Jumping numbers and ordered tree structures on the dual graph
aa r X i v : . [ m a t h . A C ] M a r JUMPING NUMBERS AND ORDERED TREESTRUCTURES ON THE DUAL GRAPH
EERO HYRY AND TARMO J ¨ARVILEHTO
Dedicated to the memory of Professor Olli Jussila
Abstract.
Let R be a two-dimensional regular local ring having analgebraically closed residue field and let a be a complete ideal of finitecolength in R . In this article we investigate the jumping numbersof a by means of the dual graph of the minimal log resolution ofthe pair ( X, a ). Our main result is a combinatorial criterium for apositive rational number ξ to be a jumping number. In particular,we associate to each jumping number certain ordered tree structureson the dual graph. Introduction
Multiplier ideals have in recent years emerged as an important toolin algebraic geometry. Given a closed subscheme of a smooth complexvariety, there is a nested sequence of multiplier ideals parametrized by thepositive rational numbers. A jumping number is a value of the rationalparameter at which the multiplier ideal makes a jump. Jumping numbersform a discrete set of invariants, which contains important informationabout the singularities of the subscheme in question.Jumping numbers are defined by using an embedded resolution of thesubcheme. They depend on the exceptional divisors appearing in theresolution. The purpose of this article is to look at jumping numbers fromthe point of view of the combinatorics of exceptional divisors in the caseof a smooth surface. In particular, we associate to each jumping numbercertain ordered tree structures on the dual graph of the resolution. Thesestructures generalize the one discovered by Veys in [11] while he wasstudying poles of the topological zeta functionTo describe our work in more detail, let a be a complete ideal of finitecolength in a two-dimensional regular local ring R having an algebraicallyclosed residue field. Let X −→ Spec R be a minimal log resolution of thepair ( R, a ). Let D be the divisor on X such that O X ( − D ) = a O X . Let E , . . . , E N be the exceptional divisors. Recall that a divisor F = f E + . . . + f N E N is called antinef if F · E γ ≤ γ = 1 , . . . , N , where F · E γ denotes the intersection product. Our starting point is the observationmade in [7] that jumping numbers of a can be parametrized by the antinef ivisors. More precisely, the jumping number corresponding to F is ξ F := min γ f γ + k γ + 1 d γ , where D = d E + . . . + d N E N and K = k E + . . . + k N E N is the canonicaldivisor.Let Γ denote the dual graph of X . Recall that the vertices of the dualgraph correspond to the exceptional divisors and that two vertices areadjacent if and only if the corresponding exceptional divisors intersect.The above considerations motivate us to investigate the function λ ( f γ , γ )on | Γ | , where λ ( a, γ ) := a + k γ + 1 d γ for any a ∈ Z and γ ∈ | Γ | . By the above the minimum value of λ ( f γ , γ )is now the jumping number ξ = ξ F . We call the set S F := { γ ∈ | Γ | | λ ( f γ , γ ) = ξ } the support of ξ with respect to the antinef divisor F .Our main result Theorem 4.3 is a combinatorial criterium for a positiverational number ξ to be a jumping number. We observe that it is possibleto choose the divisor F in such a way that λ ( f γ , γ ) strictly increases alongevery path away from the support. By assigning the number λ ( f γ , γ ) toeach vertex γ , we can make the dual graph an ordered tree. An endof S F must either have at least three adjacent vertices or correspond tosome Rees valuation of the ideal. Moreover, one may assume that S F isa chain such that the vertices corresponding to Rees valuations do notoccur at the non-ends of S F .We also want to understand how the so called contributing divisorsarise. The notion of contribution to a jumping number by a divisor wasintroduced by Smith and Thompson in [9], and the investigation hasthen been continued by Tucker in [10]. It turns out in Theorem 4.16that every critically contributing divisor, in the sense of Tucker, is of thetype P γ ∈ S F E γ . However, the converse is not true. Therefore we give inTheorem 4.19 a necessary and sufficient condition for a reduced divisorto be a critically contributing one.Using Theorem 4.3 we will also show in Corollary 4.10 that given avertex with at least three adjacent vertices or a vertex corresponding to aRees valuation of the ideal, there is always a jumping number supportedexactly at this vertex. Note that a support of a jumping number alwayscontains vertices of this type. Moreover, by means of Theorem 4.3 wecan in Proposition 4.12 construct from a given jumping number certainnew jumping numbers having the same support as the original one.Our main technical tool is Lemma 3.6 which helps us to constructsuitable antinef divisors F . This is inspired by the work of Loeser and eys concerning the numerical data associated to the exceptional divisorsof the resolution (see [6] and [11]).Finally, we would like to refer to the book of Favre and Jonsson ([3])for related topics. It would be interesting to know whether our resultscan be interpreted in their ‘tree language’.2. Preliminaries
We begin by fixing notation and recalling some basic facts from theZariski-Lipman theory of complete ideals. For more details, we referto [4], [1], [5] and [7].Throughout this article R denotes a regular local ring of dimension twohaving an algebraically closed residue field. Let a be a complete ideal offinite colength in R . Let π : X → Spec( R ) be a minimal principalizationof a . Then X is a regular scheme and a O X = O X ( − D ) for an effectiveCartier divisor D . Note that π is a log resolution of the pair (Spec R, a ),i.e., the divisor D + Exc( π ) has simple normal crossing support, whereExc( π ) denotes the sum of the exceptional divisors of π .The morphism π is a composition of point blowups of regular schemes π : X = X N +1 π N −→ · · · π −→ X π −→ X = Spec R, where π µ : X µ +1 → X µ is the blowup of X µ at a closed point x µ ∈ X µ for every µ = 1 , . . . , N . Let E µ and E ∗ µ , respectively, be the strictand total transform of the exceptional divisor π − µ { x µ } on X for every µ = 1 , . . . , N . We denote by v µ the discrete valuation associated to thediscrete valuation ring O X,E µ , in other words, v µ is the m X µ ,x µ -adic ordervaluation.Recall that a point x µ is said to be infinitely near to a point x ν , if theprojection X µ → X ν maps x µ to x ν . This relation gives a partial orderon the set { x , . . . , x N } . A point x µ is proximate to the point x ν , denotedby µ ≻ ν , if and only if x µ lies on the strict transform of π − ν { x ν } on X µ . Following [1, Definition-Lemma 1.5], the proximity matrix is P := ( p µ,ν ) N × N , where p µ,ν = , if µ = ν ; − , if µ ≻ ν ;0 , otherwise.Note that this is the transpose of the proximity matrix given in [4, p. 6].We set Q = ( q µ,ν ) N × N := P − . The equation P Q = 1 immediately givesthe formula(1) q µ,ν = X µ ≻ ρ q ρ,ν + δ µ,ν . If x µ is infinitely near to x ν , then q µ,ν > q µ,ν = 0 otherwise.Clearly q µ,µ = 1 for all µ = 1 , . . . , N .We denote by Γ the dual graph associated to our principalization. It iswell known that Γ is a tree. Let | Γ | be the corresponding set of vertices. ecall that there is a vertex ν corresponding to each exceptional divisor E ν weighted by the number w Γ ( ν ) := − E ν . Note that w Γ ( ν ) = 1 + { µ ∈ | Γ | | µ ≻ ν } . Two vertices µ and ν are called adjacent if they can be joined by an edge.This is the case if and only if the corresponding exceptional divisors E µ and E ν intersect. We write µ ∼ ν . Then either µ ≺ ν or µ ≻ ν .Suppose, for example, that µ ≻ ν . Then ν ∼ µ in fact means that x µ isa maximal element in the set of infinitely near points proximate to x ν .The valence v Γ ( ν ) of a vertex ν means the number of vertices adjacentto it. If v Γ ( ν ) ≥
2, then ν is called a star . A vertex τ with v Γ ( τ ) = 1 isan end . The distance between two vertices µ, ν ∈ | Γ | is defined as d ( ν, µ ) := min { r | ν = ν ∼ · · · ∼ ν r = µ, where ν , . . . , ν r ∈ | Γ |} . Furthermore, if S ⊂ | Γ | , we set d ( ν, S ) := min { d ( ν, µ ) | µ ∈ S } . If d ( ν, S ) = 1, then we write ν ∼ S .We consider the lattice Λ := Z E ⊕ . . . ⊕ Z E N of exceptional divisorson X . The lattice Λ has two other convenient bases besides { E µ | µ =1 , . . . , N } , namely { E ∗ µ | µ = 1 , . . . , N } and { b E µ | µ = 1 , . . . , N } , where E µ · b E ν = − δ µ,ν for µ, ν = 1 , . . . , N . For any G ∈ Λ, we write G = g E + . . . + g N E N = g ∗ E ∗ + . . . + g ∗ N E ∗ N = b g b E + . . . + b g N b E N . The following base change formulas now hold:(2) g ∗ = gP t and b g = gP t P = g ∗ P, where g, g ∗ and b g denote row vectors in Z n . Here ( P t P ) µ,ν = − E µ · E ν .In particular, note the formulas(3) b g µ = g ∗ µ − X ν ≻ µ g ∗ ν = w Γ ( γ ) g µ − X ν ∼ µ g ν ( µ = 1 , . . . , N ) . The support of a divisor G ∈ Λ is | G | := { γ ∈ | Γ | | g γ = 0 } .Recall that a divisor F ∈ Λ is antinef if b f ν = − F · E ν ≥ ν = 1 , . . . , N . Equivalently, the proximity inequalities (4) f ∗ µ ≥ X ν ≻ µ f ∗ ν ( µ = 1 , . . . , N )hold. Note that they can also be expressed in the form(5) w Γ ( µ ) f µ ≥ X ν ∼ µ f ν ( µ = 1 , . . . , N ) . In fact, if F = 0 is antinef, then also f ν > ν = 1 , . . . , N . Thereis a one to one correspondence between the antinef divisors in Λ and thecomplete ideals of finite colength in R generating invertible O X -sheaves,given by F ↔ Γ( X, O X ( − F )). For a divisor G ∈ Λ, there exists a inimal one among the antinef divisors F satisfying F ≥ G . This iscalled the antinef closure of G and denoted by G ∼ . We haveΓ( X, O X ( − G )) = Γ( X, O X ( − G ∼ ))for any divisor G ∈ Λ.Recall that an ideal is called simple if it cannot be expressed as a prod-uct of two proper ideals. By the famous result of Zariski, every completeideal factorizes uniquely into a product of simple complete ideals. Moreprecisely, a = p b d · · · p b d N N , where p µ ⊂ R denotes the simple complete ideal of finite colength cor-responding to the exceptional divisor E µ and b d µ > v µ is a Rees valuation of a . We have p µ O X = O X ( − b E µ ) so that p µ =Γ( X, O X ( − b E µ )). By (2)(6) b E µ = X ν q µ,ν E ∗ ν = X ν,ρ q ν,ρ q µ,ρ E ν . In particular, we observe the reciprocity formula (7) v ν ( p µ ) = X ρ q ν,ρ q µ,ρ = v µ ( p ν ) ( µ, ν = 1 , . . . , N ) . For µ = ν , the proximity inequalities now become equalities q µ,ν = X ρ ≻ ν q µ,ρ . We will next recall the definition of jumping numbers. A general ref-erence for jumping numbers is the fundamental article [2]. Recall firstthat the canonical divisor is K = P ν E ∗ ν . The formulas (2) now give(8) k ν = X µ q ν,µ and b k ν = E ν + 2 ( ν = 1 , . . . , N ) . By (3) we then obtain an important relation(9) w Γ ( ν ) k ν = 2 − w Γ ( ν ) + X µ ∼ γ k µ ( ν = 1 , . . . , N ) . For a nonnegative rational number ξ , the multiplier ideal J ( a ξ ) isdefined to be the ideal J ( a ξ ) := Γ ( X, O X ( K − ⌊ ξD ⌋ )) ⊂ R, where ⌊ ξD ⌋ denotes the integer part of ξD . It is now known that thereis an increasing discrete sequence0 = ξ < ξ < ξ < · · · of rational numbers ξ i characterized by the properties that J ( a ξ ) = J ( a ξ i ) for ξ ∈ [ ξ i , ξ i +1 ), while J ( a ξ i +1 ) ( J ( a ξ i ) for every i . The numbers ξ , ξ , . . . , are called the jumping numbers of a . Note that contrary to [2, efinition 1.4], we don’t consider 0 as a jumping number. Clearly, this isno restriction. The following Proposition 2.1, which is fundamental forthe rest of this article, results from [7, Proposition 6.7 and Proposition7.2]. Proposition 2.1.
Let R be a two-dimensional regular local ring and let a ⊂ R be a complete ideal of finite colength. Then ξ is a jumping numberof a if and only if there exists an antinef divisor F ∈ Λ such that ξ = ξ F := min ν f ν + k ν + 1 d ν . Moreover, if b is the complete ideal corresponding to F , then ξ = inf { c ∈ Q > | J ( a c ) + b } . Notation 2.2.
We write for any integer a and for any vertex νλ ( a, ν ) := a + k ν + 1 d ν and call the set S F := { ν ∈ | Γ | | λ ( f ν , ν ) = ξ } the support of the jumping number ξ with respect to the antinef divisor F . Relations between numerical data associated toexceptional divisors
Let ξ be a jumping number of the ideal a . In order to define an or-dered tree structure on the dual graph as described in the introduction,we must be able to construct an antinef divisor F such that ξ = ξ F andthat λ ( f γ , γ ) increases along every path away from the support S F . Pro-ceeding inductively, suppose that we are given a vertex γ and a vertex η ∼ γ such that d ( γ, S F ) > d ( η , S F ). Suppose, furthermore, that num-bers f γ and f η have been defined in such a way that λ ( f γ , γ ) > λ ( f η , η ).The key issue is to find for vertices η = η ∼ γ suitable numbers f η withthe property that λ ( f η , η ) > λ ( f γ , γ ). This problem will be addressed inLemma 3.5, which is the main result of this section. Other details of theabove construction will be postponed till Lemma 4.1 in the next section.We begin with the following lemma: Lemma 3.1.
Let F ∈ Λ be a divisor. Let γ ∈ | Γ | be a vertex such that b f γ ≥ and ξ := λ ( f γ , γ ) ≤ λ ( f η , η ) when η ∼ γ . Then ξ b d γ ≥ b f γ − v Γ ( γ ) + 2 . In particular, this implies the following: a) If v Γ ( γ ) ≤ and b d γ = 0 , then there are exactly two vertices η adjacent to γ and λ ( f η , η ) = ξ for both of those. Furthermore b f γ = 0 . ) If v Γ ( γ ) = 1 , then ξ b d γ ≥ b f γ + 1 . Especially, b d γ > .Proof. By the formulas (3) and (9) we obtain ξ = w Γ ( γ )( f γ + k γ + 1) w Γ ( γ ) d γ = P η ∼ γ ( f η + k η + 1) + b f γ − v Γ ( γ ) + 2 P η ∼ γ d η + b d γ . Since ξ ≤ f η + k η + 1 d η , we must have ξ b d γ ≥ b f γ − v Γ ( γ ) + 2.When v Γ ( γ ) ≤
1, this immediately gives b d γ >
0. Then suppose that b d γ = 0 and v Γ ( γ ) = 2. Now b f γ − v Γ ( γ ) + 2 ≤ b f γ = v Γ ( γ ) − . But then ξ = P η ∼ γ ( f η + k η + 1) P η ∼ γ d η , which implies that ξ = λ ( f η , η ) for both η ∼ γ . (cid:3) For any two vertices ν, γ ∈ | Γ | , set α γ,ν := k ν + 1 − k γ + 1 d γ d ν = d ν ( λ (0 , ν ) − λ (0 , γ )) . The numbers α γ,ν were first investigated by Loeser in [6] in the case of anembedded resolution of a curve. Van Proeyen and Veys generalized hisresults to the ideal case in [8]. The following Lemma 3.2 and Lemma 3.3are due to them ([8, Proposition 3.1 and Corollary 3.2]). Since we haveslightly modified the statements, we include the proofs here for the con-venience of the reader. Moreover, working in a more algebraic context,we also prefer to prove these results directly without utilizing the resultsof Loeser. Lemma 3.2. If γ ∈ | Γ | , then X ν ∼ γ α γ,ν = v Γ ( γ ) − b d γ k γ + 1 d γ . Proof.
By the relation (9) X ν ∼ γ k ν = w Γ ( γ ) k γ + w Γ ( γ ) − X ν ∼ γ d ν = w Γ ( γ ) d γ − b d γ . t follows that X ν ∼ γ α γ,ν = X ν ∼ γ k ν + v Γ ( γ ) − X ν ∼ γ d ν k γ + 1 d γ = w Γ ( γ ) ( k γ + 1) + v Γ ( γ ) − − (cid:16) w Γ ( γ ) d γ − b d γ (cid:17) k γ + 1 d γ = v Γ ( γ ) − b d γ k γ + 1 d γ . (cid:3) Lemma 3.3.
Let γ ∈ | Γ | . a) For every η ∼ γ , either | α γ,η | < , or b d γ = 0 and η is the only vertex adjacent to γ , in which case α γ,η = − . b) For all except at most one η ∼ γ , α γ,η ≥ . More precisely, if α γ,η ≤ for some η ∼ γ , then α γ,ν > for all η = ν ∼ γ , unless α γ,η = 0 , b d γ = 0 and there are only two vertices η ′ ∼ γ ∼ η , inwhich case also α γ,η ′ = 0 .Proof. a) Consider the sequence of point blowups(10) π : X = X N +1 π N −→ · · · π −→ X π −→ X = Spec R, where π µ : X µ +1 → X µ is the blowup of X µ at the closed point x µ ∈ X µ for every µ = 1 , . . . , N . We shall proceed by induction on N , the case N = 1 being trivial. Consider the exceptional divisor E N arising in thelast blowup. Depending on whether x N lies in an intersection of twoexceptional divisor or not, E N intersects one or two exceptional divisors.In other words, the vertex N is adjacent to one or two vertices.Suppose first that the vertex N is adjacent to only one vertex β . Weconsider the ideal a ′ := p b d N β Y ν = N p b d ν ν . It has the minimal principalization(11) X N π N − −−−→ · · · π −→ X π −→ X = Spec R The associated proximity matrix and its inverse are clearly restrictionsof those of (10). So k ′ ν = k ν for every ν = N . Because N is adjacentonly to β and E N = −
1, the proximity equation for the ideal p ν gives v N ( p ν ) = v β ( p ν ). By the reciprocity formula (7) we then obtain v ν ( p N ) = v N ( p ν ) = v β ( p ν ) = v ν ( p β ) . Hence d ′ ν = b d N v ν ( p β ) + X µ = N b d µ v ν ( p µ ) = b d N v ν ( p N ) + X µ = N b d µ v ν ( p µ ) = d ν or all ν = N . By the induction hypothesis, the claim therefore holds if γ, η = N . It remains to consider the numbers α β,N and α N,β .We first observe that N being proximate only to β , we have q N,ν = q β,ν for all ν = N by (1). By (8) we thus get k N = X ν q N,ν = X ν = N q β,ν + 1 = k β + 1 . Moreover, the base change formula (2) gives b d N = X ν d ν ( − E ν · E N ) = d N − d β . We then get α β,N = k β + 2 − k β + 1 d β ( d β + b d N ) = 1 − b d N k β + 1 d β = 1 − b d N k β + 1 d ′ β . Using (7), we have v β ( p β ) = X ν q β,ν ≥ X ν q β,ν = k β . Thus d ′ β = b d N v β ( p β ) + X ν = N b d ν v β ( p ν ) ≥ b d N k β + X ν = N b d ν v β ( p ν )so that 0 < b d N k β + 1 d ′ β ≤ k β + 1 k β + P ν = N ( b d ν / b d N ) v β ( p ν ) ≤ k β + 1 k β ≤ . Therefore − ≤ α β,N <
1. Moreover, by the above the equality α β,N = − k β = 1 and b d ν = 0 for all ν = N . This meansthat we must have β = 1 and a = p b d N N . In particular, N is the only vertexadjacent to β and b d β = 0. Thus a) holds for α β,N . In order to show thata) holds for α N,β , too, we note that α N,β = − α β,N d β d N = − α β,N d β d β + b d N . As necessarily b d N >
0, we see that | α N,β | < N is adjacent to twovertices, say β and β ′ . Now let a ′ := p b d N β p b d N β ′ Y ν = N p b d ν ν . The ideal a ′ has a similar minimal principalization as in the preceedingcase. Analogously one obtains d ′ ν = d ν , k ′ ν = k ν for ν = N , d N = d β + d β ′ + b d N and k N = k β + k β ′ + 1. By induction we only need to provethe claim for α N,β , α N,β ′ and α β,N , α β ′ ,N . We consider here only α N,β and α β,N , the proof for α N,β ′ and α β ′ ,N being similar. et m := v Γ ( β ). Note that β ′ ∼ Γ ′ β , where Γ ′ denotes the dual graphof the minimal principalization of a ′ . Therefore we have m = v Γ ′ ( β ), too.By Lemma 3.2 we get X ν ∼ Γ β α β,ν = m − b d β k β + 1 d β and X ν ∼ Γ ′ β α β,ν = m − b d β + b d N ) k β + 1 d β . Together the above equations imply that α β,N = α β,β ′ − b d N k β + 1 d β . By the induction hypothesis, α β,β ′ <
1. Thereby also α β,N <
1. We stillneed to show that α β,N ≥ −
1. As a β,ν < N = ν ∼ β , we seethat X N = ν ∼ Γ β α β,ν ≤ m − , where the equality holds if and only if m = 1. Hence α β,N = X ν ∼ Γ β α β,ν − X N = ν ∼ Γ β α β,ν ≥ − b d β k β + 1 d β ≥ − . We conclude that | α β,N | < m = 1 and b d β = 0, in which case α β,N = −
1. Finally, we have α N,β = − α β,N d β d N = − α β,N d β d β + d β ′ + b d N . Because | α β,N | ≤ b d N >
0, we obtain | α N,β | < γ has only one adjacent vertex.Thus we may assume that m := v Γ ( γ ) ≥
2. Suppose that there are twovertices adjacent to γ , say η and η ′ , with α γ,η , α γ,η ′ ≤
0. It then followsfrom Lemma 3.2 that X η,η ′ = ν ∼ γ α γ,ν ≥ m − . On the other hand, α γ,ν < ν ∼ γ by a). Subsequently, if m >
2, then m − > X η,η ′ = ν ∼ γ α γ,ν . Hence necessarily m = 2, and further, by Lemma 3.2 we also have α γ,η =0 = α γ,η ′ and b d γ = 0. (cid:3) otation 3.4. Let γ ∈ | Γ | and let a γ be a nonnegative integer. Supposethat { η | η ∼ γ } = { η , . . . , η m } . Set ∆ j := d η j λ ( a γ , γ ) − k η j − j = 1 , . . . , m ) so that λ (∆ j , η j ) = λ ( a γ , γ ) . Also write ∆ j = ⌊ ∆ j ⌋ + δ j where ≤ δ j < . Lemma 3.5.
With the preceeding notation, we have m X j =1 ∆ j + b d γ λ ( a γ , γ ) + m − w Γ ( γ ) a γ . In particular, m X j =1 ⌊ ∆ j ⌋ + m − w Γ ( γ ) a γ − ϕ, where ϕ := m X j =1 δ j + b d γ λ ( a γ , γ ) is a nonnegative integer.Proof. By the formulas (3) and (9) λ ( a γ , γ ) = w Γ ( γ )( a γ + k γ + 1) w Γ ( γ ) d γ = w Γ ( γ ) a γ + P mj =1 k η j + 2 P mj =1 d η j + b d γ . Therefore m X j =1 ∆ j + b d γ λ ( a γ , γ ) + m − w Γ ( γ ) a γ . The last statement is now obvious. (cid:3)
The following lemma, which can be considered as a generalization ofLemma 3.3, will play a crucial role in the sequel:
Lemma 3.6.
Given any vertex γ ∈ | Γ | and any nonnegative integer a γ ,we may choose for every vertex η ∼ γ a nonnegative integer a η so that w Γ ( γ ) a γ = X η ∼ γ a η and λ ( a η , η ) ≥ λ ( a γ , γ ) , where the latter inequality holds for each η ∼ γ except at most one. Moreprecisely, if { η | η ∼ γ } = { η , . . . , η m } , where m > , then the following is true: ) If it is possible to find a nonnegative integer a η with λ ( a η , η ) = λ ( a γ , γ ) , then one may choose the other integers a η j so that λ ( a η j , η j ) ≥ λ ( a γ , γ ) for all < j ≤ m . Moreover, this choice can be done in sucha way that the strict inequality holds for all except at most one < j ≤ m . In the case we already have a nonnegative integer a η with λ ( a η , η ) = λ ( a γ , γ ) , we can assume that the inequalityis strict for all < j ≤ m . If it is possible to find a nonnegative integer a η with λ ( a η , η ) <λ ( a γ , γ ) or, in the case b d γ > , λ ( a η , η ) = λ ( a γ , γ ) , then onecan choose the other integers a η in such a way that λ ( a η j , η j ) > λ ( a γ , γ ) holds for every < j ≤ m .Proof. Obviously we may assume that m > a γ = 0, we set a η = 0 for every η ∼ γ . Since λ ( a η , η ) − λ ( a γ , γ ) = α γ,η d η , the claim follows from Lemma 3.3.Suppose thus that a γ >
0. By Lemma 3.3 α η j ,γ ≥ −
1. Then α γ,η j = − d η j d γ α η j ,γ ≤ d η j d γ . As a γ >
0, this implies that∆ j = a γ d η j d γ − α γ,η j ≥ . By Lemma 3.5 ⌊ ∆ ⌋ + m X j =2 ( ⌊ ∆ j ⌋ + 1) = w Γ ( γ ) a γ + 1 − ϕ, where ϕ = m X j =1 δ j + b d γ λ ( a γ , γ )is a nonnegative integer.We can assume that for some j there exists a nonnegative integer a η j with λ ( a η j , η j ) ≤ λ ( a γ , γ ). If this is not the case, then choose anynonnegative integers a η j satisfying m X j =1 a η j = w Γ ( γ ) a γ . Then λ ( a η j , η j ) > λ ( a γ , γ ) for all j = 1 , . . . , m by the above assumption.But this means that we have proven the claim. uppose thus that, for example, λ ( a η , η ) ≤ λ ( a γ , γ ). We will firstconsider the case λ ( a η , η ) < λ ( a γ , γ ). Because λ ( a γ , γ ) = λ (∆ η , η ),this implies that a η < ∆ . Then either a η ≤ ⌊ ∆ ⌋ − a η = ⌊ ∆ ⌋ and δ >
0. In the first case we write ⌊ ∆ ⌋ − m X j =2 ( ⌊ ∆ j ⌋ + 1) = m X j =1 ⌊ ∆ j ⌋ + m − ≤ w Γ ( γ ) a γ , whereas in the latter case we have ϕ ≥ ⌊ ∆ ⌋ + m X j =2 ( ⌊ ∆ j ⌋ + 1) = m X j =1 ⌊ ∆ j ⌋ + m − w Γ ( γ ) a γ + 1 − ϕ ≤ w Γ ( γ ) a γ . It comes therefore out that it is possible to find numbers a η j ≥ ⌊ ∆ j ⌋ +1 > ∆ j for j = 2 , . . . , m such that m X j =1 a η j = w Γ ( γ ) a γ . Because λ ( a γ , γ ) = λ (∆ j , η j ), a η j > ∆ j implies λ ( a η j , η j ) > λ ( a γ , γ ).Consider then the case λ ( a η , η ) = λ ( a γ , γ ). Now a η = ∆ = ⌊ ∆ ⌋ .We immediately observe that the above argument works if b d >
0. Thisis the case also if some δ j >
0. We can therefore assume that δ = 0.Then ∆ = ⌊ ∆ ⌋ is an integer. Take a η = ∆ . We can now write ⌊ ∆ ⌋ + ⌊ ∆ ⌋ + m X j =3 ( ⌊ ∆ j ⌋ +1) = m X j =1 ⌊ ∆ j ⌋ + m − ≤ w Γ ( γ ) a γ − ϕ ≤ w Γ ( γ ) a γ . Finally note that λ ( a η , η ) = λ ( a γ , γ ) of course implies a η = ∆ . (cid:3) Main results
We first want to give a criterium for a positive rational number to bea jumping number. We begin with two lemmata. In the first one anantinef divisor is constructed for an ordered tree structure on the dualgraph:
Lemma 4.1.
Let S ⊂ | Γ | be a connected set of vertices. Suppose thatthere is a collection of nonnegative integers { a ν ∈ N | d ( ν, S ) ≤ } suchthat i) λ ( a ν , ν ) > ξ = λ ( a γ , γ ) for any γ ∈ S and ν ∼ S ; ii) w Γ ( γ ) a γ ≥ X ν ∼ γ a ν for every γ ∈ S .Then there exists an antinef divisor F ∈ Λ such that f ν = a ν for all ν ∈ | Γ | with d ( ν, S ) ≤ ; b f ν = 0 for ν S unless ν is an end; For any ν ∼ µ such that d ( ν, S ) > d ( µ, S ) we have λ ( f ν , ν ) > λ ( f µ , µ ) . roof. If N = 1, then there is nothing to prove. Suppose thus that N >
1. We will define nonnegative integers f ν inductively on d ( ν, S ).First set f ν = a ν for all ν ∈ | Γ | with d ( ν, S ) ≤
1. If S = | Γ | , we are done.Suppose then that f ν has been defined for some ν ∈ | Γ | with d ( ν, S ) > S is connected and Γ contains no loops, there is a unique µ ′ ∈ | Γ | such that µ ′ ∼ ν and d ( µ ′ , S ) = d ( ν, S ) −
1. By induction we know that λ ( f ν , ν ) > λ ( f µ ′ , µ ′ ). Therefore we can use Lemma 3.6 to find for each µ ′ = µ ∼ ν a nonnegative integer f µ such that λ ( f µ , µ ) > λ ( f ν , ν ). In thecase where ν is not an end we also get b f ν = w Γ ( ν ) f ν − X µ ∼ ν f µ = 0 . When all the numbers f ν have so been defined, we can set F = P ν f ν E ν . It remains to show that F is antinef. This is equivalent to b f ν ≥ ν . We have already seen that b f ν = 0 if ν S is not an end. Moreover, b f ν ≥ ν ∈ S . In order to complete the proof we need to use thefollowing Lemma 4.2. (cid:3) Lemma 4.2.
Assume that
N > . Let τ be an end and µ the vertexadjacent to it. If G ∈ Λ is such that λ ( g τ , τ ) > λ ( g µ , µ ) , then b g τ ≥ .Proof. Note first that b g τ = w Γ ( τ ) g τ − g µ by (3). The assumption nowsays that g µ + k µ + 1 d µ < g τ + k τ + 1 d τ . By the formulas (3) and (9) w Γ ( τ ) ( g τ + k τ + 1) w Γ ( τ ) d τ = w Γ ( τ ) g τ + k µ + 2 d µ + b d τ ≤ ( w Γ ( τ ) g τ + 1) + k µ + 1 d µ . Thereby g µ + k µ + 1 d µ < ( w Γ ( τ ) g τ + 1) + k µ + 1 d µ , so that g µ < w Γ ( τ ) g τ + 1, i.e., b g τ ≥ (cid:3) We are now able to prove our first main result:
Theorem 4.3.
Let a be an ideal in a two-dimensional regular local ring R . A positive rational number ξ is a jumping number of a if and onlyif there exists a connected set of vertices S ⊂ | Γ | and a collection ofnonnegative integers { a η ∈ N | d ( η, S ) ≤ } satisfying the followingconditions: i) λ ( a η , η ) > ξ = λ ( a γ , γ ) for every γ ∈ S and every η ∼ S ; ii) w Γ ( γ ) a γ = X ν ∼ γ a ν for every vertex γ ∈ S (when N > ).The condition ii) can be replaced by the condition i’) w Γ ( γ ) a γ ≥ X ν ∼ γ a ν for every vertex γ ∈ S .When these conditions hold, there exists an antinef divisor F with ξ = ξ F such that S = S F . Finally, consider the conditions S is a chain; if γ is not an end of S , then b d γ = 0 ; if γ is an end of S , then γ is a star or b d γ > .The set S can then be chosen in such a way that conditions 1) and 2)hold while condition 3) is true for any S .Proof. The case N = 1 being trivial, we may assume that N > S ⊂ | Γ | and a collectionof nonnegative integers { a ν ∈ N | d ( ν, S ) ≤ } satisfying conditions i) andii’). Let F ∈ Λ be an antinef divisor as in Lemma 4.1. For any chain γ = ν ∼ · · · ∼ ν r = ν going away from S , where γ ∈ S , we then have λ ( f ν , ν ) > · · · > λ ( f γ , γ ) = λ ( a ν , ν ) = ξ so that ξ = min { λ ( f ν , ν ) | ν ∈ | Γ |} . Then ξ = ξ F is a jumping numberby Proposition 2.1. Note that for an end vertex γ of S , we have byLemma 3.1 either b d γ > v Γ ( γ ) ≥
3, which shows that condition 3)automatically holds for S .Conversely, if ξ is a jumping number, then by Proposition 2.1 thereexists an antinef divisor F such that ξ = ξ F . We may now choose aconnected component S of S F and set a ν := f ν for every ν with d ( ν, S ) ≤
1. Clearly, the conditions i) and ii’) are satisfied.It remains to show that if there is a connected set S ′ ⊂ | Γ | with acollection { a ′ ν ∈ N | d ( ν, S ′ ) ≤ } that satisfies conditions i) and ii’), thenwe can find a new set S ⊂ S ′ together with a collection { a ν ∈ N | d ( ν, S ) ≤ } satisfying conditions i) and ii). Moreover, also conditions 1) and 2)should hold for the set S .If there is a vertex γ ∈ S ′ which has only one adjacent vertex, say η , and w Γ ( γ ) a ′ γ > a ′ η , then we choose S := { γ } and take a γ := a ′ γ , a η := w Γ ( γ ) a ′ γ .When this is not the case, we look for chains consisting of vertices γ ∈ S ′ such that b d γ = 0 if γ is not an end of the chain. We now take ourset S to be any maximal chain of this type. Obviously conditions 1) and2) then hold.We will next define the integers a ν for all ν ∈ | Γ | with d ( ν, S ) ≤
1. Tobegin with, set a ν := a ′ ν if ν ∈ S . Then take any γ ∈ S and look at thevertices η ∼ γ . Note that if there is only one vertex η adjacent to γ , thenwe now necessarily have w Γ ( γ ) a ′ γ = a ′ η . By the maximality of S , η ∈ S ′ implies η ∈ S so that a η := a ′ η will do. Thus we may assume v Γ ( γ ) ≥ uppose first that γ is an end of S such that λ ( a ′ η , η ) > ξ for all γ ∼ η / ∈ S . By assumption ii’) we have w Γ ( γ ) a γ ≥ X γ ∼ η ∈ S a η + X γ ∼ η / ∈ S a ′ η . If the equality holds, we take a η := a ′ η for every γ ∼ η / ∈ S . When this isnot the case, we may increase the a ′ η :s, η / ∈ S , to find numbers a η suchthat w Γ ( γ ) a γ = X η ∼ γ a η . For any η ∼ γ , η / ∈ S , we then have λ ( a η , η ) ≥ λ ( a ′ η , η ) > ξ .Otherwise, we can utilize Lemma 3.6 to define the numbers a η . For thiswe note that if γ now is an end of S , then necessarily, by the maximalityof S , we have b d γ > η ∈ S adjacent to γ .Furthermore, when γ is not an end of S , there are two vertices η , η ∈ S adjacent to γ . (cid:3) Remark 4.4. If ξ = ξ F where F is any antinef divisor, then it comesout from the proof of Theorem 4.3 that we can construct the set S in sucha way that S ⊂ S F and that a γ = f γ for all γ ∈ S . Recall that the first jumping number of a is the log canonical thresholdlct( a ) = min ν k ν + 1 d ν . Corollary 4.5.
Consider the set of those vertices γ ∈ | Γ | for which lct( a ) = k γ + 1 d γ . This set then satisfies the conditions 1) – 3) of Theorem 4.3.Proof.
Recall first that f γ > γ ∈ | Γ | if 0 = F ∈ Λ is an antinefdivisor. The zero divisor 0 is then the unique antinef divisor F ∈ Λ suchthat lct( a ) = ξ F . Therefore the claim follows from Theorem 4.3. (cid:3) Remark 4.6.
This was observed by Veys in [11, Theorem 3.3] in thecontext of an embedded resolution of a curve.
Example 4.7.
Conditions 1) and 2) of Theorem 4.3 do not hold for S F for an arbitrary antinef divisor F ∈ Λ . To see this, consider theconfiguration of exceptional divisors described by the dual graph ✒✑✓✏ ✒✑✓✏ ✒✑✓✏ E E E nd the proximity matrix P = − − . The dual graph is then associated to a minimal principalization of theideal a = p p . Now D = 2 E + 3 E + 3 E = b E + b E and K = E + 2 E + 2 E . Let us take F := E + E + E = b E . One now calculates that λ ( f ,
1) = 32 , λ ( f ,
2) = λ ( f ,
3) = 43 . Then ξ F = with S F = { , } . In particular, we see that S F is discon-nected. However, if F ′ := E + E + 2 E = b E , then also ξ F ′ = , but S F ′ = { } . We now observe that S F ′ satisfies theconditions 1) – 3) of Theorem 4.3. Theorem 4.3 implies that a support of a jumping number always con-tains star vertices or vertices corresponding to Rees valuations. In Corol-lary 4.10 we are now going to show the converse: this kind of verticesalways support some jumping number. First we need two lemmata:
Lemma 4.8.
Let γ ∈ | Γ | . If ν ≺ · · · ≺ ν r = η are points proximate to γ , then d η > rd γ .Proof. For every i = 1 , . . . , r , we have ν i ≻ γ and ν i ≻ ν i − , where ν = γ .Using the base change formula (2), we get d ν i = d ν i − + d γ + d ∗ ν i > d ν i − + d γ so that d η > d ν r − + d γ > d ν r − + 2 d γ > · · · > d ν + ( r − d γ > rd γ . (cid:3) Lemma 4.9.
Let γ ∈ | Γ | . a) If b d γ > , then d γ ≥ k γ b d γ . b) If v Γ ( γ ) + b d γ ≥ , then d γ − k γ ≥ .Proof. By (2) we have d ∗ ν = P µ b d µ q µ,ν ≥ b d γ q γ,ν for every ν ∈ | Γ | . Againby (2), and (8), we then obtain d γ − k γ b d γ = X ν ( d ∗ ν − b d γ ) q γ,ν ≥ b d γ X ν ( q γ,ν − q γ,ν ≥ n order to prove b), we first observe that d ∗ = d = v ( a ) ≥ v ( p γ ) = q γ, ≥ d ∗ ≥ d ∗ γ ≥ b d γ . If now q γ, ≥ b d γ ≥
3, we get d γ − k γ = X ν ( d ∗ ν − q γ,ν ≥ ( d ∗ − q γ, ≥ , as wanted.Consider then the case q γ, = 1 and b d γ ≤
2. The formula (1) impliesthat in this case γ is proximate to at most one adjacent vertex. Thenumber of adjacent vertices proximate to γ must then be v Γ ( γ ) − t , where t ∈ { , } . By using (3), we get d ∗ γ = b d γ + X ν ≻ γ d ∗ ν ≥ b d γ + v Γ ( γ ) − t ≥ − t. Assuming γ = 1, i.e., t = 1, we now obtain d γ − k γ ≥ ( d ∗ −
1) + ( d ∗ γ − ≥ . In the case γ = 1 we have t = 0, and thus d ∗ γ ≥
3. So d γ − k γ ≥ d ∗ γ − ≥ (cid:3) We are now ready to prove the promised result:
Corollary 4.10.
Let γ ∈ | Γ | . a) If b d γ is positive, then nd γ is a jumping number of a with a support { γ } for any integer n such that n b d γ > d γ . In particular, this implies that nd γ is a jumping number of a with a support { γ } for every positiveinteger n . b) If v Γ ( γ ) + b d γ ≥ , then − d γ is a jumping number of a with a support { γ } . This is the caseespecially when γ is a star.Proof. Suppose that { η | η ∼ γ } = { η , . . . , η m } . Let a γ be a nonnegativeinteger. By Lemma 3.5 we now have m X j =1 ( ⌊ ∆ j ⌋ + 1) = w Γ ( γ ) a γ + 2 − ϕ. where ϕ = m X j =1 δ j + b d γ λ ( a γ , γ ) s a nonnegative integer. It follows that if ϕ ≥
2, then we can findnonnegative integers a η j ≥ ⌊ ∆ j ⌋ + 1 for j = 1 , . . . , m such that m X j =1 a η j = w Γ ( γ ) a γ and that λ ( a η j , η j ) ≥ λ ( ⌊ ∆ j ⌋ + 1 , η j ) > λ (∆ j , η j ) = λ ( a γ , γ ) . This means that we can use Theorem 4.3 to conclude that ξ = λ ( a γ , γ )is a jumping number with a support { γ } . Let us now show that in bothcases a) and b) we can choose a γ in such a way that ϕ ≥ n such that n b d γ > d γ , set a γ = n − k γ − . By Lemma 4.9 a) a γ is positive. Moreover λ ( a γ , γ ) = nd γ . Then ϕ ≥ l b d γ λ ( a γ , γ ) m ≥ a γ = d γ − k γ −
2. Now Lemma 4.9 b)implies that a γ is nonnegative. Furthermore, λ ( a γ , γ ) = 1 − d γ . Then ∆ j = d η j λ ( a γ , γ ) − k η j − − d η j d γ + d η j − k η j − . for all j = 1 , . . . , m . Thus δ j = ∆ j − ⌊ ∆ j ⌋ = (cid:24) d η j d γ (cid:25) − d η j d γ . By the formula (3) w Γ ( γ ) d γ = m X j =1 d η j + b d γ . Hence m X j =1 δ j = m X j =1 (cid:24) d η j d γ (cid:25) − m X j =1 d η j d γ = m X j =1 (cid:24) d η j d γ (cid:25) − w Γ ( γ ) + b d γ d γ so that ϕ = m X j =1 δ j + b d γ λ ( a γ , γ ) = m X j =1 (cid:24) d η j d γ (cid:25) − w Γ ( γ ) + b d γ . Since γ ∼ η j , we have either γ ≺ η j or η j ≺ γ . In the first case wehave a maximal chain ν ≺ · · · ≺ ν r j = η j of points proximate to γ . ByLemma 4.8 this means that d η j ≥ r j d γ + 1. This certainly holds also hen η j ≺ γ if we set r j = 0 in this case. Therefore we have, for every j = 1 , . . . , m , (cid:24) d η j d γ (cid:25) ≥ r j and m X j =1 r j = { ν ∈ | Γ | | ν ≻ γ } = w Γ ( γ ) − . It follows that m X j =1 (cid:24) d η j d γ (cid:25) ≥ m − w Γ ( γ ) , and further, ϕ ≥ m − b d γ ≥ , which proves the claim. (cid:3) Remark 4.11.
In the case of a curve on a smooth surface, it was ob-served by Smith and Thompson in [9, Theorem 3.1] that b) of Corol-lary 4.10 holds for any star vertex.
We will next show how new jumping nunbers can be obtained from agiven one.
Proposition 4.12.
Suppose ξ is a jumping number of a with a support S ⊂ | Γ | . Write d = gcd { d η | d ( η, S ) ≤ } . Then, for any n ∈ N , ξ + nd is also a jumping number of a with a support S .Proof. Suppose first that S is connected. Let { a η | d ( η, S ) ≤ } be a col-lection of nonnegative integers satisfying the conditions of Theorem 4.3.For every vertex η ∈ | Γ | with d ( η, S ) ≤
1, write b η := d η d . Clearly, λ ( a η + nb η , η ) = λ ( a η , η ) + nd for any η ∈ | Γ | and n ∈ N . Hence λ ( a η + nb η , η ) > ξ + nd = λ ( a γ + nb γ , γ )for every γ ∈ S and η ∼ S . By the proximity inequality for a w γ b γ d = w γ d γ ≥ X ν ∼ γ d ν = d X ν ∼ γ b ν . We thus see that w γ ( a γ + nb γ ) ≥ X ν ∼ γ ( a ν + nb ν ) . for every γ ∈ S . The claim then results from Theorem 4.3. onsider then the general case. Let S , . . . , S r be the connected com-ponents of S . Write d i := gcd { d η | d ( η, S i ) ≤ } ( i = 1 , . . . , r ) . Then clearly d = gcd { d , . . . , d r } . By the preceeding case we now knowthat ξ + nd is a jumping number with a support S i for every i = 1 , . . . , r . Then theclaim follows from the following Lemma 4.13: (cid:3) Lemma 4.13. If S , S ⊂ | Γ | are supports of a jumping number ξ , thenso is S ∪ S .Proof. Let F , F ∈ Λ be antinef divisors such that S = S F and S = S F . Let us define a divisor F by setting f ν := min { f ,ν , f ,ν } . Since F and F are both antinef, we see that X η ∼ γ f η ≤ X η ∼ γ f i,η ≤ w Γ ( γ ) f i,γ for any γ ∈ | Γ | and i = 1 ,
2, and further, X η ∼ γ f η ≤ w Γ ( γ ) f γ . Hence F is antinef, too. Clearly, λ ( f ν , ν ) ≥ ξ , where the equality holdsif and only if ν ∈ S ∪ S . Therefore S ∪ S = S F , which proves theclaim. (cid:3) Tucker introduced in [10] the notions of contibution and critical con-tribution of a jumping number by a divisor. The contribution of a primedivisor had earlier been defined by Smith and Thompson in [9]. Considera reduced subdivisor G = E γ + . . . + E γ r ≤ D. One first says that a positive rational number ξ is a candidate jumpingnumber for G if ξd γ is an integer for all γ ∈ { γ , . . . , γ r } . The divisor G is then said to contribute the jumping number ξ if ξ is a candidatejumping number for G and J ( a ξ ) ( Γ( X, O X ( G + K − ⌊ ξD ⌋ )) . Finally, a contribution is said to be critical if, in addition, no propersubdivisor of G contributes ξ , i. e., J ( a ξ ) = Γ( X, O X ( G ′ + K − ⌊ ξD ⌋ ))for all divisors 0 ≤ G ′ < G . roposition 4.14. If a divisor G critically contributes ξ , then ξ = ξ F for an antinef divisor F with S F = | G | .Proof. Set F := ( ⌊ ξD ⌋ − K − G ⌋ ) ∼ . Because ξ is a candidate jumpingnumber for G , we have G + K − ⌊ ξD ⌋ ≤ K − ⌊ ( ξ − ǫ ) D ⌋ for small enough ǫ >
0. Let b = Γ( X, O ( − F )) be the complete idealassociated to F . By the above we get b ⊂ J ( a ξ − ǫ ). Clearly J ( a ξ ) ⊂ b ,but as G is contributing, we must have J ( a ξ ) = b . It then follows fromProposition 2.1 that ξ = ξ F . Let us now show that γ ∈ | G | is equivalentto f γ = ξd γ − k γ −
1. Since ξ = ξ F , we have f γ ≥ ξd γ − k γ − γ ∈ | Γ | . Now ⌊ ξD ⌋ − K − G ≤ F . For γ
6∈ | G | , this means that f γ ≥ ⌊ ξd γ ⌋ − k γ > ξd γ − k γ − . Suppose that we would have f γ ≥ ξd γ − k γ for some γ ∈ | G | . Write G ′ := G − E γ . Then G ′ is a proper subdivisor of G and ⌊ ξD ⌋ − K − G ≤ ⌊ ξD ⌋ − K − G ′ ≤ F = ( ⌊ ξD ⌋ − K − G ) ∼ implying that ( ⌊ ξD ⌋ − K − G ′ ) ∼ = F . But then J ( a ξ ) ( Γ( X, O ( K + G ′ − ⌊ ξD ⌋ ))which is impossible, because G critically contributes ξ . (cid:3) Lemma 4.15.
Suppose that F is an antinef divisor such that ξ = ξ F .Then G := X γ ∈ S F E γ is a contributing divisor for ξ .Proof. Because λ ( f γ , γ ) = ξ for all γ ∈ S F , ξ is a candidate jumpingnumber for G . Now f ν ≥ ξd ν − k ν − ν ∈ S F . In other words, f ν ≥ ⌊ ξd ν ⌋ − k ν if ν S F whereas f ν = ⌊ ξd ν ⌋ − k ν − ν ∈ S F . But this says that − F ≤ G + K − ⌊ ξD ⌋ implying that Γ( X, O ( − F )) ⊂ Γ( X, O X ( G + K − ⌊ ξD ⌋ )) . Because ξ = ξ F , it then follows from Proposition 2.1 that we cannot haveΓ( X, O X ( G + K − ⌊ ξD ⌋ )) = J ( a ξ ) . The following theorem now connects our approach to that of Smith,Thompson and Tucker:
Theorem 4.16.
Let a be an ideal in a two-dimensional regular local ring R . Let G be a divisor critically contributing a jumping number ξ of a .Then there exists an antinef divisor F such that b f γ = 0 unless γ is anend not in | G | , λ ( a γ , γ ) reaches its minimum ξ exactly when γ ∈ | G | andgrows strictly on every path away from | G | . Moreover, | G | is a chain; if γ ∈ | G | has more than one adjacent vertex in | G | then b d γ = 0 ; if γ is an end of | G | then γ is a star or b d γ > .Proof. According to Proposition 4.14 there is an antinef divisor F ′ suchthat λ ( f ′ γ , γ ) = ξ exactly when γ ∈ | G | . Let F be an antinef divisor asin Theorem 4.3 such that S F := { γ ∈ Γ | λ ( f γ , γ ) = ξ } ⊂ { γ ∈ Γ | λ ( f ′ γ , γ ) = ξ } = | G | (see Remark 4.4). By Lemma 4.15 the divisor P ν ∈ S F E ν is contributing.As G is critically contributing, this implies that S F = | G | , and we aredone. (cid:3) Remark 4.17.
It was discovered by Tucker in [10, Theorem 5.1] that if G is a divisor critically contributing a jumping number, then | G | satisfiesconditions 1) – 3). Unfortunately, the existence of an antinef divisor F as in Theorem 4.16does not guarantee that the contribution is critical. This will come outfrom the following example: Example 4.18.
Consider the configuration of exceptional divisors de-scribed by the dual graph ✒✑✓✏ ✒✑✓✏✒✑✓✏✒✑✓✏ ✒✑✓✏ E E E E E nd the proximity matrix P = − − − − . They are associated to the minimal principalization of the ideal a = p p p p . Now D = 20 E + 24 E + 24 E + 25 E + 27 E and K = E + 2 E + 2 E + 2 E + 2 E . Let us take F := 3 E + 3 E + 3 E + 4 E + 5 E = b E + 2 b E . One then calculates that λ ( f ,
1) = λ ( f ,
2) = λ ( f ,
3) = 14 , λ ( f ,
4) = 725 , λ ( f ,
5) = 827 . Thus ξ F = and S F = { , , } . Set G = E + E + E . We now observethat the divisor F satisfies the conditions of Theorem 4.16. However, G is not critically contributing. Indeed, set F ′ := 3 E + 4 E + 3 E + 4 E + 4 E = b E + b E + b E and F ′′ := 3 E + 3 E + 4 E + 4 E + 4 E = b E + b E + b E . Then λ ( f ′ ,
1) = 520 , λ ( f ′ ,
2) = 724 , λ ( f ′ ,
3) = 624 , λ ( f ′ ,
4) = 725 , λ ( f ′ ,
5) = 727 and λ ( f ′′ ,
1) = 520 , λ ( f ′′ ,
2) = 624 , λ ( f ′′ ,
3) = 724 , λ ( f ′′ ,
4) = 725 , λ ( f ′′ ,
5) = 727 . Therefore ξ F ′ = ξ = ξ F ′′ but S F ′ = { , } ( S F ) { , } = S F ′′ . By Lemma 4.15 both S F ′ and S F ′′ contain subsets corresponding to divi-sors contributing ξ , but then the divisor G cannot be critically contribut-ing. Moreover, it is easy to see that both E + E and E + E criticallycontribute ξ . Note that their intersection E does not contribute ξ . Ob-serve also that in this example the set S F includes a star vertex which isa non-end of S F . We next give a combinatorical criterium for the critical contribution: heorem 4.19. Let a be an ideal in a two-dimensional regular local ring R . Let G be a reduced subdivisor of D such that | G | contains at least twovertices. Then G critically contributes a jumping number ξ if and onlyif | G | is connected and there exists a collection of nonnegative integers { a ν ∈ N | d ( ν, | G | ) ≤ } such that i) λ ( a ν , ν ) > ξ = λ ( a γ , γ ) for any γ ∈ | G | and ν ∼ | G | ; ii) w Γ ( γ ) a γ = X ν ∼ γ a ν for every γ ∈ | G | ; iii) λ ( a ν − , ν ) ≤ ξ for any vertex ν ∼ | G | .Moreover, it is enough that iii) holds for vertices ν ∼ γ , where γ ∈ | G | is a star or b d γ > .Finally, if G critically contributes ξ and F is any antinef divisor suchthat ξ has support | G | with respect to F , then we may take a ν = f ν forall ν with d ( ν, | G | ) ≤ .Proof. Suppose first that | G | is connected and that there is a collectionof nonnegative integers { a ν ∈ N | d ( ν, | G | ) ≤ } satisfying conditions i) –iii). Let F ∈ Λ be an antinef divisor as in Lemma 4.1. This means that S F = { γ ∈ | Γ | | λ ( f γ , γ ) = ξ } = | G | . It then follows from Lemma 4.15 that the divisor G is contributing. Forsome subset S ⊂ | G | the divisor P ν ∈ S E ν is necessarily critically con-tributing. If S = | G | , then there must be a vertex η ∈ | G | r S adjacentto a vertex γ ∈ S . By Theorem 4.16 we can find an antinef divisor F ′ such that ξ = ξ F ′ and S = S F ′ . Then λ ( f ′ η , η ) > ξ = λ ( a η , η ) so that f ′ η > a η . Because b f ′ γ = 0, we have X ν ∼ γ f ′ ν = w Γ ( γ ) f ′ γ = w Γ ( γ ) a γ = X ν ∼ γ a ν . Thus there must be at least one vertex µ ∼ γ for which f ′ µ < a µ . So λ ( f ′ µ , µ ) < λ ( a µ , µ ). This implies µ
6∈ | G | as otherwise λ ( f ′ µ , µ ) < ξ whichis impossible. If γ is adjacent to some vertex in S , then γ must be star.If γ is not a star, then by Theorem 4.16 S = { γ } and b d γ >
0. Conditioniii) now implies that λ ( f ′ µ , µ ) ≤ λ ( a µ − , µ ) ≤ ξ. However, this is possible only if λ ( f ′ µ , µ ) = ξ . But this means that µ ∈ S ,which is a contradiction. Therefore we must have S = | G | as wanted.Suppose then that G critically contributes ξ . Let γ ∈ | G | . By Theo-rem 4.14 there then exists an antinef divisor F such that λ ( f ν , ν ) > ξ = λ ( f γ , γ ) for any ν / ∈ | G | . Set a ν = f ν for all ν ∈ | Γ | with d ( ν, | G | ) ≤ η ∼ γ such that λ ( a η − , η ) >λ ( a γ , γ ). We may choose a vertex γ ′ ∈ | G | such that γ ′ ∼ γ . Let S bea connected component of | G | \ { γ ′ } containing γ . Choose a collectionof nonnegative integers { a ′ ν ∈ N | d ( ν, S ) ≤ } so that a ′ η = a η − ′ γ ′ = a γ ′ + 1 while a ′ ν = a ν otherwise. Clearly λ ( a ′ ν , ν ) > ξ for any ν ∼ S .Also w Γ ( γ ) a ′ µ = X ν ∼ µ a ′ ν for every µ ∈ S . Therefore we can use Lemma 4.1 to find an antinefdivisor F ′ such that S = S F ′ . By Lemma 4.15 the divisor P ν ∈ S E ν isthen contributing. But this is a contradiction, because G is criticallycontributing. (cid:3) Example 4.20.
Consider the situation of Example 4.18. We now have λ ( f − ,
5) = 727 > . It therefore follows from Theorem 4.19 that E + E + E does not criticallycontribute ξ . References [1] A. Campillo, G. Gonzales-Sprinberg and M. Lejeune-Jalabert,
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