.
4. ( ˆ X, a ) is lc and has the smallest lc centre which is regular and of dimensionone.Moreover, the following hold. (i) In the cases , and , mld ˆ P ( ˆ X, a ) = mld P ( X, a i ) for any i ∈ N l afterreplacing F with a subfamily. (ii) In the case , mld ˆ P ( ˆ X, a ) is at most one.Proof. The case division follows from the existence of the smallest lc centre [18, The-orem 1.2]. The equality in (i) holds in the cases 1 and 2 by Remark 4.3(ii), and in thecase 3 by [18, Theorem 5.3]. The assertion (ii) is [18, Proposition 6.1]. q.e.d.We reduce Conjecture 4.2(iv) to the case of Q -ideals. Lemma 4.11.
Let P ∈ X be the germ of a klt variety. Suppose that for any positiveinteger n , there exists a positive integer l depending only on X and n such that if a is an ideal on X , then there exists a divisor E over X which computes mld P ( X, a /n ) and satisfies the inequality a E ( X ) ≤ l . Then Conjecture holds for P ∈ X .Proof. In Conjecture 4.2, the assertion (v) follows from (iv) for I = { r , . . . , r e } by[26]. Thus we may assume Conjecture 4.2(v) in the case when e = 1 and r = 1 /n forsome positive integer n . By Theorem 4.6, it is enough to derive the full statement of(v) from this special case. N EQUIVALENT CONJECTURES ON SMOOTH THREEFOLDS 15
We want the equality mld ˆ P ( ˆ X, a ) = mld P ( X, a i ), where a i = Q ej =1 a r j ij and a = Q ej =1 a r j j . We write m = mld ˆ P ( ˆ X, a ) for simplicity. By Remark 4.3(ii), we mayassume that m is positive. By Theorem 4.1, the set M = { mld P ( X, a ) | a ∈ { r , . . . , r e } , ( X, a ) lc } is discrete in R . Thus there exists a real number m ′ less than m such that r M forany real number m ′ < r < m .Since the set Q = { ( q , . . . , q e ) ∈ ( R ≥ ) e | ( ˆ X, Q ej =1 a q j j ) lc } is a rational polytope, the vector r = ( r , . . . , r e ) in Q is expressed as r = P s ∈ S t s q s ,where S is a finite set, all q s = ( q s , . . . , q es ) belong to Q ∩ Q e , and t s are positive realnumbers such that P s ∈ S t s = 1. By choosing q s close to r , we may assume that m ′ = mld ˆ P ( ˆ X, a ) − ( m − m ′ ) < X s ∈ S t s mld ˆ P ( ˆ X, Q ej =1 a q js j ) . Write q js = m js /n with positive integers n and m s , . . . , m es for s ∈ S . Thenmld ˆ P ( ˆ X, Q ej =1 a q js j ) = mld ˆ P ( ˆ X, ( Q ej =1 a m js j ) /n ) and the ideal Q ej =1 a m js j is the genericlimit of the sequence { Q ej =1 a m js ij } i ∈ N of ideals on X . By our assumption, the equalitymld ˆ P ( ˆ X, ( Q ej =1 a m js j ) /n ) = mld P ( X, ( Q ej =1 a m js ij ) /n ) holds for any i ∈ N l and s ∈ S after replacing F with a subfamily. Hence with Lemma 4.5(i), one has that m ′ < X s ∈ S t s mld P ( X, Q ej =1 a q js ij ) ≤ mld P ( X, Q ej =1 a r j ij ) = mld P ( X, a i ) ∈ M, which implies that mld P ( X, a i ) ≥ m by the property of m ′ . Together with Remark4.3(i), we obtain the required equality m = mld P ( X, a i ). q.e.d. Proposition 4.12.
Let P ∈ X be the germ of a klt variety and m be the maximalideal in O X defining P . Fix a finite subset I of the positive real numbers. Then thereexists a positive real number t depending only on X and I such that if a is an R -idealon X satisfying that a ∈ I and that mld P ( X, a ) is positive, then ( X, am t ) is lc.Proof. Fix r , . . . , r e ∈ I and let { a i = Q ej =1 a r j ij } i ∈ N be a sequence of R -ideals on X such that mld P ( X, a i ) is positive. It is enough to show the existence of a positive realnumber t such that ( X, a i m t ) is lc for infinitely many indices i .Following Section 3, we construct a generic limit a of { a i } i ∈ N on ˆ P ∈ ˆ X . Thenmld ˆ P ( ˆ X, a ) is positive by Remark 4.3(i), so there exists a positive real number t suchthat ( ˆ X, a ˆ m t ) is lc for the maximal ideal ˆ m in O ˆ X . By Theorem 3.10, there exists aninfinite subset N l of N such that ( X, a i m t ) is lc for any i ∈ N l . q.e.d. Corollary 4.13 ([26]) . Let P ∈ X be the germ of a klt variety and m be the maximalideal in O X defining P . Fix a finite subset I of the positive real numbers. Then thereexists a positive integer b depending only on X and I such that if a is an R -ideal on X satisfying that a ∈ I and that mld P ( X, a ) is positive, then ord E m is at most b forevery divisor E over X computing mld P ( X, a ) .Proof. Take t in Proposition 4.12. Let E be an arbitrary divisor over X which com-putes mld P ( X, a ). The log canonicity of ( X, am t ) implies thatord E m t ≤ a E ( X, a ) = mld P ( X, a ) ≤ mld P X, that is, ord E m ≤ t − mld P X . The b = ⌊ t − mld P X ⌋ is a required integer. q.e.d. Construction of canonical pairs
The objective of this section is to prove the following theorem.
Theorem 5.1.
Let P ∈ X be the germ of a smooth threefold. Fix a positive rationalnumber q . Then there exist positive integers l and c both of which depend only on q such that if a is an ideal on X satisfying that mld P ( X, a q ) is positive, then at leastone of the following holds. (i) There exists a divisor E over X which computes mld P ( X, a q ) and satisfies theinequality a E ( X ) ≤ l . (ii) There exists a birational morphism from the germ Q ∈ Y of a smooth threefoldto the germ P ∈ X such that • every exceptional prime divisor F on Y satisfies the inequalities a F ( X ) ≤ c and a F ( X, a q ) < , by which the pull-back ( Y, ∆ , a qY ) of ( X, a q ) is defined with aneffective Q -divisor ∆ , • mld Q ( Y, ∆ , a qY ) = mld P ( X, a q ) , and • mld Q ( Y, a qY ) is at least one. We recall a part of the classification of threefold divisorial contractions which willplay an important role in our argument.
Definition 5.2.
A projective birational morphism Y → X between Q -factorial ter-minal varieties is called a divisorial contraction if its exceptional locus is a primedivisor. Theorem 5.3.
Let π : Y → X be a threefold divisorial contraction which contracts itsexceptional divisor to a closed point P in X . (i) ([20]) Suppose that P is a quotient singularity of X . The spectrum of thecompletion of O X,P is the regular base change of A / Z r ( w, − w, with orbifold coor-dinates x , x , x , where w is a positive integer less than r and coprime to r . Then π is base-changed to the weighted blow-up with wt( x , x , x ) = ( w/r, ( r − w ) /r, /r ) . (ii) ([13]) Suppose that P is a smooth point of X . Then there exists a regularsystem x , x , x of parameters in O X,P and coprime positive integers w , w such that π is the weighted blow-up with wt( x , x , x ) = ( w , w , . Stepanov proved the ACC for canonical thresholds on smooth threefolds as an ap-plication of Theorem 5.3(ii).
Theorem 5.4 ([29]) . The set { t ∈ Q ≥ | P ∈ X a smooth threefold , a an ideal , mld P ( X, a t ) = 1 } satisfies the ACC.Proof. Let S denote the set in the theorem. The original statement [29, Theorem 1.7]asserts that the set T = (cid:26) t ∈ Q ≥ (cid:12)(cid:12)(cid:12)(cid:12) P ∈ X a smooth threefold , D an effective divisor , ( X, tD ) canonical but not terminal (cid:27) satisfies the ACC. It is enough to show that if t is an arbitrary element of S , then t/ T . For such t , there exists an ideal a on the germ P ∈ X of a smooththreefold such that mld P ( X, a t ) = 1. Then t ≤ mld P X = 3 and t/ S since mld P ( X, ( a ) t/ ) = 1. Thus it is sufficient to show that if t ∈ S is at most one,then t belongs to T .Take a germ P ∈ X on which t is realised by mld P ( X, a t ) = 1. Let m be the maximalideal in O X defining P . By replacing a with a + m l for a large integer l , we may assume N EQUIVALENT CONJECTURES ON SMOOTH THREEFOLDS 17 that a is m -primary. We take a log resolution Y of ( X, a ). Then a O Y = O Y ( − A ) foran effective divisor A such that − A is free over X . Thus there exists a reduced divisor H linearly equivalent to − A such that Y is also a log resolution of ( X, H X , a ), where H X is the push-forward of H . Then mld P ( X, tH X ) = 1 and ( X, tH X ) is canonical,meaning that t ∈ T . q.e.d.We shall use a consequence of the minimal model program. Definition 5.5.
Let P ∈ X be the germ of a Q -factorial terminal variety and a be an R -ideal on X such that mld P ( X, a ) equals one. A divisorial contraction to X is saidto be crepant with respect to ( P, X, a ) if its exceptional divisor computes mld P ( X, a ). Lemma 5.6.
Let P ∈ X be the germ of a Q -factorial terminal variety and a be an R -ideal on X such that mld P ( X, a ) equals one. Then there exists a divisorial contractioncrepant with respect to ( P, X, a ) .Proof. By replacing a with b in Lemma 5.7, we may assume that a is an m -primary R -ideal, where m is the maximal ideal in O X defining P , such that there exists a uniquedivisor E over X which computes mld P ( X, a ). Then by [3, Corollary 1.4.3], thereexists a projective birational morphism Y → X from a Q -factorial normal varietywhose exceptional locus is a prime divisor which coincides with E . We may assumethat the weak transform a Y on Y of a is defined. Then ( Y, a Y ) is the pull-back of( X, a ), and it is terminal by the uniqueness of E . In particular Y itself is terminal, so Y → X is a required contraction. q.e.d. Lemma 5.7.
Let P ∈ X be the germ of a Q -factorial klt variety and a be an R -idealon X such that ( X, a ) is lc. Then there exists an R -ideal b such that • b is m -primary, where m is the maximal ideal in O X defining P , • mld P ( X, a ) = mld P ( X, b ) , • there exists a unique divisor E over X which computes mld P ( X, b ) , and • E also computes mld P ( X, a ) .Proof. Writing a = Q j a r j j , if we take a large integer l , then a ′ = Q j ( a j + m l ) r j satisfiesthat mld P ( X, a ′ ) = mld P ( X, a ) and any divisor computing mld P ( X, a ′ ) also computesmld P ( X, a ). By replacing a with a ′ , we may assume that a is m -primary.Let Y be a log resolution of ( X, a ) and { E i } i ∈ I be the set of the exceptional primedivisors on Y contracting to the point P . Let A be the R -divisor on Y defined by a O Y and I ′ be the subset of I consisting of the indices i such that E i computesmld P ( X, a ). There exists an effective exceptional divisor F such that − F is veryample and such that the minimum m of { ord E i A/ ord E i F } i ∈ I ′ attains by only oneindex, say i ∈ I ′ . One can take a small positive real number ǫ such that b i = a E i ( X, a ) + ǫ (ord E i A − m ord E i F ) is greater than mld P ( X, a ) for any i ∈ I \ { i } .Note that b i remains equal to mld P ( X, a ).Let c be the ideal on X given by the push-forward of O Y ( − F ) and set the R -ideal b = a − ǫ c ǫm . Possibly replacing ǫ with a smaller real number, we may assume that( X, b ) is lc. Y is also a log resolution of ( X, b ) and a E i ( X, b ) = b i for any i ∈ I . Thus b satisfies all the required properties but being m -primary. However, one can replace b with an m -primary R -ideal just by the argument of constructing a ′ from a . q.e.d.We consider the following algorithm in order to prove Theorem 5.1. Algorithm 5.8.
Let q be a positive rational number. Let P ∈ X be the germ of asmooth threefold and a be an ideal on X such that ( X, a q ) is lc. Let E be a divisorover X which computes mld P ( X, a q ). . Start with X = X . . Suppose that X i is given, which has only terminal quotient singularities. . If the centre c X i ( E ) on X i of E is of positive dimension, then output X i . . Suppose that c X i ( E ) is a closed point, which will be denoted by P i . Let r i bethe index of the germ P i ∈ X i . One can define the weak transform b i on P i ∈ X i of a r i and let a i be the Q -ideal b /r i i . The pair ( X i , a qi ) is lc at P i . . If P i is a smooth point of X i and mld P i ( X i , a qi ) ≥
1, then output X i . . If P i is a smooth point of X i and mld P i ( X i , a qi ) <
1, then go to . . If P i is a singular point of X i and mld P i ( X i , a qi ) >
1, then output X i . . If P i is a singular point of X i and mld P i ( X i , a qi ) ≤
1, then go to . . Let q i be the positive rational number such that mld P i ( X i , a q i i ) = 1. Fix adivisorial contraction X i +1 → X i crepant with respect to ( P i , X i , a q i i ) by Lemma5.6. Go back to and proceed with X i +1 instead of X i .In this algorithm, we fix the notation that F i is the exceptional divisor of X i +1 → X i and that F ij is the strict transform on X i of F j for j < i , and we set ∆ i = P i − j =0 (1 − a F j ( X, a q )) F ij and S i = P i − j =0 F ij . Remark . By the very definition,(i) q i ≤ q , and q i < q when q i is defined at a smooth point P i of X i , and(ii) ( X i , ∆ i , a qi ) is crepant to ( X, a q ).In order to run Algorithm 5.8, we need to verify that • X i has at worst quotient singularities in the process , and • ( X i , a qi ) is lc at P i in the process ,besides the termination. The first claim follows from Theorem 5.3. For the secondclaim, let b i = 1 − a F i ( X i , a qi ). Then ( X i +1 , b i F i , a qi +1 ) is crepant to ( X i , a qi ) and b i F i is effective since a F i ( X i , a qi ) ≤ a F i ( X i , a q i i ) = 1 by q i ≤ q . Thus the log canonicity of( X i , a qi ) follows from that of ( X, a q ) inductively. Hence Algorithm 5.8 runs up to thetermination.We prepare several basic properties of the algorithm before completing its termina-tion in Proposition 5.12. Lemma 5.10.
The following hold in Algorithm . (i) The q i form a non-decreasing sequence. (ii) a F i ( X, a q i ) ≤ . (iii) a F i ( X, a q ) < . (iv) If q i = q , then a F i ( X ) ≤ q ( q − q i ) − .Proof. Since ( X i +1 , a q i i +1 ) is crepant to ( X i , a q i i ), one has that mld P i +1 ( X i +1 , a q i i +1 ) ≥ mld P i ( X i , a q i i ) = 1. Thus q i ≤ q i +1 , which shows (i).For j < i , let b ij = 1 − a F j ( X j , a q i j ). Then ( X j +1 , b ij F j , a q i j +1 ) is crepant to ( X j , a q i j )and b ij F j is effective by q j ≤ q i in (i). Thus one has the inequality a F i ( X j , a q i j ) ≤ a F i ( X j +1 , a q i j +1 ) and inductively a F i ( X, a q i ) ≤ a F i ( X i , a q i i ) = 1, which is (ii).The (iii) follows from (ii) unless q i = q . If q i = q , then q i is defined at a singularpoint P i of X i , so q < q . Let j be the greatest integer such that q j < q . Then( X i , a qi ) is crepant to ( X j +1 , a qj +1 ). On the other hand, ( X j +1 , ∆ j +1 , a qj +1 ) is crepantto ( X, a q ). By (ii), ∆ j +1 is effective and its support coincides with S j +1 . Thus onehas that a F i ( X, a q ) = a F i ( X j +1 , ∆ j +1 , a qj +1 ) < a F i ( X j +1 , a qj +1 ) = a F i ( X i , a qi ) = 1. N EQUIVALENT CONJECTURES ON SMOOTH THREEFOLDS 19
To see (iv), suppose that q i < q . By (ii) and a F i ( X, a q ) ≥
0, one computes that a F i ( X ) = a F i ( X, a q i ) + q i ord F i a ≤ q i ord F i a = 1 + q i ( q − q i ) − ( a F i ( X, a q i ) − a F i ( X, a q )) ≤ q i ( q − q i ) − = q ( q − q i ) − . q.e.d. Lemma 5.11.
Fix a positive rational number q . Then there exists a positive rationalnumber ǫ depending only on q such that every q i defined at a smooth point P i of X i inAlgorithm satisfies the inequality q i ≤ q − ǫ .Proof. It follows from Theorem 5.4. q.e.d.
Proposition 5.12.
Algorithm terminates.Proof.
Take ǫ in Lemma 5.11. Then every divisor F i defined over a smooth point P i of X i satisfies that a F i ( X, a q − ǫ ) ≤ a F i ( X, a q i ) ≤ F i is finite because ( X, a q − ǫ ) is klt. In particular, there exists an integer e ,depending on ( X, a q ) and E , such that P i is a singular point of X i for any i > e .By Theorem 5.3(i), the r i for i > e form a strictly decreasing sequence. Hence thealgorithm must terminate. q.e.d.One can also bound r i and a F i ( X ). Lemma 5.13.
Fix a positive rational number q . Then there exists a positive integer r depending only on q such that every r i in Algorithm satisfies the inequality r i ≤ r .Proof. Take ǫ in Lemma 5.11. We shall show that any positive integer r at least qǫ − − r is one. By Theorem 5.3, the r i +1 satisfies that r i +1 < r i when P i is a singular point of X i and that r i +1 ≤ a F i ( X i ) − P i is a smooth point of X i . Thus it is enough to show that a F i ( X i ) ≤ qǫ − when P i is a smooth point of X i . Since a F i ( X i ) ≤ a F i ( X i ) + i − X j =0 ( a F j ( X ) −
1) ord F i F ij = a F i ( X ) , the required inequality follows from Lemma 5.10(iv). q.e.d. Lemma 5.14.
Fix a positive rational number q . Then there exists a positive integer c depending only on q such that every F i in Algorithm satisfies the inequality a F i ( X ) ≤ c .Proof. Take a positive integer n such that nq is integral, and take ǫ in Lemma 5.11 and r in Lemma 5.13. Fix a positive integer c at least qǫ − and define positive integers c , . . . , c r inductively by the recurrence relation c j +1 = 2 + ( c j − n for 0 ≤ j < r . We shall prove that c r is a required constant.Let e be the greatest integer such that q e is defined at a smooth point P e of X e . Thenby Lemmata 5.10(i), (iv) and 5.11, the estimate a F i ( X ) ≤ c holds for any i ≤ e , andby Lemma 5.13, if the algorithm defines P e +1 ∈ X e +1 , then r e +1 ≤ r . In particular,the algorithm terminates with the output X e + r ′ for some r ′ ≤ r by Theorem 5.3(i).Thus it is enough to show that a F e + j ( X ) ≤ c j for any j ≤ r as far as F e + j is defined.This is reduced to proving that if F i is defined at a singular point P i of X i and if a F j ( X ) is bounded from above by a positive integer c ′ for all j < i , then a F i ( X ) is atmost 2 + ( c ′ − n . Suppose that F i and c ′ are given as above. By Lemma 5.10(iii), the Q -divisor ∆ i satisfies that S i ≤ n ∆ i . Since ( X i , ∆ i , a qi ) is crepant to ( X, a q ), one has thatord F i S i ≤ n ord F i ∆ i = n ( a F i ( X i , a qi ) − a F i ( X i , ∆ i , a qi )) ≤ n ( a F i ( X i , a q i i ) − a F i ( X, a q )) = n (1 − a F i ( X, a q )) ≤ n, where the second inequality follows from q i ≤ q . Together with a F i ( X i ) = 1 + 1 /r i byTheorem 5.3(i), one computes that a F i ( X ) = a F i ( X i ) + i − X j =0 ( a F j ( X ) −
1) ord F i F ij ≤ r i + ( c ′ −
1) ord F i S i < c ′ − n. q.e.d.In order to control the log discrepancy of a divisor computing mld P ( X, a q ), we needan extra assumption that mld P ( X, a q ) is positive. Lemma 5.15.
Fix a positive rational number q . Then there exists a positive integer l depending only on q such that in Algorithm if mld P ( X, a q ) is positive and if thealgorithm terminates at the process or , then there exists a divisor E ′ over X whichcomputes mld P ( X, a ) and satisfies the inequality a E ′ ( X ) ≤ l .Proof. Step
1. We take c in Lemma 5.14. Let η be a positive rational number such thatthe exist no integers a satisfying that q < /a < q + η . Let n be a positive integer suchthat nq is integral. Since Conjecture 4.2(iv) holds for in dimension two by Theorem4.9, there exists a positive integer l ′ depending only on n such that if Q ∈ H is thegerm of a smooth surface and a H is an ideal on H , then there exists a divisor E H over H which computes mld Q ( H, a /nH ) and satisfies the inequality a E H ( H ) ≤ l ′ .Let m be the maximal ideal in O X defining P . Let E ′ be an arbitrary divisor over X which computes mld P ( X, a q ). By Corollary 4.13, there exists a positive integer b depending only on q such that ord E ′ m ≤ b . Note that b can be taken independent of thegerm P ∈ X of a smooth threefold. Indeed, E ′ also computes mld P ( X, a ′ q ) for the m -primary ideal a ′ = a + m e as far as a positive integer e satisfies that ord E ′ a ≤ e ord E ′ m .Thus by Lemma 2.7, one can take the b on the germ at origin of the affine space A .For any i , one has the estimate ord E ′ S i ≤ ord E ′ m because m r O X i is contained in O X i ( − rS i ) for a positive integer r such that rS i is Cartier. Hence,ord E ′ S i ≤ b. Supposing that the algorithm terminates at the process or , we shall bound thelog discrepancy of some divisor which computes mld P ( X, a ) in terms of q , c , η , l ′ and b . Step
2. Suppose that the algorithm terminates at the process and outputs X i .Then the centre c E ( X i ) on X i of E is either a divisor or a curve. If it is a divisor,then E = F i − and it computes mld P ( X, a q ). By Lemma 5.14, F i − satisfies that a F i − ( X ) ≤ c. Suppose that c E ( X i ) is a curve C . Let H be a general hyperplane section of X i and Q be a closed point in H ∩ C . Considering a log resolution, one has thatmld Q ( H, ∆ i | H , ( a i O H ) q ) = mld η C ( X i , ∆ i , a qi ) = mld P ( X, a q ) , N EQUIVALENT CONJECTURES ON SMOOTH THREEFOLDS 21 where the second equality holds since E computes mld P ( X, a q ). Moreover by theexpression mld Q ( H, ∆ i | H , ( a i O H ) q ) = mld Q ( H, ( a nqi O H ( − n ∆ i | H )) /n ), there exists adivisor E ′ over X i with c X i ( E ′ ) = C such that an irreducible component E ′ H of E ′ × X i H mapped to Q computes mld Q ( H, ∆ i | H , ( a i O H ) q ) and satisfies the inequal-ity a E ′ H ( H ) ≤ l ′ . The E ′ computes mld P ( X, a q ) as well as mld η C ( X i , ∆ i , a qi ), and a E ′ ( X i ) = a E ′ H ( H ) ≤ l ′ . Therefore, a E ′ ( X ) = a E ′ ( X i ) + i − X j =0 ( a F j ( X ) −
1) ord E ′ F ij ≤ l ′ + ( c −
1) ord E ′ S i ≤ l ′ + ( c − b, where the last inequality follows from Step 1. Step
3. Suppose that the algorithm terminates at the process and outputs X i . Let n be the maximal ideal in O X i defining P i . Recall that a i = b /r i i . Set b ′ i = b i + n e fora positive integer e such that ord E b i ≤ e ord E n . Since mld P i ( X i , a qi ) is greater thanone, there exists a rational number q ′ greater than q such that mld P i ( X i , ( b ′ i ) q ′ /r i ) = 1.There exists a divisorial contraction to X i crepant to ( P i , X i , ( b ′ i ) q ′ /r i ) by Lemma 5.6and it is uniquely determined by Theorem 5.3(i). Its exceptional divisor F satisfiesthat a F ( X i ) = 1 + 1 /r i . Thus q ′ ord F b ′ i = r i ( a F ( X i ) − a F ( X i , ( b ′ i ) q ′ /r i )) = r i ( a F ( X i ) −
1) = 1 , which derives that q ′ is at least q + η by the definition of η . In particular, the pair( X i , ( b ′ i ) ( q + η ) /r i ) is canonical so a E ( X i , a q + ηi ) = a E ( X i , ( b ′ i ) ( q + η ) /r i ) ≥
1. Hence onecomputes that a E ( X i ) = a E ( X i , a qi ) + q ord E a i = a E ( X i , a qi ) + qη − ( a E ( X i , a qi ) − a E ( X i , a q + ηi )) ≤ (1 + qη − ) a E ( X i , a qi ) − qη − = (1 + qη − )( a E ( X i , ∆ i , a qi ) + ord E ∆ i ) − qη − ≤ (1 + qη − )( a E ( X, a q ) + ord E S i ) − qη − and a E ( X ) = a E ( X i ) + i − X j =0 ( a F j ( X ) −
1) ord E F ij ≤ (1 + qη − )( a E ( X, a q ) + ord E S i ) − qη − + ( c −
1) ord E S i . Together with a E ( X, a q ) = mld P ( X, a q ) ≤ E S i ≤ b in Step 1, one concludesthat a E ( X ) ≤ (1 + qη − )(3 + b ) − qη − + ( c − b. Step
4. By Steps 2 and 3, any integer l at least c , l ′ + ( c − b and (1 + qη − )(3 + b ) − qη − + ( c − b satisfies the required property. q.e.d. Proof of Theorem . We shall verify that the l in Lemma 5.15 and c in Lemma 5.14satisfy the assertion. Let a be an ideal on X such that mld P ( X, a q ) is positive. RunAlgorithm 5.8 which terminates by Proposition 5.12. If the algorithm terminates atthe process or , then the property (i) holds by Lemma 5.15. If it terminates at theprocess , then let Q ∈ Y be the output P i ∈ X i . The Q ∈ Y satisfies the property(ii) by Lemmata 5.10(iii) and 5.14. q.e.d. Extraction by weighted blow-ups
Recall the classification of divisors over a smooth surface computing the minimallog discrepancy.
Theorem 6.1 ([19]) . Let P ∈ X be the germ of a smooth surface and a be an R -idealon X . (i) If ( X, a ) is lc, then every divisor computing mld P ( X, a ) is obtained by a weightedblow-up. (ii) If ( X, a ) is not lc, then some divisor computing mld P ( X, a ) is obtained by aweighted blow-up. We want to apply this theorem with the object of extracting by a weighted blow-upa divisor over a smooth threefold whose centre is a curve and which computes the lcthreshold. In order to use such extraction in the study of the generic limit of ideals,we need to formulate it for R -varieties. We let K be a field of characteristic zerothroughout this section. The purpose of this section is to prove Theorem 6.2.
Let X be the spectrum of the ring of formal power series in threevariables over K and P be its closed point. Let a be an R -ideal on X such that • mld P ( X, a ) equals one, and • ( X, a ) has an lc centre C of dimension one.Then there exist a divisor E over X computing mld η C ( X, a ) and a part x , x of aregular system of parameters in O X such that E is obtained by the weighted blow-upof X with wt( x , x ) = ( w , w ) for some coprime positive integers w , w . When a divisor over a smooth variety is given, we often realise it by a finite sequenceof blow-ups.
Definition 6.3.
Let X be a smooth variety and E be a divisor over X whose centre Z on X has codimension at least two in X . A tower on X with respect to E is afinite sequence of projective birational morphisms X i +1 → X i of smooth varieties for0 ≤ i < l such that • X = X and Z = Z , • X i +1 is about η Z i the blow-up of X i along Z i , • E i is the exceptional prime divisor on X i +1 contracting onto Z i , • Z i +1 is the centre on X i +1 of E , and • E l − = E .A tower is called the regular tower if for any i < l , the centre Z i is smooth and X i +1 is globally the blow-up of X i along Z i . Note that the regular tower is uniquelydetermined by E if it exists. Remark . Let P ∈ X be the germ of a smooth variety. Let x , . . . , x c be a part ofa regular system of parameters in O X and E be the divisor obtained by the weightedblow-up of X with wt( x , . . . , x c ) = ( w , . . . , w c ), where c is at least two. Thenone can see that the regular tower on X with respect to E exists in terms of toricgeometry. Following the notation in [12], set N = Z d with the standard basis e , . . . , e d for d = dim X . One may assume that w = ( w , . . . , w c , , . . . ,
0) is primitive in N .Construct a finite sequence of fans ( N, ∆ i ) for 0 ≤ i ≤ l such that • I i = { e , . . . , e d } ∪ { v , . . . , v i } , • ∆ i is the set of all cones spanned by a subset of I i , • J i is the smallest subset of I i such that w belongs to the cone spanned by J i , • v i +1 = P v ∈ J i v , and • J i = { w } for i < l and J l = { w } . N EQUIVALENT CONJECTURES ON SMOOTH THREEFOLDS 23
Set the toric variety T i = T N (∆ i ) and let E Ti be the exceptional divisor of T i +1 → T i .Then X has an ´etale morphism to T by corresponding e i to x i . The base changes of T i to X form the regular tower on X with respect to E , and every E i = E Ti × T X isobtained by a weighted blow-up of X .We collect basic properties of the log discrepancies in a tower which was essentiallywritten in [19, Proposition 6]. Lemma 6.5.
Notation as in Definition . Let a be an R -ideal on X and a i be itsweak transform on X i . Set a i = a E i ( X, a ) . (i) The ord Z i a i form a non-increasing sequence. (ii) If ord Z a ≤ , then a i ≥ and the a i form a non-decreasing sequence. (iii) If ord Z a < , then a i > and the a i form a strictly increasing sequence.Proof. Take a subvariety V i +1 of Z i +1 such that V i +1 → Z i is finite and dominant.Then ord Z i +1 a i +1 ≤ ord V i +1 a i +1 ≤ ord Z i a i by [11, III Lemmata 7 and 8], which is (i).The assertion (ii) is reduced to (iii) because a i is the limit of a E i ( X, a − ǫ ) when ǫ decreases to zero. Suppose that ord Z a < Z i a i by(i). In particular, a E i ( X i , a i ) = a E i ( X i ) − ord Z i a i >
1. Since ( X i , P i − j =0 (1 − a j ) E ij , a i )is crepant to ( X, a ), where E ij is the strict transform of E j , one computes that a i = a E i ( X i , a i ) + i − X j =0 ( a j −
1) ord E i E ij > i − X j =0 ( a j −
1) ord E i E ij . This derives that a i > a i > a i − again byinduction. q.e.d. Proposition 6.6.
Let P ∈ X be the germ of a smooth variety and a be an R -ideal on X . Let E be the divisor obtained by the blow-up of X at P . (i) If ord P a ≤ , then E computes mld P ( X, a ) . (ii) If ord P a < , then E is the unique divisor computing mld P ( X, a ) .Proof. It is [19, Proposition 6] exactly. Just apply Lemma 6.5(ii) and (iii) to the toweron X with respect to a divisor which computes mld P ( X, a ). q.e.d.We shall study divisors computing the minimal log discrepancy on a K -variety ofdimension two. Lemma 6.7.
Let P ∈ X be the germ at a K -point of a regular K -variety of dimen-sion two and a be an R -ideal on X . Then there exists a divisor over X computing mld P ( X, a ) which is obtained by a sequence of finitely many blow-ups at a K -point.Proof. Let m be the maximal ideal in O X defining P . By adding a high multiple of m to each component of a , we may assume that a is m -primary. Suppose that ( X, a ) isnot lc. Then there exists a positive real number t less than one such that mld P ( X, a t )is zero. Replacing a with a t , we may assume that ( X, a ) is lc.We write m = mld P ( X, a ) for simplicity. Let Y be the blow-up of X at P and E be its exceptional divisor. There is nothing to prove if E computes mld P ( X, a ). Thuswe may assume that a E = a E ( X, a ) is greater than m . Then a E = 2 − ord P a < K -varieties by Remark 2.6). Thatis, m < a E <
1. Let a Y be the weak transform on Y of a , then ( Y, (1 − a E ) E, a Y )is crepant to ( X, a ). We claim that there exists a unique point Q in Y such thatmld Q ( Y, (1 − a E ) E, a Y ) = m , and claim that Q is a K -point.Let Q be an arbitrary closed point in Y such that mld Q ( Y, (1 − a E ) E, a Y ) = m .Such Q exists since a E = m . Set the base change ¯ X = X × Spec K Spec ¯ K of X to the algebraic closure ¯ K of K . Let ¯ P , ¯ a , ¯ Y , ¯ E and ¯ a Y be the base changes of P , a , Y , E and a Y to ¯ K as well. Then every closed point ¯ Q in Q × X ¯ X satisfies thatmld ¯ Q ( ¯ Y , (1 − a E ) ¯ E, ¯ a Y ) = mld ¯ P ( ¯ X, ¯ a ). Thus our claims on Q come from those on ¯ Q ,so we may assume that K is algebraically closed.One has that mld Q ( Y, E, a Y ) ≤ mld Q ( Y, (1 − a E ) E, a Y ) − a E = m − a E <
0, whichmeans that (
Y, E, a Y ) is not lc at Q . By inversion of adjunction, ( E, a Y O E ) is notlc at Q , that is, ord Q ( a Y O E ) >
1. Hence the number of Q is less than the degree ofthe divisor on E ≃ P K defined by a Y O E , which equals ord E a = 2 − a E . Thus, theuniqueness of Q follows.While a E is greater than m , we replace P ∈ ( X, a ) with Q ∈ ( Y, O Y ( − E ) − a E · a Y )and repeat the same argument. This procedure terminates at finitely many times.Indeed, let l be the minimum of a F ( X ) for all divisors F over X computing mld P ( X, a ).Then after at most ( l −
1) blow-ups, one attains a divisor which computes mld P ( X, a ).q.e.d. Example . There may exist a divisor computing mld P ( X, a ) which is not obtainedby a sequence of blow-ups at a K -point. For example, let P ∈ A R be the germ atorigin of the affine plane over R with coordinates x , x , and H be the divisor on A R defined by x + x . Then mld P ( A R , H ) = 0. Let Y be the blow-up of A R at P and E be its exceptional divisor. Then ( Y, H Y + E ) is crepant to ( A R , H ), where H Y is the strict transform. The intersection Q of H Y and E is a C -point such thatmld Q ( Y, H Y + E ) = 0.Now we apply Theorem 6.1 to K -varieties of dimension two. Proposition 6.9.
Let P ∈ X be the germ at a K -point of a regular K -variety ofdimension two and a be an R -ideal on X . Then there exists a divisor E over X computing mld P ( X, a ) which is obtained by a weighted blow-up.Proof. We may assume the log canonicity of ( X, a ) by the argument in the first para-graph of the proof of Lemma 6.7. By Lemma 6.7, there exists a divisor E over X computing mld P ( X, a ) which is obtained by a sequence of finitely many blow-ups ata K -point. Set the base change ¯ X = X × Spec K Spec ¯ K of X to the algebraic closure¯ K of K and let ¯ P and ¯ a be the base changes of P and a to ¯ X . Since E is obtainedby finitely many blow-ups at a K -point, its base change ¯ E = E × X ¯ X is irreducible,so ¯ E is a divisor over ¯ X . Thus by Theorem 6.1, there exists a regular system x , x of parameters in O ¯ X, ¯ P such that ¯ E is obtained by the weighted blow-up of ¯ X withwt( x , x ) = ( w , w ) for some coprime positive integers w , w .We shall show that one can take x and x from O X . This is obvious when w = w = 1 because the weighted blow-up in this case is nothing but the blow-up at thepoint. Suppose that w > w . Let L be a finite Galois extension of K such that x and x belong to O X ⊗ K L . Then for any element σ of the Galois group G of L/K ,the ¯ E = ¯ E σ is obtained by the weighted blow-up with wt( x σ , x σ ) = ( w , w ). Thusone can replace x i with its trace P σ ∈ G x σi by Remark 2.3. Here one can assume that P σ ∈ G x σi ∈ m \ m , where m is the maximal ideal in O X defining P , by replacing x i with λ i x i for a general member λ i in L .Now we may assume that x and x belong to O X . Then E is obtained by theweighted blow-up of X with wt( x , x ) = ( w , w ). q.e.d. Proof of Theorem . Step
1. First of all, remark that C is the smallest lc centreof ( X, a ). The C is regular by [18, Theorem 1.2], and it is geometrically irreduciblebecause its base change to any field is again the smallest lc centre of the base changeof ( X, a ). Thus, there exists a regular system x , x , x of parameters in O X such that N EQUIVALENT CONJECTURES ON SMOOTH THREEFOLDS 25 the ideal I C in O X defining C is generated by x and x . If we consider instead of a = Q j a r j j the R -ideal b = Q j ( a j + ( x , x ) l O X ) r j for a large integer l , then C is stillthe smallest lc centre of ( X, b ) and mld P ( X, b ) ≥ mld P ( X, a ) = 1. On the other hand,mld P ( X, b ) is at most one by [18, Proposition 6.1]. Thus mld P ( X, b ) must equal one.Hence by replacing a with b , we may assume that a is the pull-back of an R -ideal a ′ on X ′ = Spec K [[ x ]][ x , x ]. Set X ′′ = Spec K (( x ))[ x , x ], where K (( x )) is thequotient field of K [[ x ]]. There exist natural morphisms X → X ′ ← X ′′ . Let P ′ be the point of X ′ defined by ( x , x , x ) O X ′ and P ′′ be the point of X ′′ definedby ( x , x ) O X ′′ .One has that mld P ′′ ( X ′′ , a ′ O X ′′ ) = mld η C ( X, a ) = 0. By Proposition 6.9, thereexists a divisor E ′′ over X ′′ computing mld P ′′ ( X ′′ , a ′ O X ′′ ) which is obtained by aweighted blow-up of X ′′ . Let E ′ be the unique divisor over X ′ such that E ′′ = E ′ × X ′ X ′′ and let E = E ′ × X ′ X . Note that C is the centre of E on X . Step
2. There exists a regular tower T ′′ on X with respect to E ′′ in Definition6.3 (which can be extended to K (( x ))-varieties). As seen in Remark 6.4, T ′′ isa finite sequence X ′′ l → · · · → X ′′ = X ′′ of blow-ups at a K (( x ))-point and theexceptional divisor F ′′ i of X ′′ i +1 → X ′′ i is obtained by a weighted blow-up of X ′′ . Notethat E ′′ = F ′′ l − . Possibly by replacing E ′′ with some F ′′ i , we may assume that F ′′ i does not compute mld P ′′ ( X ′′ , a ′ O X ′′ ) unless i = l − T ′′ is compactified over X ′ , that is, X ′′ i +1 → X ′′ i is the base change of a pro-jective birational morphism X ′ i +1 → X ′ i of regular schemes. Then the base changes X i = X ′ i × X ′ X to X form a tower T on X with respect to E . Let C i be the centreon X i of E . Since T ′′ consists of blow-ups at a K -point, C i is birational to C forany i < l . Hence C i must be isomorphic to the regular scheme C . Therefore one canreplace X i and X ′ i inductively so that T is the regular tower on X with respect to E .Let F i denote the exceptional divisor of X i +1 → X i , and set a i = a F i ( X, a ). Byour construction, every a i is positive except for i = l − a l − is zero. Let a i be the weak transform on X i of a and set the R -divisor ∆ i = P i − j =0 (1 − a j ) F ij on X i ,where F ij is the strict transform of F j . Then ( X i , ∆ i , a i ) is crepant to ( X, a ). We claimthat a i < i . This is obvious for i = l − a l − = 0. In order to seethe inequality a i < i < l − a j < j less than i . Then ∆ i is effective. Since F i does not compute mld η C ( X, a ),one has that ord F i ∆ i + ord F i a i > a i = a F i ( X i , ∆ i , a i ) = 2 − (ord F i ∆ i + ord F i a i ) < Step
3. We have that 0 < a i < i < l − a l − = 0. We let f i denotethe fibre of F i → C at P , which is isomorphic to P K . For i < l , let P i be the K -pointin C i mapped to P . We claim that for any indices i and j such that j < i < l , thecentre C i is either disjoint from F ij or contained in F ij .Indeed if C i intersected F ij properly at P i , then the morphism F i +1 j → F ij would notbe an isomorphism. Thus F i +1 j must contain the fibre f i of F i → C i . In particular, F i +1 j intersects C i +1 . On the other hand, C i +1 is not contained in F i +1 j as C i is notin F ij . Thus one obtains that C i +1 must also intersect F i +1 j properly at P i +1 , unless i + 1 = l . Repeating this argument, one would have that F lj contains f l − as well as F l − contains f l − . Now let G be the divisor obtained by the blow-up of X l along f l − . One computes that a G ( X, a ) ≤ a G ( X l , ∆ l ) ≤ − (1 − a j ) − (1 − a l − ) = a j < , which contradicts that mld P ( X, a ) = 1. Step
4. Let i be any index such that ord F i I C = 1. We shall show that there existsa part y of a regular system of parameters in O X such that C i is contained in thestrict transform H i on X i of the divisor on X defined by y . This is obvious for i = 0.The condition ord F i I C = 1 for the fixed i ≥ F i − I C = 1 sinceord F i I C = ord F i I i + i − X j =0 ord F j I C · ord F i F ij ≥ ord F i − I C for the weak transform I i on X i of I C . Hence by induction on i , we may assume theexistence of y such that C i − is contained in H i − .We extend y to a regular system y , y , x of parameters in O X in which y is ageneral member in I C . Then for any j ≤ i − F j is as a divisor over X obtainedby the weighted blow-up of X with wt( y , y ) = ( j + 1 , y /y j , y , x forma regular system of parameters in O X j ,P j . In particular, f i − ≃ P K has homogeneouscoordinates y /y i − , y . Moreover, the K -point P i ∈ f i − is not defined by [ y /y i − : y ] = [1 : 0]. This follows when i = 1 from the general choice of y , and when i ≥ C i does not intersect F ii − . Take c ∈ K such that P i ∈ f i − is defined by [ y /y i − : y ] = [ c : 1]. Replacing y with y − cy i , we mayassume that c = 0.Then y /y i , y , x form a regular system of parameters in O X i ,P i . The H i , F i − and f i − are defined at P i by y /y i , y and ( y , x ) O X i . Because the fibration F i − → C i − is isomorphic to the projection of the product P K × Spec K C i − , its section C i is definedat P i by ( y /y i + x v ( x ) , y ) O X i for some v ( x ) ∈ K [[ x ]]. After replacing y with y + y i x v ( x ), one has that C i is contained in H i . Step
5. Let e be the maximal index such that ord F e I C = 1 and choose a regularsystem y , y , x of parameters in O X such that y satisfies the condition in Step 4for i = e and y is a general member in I C . Now repeating the process in Step 1for y , y , x instead of x , x , x , we may assume that x = y and x = y . Thenby Remark 6.4, one can obtain E ′′ = F ′′ l − by a weighted blow-up with respect to thecoordinates x , x . More precisely, there exist a non-negative integer p and a positiveinteger q such that E ′′ is obtained by the weighted blow-up of X ′′ with wt( x , x ) = p ( e,
1) + q ( e + 1 , p is positive iff e + 1 < l .Therefore, we conclude that E is also obtained by the weighted blow-up of X withwt( x , x ) = p ( e,
1) + q ( e + 1 , Reduction to the case of decomposed boundaries
The objective of this section is to complete the reduction to Conjecture 1.2.
Remark . In order to prove Conjecture 1.2 or a statement of the same kind, it issufficient to find an integer l which satisfies the required property but may depend onthe germ P ∈ X of a smooth threefold, for the reason that one has only to considerthose a which are m -primary. Indeed, there exists an ´etale morphism from P ∈ X tothe germ o ∈ A at origin of the affine space. Then as it is seen in Lemma 2.7, any m -primary ideal a on X is the pull-back of some ideal b on A , and mld P ( X, a q m s )coincides with mld o ( A , b q n s ), where n is the maximal ideal in O A defining o . Thus,the bound l on the germ o ∈ A can be applied to an arbitrary germ P ∈ X .We shall make the reduction by using the generic limit of ideals. For a moment, wework in the general setting that P ∈ X is the germ of a klt variety. Let r , . . . , r e be N EQUIVALENT CONJECTURES ON SMOOTH THREEFOLDS 27 positive real numbers and S = { a i = Q ej =1 a r j ij } i ∈ N be a sequence of R -ideals on X .Let a = Q ej =1 a r j j be a generic limit of S . We use the notation in Section 3. The a is thegeneric limit with respect to a family F = ( Z l , ( a j ( l )) j , N l , s l , t l ) l ≥ l of approximationsof S , and a is an ideal on ˆ P ∈ ˆ X where ˆ X is the spectrum of the completion of thelocal ring O X,P ⊗ k K .Let ˆ f : ˆ Y → ˆ X be a projective birational morphism isomorphic outside ˆ P . Supposethat ˆ Y is klt and the exceptional locus of ˆ f is a Q -Cartier prime divisor ˆ F . Let ˆ C bea closed proper subset of ˆ F . As in Remark 3.8, after replacing F with a subfamily,for any l ≥ l the ˆ f is descended to a projective morphism f l : Y l → X × Z l from aklt variety whose exceptional locus is a Q -Cartier prime divisor F l . One may assumethat for any i ∈ N l , the fibre f i : Y i → X at s l ( i ) ∈ Z l is a morphism from a klt varietywhose exceptional locus is a Q -Cartier prime divisor F i . Refer to [6, Section B] for theproperties of a family of normal Q -Gorenstein rational singularities. We may assumethat ˆ C is descended to a closed subset C l in F l . The f i , F i and C i = C l × Y l Y i areindependent of l because they are compatible with t l . Lemma 7.2.
Notation and assumptions as above. Suppose that a ˆ F ( ˆ X, a ) is at mostone and that the intersection of ˆ F and the non-klt locus on ˆ Y of ( ˆ X, a ) is containedin ˆ C . Then there exists a positive integer l depending only on a and ˆ f such thatafter replacing F with a subfamily, for any i ∈ N l if a divisor E over X computes mld P ( X, a i ) and has centre c Y i ( E ) not contained in C i , then a E ( X ) ≤ l .Proof. Let r be a positive integer such that r ˆ F is Cartier. We may assume that rF l is Cartier. By replacing a ij with ( a ij ) r and r j with r j /r , we may assume that a ij is an ideal to the power of r and so is a j . Thus one can define the weak transform a iY = Q j ( a ijY ) r j on Y i of a i , as well as the weak transform a Y on ˆ Y of a . Wemay assume that ord ˆ F a j = ord F l a j ( l ) = ord F i a ij < l for any i ∈ N l and j . Set a jY ( l ) = a j ( l ) O Y l ( a j F l ) and a Y ( l ) = Q j ( a jY ( l )) r j for a j = ord ˆ F a j , which is divisibleby r .One can fix a positive real number t such that the intersection of ˆ F and thenon-klt locus on ˆ Y of ( ˆ X, a t ) is still contained in ˆ C . Set the real number b sothat ( ˆ Y , b ˆ F , a tY ) is crepant to ( ˆ X, a t ), then 0 ≤ b < a ˆ F ( ˆ X, a ) ≤
1. Then( Y l , bF l , a Y ( l ) t ) is crepant to ( X × Z l , a ( l ) t ) while ( Y i , bF i , ( a iY ) t ) is crepant to( X, a ti ). One may assume that ( Y l , bF l , a Y ( l ) t ) is klt about F l \ C l .Apply Corollary 3.12 to the family Y l \ C l → Z l . Since a ijY + m l O Y i ( a j E i ) = a jY ( l ) O Y i , one has that ( Y i , bF i , ( a iY ) t ) is lc about F i \ C i for any i ∈ N l afterreplacing F with a subfamily. Thus if a divisor E over X satisfies that c Y i ( E ) C i ,then a E ( X, a ti ) ≥
0, that is, t ord E a i ≤ a E ( X, a i ). Hence, a E ( X ) = a E ( X, a i ) + ord E a i ≤ (1 + t − ) a E ( X, a i ) . In addition if E computes mld P ( X, a i ), then a E ( X ) ≤ (1 + t − ) mld P ( X, a i ) ≤ (1 + t − ) mld P X. Hence any integer l at least (1 + t − ) mld P X satisfies the required property. q.e.d.We provide a meta theorem which connects statements involving the maximal ideal m to those involving m -primary ideals. For the property P in the theorem, one cantake for example empty or being terminal. Theorem 7.3.
Let P ∈ X be the germ of a smooth threefold and m be the maximalideal in O X defining P . Fix a positive rational number q . Let P be a property ofcanonical pairs ( X, a q ) for ideals a on X . Then the following statements are equivalent. (i) Fix a non-negative rational number s . Then there exists a positive integer l depending only on q and s such that if a is an ideal on X satisfying that ( X, a q ) iscanonical and has the property P , then there exists a divisor E over X which computes mld P ( X, a q m s ) and satisfies the inequality a E ( X ) ≤ l . (ii) Fix a non-negative rational number s and a positive integer b . Then there existsa positive integer l depending only on q , s and b such that if a and b are ideals on X satisfying that ( X, a q ) is canonical and has the property P and that b contains m b ,then there exists a divisor E over X which computes mld P ( X, a q b s ) and satisfies theinequality a E ( X ) ≤ l .Proof. Step
1. The (i) follows from the special case of (ii) when b = 1. It is necessaryto derive (ii) from (i). Let S = { ( a i , b i ) } i ∈ N be an arbitrary sequence of pairs ofideals on X such that ( X, a qi ) is canonical and has the property P and such that b i contains m b . Assuming the (i), it is sufficient to find an integer l such that forinfinitely many i , there exists a divisor E i over X which computes mld P ( X, a qi b si ) andsatisfies the inequality a E i ( X ) ≤ l . Note Remark 7.1. By Theorem 4.8, we may assumethat mld P ( X, a qi b si ) is positive. Then by Corollary 4.13, there exists a positive integer b depending only on q and s such that ord G i m ≤ b for every divisor G i over X computing mld P ( X, a qi b si ). Step
2. We construct a generic limit ( a , b ) of S using the notation in Section 3. The( a , b ) is the generic limit with respect to a family F = ( Z l , ( a ( l ) , b ( l )) , N l , s l , t l ) l ≥ l ofapproximations of S . The a and b are ideals on ˆ P ∈ ˆ X where ˆ X is the spectrum ofthe completion of the local ring O X,P ⊗ k K . We let ˆ m denote the maximal ideal in O ˆ X . Note that ˆ m b ⊂ b by m b ⊂ b i . By Lemma 4.7 and Remark 4.3(i), the existenceof l is reduced to the inequality mld ˆ P ( ˆ X, a q b s ) ≤ mld P ( X, a qi b si ) for any i ∈ N l afterreplacing F with a subfamily. By Theorem 4.10, we may assume that ( ˆ X, a q b s ) hasthe smallest lc centre ˆ C which is regular and of dimension one.Since b is ˆ m -primary, ˆ C is also the smallest lc centre of ( ˆ X, a q ). In particular,mld ˆ P ( ˆ X, a q ) ≤ ˆ P ( ˆ X, a q ) ≥ ˆ P ( ˆ X, a q ) = 1. Step
3. We apply Theorem 6.2 to ( ˆ X, a q ). There exist a divisor ˆ E over ˆ X computingmld η ˆ C ( ˆ X, a q ) and a regular system x , x , x of parameters in O ˆ X such that ˆ E isobtained by the weighted blow-up of ˆ X with wt( x , x ) = ( w , w ) for some coprimepositive integers w , w . We take x generally from ˆ m so that ord x a is zero, whereord x stands for the order along the divisor on ˆ X defined by x . Note that ˆ C isgeometrically irreducible.We fix a positive integer j such that j > bb . Let ˆ f : ˆ Y → ˆ X be the weighted blow-up with wt( x , x , x ) = ( jw , jw ,
1) and ˆ F be its exceptional divisor. By Remark 2.3, there exists a regular system y , y , y ofparameters in O X,P ⊗ k K such that ˆ f is also the weighted blow-up with wt( y , y , y ) =( jw , jw ,
1) (after regarding y , y , y as elements in O ˆ X ).Discussed in the paragraph prior to Lemma 7.2, after replacing F with a subfamily,ˆ f is descended to a projective morphism f l : Y l → X × Z l for any l ≥ l . One canassume that y , y , y come from O X,P ⊗ k O Z l and that for any i ∈ N l , their fibres N EQUIVALENT CONJECTURES ON SMOOTH THREEFOLDS 29 y i , y i , y i at s l ( i ) ∈ Z l form a regular system of parameters in O X,P . Y l is klt andthe exceptional locus of f l is a Q -Cartier prime divisor F l . The fibre f i : Y i → X of f l at s l ( i ) is the weighted blow-up of X with wt( y i , y i , y i ) = ( jw , jw ,
1) whoseexceptional divisor is F i = F l × Y l Y i .Since ( jw , jw ,
1) = j ( w , w ,
0) + (0 , , ˆ F a q ≥ j ord ˆ E a q + ord x a q = j ( w + w )using ord ˆ E a q = a ˆ E ( ˆ X ) − a ˆ E ( ˆ X, a q ) = w + w . Equivalently, a ˆ F ( ˆ X, a q ) = a ˆ F ( ˆ X ) − ord ˆ F a q ≤
1. Hence a ˆ F ( ˆ X, a q ) = 1 by mld ˆ P ( ˆ X, a q ) = 1 in Step 2. Step
4. Let ˆ Q be the closed point in ˆ F which lies on the strict transform of ˆ C .For i ∈ N l , let Q i be the closed point in F i which lies on the strict transform of thecurve on X defined by ( y i , y i ) O X . Applying Lemma 7.2 to ( ˆ X, a q b s ), ˆ f , and ˆ Q , onehas only to treat the case when mld P ( X, a qi b si ) is computed by a divisor G i such that c Y i ( G i ) = Q i .For such G i , the inequality ord G i m ≤ b holds by Step 1. Thus,ord G i b i ≤ ord G i m b ≤ bb < j ≤ jw = ord F i ( y i , y i ) O X ≤ ord F i ( y i , y i ) O X · ord G i F i ≤ ord G i ( y i , y i ) O X , whence ord G i b i = ord G i ( b i + ( y i , y i ) O X ).By ˆ m b ⊂ b , there exists a non-negative integer b ′ at most b satisfying that b + ( y , y ) O ˆ X = ˆ m b ′ + ( y , y ) O ˆ X . Then one can assume that b ( l ) + ( y , y ) O X × Z l = ( m b ′ + m l ) O X × Z l + ( y , y ) O X × Z l forany l ≥ l , which derives the inclusion m b ′ ⊂ b i + m l + ( y i , y i ) O X . One may assumethat l ≥ b , then m b ′ ⊂ b i + ( y i , y i ) O X . Thus, ord G i b i = ord G i ( b i + ( y i , y i ) O X ) ≤ ord G i m b ′ . In particular,mld P ( X, a qi m sb ′ ) ≤ a G i ( X, a qi m sb ′ ) ≤ a G i ( X, a qi b si ) = mld P ( X, a qi b si ) . Step
5. We want the inequality mld ˆ P ( ˆ X, a q b s ) ≤ mld P ( X, a qi b si ), as seen in Step 2.Applying our assumption (i) in the case when the exponent of m is one of 0 , s, s, . . . , bs ,there exists a positive integer l ′ depending only on q , s and b such that mld P ( X, a qi m sb ′ )is computed by a divisor E i satisfying the inequality a E i ( X ) ≤ l ′ . One has thatord E i a i = q − ( a E i ( X ) − a E i ( X, a qi )) ≤ q − l ′ , so E i computes mld P ( X, ( a i + m e ) q m sb ′ )for any integer e at least q − l ′ , which equals mld P ( X, a qi m sb ′ ). Together with Lemma3.7, one obtains that mld ˆ P ( ˆ X, a q ˆ m sb ′ ) = mld P ( X, a qi m sb ′ ) for any i ∈ N l after re-placing F with a subfamily. Hence by Step 4, the problem is reduced to showing theequality mld ˆ P ( ˆ X, a q b s ) = mld ˆ P ( ˆ X, a q ˆ m sb ′ ) . The equality b + ( y , y ) O ˆ X = ˆ m b ′ + ( y , y ) O ˆ X tells that b + ( y , y , y j ) O ˆ X = ˆ m b ′ + ( y , y , y j ) O ˆ X . Since ( y , y , y j ) O ˆ X = ˆ f ∗ O ˆ Y ( − j ˆ F ) = I ˆ C + ˆ m j , where I ˆ C is the ideal sheaf of ˆ C , oneconcludes that b + I ˆ C = ˆ m b ′ + I ˆ C , using ˆ m j ⊂ ˆ m b ⊂ b and ˆ m j ⊂ ˆ m b ′ . Therefore, therequired equality follows from the precise inversion of adjunction, Corollary 7.8. q.e.d. Precise inversion of adjunction compares the minimal log discrepancy of a pair andthat of its restricted pair by adjunction. Let P ∈ X be the germ of a normal varietyand S + B be an effective R -divisor on X such that S is a normal prime divisor whichdoes not appear in B . Suppose that they form a pair ( X, S + B ), then one has theadjunction K X + S + B | S = K S + B S in which B S is the different on S of B . Conjecture 7.4 (Precise inversion of adjunction) . Notation as above. Then one hasthat mld P ( X, S + B ) = mld P ( S, B S ) . This conjecture is regarded as the more precise version of Theorem 2.5. At presentwe know two cases when it holds. One is when X is smooth [9] or more generally haslci singularities [8]. The other is when the minimal log discrepancy is at most one [3],that is, Theorem 7.5.
Conjecture holds when ( X, ∆) is klt for some boundary ∆ and mld P ( X, S + B ) is at most one.Proof. It is enough to show the inequality mld P ( X, S + B ) ≥ mld P ( S, B S ). By in-version of adjunction, we may assume that 0 < mld P ( X, S + B ) ≤
1. Then by [3,Corollary 1.4.3], there exists a projective birational morphism π : Y → X from a Q -factorial normal variety such that the divisorial part of its exceptional locus is aprime divisor E computing mld P ( X, S + B ). Let S Y and B Y denote the strict trans-forms of S and B . Then the pull-back of ( X, S + B ) is ( Y, S Y + B Y + bE ) where b = 1 − mld P ( X, S + B ) ≥
0. Let C be an arbitrary irreducible component of E ∩ S Y .By mld P ( X, S + B ) >
0, the (
Y, S Y + B Y + bE ) is plt about the generic point η C of C . By adjunction, one can write K Y + S Y + B Y + bE | S Y = K S Y + B S Y about η C . By [28, Corollary 3.10], C has coefficient at least b in B S Y . Hence, oneobtains that mld P ( S, B S ) ≤ a C ( S Y , B S Y ) ≤ − b = mld P ( X, S + B ). q.e.d. Lemma 7.6.
Let P ∈ X be the germ of a klt variety and S be a prime divisor on X such that ( X, S ) is plt. Let ∆ be the different on S defined by K X + S | S = K S + ∆ .Let ˆ X be the spectrum of the completion of the local ring O X,P and ˆ P be its closedpoint. Set ˆ S = S × X ˆ X and ˆ∆ = ∆ × X ˆ X . Let a be an R -ideal on ˆ X such that mld ˆ P ( ˆ X, ˆ S, a ) ≤ . Then mld ˆ P ( ˆ X, ˆ S, a ) = mld ˆ P ( ˆ S, ˆ∆ , a O ˆ S ) .Proof. Adding a high multiple of the maximal ideal ˆ m in O ˆ X to each componentof a , we may assume that a is ˆ m -primary. Then a is the pull-back of an R -ideal a on X . By Remark 2.6, the assertion is reduced to the precise inversion of ad-junction mld P ( X, S, a ) = mld P ( S, ∆ , a O S ) for varieties, which follows from Theorem7.5. q.e.d. Proposition 7.7.
Let X be the spectrum of the ring of formal power series in threevariables over a field K of characteristic zero and P be its closed point. Let x , x be apart of a regular system of parameters in O X and w , w be coprime positive integers.Let Y → X be the weighted blow-up of with wt( x , x ) = ( w , w ) , E be its exceptionaldivisor, and f be the fibre of E → X at P . Let ∆ be the different on E defined by K Y + E | E = K E + ∆ . Let a be an R -ideal on X whose weak transform a Y on Y isdefined. Suppose that a E ( X, a ) is zero. Then mld P ( X, a ) = mld f ( E, ∆ , a Y O E ) .Proof. By the regular base change, we may assume that K is algebraically closed.Since ( Y, E, a Y ) is crepant to ( X, a ), it is enough to prove that mld f ( Y, E, a Y ) =mld f ( E, ∆ , a Y O E ). N EQUIVALENT CONJECTURES ON SMOOTH THREEFOLDS 31
Extend the x , x to a regular system x , x , x of parameters in O X and set X ′ = Spec K [ x , x , x ]. Then Y is the base change of the weighted blow-up of X ′ with wt( x , x ) = ( w , w ). Thus, the equality mld η f ( Y, E, a Y ) = mld η f ( E, ∆ , a Y O E )follows from Lemma 7.6 by cutting by the strict transform of the divisor on X de-fined by x w + λx w for a general member λ in K . Together with mld f ( Y, E, a Y ) ≤ mld η f ( Y, E ) = 1, it is sufficient to verify that mld Q ( Y, E, a Y ) = mld Q ( E, ∆ , a Y O E ) forany closed point Q in f such that mld Q ( Y, E, a Y ) ≤
1, which follows from Lemma 7.6again. q.e.d.
Corollary 7.8.
Let X be the spectrum of the ring of formal power series in threevariables over a field K of characteristic zero and P be its closed point. Let a , b and c be R -ideals on X . Suppose that mld P ( X, a ) equals one and that ( X, a ) has an lccentre C of dimension one on which b O C = c O C . Then mld P ( X, ab ) = mld P ( X, ac ) .Proof. We may assume that C is not contained in the cosupport of bc , because other-wise mld P ( X, ab ) = mld P ( X, ac ) = −∞ . By Theorem 6.2, there exist a divisor E over X computing mld η C ( X, a ) = 0 and a part x , x of a regular system of parameters in O X such that E is obtained by the weighted blow-up Y of X with wt( x , x ) = ( w , w )for some w , w . We may assume that the weak transform a Y on Y of a is defined.Then, the assertion follows from Proposition 7.7 by b O E = c O E . q.e.d. Proof of Theorem . It is sufficient to derive Conjectures A, B, C and D from Con-jecture 1.2. All R -ideals on the germ P ∈ X of a smooth threefold in Conjectures A toD may be assumed to be m -primary, where m is the maximal ideal in O X defining P .There exists an ´etale morphism from P ∈ X to the germ o ∈ A at origin of the affinespace, by which any m -primary R -ideal a on X is the pull-back of some R -ideal b on A by Lemma 2.7. Thus for Conjectures A to D, one has only to consider R -ideals onthe fixed germ P ∈ X of a smooth threefold.By Lemma 4.11, these conjectures are reduced to the case I = { /n } of ConjectureD, that is, for a fixed positive integer n , it is enough to find an integer l such that if a is an ideal on X , then there exists a divisor E over X which computes mld P ( X, a /n )and satisfies the inequality a E ( X ) ≤ l . By Theorem 4.8, we have only to considerthose a for which mld P ( X, a /n ) is positive. Then by Corollary 4.13, there exists apositive integer b depending only on n such that ord E m ≤ b for every divisor E over X computing mld P ( X, a /n ).Set q = 1 /n and apply Theorem 5.1. It is enough to bound a E ( X ) for those a in the case (ii) of Theorem 5.1. Suppose this case and use the notation in Theorem5.1. Let E be an arbitrary divisor over Y which computes mld Q ( Y, ∆ , a qY ), that equalsmld P ( X, a q ). Then, a E ( X ) = a E ( Y ) + X F ( a F ( Y ) −
1) ord E F ≤ a E ( Y ) + ( c − X F ord E F, in which the summation takes over all exceptional prime divisors on Y , and X F ord E F ≤ ord E m ≤ b. Thus the boundedness of a E ( X ) is reduced to that of a E ( Y ). In other words, it issufficient to treat the divisors computing mld Q ( Y, ∆ , a qY ). The ideal b = O Y ( − n ∆) in O Y is defined since n ∆ is integral, for which ( Y, a qY b q ) is crepant to ( Y, ∆ , a qY ). The b satisfies that ord E b ≤ n P F ord E F ≤ nb by O Y ( − n P F ) ⊂ b . In particular, E computes mld Q ( Y, a qY ( b + n nb ) q ) as well as mld Q ( Y, ∆ , a qY ) for the maximal ideal n in O Y defining Q . Replacing the notation ( Y, a qY ( b + n nb ) q ) with ( X, a q b q ) and n with m , Conjectures Ato D follow from the boundedness of a E ( X ) for some divisor E over X which computesmld P ( X, a q b q ) such that mld P ( X, a q ) ≥ m nb ⊂ b . One may assumethat a is m -primary. Then one can apply Theorem 7.3 with the property P beingempty, which reduces the boundedness of a E ( X ) to Conjecture 1.2. q.e.d.8. Boundedness results
In this section, we shall prove Conjecture 1.2 in several cases. First we treat thecase when either ( X, a q ) is terminal or s is zero. Proof of Theorem . Let S = { a i } i ∈ N be an arbitrary sequence of ideals on X suchthat ( X, a qi ) is terminal. We construct a generic limit a of S . We use the notation inSection 3, so a is the generic limit with respect to a family F = ( Z l , a ( l ) , N l , s l , t l ) l ≥ l of approximations of S , and a is an ideal on ˆ P ∈ ˆ X . We let ˆ m denote the maximalideal in O ˆ X . To see the assertion (i), by Lemma 4.7 and Remark 7.1, it is enoughto show the equality mld ˆ P ( ˆ X, a q ˆ m s ) = mld P ( X, a qi m s ) for any i ∈ N l after replacing F with a subfamily. One has that mld ˆ P ( ˆ X, a q ) > X, a q )satisfies the case 1, 2 or 3 in Theorem 4.10, which derives that ( ˆ X, a q ˆ m s ) does nothave the smallest lc centre of dimension one. Hence the required equality holds byTheorem 4.10.For (ii), starting instead with S such that ( X, a qi ) is canonical, we need to showthe equality mld ˆ P ( ˆ X, a q ) = mld P ( X, a qi ). This holds in the cases other than the case4 in Theorem 4.10, so we may assume the case 4, in which mld ˆ P ( ˆ X, a q ) ≤
1. Bymld P ( X, a qi ) ≥
1, the required equality follows from Remark 4.3(i). q.e.d.We shall study the case in Theorem 1.4(i) when the lc threshold of the maximalideal is at most one-half. We prepare a useful criterion for identifying a divisor over avariety.
Lemma 8.1.
Let P ∈ X be the germ of a smooth variety and E be a divisor over X .Let x , . . . , x c be a part of a regular system of parameters in O X,P and w , . . . , w c be positive integers. Let Y → X be the weighted blow-up with wt( x , . . . , x c ) =( w , . . . , w c ) , F be its exceptional divisor, and H i be the strict transform of the di-visor on X defined by x i . Suppose that • c X ( E ) coincides with c X ( F ) , and • the vector ( w , . . . , w c ) is parallel to (ord E x , . . . , ord E x c ) .Then the centre on Y of E is not contained in the union S ci =1 H i .Proof. The idea has appeared already in [14, Lemma 6.1]. We may assume that w , . . . , w c have no common divisors. One computes thatord E x i = ord E H i + ord F x i · ord E F = ord E H i + w i ord E F. The ord E H i is positive iff the centre c Y ( E ) lies on H i . Since the intersection T ci =1 H i is empty, at least one of ord E H i is zero. Because (ord E x , . . . , ord E x c ) is parallel to( w , . . . , w c ), one concludes that ord E H i is zero for every i , which proves the assertion.q.e.d.The next lemma plays a central role in the proof of Theorem 1.4(i). Lemma 8.2.
Let C be the spectrum of the ring of formal power series in one variableover k and P be its closed point. Let X → C be a smooth projective morphism ofrelative dimension one and f be its fibre at P . Let ∆ be an effective R -divisor on X , Q be a closed point in f , and t be a positive real number. Suppose that N EQUIVALENT CONJECTURES ON SMOOTH THREEFOLDS 33 • ( K X + ∆) · f = 0 , • mld f ( X, ∆) = 1 , and • mld Q ( X, ∆ + tf ) = 0 .Then t is at least one-half. Moreover if t equals one-half, then mld Q ( X, ∆+ sf ) = 1 − s for any non-negative real number s at most one-half.Proof. Step
1. Let E be a divisor over X which computes mld Q ( X, ∆ + tf ) = 0. Wedefine the coprime positive integers w and w so that the vector ( w , w ) is parallelto (ord E f, ord E n ), where n is the maximal ideal in O X defining Q . Take a regularsystem x , x of parameters in O X,Q such that x defines f and x is a general memberin n . We claim that the divisor F obtained by the weighted blow-up Y → X withwt( x , x ) = ( w , w ) computes mld Q ( X, ∆ + tf ).This claim can be verified in the same way as in [19]. Assuming that a F ( X, ∆ + tf )is positive, we shall derive a contradiction. For i = 1 ,
2, let H i be the strict transformof the divisor defined on X by x i . By Lemma 8.1, the centre on Y of E would be aclosed point R in F \ ( H + H ). The pull-back of ( X, ∆ + tf ) is ( Y, bF + ∆ Y + tH ) inwhich ∆ Y is the strict transform of ∆ and b = 1 − a F ( X, ∆+ tf ) <
1. Thus (
Y, F +∆ Y )is not lc about R , so ( F, ∆ Y | F ) is not lc about R by inversion of adjunction. Remarkthat this inversion of adjunction on R ∈ Y holds by Lemma 7.6 because Y → X isthe base change of the weighted blow-up of Spec k [ x , x ] with wt( x , x ) = ( w , w ).This means that ord R (∆ Y | F ) is greater than one.One computes that1 = − ( K Y + bF + ∆ Y + tH ) · F + 1 ≤ − ( K Y + bF + tH ) · F − ord R (∆ Y | F ) + 1 < (( w + w −
1) + b − tw )( − F ) = 1 w + 1 − tw − − bw w . Together with w ≥ w and b <
1, one would obtain that w = 1 and tw < b . Butthen a F ( X, ∆) = a F ( X, ∆ + tf ) + t ord F f = (1 − b ) + tw <
1, which contradicts thatmld f ( X, ∆) = 1. Step
2. We have seen that F computes mld Q ( X, ∆ + tf ) = 0. Then a F ( X, ∆) = t ord F f = tw . Since ord Q ∆ ≤ f ( X, ∆) = 1, one has that ord F ∆ ≤ w .Thus, w ≤ w + w − ord F ∆ = a F ( X, ∆) = tw . By ( K X + ∆) · f = 0, one has that (∆ · f ) = 2. Hence w − ord F ∆ = (ord F ∆)( F · H ) ≤ (∆ Y + (ord F ∆) F ) · H = (∆ · f ) = 2 , where the inequality follows from the fact that f does not appear in ∆ by mld f ( X, ∆) =1. Thus ord F ∆ ≤ w and w − w ≤ w + w − ord F ∆ = a F ( X, ∆) = tw , that is, (1 − t ) w ≤ w .We have obtained that (1 − t ) w ≤ w ≤ tw . Therefore t ≥ /
2, and moreover if t = 1 /
2, then w = 2 w so ( w , w ) = (2 , Step
3. Suppose that t = 1 /
2. Let s be a non-negative real number at most one-half.It is necessary to show that mld Q ( X, ∆ + sf ) = 1 − s . One has that mld Q ( X, ∆ +(1 / f ) = 0 and it is computed by F . In particular, a F ( X, ∆) = 2 − ord F f = 1. By mld f ( X, ∆) = 1, one obtains that mld Q ( X, ∆) = 1 and it is also computed by F .Then Lemma 4.5(ii) provides thatmld Q ( X, ∆ + sf ) = (1 − s ) mld Q ( X, ∆) + 2 s mld Q ( X, ∆ + (1 / f ) = 1 − s. q.e.d. Proposition 8.3.
Let X be the spectrum of the ring of formal power series in threevariables over a field K of characteristic zero and P be its closed point. Let a bean R -ideal such that mld P ( X, a ) equals one and such that ( X, a ) has an lc centre ofdimension one. Then one of the following holds for the maximal ideal m in O X . (i) The mld P ( X, am s ) equals − s for any non-negative real number s at mostone-half. (ii) The mld P ( X, am / ) is positive.Proof. We may assume that K is algebraically closed. By Theorem 6.2, there exista divisor E over X and a part x , x of a regular system of parameters in O X suchthat a E ( X, a ) = 0 and such that E is obtained by the weighted blow-up Y → X withwt( x , x ) = ( w , w ) for some w , w . We may assume that the weak transform a Y of a is defined. Let f be the fibre of E → X at P and ∆ be the different on E definedby K Y + E | E = K E + ∆. Take an R -divisor A X = e − P ei =1 A i on X for large e inwhich A i are defined by general members in a . Let A be the strict transform on Y of A X and set A E = A | E . By Proposition 7.7, one obtains thatmld P ( X, am s ) = mld f ( E, ∆ + A E + sf )for any non-negative real number s . This equality for s = 0 supplies that mld f ( E, ∆ + A E ) = 1. In particular, f does not appear in ∆ + A E .Let t be the positive real number such that mld P ( X, am t ) = 0. If t > /
2, thenthe case (ii) holds. Suppose that t ≤ /
2. Then mld f ( E, ∆ + A E + tf ) = 0 butmld η f ( E, ∆ + A E + tf ) = 1 − t >
0, so there exists a closed point Q in f such thatmld Q ( E, ∆ + A E + tf ) = 0. With ( K E + ∆ + A E ) · f = 0, one can apply Lemma 8.2 to( E, ∆ + A E ), which derives that t = 1 / f ( E, ∆ + A E + sf ) = 1 − s . Hencethe case (i) holds. q.e.d. Proof of Theorem . Let S = { a i } i ∈ N be an arbitrary sequence of ideals on X suchthat ( X, a qi ) is canonical and such that mld P ( X, a qi m / ) is not positive. We constructa generic limit a of S . We use the notation in Section 3, so a is the generic limit withrespect to a family F = ( Z l , a ( l ) , N l , s l , t l ) l ≥ l of approximations of S , and a is an idealon ˆ P ∈ ˆ X . We let ˆ m denote the maximal ideal in O ˆ X . By Lemma 4.7 and Remarks4.3(i) and 7.1, it is enough to show the inequality mld ˆ P ( ˆ X, a q ˆ m s ) ≤ mld P ( X, a qi m s )for any i ∈ N l after replacing F with a subfamily. By Theorem 4.10, we may assumethat ( ˆ X, a q ) has the smallest lc centre of dimension one, in which mld ˆ P ( ˆ X, a q ) ≤
1. ByRemark 4.3(i) again, one obtains that mld ˆ P ( ˆ X, a q ) = 1 from the canonicity of ( X, a qi ).Let t be the positive rational number such that mld ˆ P ( ˆ X, a q ˆ m t ) = 0. By Theorem4.8, we may assume that s < t . By Remark 4.3(ii), one has that mld P ( X, a qi m t ) =0 for any i ∈ N l after replacing F with a subfamily. In particular, t ≤ / P ( X, a qi m / ) ≤
0. Applying Proposition 8.3 to ( ˆ X, a q ), one obtains that t = 1 / ˆ P ( ˆ X, a q ˆ m s ) = 1 − s . Thus mld P ( X, a qi m / ) = 0. By Lemma 4.5(i), oneobtains that mld P ( X, a qi m s ) ≥ (1 − s ) mld P ( X, a qi ) + 2 s mld P ( X, a qi m / ) ≥ − s = mld ˆ P ( ˆ X, a q ˆ m s ) . N EQUIVALENT CONJECTURES ON SMOOTH THREEFOLDS 35 q.e.d.
Proof of Corollary . We may assume that a is m -primary. By Theorems 1.3(i) and1.4(i), we have only to consider ideals a such that mld P ( X, a q ) equals one and suchthat mld P ( X, a q m / ) is positive. By Lemma 4.5(i), such a satisfies thatmld P ( X, a q m /n ) ≥ (cid:16) − n (cid:17) mld P ( X, a q ) + 2 n mld P ( X, a q m / ) > − n , whence mld P ( X, a q m /n ) ≥ − /n since mld P ( X, a q m /n ) belongs to n − Z .Let E be an arbitrary divisor over X which computes mld P ( X, a q ). Then1 − n ≤ mld P ( X, a q m /n ) ≤ a E ( X, a q m /n ) = a E ( X, a q ) − n ord E m ≤ − n , so mld P ( X, a q m /n ) = 1 − /n and it is computed by E . By Lemma 4.5(ii), E alsocomputes mld P ( X, a q m s ) for any non-negative real number s at most 1 /n . ThusCorollary 1.5 follows from Theorem 1.3(ii). q.e.d.By a similar argument, one can prove Conjecture 1.2 in the opposite case when thelc threshold of the maximal ideal is at least one. Proof of Theorem . Let S = { a i } i ∈ N be an arbitrary sequence of ideals on X such that ( X, a qi ) is canonical and such that ( X, a qi m ) is lc. It is enough to show theexistence of a positive integer l such that for infinitely many indices i , there exists adivisor E i over X which computes mld P ( X, a qi m s ) and satisfies the equality a E i ( X ) = l .Note Remark 7.1. As in the proof of Theorem 1.4(i), we construct a generic limit a on ˆ P ∈ ˆ X of S with respect to a family F = ( Z l , a ( l ) , N l , s l , t l ) l ≥ l of approximationsof S . By Lemma 4.7 and Theorem 4.10, we may assume that ( ˆ X, a q ) has the smallestlc centre of dimension one, in which mld ˆ P ( ˆ X, a q ) ≤
1. By Remark 4.3(i), one has thatmld ˆ P ( ˆ X, a q ) = 1.Let ˆ E be a divisor over ˆ X which computes mld ˆ P ( ˆ X, a q ). As in Remark 3.8(i),replacing F with a subfamily, one can descend ˆ E to a divisor E l over X × Z l forany l ≥ l . Writing E i for a component of the fibre of E l at s l ( i ) ∈ Z l , one mayassume that a E i ( X ) = a ˆ E ( ˆ X ) and a E i ( X, a qi ) = 1 for any i ∈ N l . Then for any i ∈ N l , mld P ( X, a qi ) = 1 by the canonicity of ( X, a qi ) and it is computed by E i .By the log canonicity of ( X, a qi m ), the ord E i m must equal one and E i also computesmld P ( X, a qi m ) = 0. Therefore, E i computes mld P ( X, a qi m s ) by Lemma 4.5(ii). q.e.d.9. Rough classification of crepant divisors
By Theorems 1.3(i) and 1.4(i), for Conjecture 1.2 one has only to consider ideals a such that mld P ( X, a q ) equals one and such that mld P ( X, a q m / ) is positive. Thenevery divisor E over X computing mld P ( X, a q ) satisfies that ord E m equals one. Weclose this paper by providing a rough classification of E . Theorem 9.1.
Let P ∈ X be the germ of a smooth threefold and m be the maximalideal in O X defining P . Let a be an R -ideal on X such that mld P ( X, a ) equals one.Let E be a divisor over X which computes mld P ( X, a ) such that ord E m equals one.Then there exist a regular system x , x , x of parameters in O X,P and positive integers w , w with w ≥ w such that for the weighted blow-up Y of X with wt( x , x , x ) =( w , w , , one of the following cases holds by identifying the exceptional divisor F with P ( w , w , with weighted homogeneous coordinates x , x , x . E equals F as a divisor over X . The centre c Y ( E ) is the curve on F defined by x x p + x q for some positiveintegers p and q satisfying that w + p = qw ≤ w + w . The centre c Y ( E ) is the curve on F defined by x x + x w + w .Proof. Step
1. Let w be the maximum of ord E x for all elements x in m \ m . To seethe existence of this maximum, let Z → X be the birational morphism from a smooththreefold Z on which E appears as a divisor. Applying Zariski’s subspace theorem[1, (10.6)] to O X,P ⊂ O Z,Q for a closed point Q in E , one has an integer w such that O Z ( − wE ) Q ∩ O X,P ⊂ m . Then ord E x is less than w for any x ∈ m \ m , so w exists. Fix x for which ord E x attains the maximum w .Then let w be the maximum of ord E x for those x such that x , x form a partof a regular system of parameters in O X,P . Note that w ≥ w . Fix x for whichord E x attains the maximum w , and take a general member x in m . Note thatord E x = ord E m = 1. The x , x , x form a regular system of parameters in O X,P .Let Y be the weighted blow-up of X with wt( x , x , x ) = ( w , w ,
1) and F be itsexceptional divisor. We identify F with P ( w , w ,
1) with weighted homogeneouscoordinates x , x , x . Step
2. Let L be an arbitrary locus in F defined by a weighted homogeneouspolynomial of form either u , u or x where • u = x + P ⌊ w /w ⌋ i =0 λ i x i x w − iw for some λ i ∈ k , • u = x + λx w for some λ ∈ k .We claim that the centre c Y ( E ) is not contained in any such L . Indeed by Lemma8.1, c Y ( E ) is not contained in the locus defined by x x x . In particular, ord E F =ord E m = 1. If L is defined by u i for i = 1 or 2, then the u i , as an element in O X ,satisfies that w i ≥ ord E u i ≥ ord E L + ord F u i · ord E F = ord E L + w i . Thus ord E L = 0, meaning that c Y ( E ) L . One also has that ord E x i = ord E u i . ByRemark 2.3, we are free to replace x with u as well as x with u for the regularsystem x , x , x of parameters constructed in Step 1. Step
3. Since an arbitrary closed point in F lies on some L in Step 2, the c Y ( E ) iseither a curve or F itself. The case c Y ( E ) = F is nothing but the case 1. We shallinvestigate the case when c Y ( E ) is an irreducible curve C other than any L .Let d be the weighted degree of C in F ≃ P ( w , w , a Y of a is defined, so ( Y, bF, a Y ) is the pull-back of ( X, a ) where b = 1 − a F ( X, a ) ≤
0. Since a E ( Y, F, a Y ) = a E ( Y, bF, a Y ) − (1 − b ) ord E F ≤ a E ( X, a ) − (1 − b ) = b ≤ , the ( Y, F, a Y ) is not plt about η C . Thus ( F, a Y O F ) is not klt about η C by inversion ofadjunction, that is, ord C ( a Y O F ) ≥
1. Thus the strict transform A Y of the R -divisor on X defined by a general member in a satisfies the inequality A Y | F ≥ C . One computesthat d = w w C · ( − F ) ≤ dw w A Y | F · ( − F ) = w w (ord F a ) F = ord F a = w + w + 1 − a F ( X, a ) ≤ w + w . Step
4. Let f be the weighted homogeneous polynomial in x , x , x defining C ,which has weighted degree d ≤ w + w . Since any weighted homogeneous polynomialin x , x is decomposed as the product of polynomials of form x or x + λx w for some λ ∈ k , by Step 2 and the irreducibility of C , there exists a monomial which involves x and appears in f . N EQUIVALENT CONJECTURES ON SMOOTH THREEFOLDS 37
Suppose that d < w + w . Then x x p is the only monomial of weighted degree d involving x , in which p = d − w ≥
0. The p must be positive by Step 2, so1 ≤ p < w . The f is, up to constant, written as f = x x p + q X i =0 λ i x i x w + p − iw = (cid:16) x + q − X i =0 λ i x i x w − iw (cid:17) x p + λ q x q x w + p − qw for some λ i ∈ k , where q = ⌊ ( w + p ) /w ⌋ . Since f is irreducible, one has that λ q = 0and w + p = qw . Replacing f with λ − q f and x with λ − q ( x + P q − i =0 λ i x i x w − iw ), f is expressed as x x p + x q , which is the case 2.Suppose that d = w + w and w > w . Then x x and x x w are the onlymonomials of weighted degree d involving x . If only x x w appears in f , then thecase 2 holds by the same discussion as in the case d < w + w . If x x appears in f ,then the part in f involving x is, up to constant, written as x ( x + λx ) for some λ ∈ k . Replacing x with x + λx , one may write f as f = x x + q X i =0 λ i x i x w + w − iw = (cid:16) x + q X i =1 λ i x i − x w + w − iw (cid:17) x + λ x w + w for some λ i ∈ k , where q = ⌊ w /w ⌋ +1. One has that λ = 0 by the irreducibility of f .Replacing f with λ − f and x with λ − ( x + P qi =1 λ i x i − x w + w − iw ), f is expressedas x x + x w + w , which is the case 3.Finally, suppose that d = 2 w and w = w = w for some w . If w = 1, then C must be a conic in F ≃ P , so the case 3 holds after replacing x , x , x . If w ≥ x , x with their suitable linear combinations, we may assumethat the part in f not involving x is either x x or x . In the first case, f is writtenas f = ( x + λ x w )( x + λ x w ) + λ x w for some λ , λ , λ ∈ k . Then λ = 0 by theirreducibility. Replacing f with λ − f , x with λ − ( x + λ x w ), and x with x + λ x w , f is expressed as x x + x w , which is the case 3. In the second case, f is writtenas f = ( λ x + λ x + λ x w ) x w + x for some λ , λ , λ ∈ k . Then λ = 0 by theirreducibility. Replacing x with λ x + λ x + λ x w , f is expressed as x x w + x ,which is the case 2. q.e.d.One can compute the lc threshold of the maximal ideal in the case 3. Proposition 9.2.
Suppose the case in Theorem . Then ( X, am ) is lc.Proof. We keep the notation in Theorem 9.1. Y is the weighted blow-up of X withwt( x , x , x ) = ( w .w ,
1) and F is its exceptional divisor. For i = 1 , ,
3, let H i bethe strict transform of the divisor defined by x i . Let Q i be the closed point in F whichlies on H j ∩ H k for a permutation { i, j, k } of { , , } . Let C denote the centre on Y of E , which is the curve defined in F ≃ P ( w , w ,
1) by x x + x w w of weighted degree d = w + w in our case 3. Let A be the R -divisor defined by a general member in a and A Y be its strict transform on Y .We have seen in Step 3 of the proof of Theorem 9.1 that d = w w C · ( − F ) ≤ dw w A Y | F · ( − F ) = w + w + 1 − a F ( X, a ) ≤ w + w . Since d = w + w , the above inequalities are eventually equalities, whence a F ( X, a ) = 1and A Y | F = C .The triple ( Y, F + A Y + H ) is crepant to ( X, A + H X ), where H X is the divisordefined by x . Thus it is enough to show the log canonicity of ( Y, F + A Y + H ). Let∆ be the different on F defined by K Y + F | F = K F + ∆. By inversion of adjunction,the log canonicity of ( Y, F + A Y + H ) is equivalent to that of ( F, ∆ + A Y | F + H | F ). Let g be the greatest common divisor of w and w . Let L ≃ P be the line in F defined by x . Since Y has a quotient singularity of type g (1 , −
1) at η L , the different∆ equals (1 − g − ) L as in Example 2.4. Together with A Y | F = C and H | F = g − L ,one has that ∆ + A Y | F + H | F = C + L .The assertion is reduced to the log canonicity of ( F, C + L ), which can be checkeddirectly by using the explicit expression ( x x + x w + w ) x of the defining weightedpolynomial of C + L . Along L , one can use inversion of adjunction again, which tellsthat K F + C + L | L = K L + Q + Q . q.e.d. Acknowledgements
I should like to thank Professor M. Mustat¸˘a for introducing me to the approach byusing the generic limit of ideals to the ACC conjecture for minimal log discrepancies.I should also like to thank Dr. Y. Nakamura for the discussions on the relationshipbetween (iii), (iv) and (v) in Conjecture 4.2.
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