aa r X i v : . [ m a t h . AG ] J a n SMOOTH 3-DIMENSIONAL CANONICAL THRESHOLDS
D. A. STEPANOV
Dedicated to the memory of my advisorVasilii Alekseevich Iskovskikh
Abstract. If X is an algebraic variety with at worst canonical singu-larities and S is a Q -Cartier hypersurface in X , the canonical thresholdof the pair ( X, S ) is the supremum of c ∈ R such that the pair ( X, cS )is canonical. We show that the set of all possible canonical thresholdsof the pairs (
X, S ), where X is a germ of smooth 3-dimensional variety,satisfies the ascending chain condition. We also deduce a formula forthe canonical threshold of ( C , S ), where S is a Brieskorn singularity. Introduction
Let P ∈ X be a germ of a complex algebraic variety X with at worstcanonical singularities. Let S be a hypersurface (not necessarily irreducibleor reduced) in X which is Q -Cartier, i. e., for some integer r the divisor rS can locally be defined on X by one equation. Definition 1.1.
The canonical threshold of the pair (
X, S ) isct P ( X, S ) = sup { c ∈ R | the pair ( X, cS ) is canonical } . If we require in the above definition the variety X and the pair ( X, cS )to be log canonical, we get the analogous notion of log canonical threshold lct P ( X, S ) which is perhaps better known (see, e. g., [7], Sections 8, 9, 10).In the same way as it is done for the log canonical threshold, considering anappropriate resolution of singularities of (
X, S ) one shows that the numberct P ( X, S ) is rational and ct P ( X, S ) ∈ [0 , ∩ Q . Definition 1.2.
The set of n -dimensional canonical thresholds is the set T can n = { ct P ( X, S ) | dim X = n } , where ( X, S ) varies over all pairs satisfying conditions of Definition 1.1.We shall denote by T lc n the corresponding set of n-dimensional log canon-ical thresholds . The following conjecture is very important for the MinimalModel Program. Its first part was formulated by V. V. Shokurov and thesecond by J. Koll´ar. Conjecture 1.3. (i):
The set T lc n satisfies the ascending chain condi-tion (ACC); (ii): the set of accumulation points of T lc n is T lc n − \ { } . The research was supported by the Russian Grant for Scientific Schools 1987.2008.1and by the Russian Program for Development of Scientific Potential of the High School2.1.1/227.
An analog of Conjecture 1.3 (i) for canonical thresholds is
Conjecture 1.4.
The set T can n satisfies ACC. Conjecture 1.4 is interesting for applications to birational geometry ([3]).It is also a particular case of conjecture of C. Birkar and V. V. Shokurovabout a -lc thresholds ([1], Conjecture 1.7). We can not estimate whetherthe analog of Conjecture 1.3 (ii) for canonical thresholds would be plausible. Remark . The log canonical threshold lct P ( X, S ) can be studied not onlyfrom algebraic geomtry point of view. It has interpretations in terms of con-vergence of some integrals, Bernstein-Sato polynomials etc. (see [7], [8]). Itwould be interesting to find similar interpretations for the canonical thresh-old.As far as we know, all the conjectures are still open in their generalform. However, some important cases have been established. Let us denoteby T lc n, smooth the set { lct P ( X, S ) | dim X = n, X is smooth } of smooth n -dimensional log canonical thresholds . T. de Fernex and M. Mustat¸˘a ([4]) andJ. Koll´ar ([8]) proved the analog of Conjecture 1.3, (ii) for the set T lc n, smooth .A theorem of T. de Fernex, L. Ein and M. Mustat¸˘a ([5]) states ACC for theset T lc n, smooth for any n (actually some rectricted types of singularities of X are allowed). Concerning canonical thresholds, there is a theorem of Yu. G.Prokhorov ([12], Theorem 1.4) describing the upper part of the set T can3 of3-dimensional canonical thresholds. Theorem 1.6 (Prokhorov) . If c = ct P ( X, S ) for some 3-dimensional vari-ety X and c = 1 , then c ≤ / and the bound is attained. Moreover, if X issingular, then c ≤ / and the bound is attained. In this paper we also restrict ourselves to a particular case of Conjec-ture 1.4. Namely, we prove it for smooth 3-dimensional germs P ∈ X . Let T can3 , smooth = { ct P ( X, S ) | dim X = 3 , X is smooth } be the set of smooth 3-dimensional canonical thresholds. Our main result isthe following. Theorem 1.7.
The set T can , smooth satisfies the ascending chain condition. It is not difficult to check that T can n − , smooth ⊆ T can n, smooth , so if we proveACC in some dimension, we automatically have it in all smaller dimen-sions. The proof of Theorem 1.7 is contained in Section 2. Its main point isM. Kawakita’s classification of 3-dimensional contractions to smooth points([6]). In that section we assume some familiarity of the reader with the Mini-mal Model Program ([9]). In Section 3 we deduce a formula for the canonicalthreshold of a 3-dimensional Brieskorn singularity. It is interesting that itturns out to be much more cumbersome than the corresponding formulafor the log canonical threshold. In Section 4 we slightly strengthen The-orem 1.6 by showing that there are no 3-dimensional canonical thresholdsbetween 4 / / ANONICAL THRESHOLDS 3 Ascending chain condition for smooth 3-dimensionalcanonical thresholds
Reduction to extremal contraction.
Let P ∈ X be a germ of ter-minal Q -factorial complex 3-dimensional algebraic variety and S an effectiveinteger divisor on X such that the pair ( X, S ) is not terminal. Let g : e X → X be an embedded resolution of the pair ( X, S ). We denote by E i , i ∈ I , theprime exceptional divisors of g and by e S the strict transform of S . Then wecan write the relations K e X = g ∗ K X + X i ∈ I a i E i , g ∗ S = e S + X i ∈ I b i E i for some rational numbers a i , b i . Here K e X and K X stand for the canonicalclasses of e X and X respectively. Now let c ∈ Q be the canonical thresholdof the pair ( X, S ). This means that the pair (
X, cS ) is canonical, i. e., if wewrite K e X + c e S = g ∗ ( K X + cS ) + X i ∈ I ( a i − cb i ) E i , then a i − cb i ≥ i ∈ I . This implies the estimate c ≤ a i b i for all i ,and, since we assume that c is the threshold, the equality c = ct P ( X, S ) = min i ∈ I a i b i . (We can always assume that the resolution g has at least 1 exceptional divisorwith discrepancy 0 over ( X, cS ); this follows from [7], Corollary 3.13.) Thisshows, in particular, that c is indeed rational. If for a divisor E i we have a i − cb i = 0, we shall say that E i (considered as a discrete valuation of thefield C ( X ) of rational functions on X ) realizes the canonical threshold forthe pair ( X, S ). Lemma 2.1 (cf. [11], Section 3) . Let c be the canonical threshold of a pair ( X, S ) with terminal Q -factorial 3-dimensional germ P ∈ X . Then thereexists an extremal divisorial contraction g ′ : X ′ ⊃ E ′ → X ∋ P such that itsexceptional divisor E ′ realizes the threshold c . (We shall also say that thethreshold c is achieved on the contraction g ′ ).Proof. We use the notation introduced before the lemma. Let us apply to e X the ( K e X + c e S )-Minimal Model Program (MMP) relative over X (see [9],Ch. 11). It stops with a Q -factorial variety b X such that the pair ( b X, c b S ),where b S is the strict transform of e S , is terminal. Actually MMP contracts allthe exceptional divisors of g which have positive discrepancies over ( X, cS ).Next we apply K b X -MMP over X to b X . It contracts all the exceptional divi-sors which remain in b X and stops with the variety X . Since X was supposedto be Q -factorial, the last step of K b X -MMP is an extremal divisorial con-traction g ′ : X ′ → X from some Q -factorial terminal variety X ′ . Let E ′ bethe exceptional divisor of g ′ . All the divisors that we contract on this stagehave discrepancy 0 over ( X, cS ), thus E ′ realizes the canonical threshold of( X, S ). (cid:3) D. A. STEPANOV
For the rest of this section we assume X to be smooth. The variety X ′ obtained in Lemma 2.1 is Q -factorial, so we again can write(1) K X ′ = g ′∗ K X + a ′ E ′ , g ′∗ S = S ′ + b ′ E ′ , and since the canonical threshold c is realized by E ′ , we have c = a ′ /b ′ .This reduces the calculation of the canonical threshold in the smooth 3-dimensional case to an extremal divisorial contraction , i. e., to a morphismwith connected fibers g ′ : X ′ → X subject to the following conditions: (i): X ′ is Q -factorial with only terminal singularities; (ii): the exceptional locus of g ′ is a prime divisor; (iii): − K X ′ is g ′ -ample; (iv): the relative Picard number of g ′ is 1.In the 3-dimensional situation g ′ can contract the divisor E ′ either ontoa curve C ⊂ X or to the point P ∈ X . In the first case it follows fromMori’s classification of smooth extremal contractions ([10]) that at a genericpoint of C the morphism g ′ is isomorphic to an ordinary blow up of X atthe curve C . Then it follows from (1) that ct P ( X, S ) = 1 / mult C ( f ), wheremult C ( f ) is the multiplicity of the defining function f of S at a generic pointof C . Thus the set T can3 , smooth contains the subset { /n | n ∈ N } which satisfiesACC.Now suppose that g ′ contracts the divisor E ′ to the smooth point P ∈ X .In this case we have an important result of M. Kawakita ([6], Theorem 1.2)classifying extremal divisorial contractions to smooth points. Theorem 2.2 (Kawakita) . Let Y be a 3-dimensional Q -factorial varietywith only terminal singularities, and let g : ( Y ⊃ E ) → X ∋ P be an alge-braic germ of an extremal divisorial contraction which contracts its excep-tional divisor E to a smooth point P . Then we can take local parameters x , y , z at P and coprime positive integers a and b such that g is the weightedblow up of X with its weights (1 , a, b ) . So further we may assume that the canonical threshold of the pair (
X, S )is realized by some weighted blow up. Also, since X is smooth and thecanonical threshold can be defined and calculated using analytic germs aswell, we assume in the sequel that P ∈ X is isomorphic to 0 ∈ C n and S isdetermined by a convergent power series f .2.2. Canonical threshold and weighted blow ups.
The affine space A n C can be given a structure of a toric variety X ( τ, N ) where N = Z n and thecone τ is the positive octant of the real vector space R n ≃ N N R . Let w = ( w , . . . , w n ) ∈ N ∩ τ be a primitive vector. The weighted blow up σ w of the space A n C ≃ C n with the weight vector w is the toric morphism σ w : X (Σ w , N ) → C n ≃ X ( τ, N )given by the natural subdivision Σ w of the cone τ with a help of the vec-tor w . Certainly, a weighted blow up depends not only on its weights w but also on the choice of a toric structure A n C ≃ X ( τ, N ). The variety X (Σ w , N ) is Q -factorial and can be covered by n affine charts. The i thchart is isomorphic to C n / Z w i where the cyclic group acts with weights ANONICAL THRESHOLDS 5 ( − w , . . . , − w i − , , − w i +1 , . . . , − w n ), and the morphism σ w is given in thischart by the formulae x i = y w i i , x j = y j y w j i , j = i where x , . . . , x n are the coordinates on the target and y , . . . , y n on thesource space (see, e. g., [11], 3.7).Given a hypersurface S = { f = 0 } in C n , we can estimate the canonicalthreshold ct ( C n , S ) with a help of weighted blow ups. Namely, for anyweight vector w (not equal to 0 or to a vector e i of the standard basis) wehave ct ( C n , S ) ≤ w + · · · + w n − w ( f )where w ( f ) is the least weight of a monomial appearing in f with respect tothe weights w , . . . , w n (see [12]). Moreover, if we know that the canonicalthreshold of ( C n , S ) is achieved on some weighted blow up, we can calculateit as(2) ct ( C n , S ) = min w =0 ,e i ,i =1 ,...,n w + · · · + w n − w ( f ) . where the minimum is taken over all integer vectors in τ . It is possible thatthe denominator in (2) is 0 for some weights w ; such fractions should betreated as + ∞ .Recall that the extended Newton diagram Γ + ( f ) of a polynomial (or aseries) f = P m a m x m is the convex hull in R n of the set { m + R n ≥ | a m = 0 } .Note that the denominator w ( f ) in the above formulae depends only onΓ + ( f ) but not on f itself. Let us introduce a new set T can n, smooth,w = the setof smooth n -dimensional canonical thresholds which can be realized by someweighted blow up. Remark . The set T can n, smooth,w may look a bit artificial. However, it con-tains, for example, the set of smooth 3-dimensional canonical thresholds(for n = 3; see subsection 2.1), or the set of canonical thresholds achievedon hypersurfaces S defined by series f nondegenerate with respect to theirNewton diagrams.Now Theorem 1.7 follows from the next result. Theorem 2.4.
The set T can n, smooth,w satisfies ACC. The proof of Theorem 2.4 will follow from the 2 lemmas below.Let us denote by N + n the set of all possible extended Newton diagramsin R n . Clearly this set is ordered with respect to the inclusion relation ⊆ .The next lemma is perhaps well known for specialists on singularity theoryor on Gr¨obner bases. But since we do not know a good reference, we stateit with a proof. Lemma 2.5.
Any infinite sequence of elements of the ordered set N + n con-tains a monotonous non increasing subsequence. In particular, the set N + n satisfies ACC.Proof. It is more convinient to use the ideals of semigroup ( Z n ≥ , +) (see [2], § + we can associate the ideal generated by its vertices. From any ideal D. A. STEPANOV we can construct an extended Newton diagram by taking its convex hull in R n ≥ . Despite this correspondence is not one-to-one, it should be clear thatit suffices to prove the lemma for ideals.The proof goes by induction on the dimension n . For n = 1 the statementis obvious. Let Γ +1 , Γ +2 , . . . be a sequence of ideals of the semigroup Z n ≥ .We can project every ideal Γ + i to the coordinate axes of the space R n . Everyprojection gives an ideal in the semigroup Z ≥ . First let us suppose thatfor some axis, say, the last one, the so obtained sequence of projections isunbounded, i. e., for any M > k such that the projectionof the ideal Γ + k to the last axis lies above M . Choosing a subsequencewe can assume that in fact the projection of any ideal Γ + i lies above M for i > k . The projection of every ideal Γ + i to the coordinate hyperplanecontaining the first n − Z n − ≥ .By the induction hypothesis we can choose a subsequence such that thecorresponding sequence of projections to the given hyperplane is monotonousnon increasing. To simpify the notation we assume that already the sequenceΓ +1 , Γ +2 , . . . has this property. Consider the ideal Γ +1 . Let us denote by M themaximal of the last coordinates of its vertices (we use here the fact that anyideal of Z n ≥ is a finite union of the shifted coordinate octants; this is calledthe first finiteness property of the semigroup Z n ≥ , see [2], § k all the ideals Γ + i lie above M with respect tothe last coordinate, and thus they all are contained in Γ +1 . Going on in thesame way we construct the needed non increasing subsequence of ideals.Now suppose that the projections of the sequence Γ +1 , Γ +2 , . . . to all thecoordinate axes are bounded. This means that all the ideals Γ + i have a vertexinside some fixed polytope. But every bounded polytope has only finitenumber of integer points, thus, again choosing a subsequence if necessary,we can assume that all Γ + i have a common vertex m . It follows that allthe ideals Γ + i contain also a common shifted coordinate octant m + R n ≥ .Consider the complement to this octant in Z n ≥ . It is clear that it is a unionof finite number of shifted coordinate subsemigroups of the semigroup Z n ≥ (this is a particular case of the second finiteness property of the semigroup Z n ≥ , [2], § n −
1, and the intersection of any ideal Γ + i with such a semigroupis an ideal of it. Applying the induction hypothesis and repeatedly (butfinite number of times!) choosing a subsequence, we conclude that outsidethe octant m + R n ≥ our sequence Γ +1 , Γ +2 , . . . can be assumed to be nonincreasing. But taking the union with the same octant obviously does notchange this property. This finishes the proof. (cid:3) The formula (2) formally defines the canonical threshold of an extendedNewton diagram Γ + which we shall denote by ct(Γ + ). We can consider itas a map ct : N + n → R . Lemma 2.6.
The map ct is monotonous, i. e., Γ +1 ⊆ Γ +2 ⇒ ct(Γ +1 ) ≤ ct(Γ +2 ) . ANONICAL THRESHOLDS 7
Proof.
Let w be a vector in Z n ≥ , w = 0 , e , . . . , e n . It defines a rationalfunction w : R n − → R w ( x ) = w + · · · + w n − w x + · · · + w n x n . The level set w = c of this function ( w is fixed, x varies) is a hyperplanewith a non negative normal vector w . The smaller c > w = c is.Suppose that the canonical threshold c = ct(Γ +2 ) of the diagram Γ +2 isrealized by the weight vector w and this threshold is attained on a vertex m of the diagram Γ +2 , c = w ( m ). The minimum c ′ of the function w onthe diagram Γ +1 is not greater than c because Γ +1 is situated “above” thediagram Γ +2 . The threshold ct(Γ +1 ) can only be less or equal to c ′ . (cid:3) To finish the proof of Theorem 2.4, suppose that there exists a strictly in-creasing sequence c < c < . . . of canonical thresholds from T can n, smooth,w .Let Γ +1 , Γ +2 , . . . be a sequence of extended Newton diagrams such thatct(Γ + k ) = c k . We can not have an inclusion Γ + i ⊇ Γ + j for any i < j be-cause of Lemma 2.6. But this contradicts Lemma 2.5.3. Canonical threshold for Brieskorn singularities in C A Brieskorn singularity is a hypersurface singularity S is C n given by theequation x a + x a + · · · + x a n n = 0 . For n = 3 we shall assume that S is given by(3) x a + y b + z c = 0 , where 2 ≤ a ≤ b ≤ c . The log canonical threshold of the pair ( C n , S ) can bedetermined by the formula ([7], 8.15)lct ( C n , S ) = min { a + 1 a + · · · + 1 a n , } . In this section we calculate the 3-dimensional canonical threshold ct ( C , S ).Brieskorn singularities are nondegenerate with respect to their Newton di-agrams, thus they admit embedded toric resolutions (for the definition ofnondegeneracy and construction of embedded toric resolution see [13]). Itfollows, in particular, that their canonical thresholds are realized by weightedblow ups and we can apply formula (2) from subsection 2.2. In the case of3-dimensional Brieskorn singularities it takes the form(4) ct ( C , S ) = min w = e ,e ,e , w + w + w − { aw , bw , cw } , where the minimum is taken over all vectors w from Z ≥ . Remark . Formula (4) is not a direct consequence of Theorem 2.2 andsubsection 2.1. Indeed, the theorem states that there exist some coordi-nates in which the extremal contraction realizing the canonical threshold isa weighted blow up. But in those coordinates the equation of our singularitymust not be of Brieskorn type or even nondegenerate.
D. A. STEPANOV
Lemma 3.2.
Let S ⊂ C be a Brieskorn singularity given by equation (3) .Suppose that a weight vector w = ( p, q, r ) realizes the minimum in (4) . Then p ≥ q ≥ r .Proof. It is clear that the vector realizing the canonical threshold satisfies p, q, r = 0. Assume, for example, that p < q . Thenct ( C , S ) = p + q + r − { ap, cr } and q ≥
2. But in this case we could take w ′ = ( p, q − , r ) instead of w and w ′ would give strictly smaller canonical threshold, a contradiction. Otherinequalities can be considered similarly. (cid:3) Lemma 3.3.
A weight vector w giving the canonical threshold of a Brieskornsingularity (3) can always be chosen in the from w = ( p, q, , where p and q are coprime positive integers.Proof. Consider a piecewise rational function h determined on R > by theformula h ( w ) = w + w + w − { aw , bw , cw } . Its level set h ( w ) = s , s ≥
0, coincides with the lateral surface of a tetra-hedron ∆ s with vertices (1 , , , , , ,
1) (forming the base face ofthe tetrahedron) and with the last vertex11 /a + 1 /b + 1 /c − s (cid:18) a , b , c (cid:19) on the line aw = bw = cw . If s < s ′ , then ∆ s ⊂ ∆ s ′ . We see that thecanonical threshold of a Brieskorn singularity (3) can be found with a helpof the following process. For every s ≥ s and find the minimal s for which ∆ s contains an integer point ( p, q, r ) with p, q, r > ( C , S ) = s and the threshold isrealized by the weight vector ( p, q, r ).Now suppose that ct ( C , S ) = s = h ( p, q, r ) and r ≥
2. From Lemma 3.2we know that p ≥ q ≥ r . Consider also a tetrahedron ∆ w with vertices(1 , , , , , , p, q, r ). Obviously ∆ w ⊂ ∆ s and we proveour lemma if we show that the intersection of ∆ w with the plane w = 1contains a positive integer point. Indeed, if ( p ′ , q ′ ,
1) is such a point and d is the greatest common divisor of p ′ and q ′ , the point ( p ′ /d, q ′ /d,
1) alsolies in ∆ w and gives smaller value of h . The intersection ∆ w ∩ { w = 1 } is a triangle with vertices (0 , /r ( p + r − , q ), 1 /r ( p, q + r −
1) (thethird coordinate w = 1 is omitted here). It is a bit more convenient tomultiply everything by r and to show that the triangle OP Q , O = (0 , P = ( p, q + r − Q = ( p + r − , q ), contains a positive integer pointwith coordinates 0 mod r (see Figure 1). Other points in Figure 1 have thefollowing meaning. E is the intersection point of the lines OP and w = q .Its coordinates are ( pq/ ( q + r − , q ). The segment BC is the middle lineof the triangle EP Q and the segment AD lies on the line w = q − r/ Q lies under the diagonal w = w . Thus if the point P lies above the diagonal, then the triangle OP Q contains already the point( r, r ). So let us assume p > q + r −
1. Choose an integer k ≥ ANONICAL THRESHOLDS 9 ✲ w ✻ w O ✦✦✦✦✦✦✦✦✦✦✦✦✦✦✦❅❅❅✚✚✚✚✚✚✚✚✚✚✚✚ Q ( p + r − , q ) P ( p, q + r − A E B CD
Figure 1.
Integer points in the triangle( k − q + r − < p ≤ k ( q + r − q ≤ r − q ≥ r . Suppose first that q ≤ r −
1. Consider a transformed triangle OP ′ Q ′ obtained from OP Q with a help of the unimodular transformation (cid:18) − ( k − (cid:19) . The point P ′ ( p − ( k − q + r − , q + r −
1) lies above the diagonal.Indeed, p − ( k − q + r − ≤ q + r −
1. On the other hand, the point Q ′ ( p − ( k − q + r − , q ) lies under the diagonal: p − ( k − q + r − > ( k − q + r − − ( k − q + r − r − k − r − ( k − ≥ q + ( k − r − ( k − ≥ q − r , k ≥
2. It follows that the triangle OP ′ Q ′ contains the point ( r, r ).But then the triangle OP Q also has a positive integer point 0 mod r .Now suppose that q ≥ r . Note that the length of EQ is p + r − − pqq + r − r − p + q + r − q + r − > r − BC > r − AD ≥ (3 / EQ > (3 / r − r for r ≥
3. If we have a segment [ x, y ] with integer x or y onthe real line R , and if this segment has length ≥ r −
1, then it necessarilycontains a point 0 mod r (perhaps as one of its border points). If the lengthof the segment is ≥ r , it always contains a point 0 mod r no matter whether x or y are integers. From this observations it easily follows that for r ≥ ABCQD contains a point 0 mod r . We leave thedetails to the reader.So it remains to consider the case when r = 2 and q ≥
4. Moreover, wecan assume that p and q are odd because otherwise already one of the points P , Q , or ( p, q ) will have coordinates 0 modulo r . Let us check if the point( p − , q −
1) lies in the triangle
OP Q . This holds if q − p − ≥ qp + 1 , or p ≤ q −
1. On the other hand, AD = q − q · p + q + 1 q + 1 ≥ p ≥ ( q + 1) q − > q + 1 . Two inequalities p ≤ q − p > q + 1 cover all possibilities for p and q .Thus the triangle OP Q always contains the desired point. (cid:3)
Remark . Again Lemma 3.3 is not a direct consequence of Kawakita’sTheorem 2.2, see Remark 3.1Now we are ready to deduce a formula for the canonical threshold ofa Brieskorn singularity. To do this, let us introduce some new notation.Denote by L the point ( c/a, c/b,
1) of the intersection of the line aw = bw = cw with the plane w = 1. Fix a real number s and consider the intersectionof the tetrahedron ∆ s with the plane w = 1 (see the proof of Lemma 3.3).For s < /a +1 /b this intersection is empty; for s = 1 /a +1 /b it is the segment OL ; for s > /a + 1 /b it is a triangle OM N where M = ( c/a, sc − c/a, N = ( sc − c/b, c/b,
1) (see Figure 2). This almost immediately implies ✲ w ✻ w O ✟✟✟✟✟✟✟✟✟✟✟✟❅❅❅(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)✑✑✑✑✑✑✑✑✑ r L M Nk r Figure 2.
Minimizing the function h Lemma 3.5.
Let S be a Brieskorn singularity of the form (3) . Then ct ( C , S ) ≥ /a + 1 /b . Moreover, if c ≥ l . c . m . ( a, b ) , where l . c . m . is theleast common multiple, then ct ( C , S ) = 1 /a + 1 /b .Proof. Only the “moreover” part of the lemma needs a proof. If c ≥ m = l . c . m . ( a, b ), then the segment OL contains the point ( m/a, m/b, h ( m/a, m/b,
1) = 1 /a + 1 /b , but we know from the first part of the lemmathat 1 /a + 1 /b is the least possible value of the canonical threshold. Theproof is completed. (cid:3) When c < l . c . m . ( a, b ), we have a kind of integer programming problem:minimize the piecewise rational function h on positive integer points of theplane w = 1. Equivalently, we have to determine the minimal s such thatthe triangle OM N contains a positive integer point. The point w on which h attains its minimum is on the following list: ANONICAL THRESHOLDS 11 (i): w is the nearest to OL integer point lying on the horizontal line w = k , 1 ≤ k ≤ ⌊ c/b ⌋ to the right from the segment OL (seeFigure 2); in this case w = ( ⌈ kb/a ⌉ , k, ⌊·⌋ denotes the lowerand ⌈·⌉ the upper integer; (ii): w is the nearest to OL integer point lying on the vertical line w = k , 1 ≤ k ≤ ⌊ c/a ⌋ above the segment OL ; in this case w =( k, ⌈ ka/b ⌉ , (iii): w is “the first” integer point in the triangle LM N ; then w =( ⌈ c/a ⌉ , ⌈ c/b ⌉ , s = min ≤ k ≤⌊ c/b ⌋ { h ( ⌈ kb/a ⌉ , k, } = min ≤ k ≤⌊ c/b ⌋ (cid:26) b + 1 kb (cid:24) kba (cid:25)(cid:27) ,s = min ≤ k ≤⌊ c/a ⌋ { h ( k, ⌈ ka/b ⌉ , k, } = min ≤ k ≤⌊ c/a ⌋ (cid:26) a + 1 ka (cid:24) kab (cid:25)(cid:27) ,s = h ( ⌈ c/a ⌉ , ⌈ c/b ⌉ ,
1) = ⌈ c/a ⌉ + ⌈ c/b ⌉ c . We summarize what we did in this section in the following result.
Theorem 3.6.
Let S be a Brieskorn singularity (3) . If l . c . m . ( a, b ) ≤ c , then ct ( C , S ) = 1 /a + 1 /b ; otherwise ct ( C , S ) = min { s , s , s , } (notationas above).Example . Consider a Brieskorn singularity x + y + z = 0 . Using our formulae we get s = min ≤ k ≤ { / / · } = 47 ,s = min ≤ k ≤ { / / k ) ⌈ (3 k ) / ⌉} == min { / / , / / , / / } = 12 ,s = (4 + 2) /
11 = 611 . It follows that ct ( C , S ) = 1 / s and it is achieved on the weighted blowup with weights (2 , , Example . Let S be a Brieskorn singularity x + y + z = 0 . We have s = min ≤ k ≤ { / / (6 k ) ⌈ (6 k ) / ⌉} == min { / / , / / , / / , / / } = 38 ,s = min ≤ k ≤ { / / (5 k ) ⌈ (5 k ) / ⌉} = 25 ,s = (6 + 5) /
29 = 1129 . It follows that ct ( C , S ) = 3 / s and it is achieved on the weighted blowup (5 , , Example . Now let S be a Brieskorn singularity x + y + z = 0 . We get s = min ≤ k ≤ { /
18 + 1 / } = 16 ,s = min ≤ k ≤ { /
12 + 1 / (12 k ) ⌈ (2 k ) / ⌉} = 16 ,s = (3 + 2) /
35 = 17 . It follows that ct ( C , S ) = 1 / s and it is achieved on the weighted blowup (3 , , The upper part of the canonical set
In this section we strengthen Theorem 1.6 of Yu. G. Prokhorov describingthe upper part of the set T can3 of 3-dimensional canonical thresholds. Theorem 4.1.
The intersection T can ∩ [4 / , is precisely { / , / , } .Proof. Recall that if X is singular, then ct P ( X, S ) ≤ / S has non isolated singularities in a neighborhood of P , then ct P ( X, S ) ≤ / S is a hypersurface in C with isolated singularity at the origin. Then by Kawakita’s Theorem 2.2and subsection 2.1 the canonical threshold ct ( C , S ) is achieved on someweighted blow up.Let S be given in C by an equation f = 0. We shall analyze the Newtondiagram Γ( f ) of f and show that the canonical threshold of S can be 1, 5 / / f lies above theplane α + β + γ = 3, then ct ( C , S ) ≤ /
3. Indeed, in this case Γ + ( f ) iscontained in the extended Newton diagram of the singularity x + y + z = 0which has canonical threshold 2 /
3. Thus by Lemma 2.6 ct ( C , S ) ≤ / f necessarily has monomials of degree 2. Ifthe second differential of f has rank 2 or 3, f is isomorphic to a Du Valsingularity of type A n . In this case its canonical threshold is 1. But thesame holds even if the second differential of f has rank 1 and f has at least2 monomials of degree 2. Indeed, recall that the canonical threshold of S depends only on the Newton diagram Γ( f ). But then we can perturb thecoefficients of f in such a way that the second differential becomes of rank ≥
2. It follows that we can assume that f has the form f = x + terms of degree ≥ . Moreover, making a substitution x ′ = x √ . . . (it does not violate theproperty that the canonical threshold of f is achieved on a weighted blowup) we can assume that x is the only monomial of f containing x withdegree ≥ ANONICAL THRESHOLDS 13
Further, let us compare f with the Brieskorn singularity x + y + z = 0 . Its canonical threshold is 3 / f ) liesabove the plane 2 α + β + γ = 4, then ct ( C , S ) ≤ /
4. Therefore we cansuppose that f has a monomial of degree 3. If this monomial is y z , weagain conclude that f is a Du Val singularity (this time of type D n ) andct ( C , S ) = 1. Thus we assume f = x + y + other terms of degree ≥ . Next we compare f with the Brieskorn singularity x + y + z = 0 . Its canonical threshold is 5 /
6. Suppose that f contains monomials x α y β z γ lying below the plane 3 α + 2 β + γ = 6. Possible monomials are z , z , z , yz , yz , xz . In this case f is a Du Val singularity of type D n or E n andits canonical threshold is 1. Therefore it remains to consider the case whenΓ( f ) lies above the plane 3 α + 2 β + γ = 6 and ct ( C , S ) ≤ /
6. Let us showthat in fact ct ( C , S ) = 5 / f defines an isolated singularity, it has monomialsof the form z n , xz n , or yz n . Hence we can estimate the canonical thresholdof S from below comparing it with the nondegenerate singularities x + y + z n = 0 , n ≥ ,x + y + xz n = 0 , n ≥ , or x + y + yz n = 0 , n ≥ . The first and the second singularity are easily seen to be isomorphic toBrieskorn singularities with canonical threshold 5 /
6. Let us prove by a directcomputation that the canonical threshold of the third singularity S ′ ⊂ C is also 5 / σ of C with weights (3 , , f C is covered by 3 affine charts (see subsection 2.2). In the firstisomorphic to C / Z (1 , ,
2) the strict transform S ′ f C of S ′ is isomorphic to1 + y + x n − yz n = 0and is nonsingular. In the second chart C / Z (1 , ,
1) the strict transform x + 1 + y n − z n = 0is again nonsingular. Note that the quotient singularities of the first 2charts are terminal and S ′ f C does not pass through them. The third chart isisomorphic to C and the strict transform to x + y + yz n − = 0 , i. e., to a singularity of the same form but with smaller n . Computingdiscrepancy we get K f C + (5 / S ′ f C = σ ∗ ( K C + (5 / S ′ ) . Therefore by induction we show that ct ( C , S ′ ) = 5 / (cid:3) References [1] Birkar, C., Shokurov, V. V.
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