A bound of the number of weighted blow-ups to compute the minimal log discrepancy for smooth 3-folds

We study a pair consisting of a smooth 3-fold defined over an algebraically closed field and a general real ideal. We show that the minimal log discrepancy of every such a pair is computed by a prime divisor obtained by at most two weighted blow-ups. This bound is regarded as a weighted blow-up version of Mustata-Nakamura Conjecture. We also show that if the mld of such a pair is not less than 1, then it is computed by at most one weighted blow-up. As a consequence, ACC of mld holds for such pairs.


Introduction
Throughout this paper, the base field k of varieties is an algebraically closed field of arbitrary characteristic.We study pairs (A, a) consisting of a smooth variety A of dimension N > 1 and a "R-ideal" a which means a = a e 1 1 • • • a er r , where a i 's are non-zero coherent ideal sheaves on A and e = (e 1 , . . ., e r ) ∈ R r >0 .We fix a closed point 0 ∈ A.
The minimal log discrepancy ("mld" for short) mld(0; A, a) is an important invariant to measure the singularity of the pair (A, a) at 0 and plays important roles in birational geometry.We consider every prime divisor over A with the center at 0 and construct a "good model" of the divisor to approximate the mld.The prototype is as follows: Theorem 1.1 ([9], [6]).Assume N = 2.For every prime divisor E over A with the center at 0, there exists a prime divisor F obtained by one weighted blow-up with the center at 0 satisfying a(E; A, a) ≥ a(F ; A, a), for every R-ideal a such that a(E; A, a) ≥ 0.
The inequality in the theorem implies that F is a better divisor to approximate the mld.Therefore the theorem states that every prime divisor over A with the center at 0 has a better divisor which is 1 The author is partially supported by JSPS 19K03428 obtained in a simple procedure.Here, we note that F is constructed from E and does not depend on the choice of an R-ideal a.
Actually, in the paper [9]and [6], the main theorem is not stated in this form, but its proof shows Theorem 1.1.The paper [9] is for char k = 0, and the paper [6] is for char k = p > 0 and the main statements of both papers are in the following form: Corollary 1.2 ([9], [6]).Assume N = 2.Then, for every pair (A, a), the minimal log discrepancy mld(0; A, a) is computed by a prime divisor obtained by one weighted blow-up.
The corollary follows from the theorem immediately.See, for example, the proof of Corollary 1.9 in the section 5.
When we consider the case N = 3, we can see that one weighted blow-up is not sufficient to obtain a prime divisor computing the mld (see Example 3.3).On the other hand, in the example we can also show that the mld is computed by a prime divisor obtained by two weighted blow-ups.So it is natural to expect the following conjecture: Conjecture 1.3.Assume N ≥ 3.For every prime divisor E over A with the center at 0, there exists a prime divisor F centered at 0 obtained by at most N − 1 weighted blow-ups satisfying a(E; A, a) ≥ a(F ; A, a), for every R-ideal a such that a(E; A, a) ≥ 0.
As an immediate consequence of the conjecture, we obtain the following: Conjecture 1.4 (Corollary of Conjecture 1.3).Assume N ≥ 3.Then, for every pair (A, a), the minimal log discrepancy mld(0; A, a) is computed by a prime divisor obtained by at most N − 1 weighted blow-ups.
One of the motivations of the conjectures is that it is considered as a "weighted blow-up version" of Mustaţǎ-Nakamura Conjecture (MN-Conjecture for short): Conjecture 1.5 (MN-Conjecture [13]).Fix N and the exponent e of R-ideals.Then, there exists a number ℓ N,e ∈ N depending only on N and e such that for any R-ideal a with the exponent e the minimal log discrepancy mld(0; A, a) is computed by a prime divisor obtained by at most ℓ N,e times blow-ups.Here, the blow-up means the "usual blow-up", i.e., blow-up with the center at an irreducible reduced closed subset.
If this conjecture holds, then ACC Conjecture for these pairs holds ( [13]), so it seems to be a significant conjecture.On the other hand, MN-Conjecture is equivalent to a reasonable conjecture on arc spaces ( [5]), so it makes sense to study it.Note that MN-Conjecture requires to fix an exponent e, while the weighted blow-up versions (Conjecure 1.3, 1.4) do not require it.Assume Conjecture 1.3 holds, it is also an interesting question whether the weights of the blow-ups can be bound uniformly in terms of exponents.This will strengthen the MN-Conjecture.
Another motivation of Conjecture 1.3 is for the project to bridge between positive characteristic and characteristic 0 ( [5]).In [5], we have: Lemma 1.6.Let a be an R-ideal on a smooth variety A k over k (char k = p > 0) and E a prime divisor over (A k , 0 k ) computing mld(0 k ; A k , a).
If there exists an R-ideal a on a smooth variety A C over C and a prime divisor E over (A C , 0 C ), where 0 C ∈ A C such that 1. a(mod p) = a (see [5] for the definition of (mod p)) Remark 1.7.In particular, if such a and E exist for every a and E and assume that mld(0 k ; A k , a) is computed by a divisor, then the set of mld(0 Therefore, if we fix the exponent e and the dimension N of A k , then the number of the values Λ e := {mld(0 k , A k , a) | a is a R-ideal with the exponent e} is finite for char k > 0, because it is proved to be finite in characteristic 0 by [8].Similarly, if ACC holds in characteristic 0, then it also holds in positive characteristic.
Now, the problem is to construct appropriate E and a for given E and a.If Conjecture 1.3 holds, we can reduce this problem to a divisor F of special type (i.e., obtained by at most N − 1 weighted blow-ups), which seems easier to handle.
The main results of this paper are the following: Theorem 1.8.Assume N = 3.For every prime divisor E over A with the center at 0, there exists a prime divisor F centered at 0 obtained by at most two weighted blow-ups satisfying a(E; A, a) ≥ a(F ; A, a), for every "general" R-ideal a for E such that a(E; A, a) ≥ 0.
The terminology "general" will be defined in Definition 4.9.The weighted blow-ups will be constructed by "squeezed" blow-ups (see, Definition 4.4) depending only on E and it works for every general ideal.Here, "general" is necessary, because there exists an example of non-general ideal such that two squeezed blow-ups do not give the required divisor in the theorem (cf.Example 5.5).But it does not give a counter example for Conjecture 1.3, indeed for the example there exists another sequence of weighted blow-ups to obtain the required divisor (see, also Example5.5).
As a corollary we obtain: Corollary 1.9.Assume N = 3.Then, for every pair (A, a) with a "general" R-ideal a, the minimal log discrepancy mld(0; A, a) is computed by a prime divisor obtained by at most two weighted blowups.
It is known as the Zariski's sequence that every prime divisor E over A with the center at 0 is obtained by successive usual blow-ups from A, such that the centers of blow-ups are the center of E on each step ([11,VI,1.3]).The following corollary shows that in some cases, we obtain the two weighted blow-ups to compute the mld by just looking at the center of the second blow-up in the Zariski's sequence.
Corollary 1.10 (Corollary 5.9).Assume N = 3.Let E be a prime divisor over A computing mld(0; A, a) for a pair (A, a).Let A 1 −→ A be the first usual blow-up with the center at 0 in the Zariski's sequence.Assume that the center C ⊂ A 1 of E is a curve of degree ≥ 2 in the exceptional divisor E 1 ≃ P2 .Then a weighted blow-up which is called " squeezed blow-up" at C gives a divisor computing mld(0; A, a).
Note that in this case the first blow-up is also a squeezed blow-up.Example 3.3 is just in this case.In Section 5, we show a more general corollary.On the other hand, if we restrict to the case mld ≥ 1, then we have the following: Theorem 1.11.Assume N = 3.Then, for every general pair (A, a) with mld(0; A, a) ≥ 1, the minimal log discrepancy is computed by a prime divisor obtained by one weighted blow-up.This paper is organized as follows: in Section 2 we prepare basic terminologies which will be used in this paper.In Section 3 we discuss about weighted blow-up at a (not necessarily closed) smooth point and basic formula on weighted projective space, that is the exceptional divisor appearing in a weighted blow-up.In Section 4 we construct an appropriate regular system of parameter (RSP for short) with the weight, in order to make a weighted blow-up.In Section 5 we give the proofs of the main results.Let a be an R-ideal on A as in the beginning of the first section and e i 's are the exponents.The log discrepancy of the pair (A, a) at E is defined as and the minimal log discrepancy of the pair at a closed point 0 is defined as mld(0; A, a) := inf{a(E; A, a) | E prime divisor over A with the center at 0} It is known that for N ≥ 2, either mld(0; A, a) ≥ 0 or mld(0; A, a) = −∞ holds.For N = 1, we define mld(0; A, a) = −∞ if the lefthand side is negative, by abuse of notation, because it is convenient to describe the Inversion of Adjunction.
Definition 2.3.We say that a prime divisor E over A with the center at 0 computes mld(0; A, a) if either a(E; A, a) = mld(0; A, a) (when the right hand side is ≥ 0) or a(E; A, a) < 0 (when the mld is −∞).
Remark 2.4.Assume there exists a log resolution of the pair (A, am 0 ), where m 0 is the maximal ideal defining 0 ∈ A. If mld(0; A, a) ≥ 0, then, on every such resolution there is a prime divisor computing mld(0; A, a).If mld(0; A, a) = −∞ and Z(a) ⊂ A contains an irreducible component of codimension one, there may not exist a prime divisor computing the mld among the exceptional divisors appearing in a given log resolution (cf.[3,Proposition 7.2]).But in this case, if we construct an appropriate log resolution of (A, am 0 ) by taking more blowing-ups from the given one, a prime divisor computing mld(0; A, a) appears on that.Therefore, for char k = 0 or N ≤ 3, every pair (A, a) has a prime divisor computing mld(0; A, a), since there is a log resolution for every pair.

Weighted blow-ups and weighted projective spaces
In this section A is always a smooth variety of dimension N ≥ 2 defined over an algebraically closed field k and P ∈ A is a (not necessarily closed) point.
Definition 3.1.Let x 1 , . . ., x c be an RSP of a regular local ring R with the algebraically closed residue field and w 1 , . . ., w c be positive integers with gcd(w 1 , . . ., w c ) = 1.For n ∈ N, denote by The weighted blow-up of Spec R with wt w (x 1 , . . ., x c ) = (w 1 , . . ., w c ) is the canonical projection: The exceptional divisor E for the weighted blow-up is called a prime divisor obtained by a weighted blow-up of A at P .More generally, let P ∈ A be a smooth point with the not-necessarily-algebraically closed residue field K. Let K be the algebraic closure of the residue field of O A,P .A weighted blow-up of A at the point P is the canonical morphism induced from a weighted blow-up A −→ Spec K O A,P for some RSP x 1 , . . ., x c of K O A,P with wt w (x 1 , . . ., x c ) = (w 1 , . . ., w c ) for some (w 1 , . . ., w c ) ∈ Z c >0 , where K O A,P is the extension of the formal power series ring O A,P over K to the one over K. Let E be the prime divisor obtained by the weighted blow-up A −→ Spec K O A,P .The prime divisor E over A with the center at P corresponding to E is called a prime divisor obtained by a weighted blow-up of A at P .Note that if E gives a valuation v and the valuation ring O v , the prime divisor E corresponds to the valuation v whose valuation ring is Note that weighted blow-ups are only defined at smooth points.
Here, we show a 3-dimensional example that the minimal log discrepancy is not computed by a divisor obtained by only one weighted blow-up, but computed by a divisor obtained by two weighted blow-ups .
The following are well known, for example see [10, Remark 2.6, Lemma 2.7].
Remark 3.2.Let P ∈ A be a point of a smooth variety with the residue field K.
1.The set of prime divisors over A with the center at P corresponds bijectively to the set of prime divisors over A := Spec O A,P with the center at the closed point.Moreover, if prime divisors E and E correspond under the above bijection, then for every R-ideal a on A we have v E (a) = v E (a) and also a(E; A, a) = a( E, A, aO A ).
2. Let K ′ ⊃ K be a field extension and A ′ := Spec K ′ O A,P .Then, there is a surjective map from the set of prime divisors over A ′ with the center at the closed point to the set of prime divisors over A with the center at P .If prime divisors E ′ and E correspond by the above surjective map, then it follows a(E , where Then, a divisor computing mld(0; A, a) = 0 is not obtained by one weighted blow-up ([12, Ex.

6.45]).
On the other hand, there is a sequence of weighted blow-ups where ϕ 1 is the usual blow-up at 0 and ϕ 2 is a weighted blow-up with weight (1, 2) at the generic point of the curve Here, E 1 is the exceptional divisor for ϕ 1 .The exceptional divisor E 2 for ϕ 2 computes mld(0; A, a) = 0 The following lemma for a weighted projective space with a special weight is used for our main results.The statement is easily generalized to higher dimensional case, but for simplicity of notation we state here only for 2-dimensional case.Lemma 3.4.Let r ≤ s be positive integers such that gcd(r, s) = 1.Let g ∈ k[x 1 , x 2 , x 3 ] be a weighted homogeneous polynomial with respect to the weight w = (w(x 1 ), w(x 2 ), w(x 3 )) = (r, r, s) and where L ⊂ P k (r, r, s) is the divisor defined by ℓ = 0 in P k (r, r, s).
, the first inequality is trivial.We will show the second inequality.Let G ⊂ P k (r, r, s) be the subscheme defined by g = 0 on P k (r, r, s).Let be the canonical covering.Then, as π * L and π * G has no common irreducible components, Bezout's theorem on P 2 implies In case char k = 0 or char k = p > 0 and p |r • s, the morphism π is étale around Q. Therefore, • s} whose analytic neighborhoods of π * G and π * L are isomorphic to those of G and L at Q, respectively.Then, by (1) we obtain which yields the required inequality.
In case p|r, denote r = p e • q (gcd(p, q) = 1).Then, the fiber π −1 (Q) consists of q 2 • s closed points, as a topological space.For a closed point where m Q and m Q i are the maximal ideals of Q ∈ P(r, r, s) and of Q i ∈ P 2 , respectively.Let C ⊂ P 2 be the subscheme with the reduced structure of π * L.Then, we have where m L,Q and m C,Q i are the maximal ideals of Q ∈ L and of Q i ∈ C, respectively.Therefore, for every i = 1, . . ., q 2 • s it follows Now, there are q • s points Q i lying on C.Then, by Bezout's theorem on P 2 for C and π * G, we obtain Here noting that q • s • p e = r • s, this is the required inequality.
In case p|s, the proof is similar.

Squeezed Systems and Squeezed Blow-Ups
Let A be a variety of dimension N ≥ 2 over an algebraically closed field k.
Definition 4.1.Let P ∈ A be a smooth point (not necessarily closed), K the residue field, and E a prime divisor over A with the center at P .Denote the algebraic closure of K by K.An RSP {x 1 , . . ., x c } of K O A,P at the closed point is called a squeezed system for E at P , if where K O A,P is the extension of the coefficient field K of the formal power series ring O A,P to K, and m ⊂ K O A,P is the maximal ideal.
In this case, ) be as above.In this case, we call E a prime divisor of squeezed type v ′ .
Note that the squeezed weight for E is determined by a prime divisor but squeezed system is not uniquely determined by the prime divisor E.
Remark 4.2.For every A, P and E as in Definition 4.1, there exists a squeezed system of K O A,P .Indeed, it is obvious that there is ) among the set is proved by Zariski's subspace theorem (cf. [1, (10.6)] ).Now, we extend Then, we obtain a squeezed system {x 1 , x 2 , . . ., x c }.
Actually in [9] and [6], the proofs of Theorem 1.1 show the following: Example 4.3 (Theorem 1.1).For every prime divisor E over a smooth surface A with the center at 0 such that a(E; A, a) ≥ 0 for an R-ideal a on A. Then, the exceptional divisor E 1 obtained by a squeezed blow-up for E satisfies a(E; A, a) ≥ a(E 1 ; A, a).
Definition 4.4.Let A, P and E as above and let {x 1 , . . ., x c } be a squeezed system for E and v ′ = (v ′ 1 , . . ., v ′ r ) be the squeezed weight.We call the weighted blow-up of weight v ′ with respect to the coordinate system {x 1 , . . ., x c } a squeezed blow-up for E.
Remark 4.5.As in the definitions, a squeezed system is a RSP in the local ring with extended coefficient field.A squeezed system is not in general a RSP of the original local ring O A,P .
where K is the algebraic closure of K. Take an element a ∈ K \ K and let φ ∈ K[T ] be the minimal polynomial of a.Let ϕ 1 : A 1 −→ A K be the usual blow-up at the closed point of A K .Then the exceptional divisor E 1 is the projective line P 1 K with the homogeneous coordinates {y, z}.Denote the homogenized polynomial of φ by Φ(y, z) := z deg φ φ(y/z).Take the blow-up ϕ 2 : A 2 −→ A 1 with the center at the closed subscheme C defined by the ideal (Φ(y, z)) on E 1 .As the proper transforms of any curves defined by linear forms ℓ = cy On the other hand, take the base change ψ : Then, the proper transform of the curve defined by z ′ = 0 contains the point (a : 1) ∈ P 1 K = E 1 where E 1 is the exceptional divisor of the blow-up at the closed point of A K .As (a : 1) ∈ E 1 satisfies Φ(y, z) = 0, the proper transform of z ′ = 0 intersects the center of the second blow-up induced from ϕ 2 .One can see that v E (z ′ ) > 1, and therefore a squeezed system cannot be taken from K [[y, z]].Now we are going to define "general" ideal.Definition 4.7.Let E be a prime divisor over A of squeezed type and let E 1 be the exceptional divisor obtained by the squeezed blow-up with respect to a squeezed system {x 1 , x 2 , x 3 }.
. (In the discussions on a weighted projective space, "degree" always means degree with respect to (ii) If a bad curve exists, then it is unique in E 1 .
Proof.It is clear that if E = E 1 , then the center of E on E 1 is the generic point, so there is no bad curve on E 1 .We exclude this trivial case in the following discussions.In case the squeezed blow-up is the usual blow-up, then the exceptional divisor does not have a bad curve.Because if B is a bad curve, it is defined by linear form ℓ = i a i X i = 0 with a 3 = 0, where {X 1 , X 2 , X 3 } is the projective coordinate system on E 1 = P 2 corresponding to the squeezed system {x 1 , x 2 , x 3 } on O A,0 .This is a contradiction to the fact that (1, 1, 1) is the squeezed system , as we obtain another Here, we give the proof of this inequality, as this kind of discussion is used frequently in this paper.
Let ϕ 1 : A 1 −→ A be the squeezed blow-up and ψ : A −→ A 1 a birational morphism on which E appears.Denote the composite ϕ 1 • ψ by ϕ.Let D be the proper transform of Z(ℓ(x i )) ⊂ A in A 1 , then D ∩ E 1 contains the center of E on A 1 by the assumption.Note that we can express Here, we remind us that v E (ℓ(x i )) is the coefficient of the divisor (ϕ * ℓ(x i )) = ψ * (rE 1 + D) at the component E. The center of E on A 1 is contained in D, therefore the contribution from This shows the inequality (2).
For the case where E 1 is an exceptional divisor of a squeezed blow-up with respect to , then there is no bad curve.Because, a curve of degree v ′ 1 cannot contain a curve of degree > v ′ 1 .This gives the proof of "if" part of (i).
Assume a bad curve exists on E 1 .When the center of E on E 1 is a curve, then it should coincide with the bad curve by the definition, therefore the center should be of degree v ′ 1 .When the center of E on E 1 is a closed point P , then a bad curve should contain P .Express the point P by the homogeneous coordinates (a, b, c) with a, b, c ∈ k.Then a curve of degree v ′ 1 containing P is defined by bX 1 − aX 2 = 0. Now we obtain the uniqueness of the bad curve on E 1 .This completes the proof of "only if" part of (i) and the proof of (ii).Definition 4.9.Let E be a prime divisor over a smooth variety A with the center at a closed point 0.An R-ideal a is called general for E if there exists a squeezed blow-up A 1 −→ A for E with the exceptional divisor E 1 satisfying the following: (1) ord B a A 1 O E 1 ≤ 1, where B is the bad curve on E 1 and a A 1 is the weak transform of a at A 1 .
If there is no bad curve on E 1 , then we account it as the inequality automatically holds.
(2) In addition, if a(E; A, a) < a(E 1 ; A, a) and the center P of E on A 1 is a smooth closed point, then there exists a squeezed blow-up A 2 −→ A 1 for E at P .Let E 2 be the exceptional divisor.Then, ord , where B ′ is the bad curve on E 2 , a A 2 is the weak transform of a at A 2 and I L is the defining ideal of the intersection Here, E ′ 1 is the proper transform of E 1 on A 2 .If there is no bad curve on E 2 , then we account it as the inequality automatically holds.
We say that a pair (A, a) is general if the R-ideal a is general for a prime divisor computing mld(0; A, a).Here, the weak transform a iA 2 of an ideal a i ⊂ O A on A 2 is defined as The weak transform a A 2 of an R-ideal a on A is defined as the canonical extension of the one for an ideal of O A (see, for example [9]).
Remark 4.10.In (2), we assume smoothness of the center P of E on A 1 .But it turns out that it always holds by Lemma 5.1.
Remark 4.11.The definition of generality of an R-ideal is rather complicated.However, one can see that under a fixed exponent, the inequalities of orders at specific curves of E 1 and E 2 are open conditions in the space of regular functions of A, which is the reason why we call the ideal a "general".The following gives a sufficient condition for generality of the ideal.
Under the same symbols as in Definition 4.9, the R-ideal a is general for E if one of the following hold: (1) There is no bad curve on E 1 or E 2 .
(2) Assume the bad curves B ⊂ E 1 and

Proofs of the main results
For the proofs of the main theorems we need the following lemma which guarantees that the second weighted blow-up is possible.Lemma 5.1.Let E be a prime divisor over a smooth N -fold A (N ≥ 2) with the center at the closed point 0. Let {x 1 , . . ., x N } be a RSP at 0.
Let ϕ 1 : A 1 −→ A be the weighted blow-up with respect to {x 1 , . . ., x N } with weight v ′ .Denote the exceptional divisor of ϕ 1 by E 1 .Assume E = E 1 and let C be the center of E on A 1 and P ∈ C the generic point of C.
where X i is a homogeneous coordinate function corresponding to x i .In particular, P is smooth on A 1 and also on E 1 .
Proof.Assume that the statement does not hold, then we may assume that P is in the hyperplane defined by X 1 = 0 in E 1 = P(v ′ ).There exists at least one homogeneous coordinate function X i such that P does not lay in the hyperplane defined by X i = 0. Then we obtain This is a contradiction to the fact that The following lemma is a basic idea appeared in [9].
Lemma 5.2.Let a be an R-ideal on A with a(E; A, a) ≥ 0. Let A ′ −→ A be a proper birational morphism with normal A ′ , and D an irreducible divisor on A ′ with the same center on A as that of Assume a(D; A, a) > a(E; A, a) and the generic point P of the center of E on A ′ is smooth and not contained in the other exceptional divisors for A ′ −→ A.
Then, we have mld(P ; D, a A ′ O D ) < 0, in particular where a A ′ is a weak transform of a on A ′ .
Proof.First we express the log discrepancy at E as follows: where k E/A ′ is the coefficient of the relative canonical divisor K A/A ′ at E and I D is the defining ideal of D in A ′ .Then, by the assumption, it follows a(E; A ′ , I D • a A ′ ) < 0 and therefore we obtain By Inversion of Adjunction ([3], [7]) we obtain mld(P ; D, a Setting for the proof of Theorem 1.8. Let E be a prime divisor over a smooth 3-fold A with the center at a closed point 0. Let a be a general R-ideal on A such that a(E; A, a) ≥ 0. Let be a squeezed blow-up for E satisfying the condition (1) in Definition 4.9.Let the squeezed system {x 1 , x 2 , x 3 } and the weight 2 ).Denote the exceptional divisor for ϕ by E 1 .If a(E 1 ; A, a) ≤ a(E; A, a), then E 1 is the required prime divisor F in the theorem.Therefore, from now on, we assume that the inequalities a(E 1 ; A, a) > a(E; A, a) ≥ 0 hold.Lemma 5.3.Let A, E and E 1 be as above.If a is general for E and the inequalities a(E 1 ; A, a) > a(E; A, a) ≥ 0 hold, then we obtain the following: , where in v ′ f is the initial part of f with respect to the weight v ′ .
We divide the proof into two cases according to the dimension of the center of E on A 1 .Let P ∈ A 1 be the generic point of the center of E on A 1 .
Here, note that α > 1 follows from Lemma 5.2.As a is general, C is not a bad curve, therefore its degree is greater than (Note that such a polynomial with smallest degree is in the form ax ) Then, we have: By the assumption a(E 1 ; A, a) > a(E; A, a) ≥ 0, it follows The possibilities of Then, in this case we have (i) and (b) of (ii).
for a nonzero homogeneous polynomial h of degree n.As E has the center at the curve ℓ = 0, in the same way as the proof of (2) we have and also 0 which is a contradiction to the maximality of v E (x 3 ).Therefore, in this case also we have a(E 1 ; A, a)) < 2 + n − (n + 1) = 1, which shows (i) and (a) of (ii).

First we will show that
and also bx which is a contradiction to the maximality of v E (x 3 ).
Now we may assume that v ′ 1 ≥ 2.Then, of course v ′ 1 < v ′ 3 and the curve B defined by aX 1 − X 2 = 0 contains P .Note that B is the bad curve.

Take a general element
By generality of a, we have α ≤ 1.By Lemma 5.2, we have mld(P ; By the description (5) of the divisor defined by a general element f e , we have Hence, it follows ord P ( i Applying Lemma 3.4 to the curve B of degree v ′ 1 , we obtain Here, for the third inequality, we use Then, the only possibility of v ′ satisfying these inequalities is (2, 2, 3) and we also have v E 1 (a) = v E 1 (f e ) > 2 • 3 which completes the proof of (i) and (ii) in case dim {P } = 0.
Corollary 5.4 (Theorem 1.11).Let A be a smooth variety of dimension 3 over an algebraically closed field k.For any general pair (A, a) with mld(0; A, a) ≥ 1 the minimal log discrepancy is computed by a prime divisor obtained by one weighted blow-up.
Proof.As a(E 1 ; A, a) ≥ mld(0; A, a) ≥ 1, the inequality a(E 1 ; A, a) > a(E; A, a) does not hold by (i) in Lemma 5.3.
Proof of Theorem 1.8.
Let A 1 , E 1 be as in the setting above.Assuming 0 ≤ a(E; A, a) < a(E 1 ; A, a), we will prove that a(E; A, a) ≥ a(E 2 ; A, a) for a divisor E 2 obtained by the second "blow-up" constructed below in Case 1 and Case 2.
Let P ∈ E 1 ⊂ A 1 be the center of E. First, for every prime divisor D over A 1 with the center at P and with the inequality a(D; A, a) > a(E; A, a) ≥ 0, we observe that Here, we note that {y 1 , y 2 } is not necessarily a squeezed system on A 1K for E as is shown in Example 4.6, but it does not matter.Because we are interested only in ideals which came from A 1 and in this case a squeezed system on A 1 for E works in the same way as in [9] and [6], which one can see below: Let Ã −→ A 1 be a log resolution of (A 1 , aO A 1 ) on which E appears.Then, the base change Ã −→ A 1K by A 1K −→ A 1 is also a log resolution of (A 1K , aO A 1K ) on which the prime divisor E corresponding to E appears.Let A 2 −→ A 1 be the squeezed blow-up with respect to the squeezed system {y 1 , y 2 } and E 2 the exceptional divisor.By definition, it means that A 2K −→ A 1K is squeezed weighted blow-up with respect to the squeezed system {y 1 , y 2 } and E 2 be the exceptional divisor corresponding to E 2 .
If E = E 2 , then we have E = E 2 and we are done.So, we may assume that the center of E on A 2K is a point.Then the center Q ∈ A 2K is not on the proper transform of E 1 on A 2K .This is proved as follows: Let w = (r, s) be the weight of the squeezed system {y 1 , y 2 } on A 1 .
First, we show that r = s does not happen.Assume r = s, i.e., w = (1, 1), then we can take an expression Q = (a, b) of Q ∈ E 2 = P 1 K by homogeneous coordinates with a, b = 0. Let z := by 1 − ay 2 ∈ O A 1K .As Q is the center of E on E 2 ⊂ A 2K and satisfying bY 1 − aY 2 = 0 (Y 1 , Y 2 are the homogeneous coordinates on E 2 = P 1 K corresponding to y 1 , y 2 .), it follows which is a contradiction to the fact that {y 1 , y 2 } is a squeezed system.Now, we may assume that r < s.Let h = 0 be the defining equation of E 1 in A 1 around P , then E 1 is also defined by h = 0 and it is smooth at the closed point 0 ∈ A 1K .Therefore, we have ord y 1 ,y 2 h = 1.Then the initial part of h with respect to w is one of the following: ) which is a contradiction to the assumption that {y 1 , y 2 } is a squeezed system.Now, in any case we obtain that On the other hand, a(E; A, a) has another expression as follows: It is sufficient to show that a(E; A, a) ≥ a(E 2 ; A, a).
Assume contrary, then where a A 2K is the weak transform of a A 1 O A 1K .For the calculation of ( 7), we used Then the inequality (7) shows that a( Then, by Inversion of Adjunction ([3], [7]), it follows Let (r, s) be the squeezed weight for E at the closed point 0 ∈ A 1K , then where the last inequality follows from (6).Now we reach the situation in Theorem 1.1 and apply the argument in ( [9]) for the surface pair (A 1K , a A 1K ), we obtain where note that a A 2K = (a A 1 O A 1K ) A 2K and the third inequality follows from by (6).The possible positive intergers {r, s} satisfying (8) with gcd(r, s) = 1 are only {1, s}.In this case let z , which is a contradiction to that {y 1 , y 2 } is a squeezed system for E. Hence we obtain a(E; A, a) ≥ a(E 2 ; A, a), which completes the proof of the theorem for Case 1.Now take a squeezed blow-up A 2 −→ A 1 of weight w = (w 1 , w 2 , w 3 ) at P and let E 2 be the exceptional divisor.We may assume that the condition (2) in Definition 4.9 holds.Let Q ∈ E 2 be the center of E on A 2 .
Let E ′ 1 be the proper transform of E 1 on A 2 .Denote the defining ideals of E ′ 1 and E 2 in A 2 by I E ′ 1 and I E 2 , respectively.
Then, we have the similar expansion of a(E; A, a) as in (3) as follows: where a A 2 is the weak transform of a on A 2 and is also the weak transform of a A 1 on A 2 .
Case 2.1.dim {Q} = 0: We will prove a(E 2 ; A, a) ≤ a(E; A, a).Assume on the contrary that a(E 2 ; A, a) > a(E; A, a).Then, by (9), we obtain Let B ′ be the bad curve on E 2 (note that a bad curve exists in our case by Lemma 4.8).Then, we obtain ord Indeed, when L = B ′ , then generality of a implies that ord B ′ a A 2 O E 2 = 0, as ord B ′ I L = 1.On the other hand, when L = B ′ , then Q ∈ L and therefore generality implies ord B ′ a A 2 O E 2 ≤ 1.Now, in the same way as Case 2 in the proof of Lemma 5.3, we obtain that the weight of the second squeezed blow-up is (2, 2, 3).
We will show a contradiction under this situation.In this case, we have by applying (i) of Lemma 5.3 for (A 1 , a A 1 ), E 2 with the weight w = (2, 2, 3) and also for (A, a), E 1 with the weight v ′ = (2, 2, 3).As the squeezed system {y 1 , y 2 , y 3 } at On the other hand, applying Lemma 3.4 to E 1 = P(2, 2, 3) and a general element of a 2•3 .Note that the first inequality follows from Lemma 5.2.
In the following, we will prove a(E 2 ; A, a) ≤ a(E; A, a).Assume contrary, a(E 2 ; A, a) > a(E; A, a).The curve {Q} is not a bad curve, because if it is, then while the generality of a implies the converse inequality We also have {Q} = L.This is proved as follows: Let h ′ ∈ O A 1 define E 1 around P .As P is smooth on E 1 and also on A 1 , we have ordh ′ = 1 with respect to RSP {y 1 , y 2 , y 3 } of O A 1 at P .Then, considering of the initial term of h ′ with respect to the weight w, we see that one of the following holds: (1) L is a coordinate axis of E 2 = P(w); (2) L is defined by Y 1 + aY 2 (a ∈ k) in E 2 ; (3) L is defined by Y 3 + f (Y 1 , Y 2 ) in E 2 , where f is a homogeneous polynomial of degree d.
In the third case, the weight w must be (1, 1, d).In this case, if {Q} = L, it follows y ′ 3 := y 3 + f (y 1 , y 2 ) ∈ m A 1 ,P \ m 2 A 1 ,P and v E (y ′ 3 ) > v E (y 3 ), which is a contradiction to the maximality of v E (y 3 ).In case (1), {Q} = L because Q is not contained in the coordinate axes (Lemma 5.1).In case (2), L becomes the bad curve, therefore {Q} = L, because {Q} is not the bad curve, as we saw above.
Now we obtain Q ∈ E ′ 1 ∩ E 2 .By using this, we have By Inversion of Adjunction, we have Then, we have First we show that the squeezed weight w = (r, r, s) for E at P ∈ A 1 is (1, 1, n) for n ∈ N. Let C := {Q} be defined by ℓ = 0 in E 2 = P(r, r, s).If w = (1, 1, n), then the other possible weight w is (2, 2, 3).In this case the smallest possible value for the degree of ℓ on P(2, 2, 3) with respect to w is 6.Therefore, by Now we obtain the inequality (12).The inequalities ( 13) and ( 14) also hold in the present case.Therefore, we induce a contradiction and w must be (1, 1, n).By Lemma 5.3, deg w ℓ = 1 + n.
Let {y 1 , y 2 , y 3 } be a squeezed system at P ∈ A 1 with the weight (1, 1, n).Let {Y 1 , Y 2 , Y 3 } be the homogeneous coordinates of E 2 = P(1, 1, n) corresponding to {y 1 , y 2 , y 3 }.As ℓ is irreducible of degree 1 + n with respect to the weight (1, 1, n), we can express For simplicity, assume a = a e 1 1 and take a general element f ∈ a 1 O A,0 ⊂ k[[x 1 , x 2 , x 3 ]], where {x 1 , x 2 , x 3 } is a squeezed system for E at 0 ∈ A of weight (2,2,3).Then the weak transform f A 1 of f on A 1 is written as where ℓ ′ is weighted homogeneous and g(y) is the term with the higher weight with respect to the weight w = (1, 1, n).
Here, we may assume that P = (1, 1, 1) ∈ E 1 = P(2, 2, 3), then we can take a RSP at P ∈ A 1 by making use of the squeezed system {x 1 , x 2 , x 3 } of squeezed weight (2, 2, 3) which gives the first weighted blow-up ϕ 1 : A 1 −→ A: , z 3 = x 3 , where x 3 defines E 1 in the neighborhood of P .Take the minimal m ∈ N such that

Corollary 1 . 12 .
Assume N = 3.In Λ = {(A, a) | mld(0; A, a) ≥ 1 with general a} Mustaţǎ-Nakamura Conjecture holds and also ACC Conjecture holds for char k ≥ 0. Here, ACC Conjecture means that the set of mld(0; A, a) for the pairs in the subset Λ J ⊂ Λ consisting of R-pairs with the exponents in J ⊂ R >0 satisfies Ascending Chain Condition.Here, J is a DCC set.The corollary follows from Theorem 1.11 in the same way as in the proof of [6, Corollary 1.6], since the mld is computed by one weighted blow-up.
and it is sometimes denoted by deg v ′ .)(2) B contains the center of E. Lemma 4.8.Under the setting of Definition 4.7, the following hold: (i) A bad curve does not always exist.More precisely a bad curve does not exist if and only if one of the following holds: (a) The squeezed weight is (1, 1, 1), or (b) The squeezed weight

1 Let {y 1 , y 2 }
Indeed, we have an expression of a(D; A, a) as follows:a(D; A, a) = a(D; A 1 , a A 1 ) + v D (E 1 )(a(E 1 ; A, a) − 1).As a(D; A, a) ≥ 0 and a(E 1 ; A, a) − 1 < 0 ( Lemma 5.3 ), we have a(D; A 1 , a A 1 ) ≥ 0.Case 1. dim {P } = be a squeezed system for E on A 1 at P and E 2 the prime divisor obtained by the squeezed blow-up of A 1 at P with respect to {y 1 , y 2 }.Let K := O A 1 ,P /m A 1 ,P and K the algebraic closure of K. Let A 1K := Spec O A,P , A 1K := Spec K O A,P = Spec K[[y 1 , y 2 ]].Denote the both closed points of A 1K and of A 1K by 0.
in the zero locus of the coordinate functions, where E ′ 1 is the proper transform of E 1 on A 2K .Therefore it does not contain the center Q of E by Lemma 5.1.In case (3), it follows w
Definition 2.1.We call E a prime divisor over A, if there is a proper birational morphism ϕ : A ′ −→ A from a normal variety A ′ on which E is an irreducible divisor.The generic point P ∈ A of the image ϕ(E) is called the center of E on A. In this case, we sometimes call E a prime divisor over (A, P ).For a prime divisor E over a non-singular variety A, let ϕ : A ′ −→ A be a proper birational morphism with normal A ′ such that E appears on A ′ .Let k E (or sometimes written as k E/A ) be the coefficient of the relative canonical divisor K A ′ /A at E and v E the valuation defined by the prime divisor E. Here, note that k E (k E/A ) does not depend on the choice of A ′ .