1 Introduction
We work over an algebraically closed field of arbitrary characteristic.
The minimal log discrepancy (MLD) introduced by Vyacheslav Shokurov is an important invariant in birational geometry. Shokurov conjectured that the set of MLDs should satisfy the ascending chain condition (ACC) [Reference Shokurov18, Problem 5], and proved that the conjecture on termination of flips in the minimal model program (MMP) in characteristic
$0$
follows from two conjectures on MLDs: the ACC conjecture for MLDs and the lower-semicontinuity conjecture for MLDs [Reference Shokurov21].
The ACC conjecture for MLDs in dimension 2 was proved by Alexeev [Reference Alexeev1] and Shokurov [Reference Shokurov19] independently. We refer readers to [Reference Chen and Han5, Lemma 4.5, Theorems B.1 and B.4] and [Reference Han and Luo9, Theorem 1.5, Appendix A] for detailed proofs following Alexeev’s and Shokurov’s arguments, respectively, see also [Reference Chen, Gongyo and Nakamura6, Theorem 1.5]. The ACC conjecture for MLDs is still widely open in dimension 3 in general. We refer readers to [Reference Han, Liu and Shokurov8] and references therein for a brief history and related progress.
In order to study this conjecture for the case when
$X\ni x$
is a fixed kawamata log terminal (klt) germ, Nakamura proposed conjecture 1.1. It is proved by Mustaţǎ-Nakamura [Reference Mustaţǎ and Nakamura17, Theorem 1.5] and Kawakita [Reference Kawakita13, Theorem 4.6] that Conjecture 1.1 is equivalent to the ACC conjecture for MLDs in this case.
Conjecture 1.1 ([Reference Mustaţǎ and Nakamura17, Conjecture 1.1])
Let
${\Gamma }\subseteq [0,1]$
be a finite set and
$X\ni x$
a klt germ. Then there exists an integer N depending only on
$X\ni x$
and
${\Gamma }$
satisfying the following.
Let
$(X\ni x,B)$
be an lc germ, such that
$B\in {\Gamma }$
. Then there exists a prime divisor E over
$X\ni x$
, such that
$a(E,X,B)=\mathrm {mld}(X\ni x,B)$
and
$a(E,X,0)\le N.$
When
$\dim X=2$
, Conjecture 1.1 is proved by Mustaţǎ-Nakamura [Reference Mustaţǎ and Nakamura17, Theorem 1.3] in characteristic zero. Alexeev proved it when
${\Gamma }$
satisfies the descending chain condition (DCC) [Reference Alexeev1, Lemma 3.7] as one of key steps in his proof of the ACC for MLDs for surfaces. See [Reference Chen and Han5, Theorem B.1] for a proof of Alexeev’s result in detail. Very recently, Ishii gave another proof of Conjecture 1.1 in dimension 2 [Reference Ishii11, Theorem 1.4] that works in any characteristic. Kawakita proved Conjecture 1.1 for the case when
$\dim X=3$
, X is smooth,
${\Gamma }\subseteq \mathbb {Q}$
and
$(X\ni x,B)$
is canonical [Reference Kawakita13, Theorem 1.3] in characteristic zero.
Naturally, one may ask whether Nakamura’s conjecture holds or not when
$X\ni x$
is not necessarily fixed and
${\Gamma }$
satisfies the DCC. If the answer is yes, not only the ACC conjecture for MLDs will hold for fixed germs
$X\ni x$
as we mentioned before, but also both the boundedness conjecture of MLDs and the ACC conjecture for MLDs for terminal threefolds
$(X\ni x,B)$
will be immediate corollaries. One of main goals of this paper is to give a positive answer to this question in dimension 2.
Theorem 1.2. Let
${\Gamma }\subseteq [0,1]$
be a set which satisfies the DCC. Then there exists an integer N depending only on
${\Gamma }$
satisfying the following.
Let
$(X\ni x, B)$
be an lc surface germ, such that
$B\in {\Gamma }$
. Then there exists a prime divisor E over
$X\ni x$
, such that
$a(E,X,B)=\mathrm {mld}(X\ni x,B)$
and
$a(E,X,0)\leq N$
.
Theorem 1.2 implies that for any surface germ
$(X\ni x,B)$
, such that
$B\in {\Gamma }$
, there exists a prime divisor E, which could compute
$\mathrm {mld}(X\ni x,B)$
and is ‘weakly bounded’, in the sense that
$a(E,X,0)$
is uniformly bounded from above, among all divisorial valuations.
A natural idea to prove Theorem 1.2 is to take the minimal resolution
$\widetilde {f}:\widetilde {X}\to X$
, and apply Conjecture 1.1 in dimension 2. However, the coefficients of
$B_{\widetilde {X}}$
may not belong to a DCC set anymore, where
$K_{\widetilde {X}}+B_{\widetilde {X}}:=\widetilde {f}^*(K_X+B)$
. In order to resolve this difficulty, we need to show Nakamura’s conjecture for the smooth surface germ, while the set of coefficients
${\Gamma }$
does not necessarily satisfy the DCC. It would be interesting to ask if similar approaches could be applied to solve some questions in birational geometry in high dimensions, that is, we solve these questions in a more general setting of coefficients on a terminalisation of X.
Theorem 1.3. Let
$\gamma \in (0,1]$
be a real number. Then
$N_0:=\lfloor 1+\frac {32}{\gamma ^2}+\frac {1}{\gamma }\rfloor $
satisfies the following.
Let
$(X\ni x,B:=\sum _i b_iB_i)$
be an lc surface germ, where
$X\ni x$
is smooth, and
$B_i$
are distinct prime divisors. Suppose that
$\{\sum _i n_ib_i-1> 0\mid n_i\in \mathbb {Z}_{\geq 0}\}\subseteq [\gamma ,+\infty )$
. Then there exists a prime divisor E over X, such that
$a(E,X,B)=\mathrm {mld}(X\ni x,B)$
, and
$a(E,X,0)\leq 2^{N_0}$
.
We remark that for any DCC set
${\Gamma }$
, there exists a positive real number
$\gamma $
, such that
$\{\sum _i n_ib_i-1> 0\mid n_i\in \mathbb {Z}_{\geq 0},b_i\in {\Gamma }\}\subseteq [\gamma ,+\infty )$
(see Lemma 2.2). However, the converse is not true. For example, the set
${\Gamma }:=[\frac {1}{2}+\frac {\gamma }{2},1]$
satisfies our assumption, but it is not a DCC set. It is also worthwhile to remark that all previous works we mentioned before did not give any effective bound even when
${\Gamma }$
is a finite set.
Theorem 1.3 indicates that there are some differences between Conjecture 1.1 and the ACC conjecture for MLDs, as the latter does not hold for such kind of sets.
When we strengthen the assumption ‘
$\{\sum _i n_ib_i-1> 0\mid n_i\in \mathbb {Z}_{\geq 0}\}\subseteq [\gamma ,+\infty )$
’ to ‘
$\{\sum _i n_ib_i-1\geq 0\mid n_i\in \mathbb {Z}_{\geq 0}\}\subseteq [\gamma ,+\infty )$
’, we may even give an explicit upper bound for
$a(E,X,0)$
for all prime divisors E over
$X\ni x$
, such that
$\mathrm {mld}(X\ni x,B)=a(E,X,B)$
. Theorem 1.4 is another main result in this paper.
Theorem 1.4. Let
$\gamma \in (0,1]$
be a real number. Then
$N_0:=\lfloor 1+\frac {32}{\gamma ^2}+\frac {1}{\gamma }\rfloor $
satisfies the following.
Let
$(X\ni x,B:=\sum _i b_iB_i)$
be an lc surface germ, such that
$X\ni x$
is smooth, where
$B_i$
are distinct prime divisors. Suppose that
$\{\sum _i n_ib_i-1\geq 0\mid n_i\in \mathbb {Z}_{\geq 0}\}\subseteq [\gamma ,+\infty )$
. Then:
-
(1)
$|S|\le N_0$
, where
$S:=\{E\mid E\text { is a prime divisor over } X\ni x, a(E,X,B)=\mathrm {mld}(X\ni x,B)\}$
and -
(2)
$a(E,X,0)\le 2^{N_0}$
for any
$E\in S$
.
Theorem 1.4 is a much deeper result than Theorem 1.3. To the best of the authors’ knowledge, such types of boundedness that result for
$a(E,X,0)$
about all prime divisors which compute MLDs were never formulated in any literature before.
We remark that
${\Gamma }:=[\frac {1}{2}+\frac {\gamma }{2},1)$
satisfies the assumption in Theorem 1.4, while
${\Gamma }:=[\frac {1}{2}+\frac {\gamma }{2},1]$
does not. It is clear that
$(\mathbf {A}^2,B_1+B_2)$
satisfies the assumptions in Theorem 1.3, but
$|S|=+\infty $
, where
$B_1$
and
$B_2$
are defined by
$x=0$
and
$y=0$
, respectively. For klt germs
$(X,B)$
, we always have
$|S|<+\infty $
. In this case, Example 3.3 shows that Theorem 1.3 does not hold for the set
${\Gamma }:=\{\frac {1}{2}\}\cup \{\frac {1}{2}+\frac {1}{k+1}\mid k\in \mathbb {Z}_{\ge 0}\}$
, and Example 5.8 shows Theorem 1.4 may fail when
$b_i=\frac {1}{2}$
for any i, that is why the equality for ‘
$\sum _i n_ib_i-1\geq 0$
’ is necessary. These examples also indicate that the assumptions in Theorems 1.3 and 1.4 might be optimal.
We also give an effective bound for Theorem 1.3 (respectively, Theorem 1.4) when
$X\ni x$
is a fixed lc surface germ (respectively,
$X\ni x$
is a fixed klt surface germ), see Theorem 5.4 (respectively, Theorem 5.5).
Postscript. Together with Jihao Liu, the authors proved the ACC for MLDs of terminal threefolds [Reference Han, Liu and Luo7, Theorem 1.1]. The proof is intertwined with Theorem 1.2 for terminal threefolds. Bingyi Chen informed the authors that he improved the upper bound
$2^{N_0}$
given in Theorem 1.3 when the characteristic of the algebraically field is zero (see [Reference Chen4, Theorem 1.3]).
2 Preliminaries
We adopt the standard notation and definitions in [Reference Kollár and Mori15], and will freely use them.
2.1 Arithmetic of sets
Definition 2.1 DCC and ACC sets
We say that
${\Gamma }\subseteq \mathbb {R}$
satisfies the DCC or
${\Gamma }$
is a DCC set if any decreasing sequence
$a_{1} \ge a_{2} \ge \cdots $
in
${\Gamma }$
stabilises. We say that
${\Gamma }$
satisfies the ACC or
${\Gamma }$
is an ACC set if any increasing sequence in
${\Gamma }$
stabilises.
Lemma 2.2. Let
${\Gamma }\subseteq [0,1]$
be a set which satisfies the DCC, and n a nonnegative integer. There exists a positive real number
$\gamma $
which only depends on n and
${\Gamma }$
, such that:
$$ \begin{align*}\left\{\sum_i n_ib_i-n>0\mid b_i\in {\Gamma}, n_i\in \mathbb{Z}_{\geq 0}\right\}\subseteq [\gamma,+\infty).\end{align*} $$
Definition 2.3. Let
$\epsilon \in \mathbb {R}, I\in \mathbb {R}\backslash \{0\}$
and
${\Gamma }\subseteq \mathbb {R}$
be a set of real numbers. We define
${\Gamma }_\epsilon :=\cup _{b\in {\Gamma }} [b-\epsilon , b]$
and
$\frac {1}{I}{\Gamma }:=\{\frac {b}{I}\mid b\in {\Gamma }\}$
.
Lemma 2.4. Let
${\Gamma }\subseteq [0,1]$
be a set which satisfies the DCC. Then there exist positive real numbers
$\epsilon ,\delta \leq 1$
, such that:
$$ \begin{align*}\left\{\sum_i n_ib_i'-1>0\mid b_i'\in {\Gamma}_{\epsilon}\cap[0,1], n_i\in \mathbb{Z}_{\geq 0}\right\}\subseteq [\delta,+\infty).\end{align*} $$
Proof. We may assume that
${\Gamma }\setminus \{0\}\neq \emptyset $
, otherwise we may take
$\epsilon =\delta =1$
.
Since
${\Gamma }$
satisfies the DCC, by Lemma 2.2, there exists a real number
$\gamma \in (0,1]$
, such that
${\Gamma }\setminus \{0\}\subseteq (\gamma ,1]$
, and
$\{\sum _i n_ib_i-1>0\mid n_i\in \mathbb {Z}_{\ge 0}, b_i\in {\Gamma }\}\subseteq [\gamma , +\infty )$
. It suffices to prove that there exist
$0<\epsilon ,\delta <\frac {\gamma }{2}$
, such that the set
$\{\sum _i n_ib_i'-1\in (0,1]\mid b_i'\in {\Gamma }_{\epsilon }\cap [0,1], n_i\in \mathbb {Z}_{\geq 0}\}$
is bounded from below by
$\delta $
, or equivalently:
$$ \begin{align*}\left\{\sum_{i} n_ib_i'-1>0\mid b_i'\in {\Gamma}_{\epsilon}\cap[0,1], n_i\in \mathbb{Z}_{\ge0},\sum_{i} n_i\le \frac{4}{\gamma}\right\}\subseteq [\delta,+\infty).\end{align*} $$
We claim that
$\epsilon =\frac {\gamma ^2}{8}, \delta =\frac {\gamma }{2}$
have the desired property. Let
$b_i'\in {\Gamma }_{\epsilon }\cap [0,1]$
and
$n_i\in \mathbb {Z}_{\ge 0}$
, such that
$\sum _i n_ib_i'-1>0$
and
$\sum _{i} n_i\le \frac {4}{\gamma }$
. We may find
$b_i\in {\Gamma }$
, such that
$0\le b_i-b_i'\leq \epsilon $
for any i. In particular,
$\sum _i n_ib_i-1>0$
. By the choice of
$\gamma $
,
$\sum _i n_ib_i-1\geq \gamma $
. Thus:
$$ \begin{align*}\sum_i n_ib_i'-1=\left(\sum_i n_ib_i-1\right)-\sum_i n_i(b_i-b_i') \geq \gamma-\frac{4}{\gamma}\epsilon=\frac{\gamma}{2}. \\[-42pt] \end{align*} $$
We will use the following lemma frequently without citing it in this paper.
Lemma 2.5. Let
${\Gamma }\subseteq [0,1]$
be a set, and
$\gamma \in (0,1]$
be a real number. If
$\{\sum _i n_ib_i-1>0\mid b_i\in {\Gamma }, n_i\in \mathbb {Z}_{\geq 0}\}\subseteq [\gamma , +\infty )$
, then
${\Gamma }\setminus \{0\}\subseteq [\gamma ,1]$
.
Proof. Otherwise, we may find
$b\in {\Gamma }$
, such that
$0<b<\gamma $
. Then
$0<(\lfloor \frac {1}{b}\rfloor +1)\cdot b-1\le b<\gamma $
, a contradiction.
2.2 Singularities of pairs
Definition 2.6. A pair
$(X,B)$
consists of a normal quasiprojective variety X and an
$\mathbb {R}$
-divisor
$B\ge 0$
, such that
$K_X+B$
is
$\mathbb {R}$
-Cartier. A germ
$(X\ni x, B:=\sum _i b_iB_i)$
consists of a pair
$(X,B)$
, and a closed point
$x\in X$
, such that
$b_i>0$
, and
$B_i$
are distinct prime divisors on X with
$x\in \cap _i \operatorname {Supp} B_i$
. We call it a surface germ if
$\dim X=2$
.
$(X\ni x,B)$
is called an lc (respectively, klt, canonical, purely log terminal (plt), terminal) surface germ if
$(X\ni x,B)$
is a surface germ, and
$(X,B)$
is lc (respectively, klt, canonical, plt, terminal) near x.
Let
$X\ni x$
be a germ. A birational morphism
$f: Y\to X\ni x$
is a birational morphism from Y to X, such that all the exceptional divisors on Y are centered at x.
Definition 2.7. Let X be a normal quasiprojective surface and
$x\in X$
a closed point. Then a birational morphism
$f: Y\to X$
(respectively,
$f: Y\to X\ni x$
) is called a minimal resolution of X (respectively,
$X\ni x$
) if Y is smooth (respectively, smooth over a neighborhood of
$x\in X$
) and there is no
$(-1)$
-curve on Y (respectively, over a neighborhood of
$x\in X$
).
Note that the existence of resolutions of singularities for surfaces (see [Reference Lipman16]) and the minimal model program for surfaces (see [Reference Tanaka22] and [Reference Tanaka24]) are all known in positive characteristic. In particular, for any surface X (respectively, surface germ
$X\ni x$
), we can construct a minimal resolution
$\tilde {f}: \widetilde {X}\to X$
(respectively,
$\widetilde {f}: \widetilde {X} \to X\ni x$
).
Definition 2.8. Let
$(X\ni x,B)$
be an lc germ. We say a prime divisor E is over
$X\ni x$
if E is over X and
$\operatorname {center}_X E=x$
.
The minimal log discrepancy of
$(X\ni x,B)$
is defined as:
$$ \begin{align*}\mathrm{mld}(X\ni x,B):=\min\{a(E,X,B)\mid E \text{ is a prime divisor over } X\ni x\}.\end{align*} $$
Let
$f:Y\to X$
be a projective birational morphism, we denote it by
$f: Y\to X\ni x$
if
$\operatorname {center}_{X}E=x$
for any f-exceptional divisor E.
Let
$\widetilde {f}: \widetilde {X} \to X\ni x$
be the minimal resolution of
$X\ni x$
, and we may write
$K_{\widetilde {X}}+B_{\widetilde {X}}+\sum _{i} (1-a_i)E_i= \widetilde {f}^{*}(K_{X}+B),$
where
$B_{\widetilde {X}}$
is the strict transform of B,
$E_i$
are
$\widetilde {f}$
-exceptional prime divisors and
$a_i:=a(E_i,X,B)$
for all i. The partial log discrepancy (PLD) of
$(X\ni x,B)$
,
$\operatorname {pld}(X\ni x,B)$
, is defined as follows.
$$ \begin{align*}\operatorname{pld}(X\ni x,B):=\left\{ \begin{array}{lcl} \min_{i}\{a_i\} &{\text{if }x\in X \text{ is a singular point,}}\\ +\infty &{\text{if } x\in X \text{ is a smooth point.}} \end{array} \right.\end{align*} $$
The following result is known as the ACC for PLDs (for surfaces).
Theorem 2.9 ([Reference Alexeev1, Theorem 3.2],[Reference Han and Luo9, Theorem 2.2])
Let
${\Gamma }\subseteq [0,1]$
be a set which satisfies the DCC. Then:
$$ \begin{align*}\mathrm{Pld}(2,{\Gamma}):=\{\operatorname{pld}(X\ni x,B)\mid (X\ni x,B) \text{ is lc}, \dim X=2, B\in {\Gamma}\},\end{align*} $$
satisfies the ACC.
2.3 Dual graphs
Definition 2.10 (c.f. [Reference Kollár and Mori15, Definition 4.6])
Let
$C=\cup _i C_i$
be a collection of proper curves on a smooth surface U. We define the dual graph
${\mathcal {DG}}$
of C as follows.
-
(1) The vertices of
${\mathcal {DG}}$
are the curves
$C_j$
. -
(2) Each vertex is labelled by the negative self intersection of the corresponding curve on U, we call it the weight of the vertex (curve).
-
(3) The vertices
$C_i,C_j$
are connected with
$C_i\cdot C_j$
edges.
Let
$f: Y\to X\ni x$
be a projective birational morphism with exceptional divisors
$\{E_i\}_{1\leq i\leq m}$
, such that Y is smooth. Then the dual graph
${\mathcal {DG}}$
of f is defined as the dual graph of
$E=\cup _{1\leq i\leq m} E_i$
. In particular,
${\mathcal {DG}}$
is a connected graph.
Definition 2.11. A cycle is a graph whose vertices and edges can be ordered
$v_1,\ldots ,v_m$
and
$e_1,\ldots ,e_m (m\ge 2)$
, such that
$e_i$
connects
$v_i$
and
$v_{i+1}$
for
$1\leq i\leq m$
, where
$v_{m+1}=v_1$
.
Let
${\mathcal {DG}}$
be a dual graph with vertices
$\{C_i\}_{1\leq i\leq m}$
. We call
${\mathcal {DG}}$
a tree if:
-
(1)
${\mathcal {DG}}$
does not contain a subgraph which is a cycle and -
(2)
$C_i\cdot C_j\leq 1$
for all
$1\leq i\neq j\leq m$
.
Moreover, if C is a vertex of
${\mathcal {DG}}$
that is adjacent to more than three vertices, then we call C a fork of
${\mathcal {DG}}$
. If
${\mathcal {DG}}$
contains no fork, then we call it a chain.
Lemma 2.12. Let
$X\ni x$
be a surface germ. Let
$Y,Y'$
be smooth surfaces, and let
$f: Y\to X\ni x$
and
$f':Y'\to X\ni x$
be two projective birational morphisms, such that
$f'$
factors through f. If the dual graph of f is a tree whose vertices are all smooth rational curves, then the dual graph of
$f'$
is a tree whose vertices are all smooth rational curves.
Proof. Let
$g: Y'\to Y$
be the projective birational morphism, such that
$f\circ g=f'$
. Since g is a composition of blow-ups at smooth closed points, by induction on the number of blow-ups, we may assume that g is a single blow-up of Y at a smooth closed point
$y\in Y$
.
Let
$E'$
be the g-exceptional divisor on
$Y'$
,
$\{E_i\}_{1\leq i\leq m}$
the set of distinct exceptional curves of f on Y and
$\{E_i'\}_{1\leq i\leq m}$
their strict transforms on
$Y'$
. By assumption,
$E_i'\cdot E_j'\leq g^{*}E_i\cdot E_j'=E_i\cdot E_j\leq 1$
for
$1\leq i\neq j\leq m$
. Since
$E_i$
is smooth,
$0=g^{*}E_i\cdot E'\ge (E_i'+E')\cdot E'$
. It follows that
$E'\cdot E_i'\leq 1$
for
$1\leq i\leq m$
.
If the dual graph of
$f'$
contains a cycle, then
$E'$
must be a vertex of this cycle. Let
$E',E_{i_1}',\ldots ,E_{i_k}'$
be the vertices of this cycle,
$1\leq k\leq m$
. Then the vertex-induced subgraph by
$E_{i_1},\ldots ,E_{i_k}$
of the dual graph of f is a cycle, a contradiction.
The following lemma maybe well known to experts (c.f. [Reference Chen and Han5, Lemma 4.2]). For the reader’s convenience, we include the proof here.
Lemma 2.13. Let
$\epsilon _0 \in (0,1]$
be a real number. Let
$(X\ni x, B)$
be an lc surface germ, Y a smooth surface and
$f:Y\to X\ni x$
a projective birational morphism with the dual graph
${\mathcal {DG}}$
. Let
$\{E_k\}_{1\leq k\leq m}$
be the set of vertices of
$\mathcal {DG}$
, and
$w_k:=-E_k\cdot E_k$
,
$a_k:=a(E_k,X,B)$
for each k. Suppose that
$a_k\leq 1$
for any
$1\leq k\leq m$
, then we have the following:
-
(1)
$w_k\leq \frac {2}{a_k}$
if
$a_k>0$
, and in particular,
$w_k\leq \frac {2}{\epsilon _0}$
for
$1\leq k\leq m$
if
$\mathrm {mld}(X\ni x, B)\geq \epsilon _0$
. -
(2) If
$w_k\geq 2$
for some k, then for any
$E_{k_1},E_{k_2}$
which are adjacent to
$E_k$
, we have
$2a_k\leq a_{k_1}+a_{k_2}$
. Moreover, if the equality holds, then
$f_*^{-1}B\cdot E_k=0$
, and either
$w_k=2$
or
$a_k=a_{k_1}=a_{k_2}=0$
. -
(3) If
$E_{k_0}$
is a fork, then for any
$E_{k_1}, E_{k_2}, E_{k_3}$
which are adjacent to
$E_{k_0}$
with
$w_{k_i}\geq 2$
for
$0\leq i\leq 2$
,
$a_{k_3}\geq a_{k_0}$
. Moreover, if the equality holds, then
$w_{k_i}=2$
and
$f_*^{-1}B\cdot E_{k_i}=0$
for
$0\leq i\leq 2$
. -
(4) Let
$E_{k_0},E_{k_1},E_{k_2}$
be three vertices, such that
$E_{k_1},E_{k_2}$
are adjacent to
$E_{k_0}$
. Assume that
$a_{k_1}\geq a_{k_2}$
,
$a_{k_1}\geq \epsilon _0$
, and
$w_{k_0}\geq 3$
, then
$a_{k_1}-a_{k_0}\geq \frac {\epsilon _0}{3}$
. -
(5) If
$E_{k_0}$
is a fork, and there exist three vertices
$E_{k_1}, E_{k_2}, E_{k_3}$
which are adjacent to
$E_{k_0}$
with
$w_{k_i}\geq 2$
for
$0\leq i\leq 3$
, then
$a(E,X,B)\geq a_{k_0}$
for any vertex E of
${\mathcal {DG}}$
. -
(6) Let
$\{E_{k_i}\}_{0\leq i\leq m'}$
be a set of distinct vertices, such that
$E_{k_i}$
is adjacent to
$E_{k_{i+1}}$
for
$0\leq i\leq m'-1$
, where
$m'\geq 2$
. If
$a_{k_0}=a_{k_{m'}}=\mathrm {mld}(X\ni x, B)>0$
and
$w_{k_i}\geq 2$
for
$1\leq i\leq m'-1$
, then
$a_{k_0}=a_{k_1}=\cdots =a_{k_{m'}}$
and
$w_{k_i}=2$
for
$1\leq i\leq m'-1$
.
Proof. For (1), we may write
$K_Y+f_*^{-1}B+\sum _{1\leq i\leq m}(1-a_i)E_i=f^*(K_X+B).$
For each
$1\leq k\leq m$
, we have
$0=(K_Y+f_*^{-1}B+\sum _{1\leq i\leq m}(1-a_i)E_i)\cdot E_k,$
or equivalently,
$$ \begin{align} a_kw_k= 2-2p_a(E_k)-\sum_{i\neq k}(1-a_i)E_i\cdot E_k-f_*^{-1}B\cdot E_k. \end{align} $$
So
$a_kw_k\le 2$
, and
$w_k\le \frac {2}{a_k}$
.
For (2), by (2.1),
$2a_k\leq a_kw_k\le a_{k_1}+a_{k_2}-f_*^{-1}B\cdot E_k\leq a_{k_1}+a_{k_2}.$
If
$2a_k=a_{k_1}+a_{k_2}$
, then
$f_*^{-1}B\cdot E_k=0$
, and either
$w_k=2$
or
$a_k=a_{k_1}=a_{k_2}=0$
.
For (3), let
$k=k_i$
in (2.1) for
$i=1,2$
,
$$ \begin{align*}a_{k_i}w_{k_i}\leq 1+a_{k_0}-\left(\sum_{j\neq k_0,k_i}(1-a_j)E_j\cdot E_{k_i}+f_*^{-1}B\cdot E_{k_i}\right)\leq 1+a_{k_0},\end{align*} $$
or
$a_{k_i}\le \frac {1+a_{k_0}}{w_{k_i}}$
. Thus, let
$k=k_0$
in (2.1), we have:
$$ \begin{align*} a_{k_3}\ge a_{k_0}w_{k_0}+1-a_{k_1}-a_{k_2}+f_*^{-1}B\cdot E_{k_0} &\ge a_{k_0}\left(w_{k_0}-\frac{1}{w_{k_1}}-\frac{1}{w_{k_2}}\right)+\left(1-\frac{1}{w_{k_1}}-\frac{1}{w_{k_2}}\right) \\ &\geq a_{k_0}. \end{align*} $$
If the equality holds, then
$w_{k_i}=2$
and
$f_*^{-1}B\cdot E_i=0$
for
$0\leq i\leq 2$
.
For (4), by (2.1), we have
$a_{k_0}w_{k_0}\leq a_{k_1}+a_{k_2}-d$
, where
$d:=f^{-1}_{*}B\cdot E_{k_0}+\sum _{j\neq k_1,k_2,{k_0}}(1-a_j)E_j\cdot E_{k_0}$
. Hence:
$$ \begin{align*} a_{k_1}-a_{k_0}\ge \frac{(w_{k_0}-1) a_{k_1}-a_{k_2}+d}{w_{k_0}}\geq \frac{(w_{k_0}-2)a_{k_1}}{w_{k_0}}\geq \frac{\epsilon_0}{3}. \end{align*} $$
For (5), we may assume that
$E\neq E_{k_0}$
. There exist
$m'+1$
distinct vertices
$\{F_i\}_{0\leq i\leq m'}$
of
${\mathcal {DG}}$
, such that:
-
•
$F_0=E_{k_0}$
,
$F_{m'}=E$
and -
•
$F_i$
is adjacent to
$F_{i+1}$
for
$0\leq i\leq m'-1$
.
Denote
$a_i':=a(F_i,X,B)$
for
$0\leq i\leq m'$
. By (3), we have
$a_1'\geq a_0'$
, and by (2),
$a_{i+1}'-a_{i}'\geq a_{i}'-a_{i-1}'$
for
$1\leq i\leq m'-1$
. Thus,
$a_m'-a_0'\geq 0$
.
For (6), by (2)
$a_{k_0}\leq a_{k_1}\leq \ldots \leq a_{k_{m'-1}}\leq a_{k_{m'}}$
. Thus,
$a_{k_0}=\ldots =a_{k_{m'}}$
. By (2) again,
$w_{k_i}=2$
for
$1\leq i\leq m'-1$
.
Lemma 2.14. Let
$(X\ni x, B)$
be an lc surface germ. Let Y be a smooth surface and
$f:Y\to X\ni x$
a birational morphism with the dual graph
${\mathcal {DG}}$
. If
${\mathcal {DG}}$
contains a
$(-1)$
-curve
$E_0$
, then:
-
(1)
$E_0$
can not be adjacent to two
$(-2)$
-curves in
${\mathcal {DG}}$
, -
(2) if either
$\mathrm {mld}(X\ni x, B)\neq \operatorname {pld}(X\ni x, B)$
or
$\mathrm {mld}(X\ni x, B)>0$
, then
$E_0$
is not a fork in
${\mathcal {DG}}$
and -
(3) if
$E,E_0,\ldots ,E_m$
are distinct vertices of
${\mathcal {DG}}$
, such that E is adjacent to
$E_0$
,
$E_i$
is adjacent to
$E_{i+1}$
for
$0\leq i\leq m-1$
, and
$-E_i\cdot E_i=2$
for
$1\leq i\leq m$
, then
$m+1<-E\cdot E=w$
.
Proof. For (1), if
$E_0$
is adjacent to two
$(-2)$
-curves
$E_{k_1}$
and
$E_{k_2}$
in
${\mathcal {DG}}$
, then we may contract
$E_0$
and get a smooth model
$f':Y'\to X\ni x$
over X, whose dual graph contains two adjacent
$(-1)$
-curves, this contradicts the negativity lemma.
By [Reference Kollár and Mori15, Theorem 4.7] and the assumptions in (2), the dual graph of the minimal resolution of
$X\ni x$
is a tree. If
$E_0$
is a fork, we may contract
$E_0$
and get a smooth model
$f': Y'\to X\ni x$
, whose dual graph contains a cycle, this contradicts Lemma 2.12.
For (3), we will construct a sequence of contractions of
$(-1)$
-curve
$X_0:=X\to X_1\to \ldots X_m\to X_{m+1}$
inductively. Let
$E_{X_k}$
be the strict transform of E on
$X_k$
, and
$w_{X_k}:=-E_{X_k}\cdot E_{X_k}$
. For simplicity, we will always denote the strict transform of
$E_k$
on
$X_j$
by
$E_k$
for all
$k,j$
. Let
$f_1: X_0\to X_1$
be the contraction of
$E_0$
on
$X_0$
, then
$w_{X_1}=w-1$
, and
$E_1\cdot E_1=-1$
on
$X_1$
. Let
$f_2: X_1\to X_2$
be the contraction of
$E_1$
on
$X_1$
, then
$w_{X_2}=w_{X_1}-1=w-2$
, and
$E_2\cdot E_2=-1$
on
$X_2$
. Repeating this procedure, we have
$f_k: X_{k-1}\to X_k$
the contraction of
$E_{k-1}$
, and
$w_{X_k}=w-k$
,
$E_k\cdot E_k=-1$
on
$X_k$
for
$1\leq k\leq m+1$
. By the negativity lemma,
$w_{X_{m+1}}=w-(m+1)>0$
, and we are done.
Lemma 2.15. Let
$\gamma \in (0,1]$
be a real number. Let
$(X\ni x,B:=\sum _i b_iB_i)$
be an lc surface germ, where
$B_i$
are distinct prime divisors. Let Y be a smooth surface and
$f: Y\to X\ni x$
a birational morphism with the dual graph
${\mathcal {DG}}$
. Let
$\{E_k\}_{0\leq k\leq m}$
be a vertex-induced subchain of
${\mathcal {DG}}$
, such that
$E_k$
is adjacent to
$E_{k+1}$
for
$0\leq k\leq m-1$
, and let
$w_k:=-E_k\cdot E_k$
,
$a_k:=a(E_k,X,B)$
for all k. Suppose that
$w_0=1$
,
$E_0$
is adjacent to only one vertex
$E_1$
of
${\mathcal {DG}}$
,
$a_k\leq 1$
, and
$w_k\geq 2$
for each
$k\geq 1$
, then:
-
(1) if
$\{\sum _i n_ib_i-1>0\mid n_i\in \mathbb {Z}_{\geq 0}\}\subseteq [\gamma ,+\infty )$
, and
$a_0<a_1$
, then
$m\leq \frac {1}{\gamma }$
, -
(2) if
$\sum _i n_ib_i-1\neq 0$
for all
$n_i\in \mathbb {Z}_{\geq 0}$
, and
$a_0\leq a_1$
, then
$a_0<a_1$
and -
(3) if
$\{\sum _i n_ib_i-1\geq 0\mid n_i\in \mathbb {Z}_{\geq 0}\}\subseteq [\gamma ,+\infty )$
, and
$a_0\leq a_1$
, then
$m\leq \frac {1}{\gamma }$
.
Proof. We may write
$K_Y+f_*^{-1}B+\sum _i (1-a_i)E_i=f^*(K_X+B)$
, then:
$$ \begin{align} -2+f_*^{-1}B\cdot E_0+w_0 a_0+\sum_{i\neq 0} (1-a_i)E_i\cdot E_0=0. \end{align} $$
Since
$E_0$
is adjacent to only one vertex
$E_1$
of
${\mathcal {DG}}$
, by (2.2), we have:
$$ \begin{align*}a_1-a_0=f_{*}^{-1}B\cdot E_0-1.\end{align*} $$
For (1), since
$1\ge a_1>a_0$
, it follows that
$f_*^{-1}B\cdot E_0-1\in \{\sum _i n_ib_i-1>0\mid n_i\in \mathbb {Z}_{\geq 0}\}\subseteq [\gamma ,+\infty )$
. Thus,
$a_1-a_0\geq \gamma $
. By Lemma 2.13(2), we have
$a_{i+1}-a_i\geq a_1-a_0\geq \gamma $
for any
$0\leq i\leq m-1$
, and
$1\geq a_m\ge \gamma m$
. So
$m\leq \frac {1}{\gamma }$
.
For (2), since
$f_*^{-1}B\cdot E_0-1=\sum _i n_ib_i-1\neq 0$
for some
$n_i\in \mathbb {Z}_{\geq 0}$
,
$a_0<a_1$
,
(3) follows immediately from (1) and (2).
2.4 Sequence of blow-ups
Definition 2.16. Let
$X\ni x$
be a smooth surface germ. We say
$X_n\to X_{n-1}\to \cdots \to X_1\to X_0:=X$
is a sequence of blow-ups with the data
$(f_i,F_i,x_i\in X_i)$
if:
-
•
$f_i: X_i\to X_{i-1}$
is the blow-up of
$X_{i-1}$
at a closed point
$x_{i-1}\in X_{i-1}$
, with the exceptional divisor
$F_i$
for any
$1\leq i\leq n$
, where
$x_0:=x$
and -
•
$x_i\in F_i$
for any
$1\leq i\leq n-1$
.
In particular,
$F_n$
is the only exceptional
$(-1)$
-curve over X.
For convenience, we will always denote the strict transform of
$F_i$
on
$X_j$
by
$F_i$
for any
$n\ge j\ge i$
.
The following lemma is well known. For a proof, see for example, [Reference Han and Luo9, Lemma 3.15].
Lemma 2.17. Let
$(X\ni x,B)$
be an lc surface germ, such that
$\mathrm {mld}(X\ni x, B)>1$
, then
$\mathrm {mld}(X\ni x, B)=2-\operatorname {mult}_x B$
, and there is exactly one prime divisor E over
$X\ni x$
, such that
$a(E,X,B)=\mathrm {mld}(X\ni x,B)$
.
Lemma 2.18 ([Reference Mustaţǎ and Nakamura17, Lemma 4.2])
Let
$X\ni x$
be a smooth surface germ, and
$X_{l_0}\to \cdots \to X_1\to X_0:=X$
a sequence of blow-ups with the data
$(f_i,E_i,x_i\in X_i)$
, then
$a(E_{l_0},X,0)\leq 2^{l_0}$
.
2.5 Extracting divisors computing MLDs
Lemma 2.19. Let
$(X\ni x,B)$
be an lc surface germ. Let
$h: W\to (X,B)$
be a log resolution, and
$S=\{E_j\}$
a finite set of valuations of h-exceptional prime divisors over
$X\ni x$
, such that
$a(E_j,X,B)\leq 1$
for all j. Then there exist a smooth surface Y and a projective birational morphism
$f: Y\to X\ni x$
with the following properties.
-
(1)
$K_Y+B_Y=f^*(K_X+B)$
for some
$\mathbb {R}$
-divisor
$B_Y\ge 0$
on Y, -
(2) each valuation in S corresponds to some f-exceptional divisor on Y and
-
(3) each f-exceptional
$(-1)$
-curve corresponds to some valuation in S.
Proof.
$$ \begin{align*}K_W+B_W=h^*(K_X+B)+F_W,\end{align*} $$
where
$B_{W}\ge 0$
and
$F_W\ge 0$
are
$\mathbb {R}$
-divisors with no common components. We construct a sequence of
$(K_{W}+B_{W})$
-MMP over X as follows. Each time we will contract a
$(-1)$
-curve whose support is contained in
$F_W$
. Suppose that
$K_{W}+B_{W}$
is not nef over X, then
$F_W\neq 0$
. By the negativity lemma, there exists a h-exceptional irreducible curve
$C\subseteq \operatorname {Supp} F_W$
, such that
$F_W\cdot C=(K_{W}+B_{W})\cdot C<0$
. Since
$B_{W}\cdot C\ge 0$
,
$K_{W}\cdot C<0$
. Thus, C is a h-exceptional
$(-1)$
-curve. We may contract C and get a smooth surface
$Y_0:=W\to Y_1$
over X. We may continue this process, and finally reach a smooth model
$Y_k$
on which
$K_{Y_k}+B_{Y_k}$
is nef over X, where
$B_{Y_k}$
is the strict transform of
$B_{W}$
on
$Y_k$
. By the negativity lemma,
$F_W$
is contracted in the MMP, thus
$K_{Y_k}+B_{Y_k}=h_k^{*}(K_X+B)$
, where
$h_k:Y_k\to X$
, since
$a(E_j,X,B)\le 1$
,
$E_j$
is not contracted in the MMP for any
$E_j\in S$
.
We now construct a sequence of smooth models over X,
$Y_k\to Y_{k+1}\to \cdots $
, by contracting a curve
$C'$
satisfying the following conditions in each step.
-
•
$C'$
is an exceptional
$(-1)$
-curve over X and -
•
$C'\notin S$
.
Since each time the Picard number of the variety will drop by one, after finitely many steps, we will reach a smooth model Y over X, such that
$f:Y\to X$
and
$(Y,B_Y)$
satisfy (1)–(3), where
$B_Y$
is the strict transform of
$B_{Y_k}$
on Y.
We will need Lemma 2.20 to prove our main results. It may be well known to experts. Lemma 2.20(1)–(4) could be proved by constructing a sequence of blow-ups (c.f. [Reference Chen and Han5, Lemma 4.3]). We give another proof here.
We remark that Lemma 2.20(5) will only be applied to prove Theorem 1.2.
Lemma 2.20. Let
$(X\ni x,B)$
be an lc surface germ, such that
$1\geq \mathrm {mld}(X\ni x,B)\neq \operatorname {pld}(X\ni x,B)$
. There exist a smooth surface Y and a projective birational morphism
$f: Y\to X$
with the dual graph
${\mathcal {DG}}$
, such that:
-
(1)
$K_Y+B_Y=f^*(K_X+B)$
for some
$\mathbb {R}$
-divisor
$B_Y\ge 0$
on Y, -
(2) there is only one f-exceptional divisor
$E_0$
, such that
$a(E_0,X,B)=\mathrm {mld}(X\ni x,B)$
, -
(3)
$E_0$
is the only
$(-1)$
-curve of
${\mathcal {DG}}$
and -
(4)
${\mathcal {DG}}$
is a chain.
Moreover, if
$X\ni x$
is not smooth, let
$\widetilde {f}: \widetilde {X}\to X\ni x$
be the minimal resolution of
$X\ni x$
, and let
$g: Y\to \widetilde {X}$
be the morphism, such that
$\widetilde {f}\circ g=f$
, then:
-
(5) there exist a
$\widetilde {f}$
-exceptional prime divisor
$\widetilde {E}$
on
$\widetilde {X}$
and a closed point
$\widetilde {x}\in \widetilde {E}$
, such that
$a(\widetilde {E},X,B)=\operatorname {pld}(X\ni x, B)$
, and
$\operatorname {center}_{\widetilde {X}} E=\widetilde {x}$
for all g-exceptional divisors E.
Proof. By Lemma 2.19, we can find a smooth surface
$Y_0$
and a birational morphism
$h:Y_0\to X\ni x$
, such that
$a(E_0',X,B)=\mathrm {mld}(X\ni x, B)$
for some h-exceptional divisor
$E_0'$
, and
$K_{Y_0}+B_{Y_0}=h^*(K_X+B)$
for some
$B_{Y_0}\ge 0$
on
$Y_0$
.
We now construct a sequence of smooth models over X,
$Y_0\to Y_{1}\to \cdots $
, by contracting a curve
$C'$
satisfying the following conditions in each step.
-
•
$C'$
is an exceptional
$(-1)$
-curve over X and -
• there exists
$C''\neq C'$
over X, such that
$a(C'',X,B)=\mathrm {mld}(X\ni x, B)$
.
Since each time the Picard number of the variety will drop by one, after finitely many steps, we will reach a smooth model Y over X, such that
$f:Y\to X$
and
$(Y,B_Y)$
satisfy (1), where
$B_Y$
is the strict transform of
$B_{Y_0}$
on Y. Since
$\mathrm {mld}(X\ni x,B)\neq \operatorname {pld}(X\ni x,B)$
, by the construction of Y, there exists a curve
$E_0$
on Y satisfying (2)–(3).
For (4), by [Reference Kollár and Mori15, Theorem 4.7], the dual graph of the minimal resolution
$\widetilde {f}:\widetilde {X}\to X\ni x$
is a tree whose vertices are smooth rational curves. Since Y is smooth, f factors through
$\widetilde {f}$
. By Lemma 2.12, the dual graph
${\mathcal {DG}}$
of f is a tree whose vertices are smooth rational curves. It suffices to show that there is no fork in
${\mathcal {DG}}$
. By Lemma 2.14(2),
$E_0$
is not a fork. Suppose that
${\mathcal {DG}}$
contains a fork
$E'\neq E_0$
, by (3) and (5) of Lemma 2.13, we have
$a(E',X,B)\leq a(E_0,X,B)$
, this contradicts (2). Thus,
${\mathcal {DG}}$
is a chain.
For (5), since there exists only one f-exceptional
$(-1)$
-curve, there is at most one closed point
$\widetilde {x}\in \widetilde {X}$
, such that
$\operatorname {center}_{\widetilde {X}} E=\widetilde {x}$
for all g-exceptional divisors E. Thus, the dual graph of g, which is denoted by
${\mathcal {DG}}'$
, is a vertex-induced connected subchain of
${\mathcal {DG}}$
by all g-exceptional divisors. Since
$\mathrm {mld}(X\ni x, B)\neq \operatorname {pld}(X\ni x, B)$
, we have
${\mathcal {DG}}'\subsetneq {\mathcal {DG}}$
.
We may index the vertices of
${\mathcal {DG}}$
as
$\{E_i\}_{-n_1\leq i\leq n_2}$
for
$n_1,n_2\in \mathbb {Z}_{\geq 0}$
, such that
$E_i$
is adjacent to
$E_{i+1}$
, and
$a_i:=a(E_i,X,B)$
for all possible i. We may assume that the set of vertices of
${\mathcal {DG}} '$
is
$\{E_j\}_{-n_1'\leq j\leq n_2'}$
, where
$0\leq n_1'\leq n_1$
and
$0\leq n_2'\leq n_2$
(see Figure 1). If
$n_1>n_1'$
, then by Lemma 2.13(2),
$a_{k}-a_{-n_1'-1}\geq \min \{0,a_{-1}-a_{0}\}\geq 0$
for all
$-n_1\leq k<-n_1' $
. If
$n_2>n_2'$
, then again by Lemma 2.13(2),
$a_{k'}-a_{n_2'+1}\geq \min \{0,a_{1}-a_{0}\}\geq 0$
for all
$n_2'< k'\leq n_2$
. Set
$a_{-n_1-1}=1, E_{-n_1-1}=E_{-n_1}$
if
$n_1=n_1'$
, and set
$a_{n_2+1}=1,E_{n_2+1}=E_{n_2}$
if
$n_2=n_2'$
. Then
$\min \{a_{n_2+1}, a_{-n_1-1}\}=\operatorname {pld}(X\ni x, B)$
, and
$\widetilde {x}= g(E_{-n_1'-1})\cap g(E_{n_2+1})\in \widetilde {E}$
, where
$a(\widetilde {E},X,B)=\operatorname {pld}(X\ni x, B)$
.
The dual graph of f.
The following lemma gives an upper bound for number of vertices of a certain kind of
${\mathcal {DG}}$
constructed in Lemma 2.20, with the additional assumption that
$\mathrm {mld}(X\ni x, B)$
is bounded from below by a positive real number.
Lemma 2.21. Let
$\epsilon _0\in (0,1]$
be a real number. Then
$N_0':=\lfloor \frac {8}{\epsilon _0}\rfloor $
satisfies the following properties.
Let
$(X\ni x,B:=\sum b_iB_i)$
be an lc surface germ, such that
$\mathrm {mld}(X\ni x,B)\geq \epsilon _0$
, where
$B_i$
are distinct prime divisors. Let Y be a smooth surface, and
$f: Y\to X\ni x$
a birational morphism with the dual graph
${\mathcal {DG}}$
, such that:
-
•
$K_Y+B_Y=f^*(K_X+B)$
for some
$\mathbb {R}$
-divisor
$B_Y\geq 0$
on Y, -
•
${\mathcal {DG}}$
is a chain with only one
$(-1)$
-curve
$E_0$
, -
•
$a(E_0,X,B)=\mathrm {mld}(X\ni x,B)$
and -
•
$E_0$
is adjacent to two vertices of
${\mathcal {DG}}$
.
Then the number of vertices of
${\mathcal {DG}}$
is bounded from above by
$N_0'$
.
Proof. Let
$\{E_i\}_{-n_1\leq i\leq n_2}$
be the vertices of
${\mathcal {DG}}$
, such that
$E_i$
is adjacent to
$E_{i+1}$
for
$-n_1\leq i\leq n_2-1$
, and
$w_i:=-(E_i\cdot E_i), a_i:=a(E_i,X,B)$
for all i. We may assume that
$E_0$
is adjacent to two vertices
$E_{-1}, E_1$
of
${\mathcal {DG}}$
.
By Lemma 2.14(1), we may assume that
$w_{-1}>2$
. By (2) and (4) of Lemma 2.13,
$a_{i-1}-a_i\geq \frac {\epsilon _0}{3}$
for any
$-n_1+1\leq i\leq -1$
, and
$a_{-1}\geq \frac {\epsilon _0}{3}$
. Since
$a_{-n_1}\leq 1$
,
$n_1\leq \frac {3}{\epsilon _0}$
. Similarly,
$n_2-n'\le \frac {3}{\epsilon _0}$
, where
$n'$
is the largest nonnegative integer, such that
$w_i=2$
for any
$1\le i\le n'$
. By Lemma 2.13(1),
$w_{-1}\leq \frac {2}{\epsilon _0}$
, and by Lemma 2.14(3),
$n'<\frac {2}{\epsilon _0}-1$
. Hence,
$n_1+n_2+1$
, the number of vertices of
${\mathcal {DG}}$
is bounded from above by
$\frac {8}{\epsilon _0}$
.
3 Proof of Theorem 1.3
Lemma 3.1 is crucial in the proof of Theorem 1.3. Before providing the proof, we introduce some notations first.
Notation
$(\star ).$
Let
$X\ni x$
be a smooth surface germ, and let
$g: X_n\to X_{n-1}\to \cdots \to X_1\to X_0:=X$
be a sequence of blow-ups with the data
$(f_i,F_i,x_i\in X_i)$
. Let
$\mathcal {DG}$
be the dual graph of g, and assume that
${\mathcal {DG}}$
is a chain.
Let
$n_3\geq 2$
be the largest integer, such that
$x_i\in F_{i}\setminus F_{i-1}$
for any
$1\leq i\leq n_3-1$
, where we set
$F_0:=\emptyset $
. Let
$\{E_j\}_{-n_1\le j\le n_2}$
be the vertices of
${\mathcal {DG}}$
, such that
$E_0:=F_n$
is the only g-exceptional
$(-1)$
-curve on
$X_n$
,
$E_{n_2}:=F_1$
, and
$E_i$
is adjacent to
$E_{i+1}$
for any
$-n_1\le i\le n_2-1$
(see Figure 2).
The dual graph of g.
We define
$n_i(g):=n_i$
for
$1\le i\le 3$
,
$n(g)=n$
,
$w_j(g):=-E_j\cdot E_j$
for all j and
$W_1(g):=\sum _{j<0}w_j(g)$
and
$W_2(g):=\sum _{j>0}w_j(g)$
.
Lemma 3.1. With Notation
$(\star )$
. Then:
$$ \begin{align} (W_1(g)-n_1(g))+n_3(g)-1=W_2(g)-n_2(g). \end{align} $$
In particular,
$n(g)=n_1(g)+n_2(g)+1\leq n_3(g)+\min \{W_1(g),W_2(g)\}$
.
Proof. For simplicity, let
$n:=n(g)$
,
$n_i:=n_i(g)$
for
$1\le i\le 3$
,
$w_j:=w_j(g)=-E_j\cdot E_j$
for all j and
$W_j:=W_j(g)$
for
$j=1,2$
.
We prove (3.1) by induction on the nonnegative integer
$n-n_3$
.
If
$n=n_3$
, then
$n_1=W_1=0$
,
$n_2=n_3-1$
and
$W_2=2n_3-2$
, thus, (3.1) holds (see Figure 3). If
$n=n_3+1$
, then
$x_{n_3}\in F_{n_3}\cap F_{n_3-1}$
. In this case,
$n_1=1$
,
$W_1=2$
,
$n_2=n_3-1$
and
$W_2=2n_3-1$
, thus, (3.1) holds (see Figure 3).
The dual graph for the case
$n=n_3$
and
$n=n_3+1$
.
In general, suppose (3.1) holds for any sequence of blow-ups g, as in Notation
$(\star )$
with positive integers
$n,n_3$
satisfying
$1\leq n-n_3\leq k$
. For the case when
$n-n_3=k+1$
, we may contract the
$(-1)$
-curve on
$X_n$
, and consider
$g':X_{n-1}\to \cdots \to X_0:= X$
, a subsequence of blow-ups of g with the data
$(f_i,F_i,x_i\in X_i)$
for
$0\leq i\leq n-1$
. Denote
$n_i':=n_i(g')$
for any
$1\le i\le 3$
, and
$W_j':=W_j(g')$
for any
$1\le j\le 2$
. By Lemma 2.14(1), either
$w_{-1}=2$
or
$w_1$
=2. In the former case,
$W_1'=W_1-2, W_2'=W_2-1, n_1'=n_1-1$
,
$n_2'=n_2$
and
$n_3'=n_3$
. In the latter case,
$W_1'=W_1-1, W_2'=W_2-2, n_1'=n_1,n_2'=n_2-1$
and
$n_3'=n_3$
. In both cases, by induction,
$$ \begin{align*}W_2'-n_2'-(W_1'-n_1')=(W_2-n_2)-(W_1-n_1)=n_3-1.\end{align*} $$
Hence, we finish the induction, and (3.1) is proved.
Since
$w_j\geq 2$
for
$j\neq 0$
, we have
$W_1=\sum _{-n_1\leq j\leq -1} w_j\geq 2n_1$
and
$W_2=\sum _{1\leq j\leq n_2} w_j\geq 2n_2$
. By (3.1),
$$ \begin{align*}n_1+n_2+1\leq n_1+W_2-n_2+1=W_1+n_3,\end{align*} $$
and
$$ \begin{align*}n_1+n_2+1\leq W_1-n_1+n_2+n_3-1=W_2,\end{align*} $$
which imply that
$n=n_1+n_2+1\le n_3+\min \{W_1,W_2\}$
.
We will need Lemma 3.2 to prove Theorems 1.3 and 1.4.
Lemma 3.2. Let
$\gamma \in (0,1]$
be a real number. Let
$N_0:= \lfloor 1+\frac {32}{\gamma ^2}+\frac {1}{\gamma }\rfloor ,$
then we have the following.
Let
$(X\ni x,B:=\sum _i b_iB_i)$
be an lc surface germ, such that
$X\ni x$
is smooth and
$B_i$
are distinct prime divisors. Suppose that
$\{\sum _i n_ib_i-1> 0\mid n_i\in \mathbb {Z}_{\geq 0}\}\subseteq [\gamma ,+\infty )$
. Let Y be a smooth surface and
$f: Y\to X\ni x$
be a birational morphism with the dual graph
${\mathcal {DG}}$
, such that:
-
•
$K_Y+B_Y=f^*(K_X+B)$
for some
$B_Y\geq 0$
on Y, -
•
${\mathcal {DG}}$
is a chain that contains only one
$(-1)$
-curve
$E_0$
, -
•
$E_0$
is adjacent to two vertices of
${\mathcal {DG}}$
and -
• either
$E_0$
is the only vertex of
${\mathcal {DG}}$
, such that
$a(E_0,X,B)=\mathrm {mld}(X\ni x,B)$
, or
$a(E_0,X,B)=\mathrm {mld}(X\ni x, B)>0$
and
$\sum _i n_i b_i\neq 1$
for all
$n_i\in \mathbb {Z}_{\geq 0}$
.
Then the number of vertices of
${\mathcal {DG}}$
is bounded from above by
$N_0$
.
Proof. By Lemma 2.5,
$b_i\geq \gamma $
for all i.
If
$\mathrm {mld}(X\ni x, B)\geq \frac {\gamma }{2}$
, then by Lemma 2.21 with
$\epsilon _0=\frac {\gamma }{2}$
, the number of vertices of
${\mathcal {DG}}$
is bounded from above by
$\frac {16}{\gamma }$
.
Thus, we may assume that
$0\leq \mathrm {mld}(X\ni x,B)\leq \frac {\gamma }{2}$
. We may index the vertices of
${\mathcal {DG}}$
as
$\{E_j\}_{-n_1\leq j\leq n_2}$
for some positive integer
$n_1,n_2$
, where
$E_j$
is adjacent to
$E_{j+1}$
for
$-n_1\leq j\leq n_2-1$
. Let
$w_j:=-E_j\cdot E_j$
and
$a_j:=a(E_j,X\ni x, B)$
for all j.
For all
$-n_1\leq k\leq n_2$
, we have:
$$ \begin{align} \left(K_Y+f^{-1}_* B+\sum_j (1-a_j)E_j\right)\cdot E_k=f^*(K_X+B)\cdot E_k=0. \end{align} $$
Let
$k=0$
, (3.2) becomes
$0=-2+f^{-1}_* B\cdot E_0+(1-a_{-1})+(1-a_1)+w_0a_0$
, thus:
$$ \begin{align*}(a_1-a_0)+(a_{-1}-a_0)=f^{-1}_* B\cdot E_0-a_0.\end{align*} $$
By the last assumption in the lemma, either
$(a_{-1}-a_0)+(a_1-a_0)>0$
or
$a_0>0$
, thus,
$f_*^{-1}B\cdot E_0>0$
in both cases. Hence,
$f_*^{-1}B\cdot E_0-a_0\geq \gamma -\frac {\gamma }{2}=\frac {\gamma }{2}$
. Possibly switching
$E_j (j<0)$
with
$E_j (j>0)$
, we may assume that
$a_{-1}-a_0\geq \frac {\gamma }{4}$
.
By Lemma 2.13(2),
$a_{-j}-a_{-j+1}\geq a_{-1}-a_0\geq \frac {\gamma }{4}$
for
$1\leq j\leq n_1$
, thus,
$n_1\cdot \frac {\gamma }{4}\leq a_{-n_1}\leq 1$
, and
$n_1\leq \frac {4}{\gamma }$
. Since
$a_j\geq \frac {\gamma }{4}$
for all
$-n_1\leq j\leq -1$
, by Lemma 2.13(1),
$w_j\leq \frac {8}{\gamma }$
for all
$-n_1\leq j\leq -1$
. Thus,
$\sum _{j=-1}^{-n_1} w_j\leq n_1\cdot \frac {8}{\gamma }\leq \frac {32}{\gamma ^2}$
. Note that
$X\ni x$
is smooth and
${\mathcal {DG}}$
has only one
$(-1)$
-curve, thus,
$f: Y\to X$
is a sequence of blow-ups as in Definition 2.16. Moreover,
${\mathcal {DG}}$
is a chain, thus, by Lemma 3.1,
$1+n_1+n_2\le n_3+\frac {32}{\gamma ^2}$
, where
$n_3=n_3(f)$
is defined as in Notation
$(\star )$
.
It suffices to show that
$n_3$
is bounded, we may assume that
$n_3>2$
. By the definition of
$n_3$
, there exists a sequence of blow-ups
$X_{n_3}\to \ldots X_1\to X_0:=X$
with the data
$(f_i,F_i,x_i\in X_i)$
, such that
$x_i\in F_i\setminus F_{i-1}$
for any
$1\leq i\leq n_3-1$
. Here,
$F_0:=\emptyset $
.
Let
$B_{X_i}$
be the strict transform of B on
$X_i$
for
$0\leq i\leq n_3$
, and let
$a_i':=a(F_i,X,B)$
for
$1\leq i\leq n_3$
, and
$a_0':=1$
. Since
$x_i\in F_i\setminus F_{i-1}$
,
$a_{i}'-a_{i+1}'=\mathrm {mult}_{x_i}B_{X_i}-1$
for any
$n_3-1\ge i\ge 0$
. By Lemma 2.13(2),
$a_{i}'-a_{i+1}'\geq \min \{a_1-a_0, a_{-1}-a_0\}\geq 0$
for
$1\leq i\leq n_3-2$
(see Figure 2). Thus, by the last assumption in the lemma, either
$\min \{a_1-a_0, a_{-1}-a_0\}>0$
or
$\mathrm {mult}_{x_i}B_{X_i}-1>0$
, in both cases, we have
$a_{i}'-a_{i+1}'=\mathrm {mult}_{x_i}B_{X_i}-1>0$
. Hence,
$a_{i}'-a_{i+1}'=\mathrm {mult}_{x_i}B_{X_i}-1\geq \gamma $
for any
$1\leq i\leq n_3-2$
as
$\{\sum _i n_ib_i-1> 0\mid n_i\in \mathbb {Z}_{\geq 0}\}\subseteq [\gamma ,+\infty )$
. Therefore,
$$ \begin{align*}0\leq a_{n_3-1}'=a_0'+\sum_{i=0}^{n_3-2}(a_{i+1}'-a_i')\leq 1-(n_3-1)\gamma,\end{align*} $$
and
$n_3\leq 1+\frac {1}{\gamma }$
.
To sum up, the number of vertices of
${\mathcal {DG}}$
is bounded from above by
$\lfloor 1+\frac {32}{\gamma ^2}+\frac {1}{\gamma }\rfloor $
.
Now we are ready to prove Theorem 1.3.
Proof of Theorem 1.3
By Lemma 2.17, we may assume that
$\mathrm {mld}(X\ni x, B)\leq 1$
.
Let
$f: Y\to X\ni x$
be the birational morphism constructed in Lemma 2.20 with the dual graph
${\mathcal {DG}}$
. We claim that the number of vertices of
${\mathcal {DG}}$
is bounded from above by
$N_0:= \lfloor 1+\frac {32}{\gamma ^2}+\frac {1}{\gamma }\rfloor $
.
Assume the claim holds, then by Lemma 2.18,
$a(E,X,0)\leq 2^{N_0}$
for some exceptional divisor E, such that
$a(E,X,B)=\mathrm {mld}(X\ni X,B)$
, we are done. It suffices to show the claim.
If the f-exceptional
$(-1)$
-curve is adjacent to only one vertex of
${\mathcal {DG}}$
, then by Lemma 2.15(1), the number of vertices of
${\mathcal {DG}}$
is bounded from above by
$ 1+\frac {1}{\gamma }$
.
If the f-exceptional
$(-1)$
-curve is adjacent to two vertices of
${\mathcal {DG}}$
, then by Lemma 3.2, the number of vertices of
${\mathcal {DG}}$
is bounded from above by
$\lfloor 1+\frac {32}{\gamma ^2}+\frac {1}{\gamma }\rfloor $
. Thus, we finish the proof.
To end this section, we provide an example which shows that Theorem 1.3 does not hold if we do not assume that
$\{\sum _i n_ib_i-1>0\mid n_i\in \mathbb {Z}_{\geq 0}\}$
is bounded from below by a positive real number.
Example 3.3. Let
$\{(\mathbf {A}^2\ni 0, B_k)\}_{k\geq 2}$
be a sequence of klt surface germs, such that
$B_k:=\frac {1}{2}B_{k,1}+(\frac {1}{2}+\frac {1}{k+1})B_{k,2}$
, and
$B_{k,1}$
(respectively,
$B_{k,2}$
) is defined by the equation
$x=0$
(respectively,
$x-y^k=0$
) at
$0\in \mathbf {A}^2$
.
For each k, we may construct a sequence of blow-ups
$X_{k}\to X_{k-1}\to \cdots \to X_{1}\to X_{0}:=X$
with the data
$(f_i,F_i,x_i\in X_i)$
, such that
$x_{i-1}\in F_{i-1}$
is the intersection of the strict transforms of
$B_{k,1}$
and
$B_{k,2}$
on
$F_{i-1}$
for
$1\le i\le k$
. Let
$g_k: X_k\to X$
be the natural morphism induced by
$\{f_i\}_{1\leq i\leq k}$
, we have
$K_{X_k}+B_{X_k}=g_k^*(K_X+B_k)$
for some snc divisor
$B_{X_k}\geq 0$
on
$X_k$
, and the coefficients of
$B_{X_k}$
are no more than
$\frac {k}{k+1}$
. Thus, we will need at least k blow-ups as constructed above to extract an exceptional divisor
$F_k$
, such that
$a(F_k,\mathbf {A}^2,B_k)=\mathrm {mld}(\mathbf {A}^2\ni 0, B_k)=\frac {1}{k+1}$
, and
$a(F_k, \mathbf {A}^2,0)\geq k$
.
4 Proof of Theorem 1.2
Proof of Theorem 1.2
We may assume that
${\Gamma }\setminus \{0\}\neq \emptyset $
.
Let
$(X\ni x, B)$
be an lc surface germ with
$B\in {\Gamma }$
. By Lemma 2.17, we may assume that
$\mathrm {mld}(X\ni x, B)\leq 1$
. By Theorem 1.3, it suffices to show the case when
$X\ni x$
is not smooth.
If
$\mathrm {mld}(X\ni x, B)=\operatorname {pld}(X\ni x, B)$
, then
$a(E,X,0)\leq 1$
for some prime divisor E over
$X\ni x$
, such that
$a(E,X,B)=\mathrm {mld}(X\ni x, B)$
. So we may assume that
$\mathrm {mld}(X\ni x, B)\neq \operatorname {pld}(X\ni x, B)$
.
By Lemma 2.20, there exists a birational morphism
$f: Y\to X\ni x$
which satisfies Lemma 2.20(1)–(5). Let
$\widetilde {f}:\widetilde {X}\to X$
be the minimal resolution of
$X\ni x$
,
$g: Y\to \widetilde {X}\ni \widetilde {x}$
the birational morphism, such that
$\widetilde {f}\circ g=f$
, where
$\widetilde {x}\in \widetilde {X}$
is chosen as in Lemma 2.20(5), and there exists an
$\widetilde {f}$
-exceptional prime divisor
$\widetilde {E}$
over
$X\ni x$
, such that
$a(\widetilde {E},X,B)=\operatorname {pld}(X\ni x, B)$
and
$\widetilde {x}\in \widetilde {E}$
. Moreover, there is at most one other vertex
$\widetilde {E'}$
of
$\widetilde {{\mathcal {DG}}}$
, such that
$\widetilde {x}\in \widetilde {E'}$
.
Let
$\widetilde {{\mathcal {DG}}}$
be the dual graph of
$\widetilde {f}$
and
$\{F_i\}_{-n_1\leq i\leq n_2}$
the vertices of
$\widetilde {{\mathcal {DG}}}$
, such that
$n_1,n_2\in \mathbb {Z}_{\geq 0}$
,
$F_i$
is adjacent to
$F_{i+1}$
,
$w_i:=-F_i\cdot F_i, a_i:=a(F_i,X,B)$
for all i, and
$F_0:=\widetilde {E}, F_1:=\widetilde {E'}$
(see Figure 4). We may write
$K_{\widetilde {X}}+B_{\widetilde {X}}=\widetilde {f}^*(K_X+B)$
, where
$B_{\widetilde {X}}:=\widetilde {f}^{-1}_*{B}+\sum _i (1-a_i)F_i$
, and we define
$\widetilde {B}:=\widetilde {f}^{-1}_*{B}+\sum _{\widetilde {x}\in F_i} (1-a_i)F_i$
.
Cases when
$\widetilde {x}\in F_0\cap F_1$
and when
$\widetilde {x}\notin F_i$
for
$i\neq 0$
.
If
$\widetilde {x}\notin F_i$
for all
$i\neq 0$
, then we consider the surface germ
$(\widetilde {X}\ni \widetilde {x}, \widetilde {B}=\widetilde {f}^{-1}_* B+(1-a_0)F_0)$
, where
$\widetilde {B}\in {\Gamma }':={\Gamma }\cup \{1-a\mid a\in \mathrm {Pld}(2,{\Gamma })\}$
. By Lemma 2.9,
${\Gamma }'$
satisfies the DCC. Thus, by Theorem 1.3, we may find a positive integer
$N_1$
which only depends on
${\Gamma }$
, and a prime divisor E over
$\widetilde {X}\ni \widetilde {x}$
, such that
$a(E,\widetilde {X}, \widetilde {B})=a(E,X,B)=\mathrm {mld}(X\ni x,B)$
, and
$a(E,X,0)\leq a(E,\widetilde {X},0)\leq N_1$
.
So we may assume that
$\widetilde {x}=F_0\cap F_1$
. By Lemma 2.4, there exist positive real numbers
$\epsilon ,\delta \leq 1$
depending only on
${\Gamma }$
, such that
$\{\sum _i n_ib_i-1>0\mid b_i\in {\Gamma }^{\prime }_{\epsilon }\cap [0,1], n_i\in \mathbb {Z}_{\ge 0}\}\subseteq [\delta ,+\infty )$
. Recall that
${\Gamma }^{\prime }_\epsilon =\cup _{b'\in {\Gamma }'} [b'-\epsilon , b']$
.
If
$a_1-a_0\leq \epsilon $
, then we consider the surface germ
$(\widetilde {X}\ni \widetilde {x},\widetilde {B}=\widetilde {f}^{-1}_*B+(1-a_0)F_0+(1-a_1)F_1)$
, where
$\widetilde {B}\in {\Gamma }^{\prime }_{\epsilon }\cap [0,1]$
. By Theorem 1.3, there exist a positive integer
$N_2$
which only depends on
${\Gamma }$
, and a prime divisor E over
$\widetilde {X}\ni \widetilde {x}$
, such that
$a(E,\widetilde {X}, \widetilde {B})=a(E,X,B)=\mathrm {mld}(X\ni x,B)$
and
$a(E,X,0)\leq a(E,\widetilde {X},0)\le N_2$
.
If
$a_1-a_0\geq \epsilon $
, then we claim that there exists a DCC set
${\Gamma }''$
depending only on
${\Gamma }$
, such that
$1-a_1\in {\Gamma }''$
.
Cases when
$a_1-a_0\geq \epsilon $
.
Assume the claim holds, then we consider the surface germ
$(\widetilde {X}\ni \widetilde {x},\widetilde {B}=\widetilde {f}^{-1}_*B+(1-a_0)F_0+(1-a_1)F_1)$
, where
$\widetilde {B}\in {\Gamma }''\cup {\Gamma }'$
. By Theorem 1.3, we may find a positive integer
$N_3$
which only depends on
${\Gamma }$
, and a prime divisor E over
$\widetilde {X}\ni \widetilde {x}$
, such that
$a(E,\widetilde {X}, \widetilde {B})=a(E,X,B)=\mathrm {mld}(X\ni x,B)$
and
$a(E,X,0)\leq a(E,\widetilde {X},0)\le N_3$
. Let
$N:=\max \{N_1,N_2,N_3\}$
, and we are done.
It suffices to show the claim. By Lemma 2.13(1),
$w_i\leq \frac {2}{\epsilon }$
for any
$0<i\leq n_2$
. Since
$1\geq a_{n_2}=a_0+\sum _{i=0}^{n_2-1} (a_{i+1}-a_i)\geq n_2\epsilon $
,
$n_2\leq \frac {1}{\epsilon }$
. We may write:
$$ \begin{align*} K_{\widetilde{X}}+\widetilde{f}^{-1}_*B+\sum_{-n_1\leq i\leq n_2} (1-a_i)F_i=\widetilde{f}^*(K_X+B). \end{align*} $$
For each
$1\leq j\leq n_2$
, we have:
$$ \begin{align*}(K_{\widetilde{X}}+\widetilde{f}^{-1}_*B+\sum_{-n_1\leq i\leq n_2} (1-a_i)F_i)\cdot F_j=0,\end{align*} $$
which implies
$\sum _{-n_1\leq i\leq n_2} (a_i-1)F_i\cdot F_j=-{F_j}^2-2+{\widetilde {f}}^{-1}_{*}B \cdot F_j$
, or equivalently,
$$ \begin{align*}\begin{pmatrix} F_1\cdot F_1&\cdots &F_{n_2}\cdot F_1 \\ \vdots&\ddots & \vdots \\ F_1\cdot F_{n_2}&\cdots &F_{n_2}\cdot F_{n_2} \\ \end{pmatrix} \begin{pmatrix} a_1-1 \\ \vdots \\ a_{n_2}-1 \\ \end{pmatrix} = \begin{pmatrix} w_{1}-2+\widetilde{f}^{-1}_*B\cdot F_1+(1-a_0)\\ \vdots\\ w_{n_2}-2+\widetilde{f}^{-1}_*B\cdot F_{n_2}\\ \end{pmatrix}.\end{align*} $$
By assumption,
$w_j-2+\widetilde {f}^{-1}_*B\cdot F_j$
belongs to a DCC set, and by Lemma 2.9,
$1-a_0$
belongs to the DCC set
$\{1-a\mid a\in \mathrm {Pld}(2,{\Gamma })\}$
.
By [Reference Kollár and Mori15, Lemma 3.40],
$(F_i\cdot F_j)_{1\le i,j\le n_2}$
is a negative definite matrix. Let
$(s_{ij})_{n_2\times n_2}$
be the inverse matrix of
$(F_i\cdot F_j)_{1\le i,j\le n_2}$
. By [Reference Kollár and Mori15, Lemma 3.41],
$s_{ij}<0$
for any
$1\le i,j\le n_2$
, thus:
$$ \begin{align*}1-a_1=-s_{11}(w_{1}-2+\widetilde{f}^{-1}_*B\cdot F_1+(1-a_0))-\sum_{j=2}^{n_2}s_{1j}(w_j-2+\widetilde{f}^{-1}_*B\cdot F_j)\end{align*} $$
belongs to a DCC set.
5 Proof of Theorem 1.4
In this section, we will first show Theorem 1.4, then we will generalise Theorems 1.3 and 1.4 (see Theorems 5.4 and 5.5).
Lemma 5.1. Let
$(X\ni x, B:=\sum _i b_iB_i)$
be an lc surface germ, where
$B_i$
are distinct prime divisors. Let
$h: W\to X\ni x$
be a log resolution of
$(X\ni x,B)$
, and let
$S=\{E_j\}$
be a finite set of valuations of h-exceptional prime divisors, such that
$a(E_j,X,B)=\mathrm {mld} (X\ni x, B)$
for all j. Suppose that
$\sum _i n_ib_i\neq 1$
for any
$n_i\in \mathbb {Z}_{\geq 0}$
,
$\mathrm {mld}(X\ni x, B)\in (0,1]$
, and
$E_j$
is exceptional over
$\tilde {X}$
for some j, where
$\tilde {X}\to X$
is the minimal resolution of
$X\ni x$
. Then there exist a smooth surface Y and a birational morphism
$f: Y\to X\ni x$
with the dual graph
${\mathcal {DG}}$
, such that:
-
(1)
$K_Y+B_Y=f^*(K_X+B)$
for some
$\mathbb {R}$
-divisor
$B_Y\geq 0$
on Y, -
(2) each valuation in S corresponds to some vertex of
${\mathcal {DG}}$
, -
(3)
${\mathcal {DG}}$
contains only one
$(-1)$
-curve
$E_0$
, and it corresponds to a valuation in S and -
(4)
${\mathcal {DG}}$
is a chain.
Proof. By Lemma 2.19, there exist a smooth surface Y and a birational morphism
$f:Y\to X\ni x$
with the dual graph of
${\mathcal {DG}}$
which satisfy (1)–(2), and each f-exceptional
$(-1)$
-curve corresponds to some valuation in S.
For (3), by the assumption on S,
${\mathcal {DG}}$
contains at least one
$(-1)$
-curve
$E_0$
. If there exist two
$(-1)$
-curves
$E_0'\neq E_0$
, then
$a(E_0,X,B)=a(E_0',X,B)=\mathrm {mld}(X\ni x, B)>0$
by construction, and there exists a set of distinct vertices
$\{C_k\}_{0\leq k \leq n}$
of
${\mathcal {DG}}$
, such that
$n\ge 2$
,
$C_{n}:=E_0'$
,
$C_0$
is a
$(-1)$
-curve,
$-C_k\cdot C_k\ge -2$
for
$1\le k\le n-1$
and
$C_k$
is adjacent to
$C_{k+1}$
for
$0\leq k\leq n-1$
. Since
$a(C_0,X,B)=a(C_{n},X,B)=\mathrm {mld}(X\ni x, B)>0$
, by Lemma 2.13(6),
$-C_k\cdot C_k=2$
for
$1\leq k\leq n-1$
. Let
$E:=\sum _{k=0}^{n} C_k$
, then
$E\cdot E=0$
, which contradicts the negativity lemma.
For (4), suppose that
${\mathcal {DG}}$
contains a fork F. Since
$E_0$
is a
$(-1)$
-curve, by Lemma 2.14(2),
$E_0\neq F$
. By (3) and (5) of Lemma 2.13, we have
$a(E_0,X,B)\geq a(F,X,B)\geq \mathrm {mld}(X\ni x, B)$
, thus,
$a(E_0,X,B)= a(F,X,B)=\mathrm {mld}(X\ni x, B)>0$
. There exists a set of distinct vertices
$\{C_i'\}_{0\leq i\leq m}$
, such that
$C_0':=E_0$
,
$C_m':= F$
, and
$C_i'$
is adjacent to
$C_{i+1}'$
for
$0\leq i\leq m-1$
. We may denote
$w_i':=-C_i'\cdot C_i'$
and
$a_i':=a(C_i',X,B)$
for
$0\leq i\leq m$
. By Lemma 2.13(6), we have
$a_0'=\cdots =a_m'$
, and
$w_k'=2$
for
$1\leq k\leq m-1$
. Since
$C_{m}'$
is a fork and
$a_{m-1}'=a_m'$
, by Lemma 2.13(3),
$w_m'=2$
.
If
$C_0'$
is adjacent to only one vertex of
${\mathcal {DG}}$
, which is
$C_1'$
by our construction, then since
$\sum _i n_ib_i\neq 1$
for any
$n_i\in \mathbb {Z}_{\geq 0}$
,
$a_0'\neq a_1'$
by Lemma 2.15(2), a contradiction.
$C_0'$
is adjacent to
$C_{-1}'$
and
$C_1'$
.
Thus, we may assume that
$C_0'$
is adjacent to a vertex
$C_{-1}'$
of
${\mathcal {DG}}$
other than
$C_1'$
(see Figure 6). We may contract
$C_k'$
for
$0\leq k\leq m-1$
step by step, and will end up with a fork which is a
$(-1)$
-curve, this contradicts Lemma 2.14(2).
Lemma 5.2. Let
$(X\ni x, B:=\sum _i b_iB_i)$
be an lc surface germ, where
$B_i$
are distinct prime divisors. Suppose that
$X\ni x$
is a klt surface germ,
$\mathrm {mld}(X\ni x, B)=0$
, and
$\sum _i n_ib_i\neq 1$
for any
$n_i\in \mathbb {Z}_{\ge 0}$
. Then there is only one prime divisor E over
$X\ni x$
, such that
$a(E,X,B)=0$
.
Proof. By [Reference Birkar2, Lemma 2.7] (this holds for surfaces in any characteristic), we can find a plt blow up
$g: Y\to X\ni x$
, such that:
-
•
$K_Y+B_Y=g^*(K_X+B)$
for some
$\mathbb {R}$
-divisor
$B_Y\geq 0$
on Y, -
• there is only one g-exceptional prime divisor E,
-
•
$\operatorname {Supp} E\subseteq \lfloor B_Y \rfloor $
and -
•
$(Y, E)$
is plt.
Since the relative Kawamata-Viehweg vanishing theorem holds for birational morphisms between surfaces in any characteristic (see [Reference Tanaka23, Theorem 0.5]), by a similar argument as in [Reference Hacon and Xu10, Proposition 4.1], E is normal. By the adjunction formula ([Reference Shokurov20, §3], [14, §16]),
$K_E+B_E:=(K_Y+B_Y)|_E$
, where:
$$ \begin{align*}B_E=\sum_i \frac{m_i-1+\sum_j n_{i,j}b_j}{m_i}p_i\end{align*} $$
for some distinct closed points
$p_i$
on E and some
$m_i\in \mathbb {Z}_{>0}$
and
$n_{i,j}\in \mathbb {Z}_{\geq 0}$
, such that
$\sum _j n_{i,j} b_j\leq 1$
for all i. By assumption,
$\sum _j n_{i,j} b_j\neq 1$
, thus,
$\sum _j n_{i,j} b_j< 1$
for all i, which implies
$\lfloor B_E \rfloor =0$
, thus,
$(E, B_E)$
is klt. By the inversion of adjunction for surfaces (which is well known in any characteristic),
$(Y, B_Y)$
is plt, and we are done.
Proof of Theorem 1.4
By Lemma 2.17, we may assume that
$\mathrm {mld}(X\ni x, B)\leq 1$
.
If
$\mathrm {mld}(X\ni x, B)=0$
, since
$X\ni x$
is smooth, then by Lemma 5.2, there exists a unique prime divisor E over
$X\ni x$
, such that
$a(E,X,B)=0$
. By Theorem 1.3,
$a(E,X,0)\leq 2^{N_0}$
. Thus, we may assume that
$\mathrm {mld}(X\ni x, B)>0$
.
We may apply Lemma 5.1 to S, there exist a smooth surface Y and a birational morphism
$f: Y\to X\ni x$
with the dual graph
${\mathcal {DG}}$
, which satisfy Lemma 5.1(1)–(4). Let
$E_0$
be the unique
$(-1)$
-curve in
${\mathcal {DG}}$
. It suffices to give an upper bound for the number of vertices of
${\mathcal {DG}}$
.
If
$E_0$
is adjacent to only one vertex of
${\mathcal {DG}}$
, then by Lemma 2.15(2)–(3), the number of vertices of
${\mathcal {DG}}$
is bounded from above by
$1+\frac {1}{\gamma }$
, and
$|S|=1$
.
If
$E_0$
is adjacent to two vertices of
${\mathcal {DG}}$
, then by Lemma 3.2, the number of vertices of
${\mathcal {DG}}$
is bounded from above by
$1+\frac {32}{\gamma ^2}+\frac {1}{\gamma }$
. Thus,
$|S|\leq N_0$
, and by Lemma 2.18,
$a(E,X,0)\le 2^{N_0}$
for any
$E\in S$
.
Now we are going to introduce and prove Theorems 5.4 and 5.5. First, we need to introduce the following definition.
Definition 5.3. Let
$X\ni x$
be a klt surface germ. Let
$\widetilde {f}: \widetilde {X}\to X\ni x$
be the minimal resolution and
$\{E_i\}_{1\leq i\leq n}$
the set of
$\widetilde {f}$
-exceptional prime divisors. The determinant of
$X\ni x$
is defined by:
$$ \begin{align*}\det(X\ni x):=\left\{ \begin{array}{lcl} |\mathrm{det} (E_i\cdot E_j)_{1\leq i,j\leq n}| &{\text{if }x\in X \text{ is a singular point,}}\\ 1 &{\text{if }x\in X \text{ is a smooth point.}} \end{array} \right.\end{align*} $$
Theorem 5.4. Let
$\gamma \in (0,1]$
be a real number and I a positive integer. Then
$N_0:=\lfloor 1+\frac {32I^2}{\gamma ^2}+\frac {I}{\gamma }\rfloor $
satisfies the following.
Let
$(X\ni x,B:=\sum _{i} b_iB_i)$
be an lc surface germ, where
$B_i$
are distinct prime divisors. Suppose that
$\det (X\ni x)\mid I$
, and
$\{\sum _i n_ib_i-t>0\mid n_i\in \mathbb {Z}_{\geq 0}, t\in \mathbb {Z}\cap [1,I]\}\subseteq [\gamma ,+\infty )$
. Then there exists a prime divisor E over
$X\ni x$
, such that
$a(E,X,B)=\mathrm {mld}(X\ni x,B)$
and
$a(E,X,0)\leq 2^{N_0}$
.
Theorem 5.5. Let
$\gamma \in (0,1]$
be a real number and
$X\ni x$
a klt surface germ. Let N be the number of vertices of the dual graph of the minimal resolution of
$X\ni x$
and
$I:=\det (X\ni x)$
. Then
$N_0:=\lfloor 1+\frac {32I^2}{\gamma ^2}+\frac {I}{\gamma }\rfloor +N$
satisfies the following.
Let
$(X\ni x,B:=\sum _i b_iB_i)$
be an lc surface germ, where
$B_i$
are distinct prime divisors. Let
$S:=\{E\mid E\text { is a prime divisor over } X\ni x, a(E,X,B)=\mathrm {mld}(X\ni x,B)\}$
. Suppose that
$\{\sum _i n_ib_i-t\geq 0\mid n_i\in \mathbb {Z}_{\geq 0}, t\in \mathbb {Z}\cap [1,I]\}\subseteq [\gamma ,+\infty )$
. Then
$|S|\le N_0$
, and
$a(E,X,0)\le 2^{N_0}$
for any
$E\in S$
.
Remark 5.6. It is easy to see that Theorem 5.5 does not hold if
$X\ni x$
is not klt. Also it does not hold if we only bound
$\det (X\ni x)$
as in Theorem 5.4 (see Example 5.9).
Theorem 5.4 follows from Theorem 1.3, and Theorem 5.5 follows from Theorems 1.4 and 5.4. We will need the following lemma to prove the above theorems.
Lemma 5.7. Let
$(X\ni x,B:=\sum _i b_iB_i)$
be an lc surface germ, where
$B_i$
are distinct prime divisors. Let
$\widetilde {f}: \widetilde {X}\to X\ni x$
be the minimal resolution, and we may write
$K_{\widetilde {X}}+B_{\widetilde {X}}=\widetilde {f}^*(K_X+B)$
for some
$\mathbb {R}$
-divisor
$B_{\widetilde {X}}:=\sum _j \widetilde {b_j}\widetilde {B_j}\geq 0$
, where
$\widetilde {B_j}$
are distinct prime divisors on
$\widetilde {X}$
. Let
$I\in \mathbb {Z}_{>0}$
, such that
$\det (X\ni x)\mid I$
, then:
-
(1)
$\{\sum _j n_j'\widetilde {b_j}-1>0 \mid n_j'\in \mathbb {Z}_{\geq 0}\}\subseteq \frac {1}{I}\{\sum _i n_ib_i-t>0 \mid n_i\in \mathbb {Z}_{\geq 0}, t\leq I\}$
and -
(2)
$\{\sum _j n_j'\widetilde {b_j}-1\geq 0 \mid n_j'\in \mathbb {Z}_{\geq 0}\}\subseteq \frac {1}{I}\{\sum _i n_ib_i-t\ge 0 \mid n_i\in \mathbb {Z}_{\geq 0},\sum _i n_i>0, t\leq I\}$
.
Proof. Let
${\mathcal {DG}}$
be the dual graph of
$\widetilde {f}$
. Let
$\{E_i\}_{1 \leq i\leq m}$
be the set of
$\widetilde {f}$
-exceptional divisors and
$w_i:=-E_i\cdot E_i$
,
$a_i:=a(E_i,X\ni x,B)$
for all i.
For each
$1\leq j\leq m$
,
$(K_Y+f^{-1}_*B+\sum _{i=1}^m (1-a_i)E_i)\cdot E_{j}=0$
, which implies that
$\sum _{i=1}^m (a_i-1)E_i\cdot E_{j}=-{E_{j}}^2-2+\widetilde {f}^{-1}_* B \cdot E_{j}$
.
By [Reference Kollár and Mori15, Lemma 3.40],
$(E_i\cdot E_{j})_{1\le i,j\le m}$
is a negative definite matrix. Let
$(s_{ij})_{m\times m}$
be the inverse matrix of
$(E_i\cdot E_{j})_{1\le i,j\le m}$
. By [Reference Kollár and Mori15, Lemma 3.41] and the assumption on I, we have
$Is_{ij}\in \mathbb {Z}_{<0}$
for
$1\leq i,j\leq m$
. Thus, for all i,
$$ \begin{align} 1-a_i=\frac{1}{I}\sum_{j=1}^m (-Is_{ij})\cdot(w_{j}-2+\widetilde{f}^{-1}_* B\cdot E_{j}). \end{align} $$
Since
$w_{j}\geq 2$
for all j, (1) and (2) follow immediately from (5.1) and the equation
$B_{\widetilde {X}}=\sum _j \widetilde {b_j}\widetilde {B_j}=\widetilde {f}^{-1}_* B+\sum _{i=1}^m (1-a_i)E_i$
.
Proof of Theorem 5.4
By Lemma 2.17, we may assume that
$\mathrm {mld}(X\ni x, B)\leq 1$
.
If
$\mathrm {mld}(X\ni x, B)=\operatorname {pld}(X\ni x, B)$
, then
$a(E,X,0)\leq 1$
for some prime divisor E over
$X\ni x$
, such that
$a(E,X,B)=\mathrm {mld}(X\ni x, B)$
. So we may assume that
$\mathrm {mld}(X\ni x, B)\neq \operatorname {pld}(X\ni x, B)$
.
By Lemma 2.20, there exists a birational morphism
$f: Y\to X\ni x$
, which satisfies Lemma 2.20(1)–(4). Let
$\widetilde {f}:\widetilde {X}\to X$
be the minimal resolution of
$X\ni x$
and
$g: Y\to \widetilde {X}$
the natural morphism induced by f. We may write
$K_{\widetilde {X}}+B_{\widetilde {X}}=\widetilde {f}^*(K_X+B)$
for some
$\mathbb {R}$
-divisor
$B_{\widetilde {X}}:= \sum _j \widetilde {b_j}\widetilde {B_{j}}\ge 0$
on
$\widetilde {X}$
, where
$\widetilde {B_j}$
are distinct prime divisors.
By Lemma 5.7(1), we have
$\{\sum _j n_j'\widetilde {b_j}-1>0 \mid n_j'\in \mathbb {Z}_{\geq 0}\}\subseteq \frac {1}{I}\{\sum _i n_ib_i-t>0 \mid n_i\in \mathbb {Z}_{\geq 0}, t\leq I\}\subseteq [\frac {\gamma }{I},\infty )$
by the assumption. Since the dual graph
${\mathcal {DG}}$
of f contains only one
$(-1)$
-curve, there exists a closed point
$\widetilde {x}\in \widetilde {X}$
, such that
$\operatorname {center}_{\widetilde {X}} E=\widetilde {x}$
for any g-exceptional divisor E. Apply Theorem 1.3 to the surface germ
$(\widetilde {X}\ni \widetilde {x}, B_{\widetilde {X}})$
, we are done.
Proof of Theorem 5.5
By Lemma 2.17, we may assume that
$\mathrm {mld}(X\ni x, B)\leq 1$
.
We may assume that
$B\neq 0$
, otherwise, we may take
$\widetilde {f}:\widetilde {X}\to X\ni x$
as the minimal resolution, and
$K_{\widetilde {X}}+B_{\widetilde {X}}=\widetilde {f}^*K_X$
, where
$B_{\widetilde {X}}$
is an snc divisor. Since
$X\ni x$
is klt, all elements in S are on
$\widetilde {X}$
, thus,
$|S|\leq N$
, and
$a(E,X,0)\leq 1$
for all
$E\in S$
. For the same reason, we may assume that not all elements of S are on the minimal resolution
$\widetilde {X}$
of
$X\ni x$
.
If
$\mathrm {mld}(X\ni x, B)=0$
, then by Lemma 5.2, there exists a unique exceptional divisor E over
$X\ni x$
, such that
$a(E,X,B)=0$
. By Theorem 5.4,
$a(E,X,0)\leq 2^{\lfloor 1+\frac {32{I^2}}{\gamma ^2}+\frac {I}{\gamma }\rfloor }$
. Thus, we may assume that
$\mathrm {mld}(X\ni x, B)>0$
from now on. Since
$\sum _i n_ib_i\neq 1$
for any
$n_i\in \mathbb {Z}_{\ge 0}$
, we have
$\lfloor B\rfloor =0$
, and
$(X,B)$
is klt near
$x\in X$
, hence, by [Reference Kollár and Mori15, Proposition 2.36], S is a finite set.
Now, we may apply Lemma 5.1 to the set S, there exists a birational morphism
$f: Y\to X\ni x$
that satisfies Lemma 5.1(1)–(4). Let
$\widetilde {f}:\widetilde {X}\to X$
be the minimal resolution of
$X\ni x$
and
$g: Y\to \widetilde {X}$
the natural morphism induced by f. We may write
$K_{\widetilde {X}}+B_{\widetilde {X}}=\widetilde {f}^*(K_X+B)$
for some
$\mathbb {R}$
-divisor
$B_{\widetilde {X}}:= \sum _j \widetilde {b_j}\widetilde {B_{j}}\ge 0$
on
$\widetilde {X}$
, where
$\widetilde {B_j}$
are distinct prime divisors on
$\widetilde {X}$
.
By Lemma 5.7(2), we have
$\{\sum _j n_j'\widetilde {b_j}-1\geq 0 \mid n_j'\in \mathbb {Z}_{\geq 0}\}\subseteq \frac {1}{I}\{\sum _i n_ib_i-t\geq 0 \mid n_i\in \mathbb {Z}_{\geq 0}, \sum _i n_i>0 ,t\leq I\}\subseteq [\frac {\gamma }{I},\infty )$
. Since the dual graph
${\mathcal {DG}}$
of f contains only one
$(-1)$
-curve, there exists a closed point
$\widetilde {x}\in \widetilde {X}$
, such that
$\operatorname {center}_{\widetilde {X}} E=\widetilde {x}$
for any g-exceptional divisor E. Apply Theorem 1.4 to the surface germ
$(\widetilde {X}\ni \widetilde {x}, B_{\widetilde {X}})$
, we are done.
To end this section, we provide some examples.
The following example shows that Theorem 1.4 does not hold if
$\sum _i n_i b_i=1$
for some
$n_i\in \mathbb {Z}_{\geq 0}$
.
Example 5.8. Let
$\{b_i\}_{1\leq i\leq m}\subseteq (0,1]$
, such that
$\sum _{i=1}^m n_ib_i=1$
for some
$n_i\in \mathbb {Z}_{>0}$
.
Let
$(X\ni x):=(\mathbf {A}^2\ni 0)$
,
$D_{m,n}$
the Cartier divisor, which is defined by the equation
$x-y^m-y^{m+n}=0$
and
$B_k:=\sum _{i=1}^m b_i\sum _{j=1}^{n_i} D_{k+i,j}$
for any positive integer k, where
$D_{k+i,j}$
is defined by
$x-y^{k+i}-y^{k+i+j}=0$
. Then
$(X\ni x, B_k)$
is canonical near x for any k as
$\operatorname {mult}_{x}B_k\le 1$
.
For each k, we may construct a sequence of blow-ups
$X_{k}\to X_{k-1}\to \cdots \to X_{1}\to X_{0}:=X$
with the data
$(f_i,F_i,x_i\in X_i)$
, such that
$x_{i-1}\in F_{i-1}$
is the intersection of the strict transforms of these
$D_{k+i,j}$
on
$F_{i-1}$
for all
$i,j$
. Then
$a(F_i,X,B_k)=1=\mathrm {mld}(X\ni x, B_k)$
for any k and
$1\le i\le k$
. Thus, both
$|S_k|$
and
$\sup _{E\in S_k}\{a(E,X,0)\}$
are not bounded from above as
$|S_k|\geq k$
, and
$a(F_k,X,0)\ge k$
for each k.
The following example shows that Theorem 5.5 does not hold if we do not fix the germ, even when we bound the determinant of the surface germs.
Example 5.9. Let
$\{(X_k\ni x_k)\}_{k\geq 1}$
be a sequence of surface germs, such that each
$X_k\ni x_k$
is a Du Val singularity of type
$D_{k+3}$
([Reference Kollár and Mori15, Theorem 4.22]). We have
$\det (X_k\ni x_k)=4$
for all
$k\geq 4$
, but
$|S_k|=k+3$
for
$k\geq 1$
is not bounded from above.
6 An equivalent conjecture for MLDs on a fixed germ
Definition 6.1. Let X be a normal variety and
$B:=\sum b_iB_i$
an
$\mathbb {R}$
-divisor on X, where
$B_i$
are distinct prime divisors. We define
$||B||:=\max _i\{|b_i|\}$
. Let E be a prime divisor over X and
$Y\to X$
a birational model, such that E is on Y. We define
$\operatorname {mult}_E B_i$
to be the multiplicity of the strict transform of
$B_i$
on Y along E for each i, and
$\operatorname {mult}_E B:=\sum _i b_i\operatorname {mult}_E B_i$
.
Conjecture 6.2 could be regarded as an inversion of stability type conjecture for divisors which compute MLDs, see [Reference Birkar and Shokurov3, Main Proposition 2.1], [Reference Chen and Han5, Theorem 5.10] for other inversion of stability type results.
Conjecture 6.2. Let
${\Gamma }\subseteq [0,1]$
be a finite set, X a normal quasiprojective variety and
$x\in X$
a closed point. Then there exists a positive real number
$\tau $
depending only on
${\Gamma }$
and
$x\in X$
satisfying the following.
Assume that
$(X\ni x,B)$
and
$(X\ni x,B')$
are two lc germs, such that:
-
(1)
$B'\le B$
,
$||B-B'||<\tau ,B\in {\Gamma }$
and -
(2)
$a(E,X,B')=\mathrm {mld}(X\ni x,B')$
for some prime divisor E over
$X\ni x$
.
Then
$a(E,X,B)=\mathrm {mld}(X\ni x,B)$
.
Conjecture 6.2 does not hold for surface germs
$(X\ni x,B)$
, in general, if either
${\Gamma }$
is a DCC set (see [Reference Han and Luo9, Example 7.5]) or we do not fix
$X\ni x$
(see [Reference Han and Luo9, Example 7.6]).
Proposition 6.3. For any fixed
$\mathbb {Q}$
-Gorenstein germ
$X\ni x$
over an algebraically closed field of characteristic zero, Conjecture 6.2 is equivalent to Conjecture 1.1, thus, Conjecture 6.2 is equivalent to the ACC conjecture for MLDs.
Remark 6.4. We need to work in characteristic zero since we need to apply [Reference Kawakita12, Theorem 1.1].
Proof. Suppose that Conjecture 6.2 holds. Let
$t:=\min \{1,\tau \}>0$
. Then
$||(1-t)B-B||<\tau $
. Let E be a prime divisor over
$X\ni x$
, such that
$a(E,X,(1-t)B)=\mathrm {mld}(X\ni x,(1-t)B)$
. Then:
$$ \begin{align*}\mathrm{mld} (X\ni x,0)\ge a(E,X,(1-t)B)=a(E,X,B)+t\operatorname{mult}_{E}B\ge t\operatorname{mult}_{E}B,\end{align*} $$
and
$\operatorname {mult}_{E}B\le \frac {1}{t}\mathrm {mld}(X\ni x,0).$
By assumption,
$\mathrm {mld}(X\ni x,0)\ge \mathrm {mld}(X\ni x,B)=a(E,X,B)=a(E,X,0)-\operatorname {mult}_{E}B$
. Hence:
$$ \begin{align*}a(E,X,0)\le \left(1+\frac{1}{t}\right)\mathrm{mld}(X\ni x,0),\end{align*} $$
and Conjecture 1.1 holds.
Suppose that Conjecture 1.1 holds, then there exists a positive real number N which only depends on
${\Gamma }$
and
$X\ni x$
, such that
$a(E_0,X,0)\le N$
for some
$E_0$
satisfying
$a(E_0,X,B)=\mathrm {mld}(X\ni x,B)$
. In particular,
$\operatorname {mult}_{E_0}B=-a(E_0,X,B)+a(E_0,X,0)\le N$
.
By [Reference Kawakita12, Theorem 1.1], there exists a positive real number
$\delta $
, which only depends on
${\Gamma }$
and
$X\ni x$
, such that
$a(E,X,B)\ge \mathrm {mld}(X\ni x,B)+\delta $
for any
$(X\ni x,B)$
and prime divisor E over
$X\ni x$
, such that
$B\in {\Gamma }$
and
$a(E,X,B)>\mathrm {mld}(X\ni x,B)$
.
We may assume that
${\Gamma }\setminus \{0\}\neq \emptyset $
. Let
$t:=\frac {\delta }{2N}$
and
$\tau :=t\cdot \min \{{\Gamma }\setminus \{0\}\}$
. We claim that
$\tau $
has the required properties.
For any
$\mathbb {R}$
-divisor
$B'$
, such that
$||B'-B||<\tau $
, we have
$B-B'<tB$
. Let E be a prime divisor over
$X\ni x$
, such that
$a(E,X,B')=\mathrm {mld} (X\ni x,B')$
. Suppose that
$a(E,X,B)>\mathrm {mld}(X\ni x,B)$
. Then
$a(E,X,B)\ge \mathrm {mld}(X\ni x,B)+\delta =a(E_0,X,B)+\delta $
. Thus:
$$ \begin{align*} a(E_0,X,0)-\operatorname{mult}_{E_0}B'&=a(E_0,X,B')\ge a(E,X,B')\ge a(E,X,B)\\ &\ge a(E_0,X,B)+\delta=a(E_0,X,0)-\operatorname{mult}_{E_0}B+\delta. \end{align*} $$
Hence,
$\delta \le \operatorname {mult}_{E_0}(B-B')\le t\operatorname {mult}_{E_0} B\le tN=\frac {\delta }{2},$
a contradiction.
Proposition 6.5. For any fixed
$\mathbb {Q}$
-Gorenstein germ
$X\ni x$
over an algebraically closed field of characteristic zero, Conjecture 1.1 implies Conjecture 6.2 holds for any DCC set
${\Gamma }$
when B has only one component.
Proof. Otherwise, there exist two sequence of germs
$\{(X\ni x,t_iB_i)\}_i$
and
$\{(X\ni x,t_i'B_i)\}_i$
, and prime divisors
$E_i$
over
$X\ni x$
, such that:
-
• for each i,
$B_i$
is a prime divisor on X,
$t_i\in {\Gamma }$
is increasing and
$t_i'<t_i$
, -
•
$t:=\lim _{i\to \infty } t_i=\lim _{i\to \infty }t_i'$
, -
•
$a(E_i,X, t_i'B_i)=\mathrm {mld}(X\ni x, t_i'B_i)$
and -
•
$a(E_i,X,t_iB_i)\neq \mathrm {mld} (X\ni x,t_iB_i)$
.
By Proposition 6.3, there exists a positive real number
$\tau $
, which only depends on t and
$X\ni x$
, such that if
$|t-t_i'|<\tau $
, then
$a(E_i,X,tB_i)=\mathrm {mld}(X\ni x,tB_i)$
. Since
$a(E_i,X,t_{*}B_i)$
is a linear function with respect to the variable
$t_{*}$
, we have
$a(E_i,X,t_iB_i)=\mathrm {mld}(X\ni x,t_iB_i)$
, a contradiction.
To end this section, we ask the following:
Conjecture 6.6. Let
${\Gamma }\subseteq [0,1]$
be a finite set and
$X\ni x$
a klt germ. Then there exists a positive real number
$\tau $
depending only on
${\Gamma }$
and
$X\ni x$
satisfying the following.
Assume that:
-
(1)
$B'\le B$
,
$||B-B'||<\tau ,B\in {\Gamma }$
and -
(2)
$K_X+B'$
is
$\mathbb {R}$
-Cartier.
Then
$K_X+B$
is
$\mathbb {R}$
-Cartier.
Acknowledgements
We would like to thank Guodu Chen, Jihao Liu, Wenfei Liu, Yuchen Liu and Yusuke Nakmura for their interests and and helpful comments. We would also like to thank Paolo Cascini, Chenyang Xu and Lei Zhang for answering our questions about characteristic p. The first named author was supported by a grant from the Simons Foundation (Grant number 814268, Mathematical Sciences Research Institute (MSRI)) and Start-up Grant no. JIH1414011Y of Fudan University. We thank the referees for useful suggestions and comments.
Competing Interests
None.
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