Proof. The proof is similar to that of [Reference Budur, Fernández de Bobadilla, Lê and Nguyen7, Lemma 2.1, Lemma 2.6] for restricted contact loci. The differences in the proof are as follows. Unlike for restricted contact loci, one does not need to consider the finite unramified covers $\tilde E_i^\circ \to E_i^\circ $ from the proof of [Reference Budur, Fernández de Bobadilla, Lê and Nguyen7, Lemma 2.6]. Thus, ‘smooth variety’ in the statement of [Reference Budur, Fernández de Bobadilla, Lê and Nguyen7, Lemma 2.6] means possibly finitely many disjoint copies of an irreducible smooth variety, while in our case, we have irreducibility of ${\mathscr X}^l_{m,i}$ . Another difference is that the varieties ${\mathscr X}^l_{m,i}$ have dimension one more than the their version for restricted contact loci. The extra coordinate takes any value in ${\mathbb {C}}^*$ , this being the coefficient of $t^{k_i}$ in the last line of the proof of [Reference Budur, Fernández de Bobadilla, Lê and Nguyen7, Lemma 2.6]. This identifies the analog of the morphism $\tilde {\pi }_0^l$ from that proof, up to homotopy, with the ${\mathbb {C}}^*$ -bundle on $E_i^\circ $ obtained by removing the zero section of the normal bundle.

Proof of Proposition 1.19

It follows directly from the previous proposition.

Proof. Suppose v has center a prime divisor E on some dlt modification $(Y',\Delta ')$ of $(X,D)$ such that E lies over $\Sigma $ . Note that the dlt modification restricts to an isomorphism $Y'\setminus \Delta '\xrightarrow {\sim }X\setminus D$ by Proposition 2.9. Take m divisible by $v(D)=\mathrm {ord}_ED$ . By sufficiently blowing up the klt locus of $(Y',\Delta ')$ , we obtain an m-separating log resolution $\mu :Y\to X$ ; see [Reference Budur, Fernández de Bobadilla, Lê and Nguyen7, Proof of Lemma 2.9]. Thus, v is an m-valuation. By Lemma 2.10, $(Y',\Delta ')$ is a minimal model of $(Y,\Delta )$ over X if $\Delta =(\mu ^*D)_{red}$ .

Proof. Suppose that $E_i$ is the birational strict transform of F, a prime divisor on another m-separating log resolution $\mu _1:Y_1\to X$ of $(X,D,\Sigma )$ such that F does not get contracted on minimal models of $(Y_1,\Delta _1=\mu _1^*(D)_{red})$ over X. By performing further blowups of smooth centers, we can find a log resolution $\mu _2:Y_2\to X$ which factors through both $\mu $ and $\mu _1$ and is an isomorphism over $X\setminus D$ . By blowing up further, we can assume that $\mu _2$ is m-separating, by [Reference Budur, Fernández de Bobadilla, Lê and Nguyen7, Proof of Lemma 2.9]. Then Lemma 2.11 implies first that the strict transform of F in $Y_2$ does not get contracted on minimal models of $(Y_2,\Delta _2=\mu _2^*(D)_{red})$ , and then that $E_i$ does not get contracted on minimal models of $(Y,\Delta )$ .

Proof of Theorem 1.13

Only the first inclusion in (1.2) needs to be proven. We fix a dlt m-valuation of $(X,D,\Sigma )$ . By Lemma 3.5, it is given by a divisor $E_i$ on some m-separating log resolution Y for some fixed $i\in S_m$ , such that $E_i$ does not get contracted on any minimal model of $(Y,\Delta )$ over X, with $\Delta =\mu ^*(D)_{red}=\sum _{k\in S}E_k$ . We want to show that the Zariski closure $ \overline {{\mathscr X} _{m,i}} $ is an irreducible component of ${\mathscr X} _m$ . Suppose that it is not. Then there exists $j\in S_m\setminus \{i\}$ such that $ {\mathscr X} _{m,i}\subset \overline {{\mathscr X} _{m,j}} $ , and the latter is an irreducible component of ${\mathscr X} _m$ , by the decomposition (1.1). In particular, the codimension of ${\mathscr X} _{m,i}$ is strictly bigger than the codimension of ${\mathscr X} _{m,j}$ . This is equivalent to

(3.2) $$ \begin{align} a(E_i,X,D){m}/{N_i}> a(E_j,X,D){m}/{N_j} \end{align} $$

since $a(E_k,X,D)=\nu _k-N_k$ and $\mathrm {codim}\; {\mathscr X} _{m,k}=m\nu _k/N_k$ for all $k\in S_m$ by Proposition 3.2.

We fix a minimal model $(Y',\Delta ',\phi ,\mu ')$ of $(Y,\Delta )$ over X as in (2.1). We have that $E_i$ is not $\phi $ -exceptional. From now on, we will denote by $E^{\prime }_k$ the image $\phi _*E_k$ for $k\in S$ , and by $S'=\{k\in S\mid E^{\prime }_k\neq 0\}$ . Thus, $\Delta '=\phi _*\Delta =(\mu ')^*(D)_{red}=\sum _{k\in S'}E^{\prime }_k$ .

Pick an arc $\gamma \in {\mathscr X} _{m,i}$ . By assumption, $\gamma $ is a limit of arcs in ${\mathscr X} _{m,j}$ . Using the curve selection lemma (see [Reference Budur, Fernández de Bobadilla, Lê and Nguyen7, Proof of Lemma 2.5]), one can assume that there exists a complex analytic surface germ

$$ \begin{align*}{\alpha}: ({\mathbb{C}}^2,(0,0)) \to (X,\gamma(0)),\quad (t,s)\mapsto {\alpha}(t,s) \end{align*} $$

such that ${\alpha }_0(t):={\alpha }(t,0)=\gamma (t)$ , and ${\alpha }_s(t):={\alpha }(t,s)$ is an arc lifting through $E_j^\circ $ for all $s\neq 0$ . By assumption, the induced meromorphic map $\tilde {\alpha }:{\mathbb {C}}^2\dashrightarrow Y$ has 2-dimensional image and has indeterminacy. Since $E_i$ is not $\phi $ -exceptional, the composition $\tilde {\alpha }'=\phi \circ \tilde {\alpha }:{\mathbb {C}}^2\dashrightarrow Y'$ also has 2-dimensional image and has indeterminacy. Using a sequence of blowups, one resolves simultaneously the indeterminacy of $\tilde {\alpha }$ and $\tilde {\alpha }'$ to obtain a commutative diagram

Here, we work with a small neighborhood of the origin in ${\mathbb {C}}^2$ .

We write $L_s$ for the line $\{(t,s)\mid t\in {\mathbb {C}}\}$ in ${\mathbb {C}}^2$ . Denote by $\sigma ^{-1}(0)=\cup _{a\in A}F_a$ the irreducible components of the exceptional divisor of $\sigma $ . Then we have rationally equivalent cycles

$$ \begin{align*}\sigma^*L_s=\tilde{L}_s\; (\text{for }s\neq 0)\quad \text{and}\quad \sigma^*L_0 = \tilde{L}_0+\sum_{a\in A}b_aF_a, \end{align*} $$

for some integers $b_a> 0$ and $F_a\ \sigma $ -exceptional prime divisors, where $\tilde {L}_s$ denotes the strict transform.

Let G be the strict transform under $\sigma $ of the s-axis in ${\mathbb {C}}^2$ . Then $\beta (G)\subset E_j$ since $E_j$ is proper and $\tilde {\alpha }_s(0)\in E_j^\circ $ for generic small $s \neq 0$ . The inverse image in Z under $\beta $ of any subset of the support of $\Delta $ is contained in $(\cup _aF_a)\cup G$ . This is because ${\alpha }^{-1}(D)$ is contained in the s-axis, shrinking to a smaller neighborhood of $0\in {\mathbb {C}}^2$ if necessary, and Z factors through ${\mathbb {C}}^2\times _XY$ – similarly for $\beta ^{\prime -1}$ of subsets of the support of $\phi _*\Delta $ .

Consider the ${\mathbb {Q}}$ -divisor on $Y'$

$$ \begin{align*}W :=\sum_{k\in S'} a(E_k',X,D)E^{\prime}_k= \sum_{k\in S'} a(E_k,X,D)E^{\prime}_k. \end{align*} $$

Then W is $\mu '$ -exceptional, and there is a ${\mathbb {Q}}$ -linear equivalence of ${\mathbb {Q}}$ -Cartier ${\mathbb {Q}}$ -divisors

$$ \begin{align*}W = K_{Y'}+\Delta' -\mu^{\prime*}(K_X+D) \end{align*} $$

by the definition of log discrepancy. As above, $\beta ^{\prime *}W$ is supported on $(\cup _aF_a)\cup G$ . The intersection products

(3.3) $$ \begin{align}\sigma^*L_s\cdot \beta^{\prime*}W = \sigma^*L_0\cdot \beta^{\prime *}W\end{align} $$

are well-defined and are equal for all small s since the same is true for the intersection with the compact $F_a$ , while for the non-compact G, $\sigma ^*L_s\cdot G=1$ for all s by the projection formula. We will derive a contradiction from computing both sides. For the right-hand side, we use the nefness property of $K_{Y'}+\Delta '$ . For the left-hand side, we use the property that log discrepancies do not decrease on relative minimal models.

The right-hand side is

$$ \begin{align*}\sigma^*L_0\cdot \beta^{\prime *}W = \beta^{\prime}_*\sigma^*L_0\cdot W = \beta^{\prime}_*\tilde L_0 \cdot W + \sum_{a\in A}b_a\beta^{\prime}_*F_a\cdot W. \end{align*} $$

Since $E_i$ is not $\phi $ -exceptional, $\beta ^{\prime }_*\tilde L_0\cdot E_k' =0$ if $k\neq i$ and $\beta ^{\prime }_*\tilde L_0\cdot E_i' =\mathrm {ord}_{\gamma }(E_i)=m/N_i$ . Thus, $\beta ^{\prime }_*\tilde L_0 \cdot W=a(E_i,X,D){m}/{N_i}$ . Further, since $\beta ^{\prime }_*F_a$ is contained in the $\mu '$ -exceptional locus of $Y'$ , $\beta ^{\prime }_*F_a\cdot \mu ^{\prime *}(K_X+D)=0$ . Thus, $\beta ^{\prime }_*F_a\cdot W=\beta ^{\prime }_*F_a\cdot (K_{Y'}+\Delta ')$ , and this is $\ge 0$ by the $\mu '$ -nefness of $K_{Y'}+\Delta '$ . Since $b_a\ge 0$ , we have obtained that

(3.4) $$ \begin{align} \sigma^*L_0\cdot \beta^{\prime *}W \ge a(E_i,X,D){m}/{N_i}. \end{align} $$

However, for $s\neq 0$ ,

$$ \begin{align*}\sigma^*L_s\cdot \beta^{\prime *}W = \tilde L_s\cdot \beta^{\prime *}W. \end{align*} $$

Note that in general it is not necessarily true for divisors that $\beta ^*\circ \phi ^*=\beta ^{\prime *}$ since $\phi $ is only a birational map and not a morphism; see [Reference Kollár26, 1.20]. However, $\phi $ is a morphism in codimension 1. Hence, $\phi $ is a morphism on a neighborhood in Y of a general point of $E_j$ . We can assume the center of the arc $\tilde {\alpha }_s$ is such a general point of $E_j^\circ $ . Then

$$ \begin{align*}\tilde L_s\cdot \beta^{\prime *}W = \beta_*L_s\cdot\phi^*W.\end{align*} $$

The only contribution to this product is from $E_j$ and its coefficient in $\phi ^*W$ . We can estimate it as follows. One has by the definition of log discrepancy that

$$ \begin{align*}K_Y+\Delta = \phi^*(K_{Y'}+\Delta') + \sum_{k\in S\setminus S'} a(E_k,Y',\Delta') E_k, \end{align*} $$

where the sum is over the $\phi $ -exceptional divisors in Y and

$$ \begin{align*}K_Y+\Delta =\mu^*(K_X+D) +\sum_{k\in S}a(E_k,X,D)E_k. \end{align*} $$

Note that on divisors $\phi ^*\circ \mu ^{\prime *}=\mu ^*$ since $\mu '$ is a morphism (see [Reference Kollár26, 1.20.1]); hence. $\mu ^*(K_X+D)=\phi ^*\mu ^{\prime *}(K_X+D)$ . Thus,

$$ \begin{align*}\phi^*W = \sum_{k\in S} (a(E_k,X,D) - a(E_k,Y',\Delta'))E_k \end{align*} $$

since $a(E_k,Y',\Delta ')=0$ if $k\in S'$ . Hence,

$$ \begin{align*}\tilde L_s\cdot \beta^{\prime *}W = (a(E_j,X,D) - a(E_j,Y',\Delta')){m}/{N_j}. \end{align*} $$

By the definition of minimal models of $(Y,\Delta )$ over X, $ a(E_j,Y',\Delta ')\ge a(E_j,Y,\Delta ) =0. $ We have thus obtained that

$$ \begin{align*}\sigma^*L_s\cdot \beta^{\prime *}W\le a(E_j,X,D){m}/{N_j}. \end{align*} $$

From (3.3) and (3.4), this implies $ a(E_j,X,D){m}/{N_j} \ge a(E_i,X,D){m}/{N_i}, $ which contradicts (3.2).

Proof of Theorem 1.16

Step 1: Preparation. We can assume that $E_i$ is $\mu $ -exceptional since otherwise the claim is true. It is enough to show that $E_i$ does not get contracted on a minimal model of $(Y,\Delta )$ over X, where $\Delta =\mu _*^{-1}D + \mathrm {Ex}(\mu )$ . Hence, it is enough to show that $E_i$ does not get contracted after any step of the minimal model program for $(Y,\Delta )$ over X. Here, by a step of the minimal model program, we mean an extremal-ray contraction followed by a flip if necessary. Let $\mu ':(Y',\Delta ')\to X$ be obtained after a succession of steps, with $\Delta '=\phi _*\Delta ={\mu ^{\prime }_*}^{-1}D+\mathrm { Ex}(\mu ')$ and $\phi :Y \dashrightarrow Y'$ is the natural birational map. Assume that $(Y',\Delta ')$ is the last step before $E_i$ gets contracted. Thus, $E_i'=\phi _*(E_i)\neq 0$ contains an open dense subset covered by curves of numerical class C such that $(K_{Y'}+\Delta ')\cdot C< 0$ . We will derive a contradiction.

We do not need Y anymore, but we need some log resolution of $(X,D,\Sigma )$ which factors through $\mu '$ and on which the valuation defined by $E_i$ has a prime center to compare contributions to jumping numbers. To avoid introducing new notation, by taking a common log resolution, we can assume that $\phi :Y\to Y'$ is a birational projective morphism.

Step 2: Multiplier ideals are computed by $\mu '$ . Since $(Y',\Delta ')$ is a log canonical pair, the multiplier ideals ${\mathcal {J}}(X,{\lambda } D)$ are computed via $\mu '$ by a result of Smith-Tucker [Reference Baumers and Veys3, Theorem A.2] – that is,

(3.5) $$ \begin{align} {\mathcal{J}}(X,{\lambda} D) =\mu^{\prime}_*{\mathcal{O}}_{Y'}(K_{Y'/X}-\lfloor {\lambda}\mu^{\prime *}D\rfloor) \end{align} $$

for every ${\lambda } \in {\mathbb {R}}_{>0}$ . Here, $K_{Y'/X}$ is the unique exceptional divisor ${\mathbb {Q}}$ -linearly equivalent to $K_{Y'}-\mu ^{\prime *}K_X$ . It has integral coefficients since X is smooth. We recall the proof since we will need an intermediate step. Since there is an equality of divisors $ \phi _*(K_{Y/X}-\lfloor {\lambda }\mu ^*D\rfloor )=K_{Y'/X}-\lfloor {\lambda }\mu ^{\prime *}D\rfloor , $ there is an inclusion of sheaves $ \phi _*{\mathcal {O}}_Y(K_{Y/X}-\lfloor {\lambda }\mu ^*D\rfloor )\subset {\mathcal {O}}_{Y'}(K_{Y'/X}-\lfloor {\lambda }\mu ^{\prime *}D\rfloor ). $ We will show

(3.6) $$ \begin{align} \phi_*{\mathcal{O}}_Y(K_{Y/X}-\lfloor {\lambda}\mu^*D\rfloor) = {\mathcal{O}}_{Y'}(K_{Y'/X}-\lfloor {\lambda}\mu^{\prime *}D\rfloor). \end{align} $$

Taking direct image under $\mu '$ , we then get (3.5).

Let g be a section of ${\mathcal {O}}_{Y'}(K_{Y'/X}-\lfloor {\lambda }\mu ^{\prime *}D\rfloor )$ – that is,

(3.7) $$ \begin{align} \operatorname{\mathrm{div}}(g) +K_{Y'/X}-\lfloor {\lambda}\mu^{\prime *}D\rfloor\ge 0. \end{align} $$

Pick ${\epsilon }\in {\mathbb {Q}}_{>0}$ small enough such that $\Delta -{\epsilon }\mu ^*({\lambda } D)\ge \{\mu ^*({\lambda } D)\}$ . Here, $\{\_\}$ means taking the fractional part of each coefficient. This also implies that $\Delta ' -{\epsilon }\mu ^{\prime *}({\lambda } D)\ge \{\mu ^{\prime *}({\lambda } D)\}$ by applying $\phi _*$ . Then $\Delta -{\epsilon }\lfloor \mu ^*({\lambda } D)\rfloor - (1+{\epsilon })\{\mu ^*({\lambda } D)\}\ge 0, $ and so

(3.8) $$ \begin{align} \Delta' -{\epsilon}\lfloor \mu^{\prime *}({\lambda} D)\rfloor - (1+{\epsilon})\{\mu^{\prime *}({\lambda} D)\}\ge 0. \end{align} $$

Adding (3.7) and (3.8), we get $ \operatorname {\mathrm {div}}(g) + K_{Y'/X}+\Delta ' - (1+{\epsilon })\mu ^{\prime *}({\lambda } D)\ge 0. $ The left-hand side is ${\mathbb {Q}}$ -Cartier; hence, we can pull it back via $\phi $ . We obtain

$$ \begin{align*}\operatorname{\mathrm{div}}(g\circ\phi) - \mu^*K_X +\phi^*(K_{Y'}+\Delta')- (1+{\epsilon})\mu^*({\lambda} D)\ge 0. \end{align*} $$

Since $(Y',\Delta ')$ has log canonical singularities, $K_{Y}+\Delta \ge \phi ^*(K_{Y'}+\Delta ')$ , and hence,

$$ \begin{align*}\operatorname{\mathrm{div}}(g\circ\phi) - \mu^*K_X +K_{Y}+\Delta- (1+{\epsilon})\mu^*({\lambda} D)\ge 0. \end{align*} $$

Taking integral parts, it follows that $ \operatorname {\mathrm {div}}(g\circ \phi ) + K_{Y/X}-\lfloor \mu ^*({\lambda } D)\rfloor \ge 0, $ – that is, g is a section of $\phi _*{\mathcal {O}}_Y(K_{Y/X}-\lfloor {\lambda }\mu ^*D\rfloor ) $ . This proves (3.6).

Step 3: Contribution to jumping numbers can be read from $\mu '$ . Assume now that $E_i$ contributes with the jumping number ${\lambda }$ of $(X,D)$ . We show that $E_i'$ also does – that is,

(3.9) $$ \begin{align} {\mathcal{J}}(X,{\lambda} D) \subsetneq \mu^{\prime}_*{\mathcal{O}}_{Y}(K_{Y'/X}-\lfloor {\lambda}\mu^{\prime *}D\rfloor +E_i'). \end{align} $$

Since there is an equality of divisors $ \phi _*(K_{Y/X}-\lfloor {\lambda }\mu ^*D\rfloor +E_i)=K_{Y'/X}-\lfloor {\lambda }\mu ^{\prime *}D\rfloor +E_i', $ there is an inclusion of sheaves $ \phi _*{\mathcal {O}}_Y(K_{Y/X}-\lfloor {\lambda }\mu ^*D\rfloor +E_i)\subset {\mathcal {O}}_{Y'}(K_{Y'/X}-\lfloor {\lambda }\mu ^{\prime *}D\rfloor +E_i'). $ Taking direct image under $\mu '$ , we obtain $ \mu _*{\mathcal {O}}_Y(K_{Y/X}-\lfloor {\lambda }\mu ^*D\rfloor +E_i) \subset \mu ^{\prime }_*{\mathcal {O}}_{Y'}(K_{Y'/X}-\lfloor {\lambda }\mu ^{\prime *}D\rfloor +E_i'). $ By assumption, the first sheaf strictly contains ${\mathcal {J}}(X,{\lambda } D)$ . Then (3.9) follows.

Step 4: Local vanishing for $\mu '$ . Since $\mu =\mu '\circ \phi $ , the Grothendieck spectral sequence implies that there is a injective morphism of sheaves of ${\mathcal {O}}_X$ -modules

$$ \begin{align*}R^1\mu^{\prime}_*(\phi_*{\mathcal{O}}_Y(K_{Y/X}-\lfloor {\lambda}\mu^*D\rfloor))\subset R^1\mu_* {\mathcal{O}}_Y(K_{Y/X}-\lfloor {\lambda}\mu^*D\rfloor).\end{align*} $$

By local vanishing [Reference Kawamata, Matsuda and Matsuki24, Theorem 1.2.3], the last sheaf is $0$ ; hence, the first sheaf is also $0$ . By (3.6), this means that

(3.10) $$ \begin{align} R^1\mu^{\prime}_*{\mathcal{O}}_{Y'}(K_{Y'/X}-\lfloor {\lambda}\mu^{\prime *}D\rfloor) =0. \end{align} $$

Step 5: Non-vanishing. There is a short exact sequence of sheaves of ${\mathcal {O}}_{Y'}$ -modules

$$ \begin{align*}0\to {\mathcal{O}}_{Y'}(K_{Y'/X}-\lfloor {\lambda}\mu^{\prime *}D\rfloor) \to {\mathcal{O}}_{Y'}(K_{Y'/X}-\lfloor {\lambda}\mu^{\prime *}D\rfloor +E_i')\to {\mathcal{Q}}\to 0 \end{align*} $$

for some sheaf ${\mathcal {Q}}$ . Applying $\mu _*'$ and using (3.5), (3.9) and (3.10), we get a short exact sequence of sheaves of ${\mathcal {O}}_X$ -modules

$$ \begin{align*}0\to {\mathcal{J}}(X,{\lambda} D) \to \mu^{\prime}_*{\mathcal{O}}_{Y'}(K_{Y'/X}-\lfloor {\lambda}\mu^{\prime *}D\rfloor +E_i')\to \mu^{\prime}_*{\mathcal{Q}} \to 0 \end{align*} $$

such that the last term is nonzero. Assuming that X is affine, which we can since the problem is local on X, this implies that $ H^0(Y',{\mathcal {Q}})\neq 0 $ and

$$ \begin{align*}H^0(Y',{\mathcal{O}}_{Y'}(K_{Y'/X}-\lfloor {\lambda}\mu^{\prime *}D\rfloor)) \subsetneq H^0(Y',{\mathcal{O}}_{Y'}(K_{Y'/X}-\lfloor {\lambda}\mu^{\prime *}D\rfloor +E_i')). \end{align*} $$

Thus, there exists a rational function g on $Y'$ such that

$$ \begin{align*}P=\operatorname{\mathrm{div}}(g)+K_{Y'/X}-\lfloor {\lambda}\mu^{\prime *}D\rfloor+E_i'\ge 0,\end{align*} $$

but $P-E_i'$ is not effective. Therefore, $E_i'$ has coefficient $0$ in P, and P is linearly equivalent to $K_{Y'/X}-\lfloor {\lambda }\mu ^{\prime *}D\rfloor +E_i'$ .

Step 6: The contradiction. We have that $K_{Y'}+\Delta ' $ is ${\mathbb {Q}}$ -linearly equivalent over X to

$$ \begin{align*}K_{Y'/X}-\mu^{\prime *}({\lambda} D) +\Delta' = (K_{Y'/X}-\lfloor \mu^{\prime *}({\lambda} D)\rfloor +E_i') + (\Delta' -E_i'-\{\mu^{\prime *}({\lambda} D)\}). \end{align*} $$

Note that $R=\Delta ' -E_i'-\{\mu ^{\prime *}({\lambda } D)\}$ is effective, and its support does not contain $E_i'$ . By Step 1, we have $ 0>C\cdot (K_{Y'}+\Delta ')$ . Using the splitting from above and ${\mathbb {Q}}$ -factoriality of $Y'$ , we get $0>C\cdot (P+R)|_{E_i'},$ where P is as in Step 5. Note that the restriction of $P+R$ to $E_i'$ is a well-defined effective ${\mathbb {Q}}$ -Cartier ${\mathbb {Q}}$ -divisor. But the curves in $E_i'$ in the numerical equivalence class C cover an open subset of $E_i'$ ; hence, $C\cdot (P+R)|_{E_i'}\ge 0$ . This is a contradiction.

Proof. Let $\mu :Y\to X$ be the minimal log resolution of $(X,D,\Sigma )$ . If $E_i$ and $E_j$ denote two distinct irreducible components of $\mu ^{-1}(\operatorname {\mathrm {Supp}}(D))$ intersecting in a point P and such that $N_i+N_j\le m$ , then blowup the point P. The multiplicity of the new exceptional divisor is $N_i+N_j$ . Since $N_i, N_j$ are strictly positive, $N_i+N_j>\max \{N_i,N_j\}$ . We repeat this process until we obtain for the first time an m-separating log resolution $\mu _0:Y_0\to X$ . To show that the result is a minimal m-separating log resolution, take $\mu _1:Y_1\to X$ another m-separating log resolution. Let $\pi :Y_1\to Y$ be the morphism through which $\mu _1$ factors. Consider $\pi ^{-1}(P)$ . This is necessarily a divisor in $Y_1$ , connecting the strict transforms of $E_i$ and $E_j$ since $\mu _1$ is m-separating. Since $\pi $ is a succession of blowups at smooth points and since $\pi ^{-1}(P)$ is an exceptional divisor, it follows that $\pi $ factors through the blowup of P. Repeating the argument, it follows that $\mu _1$ factors through $\mu _0$ .

Notation 6.3. (Resolution graphs.) We denote by $\Gamma $ the resolution graph of $(X,D, \Sigma )$ ; see [Reference Brieskorn and Knörrer5]. Recall that $\Gamma $ is the dual graph of the divisor $(\mu ^*D)_{red}$ , where $\mu $ is the minimal log resolution of $(X,D,\Sigma )$ , together with certain data at each vertex which remembers the resolution process, and in particular, the orders of vanishing $N_i$ of each $E_i$ as in Definition 1.7.

For $m\ge 1$ , we denote by $\Gamma _m$ the resolution graph obtained from the minimal m-separating log resolution of $(X,D,\Sigma )$ .

We say that a resolution graph refines another resolution graph if it is obtained only by inserting inductively vertices $E_k$ on edges $E_iE_j$ with $i\neq j$ and such that $N_k=N_i+N_j$ .

The valence of a vertex in a graph is the number of edges connected to the vertex. A path in a graph is a connected sequence of edges with distinct end points. We say that a vertex lies on a path if it is any of the vertices on the path, end points included. A simple path is a path that does not pass twice through the same vertex.

Proof. (a) This is Grauert’s Contraction Theorem [Reference Barth, Peters and Van de Ven2, Theorem III.2.1] together with the standard description of Hirzebruch-Jung singularities; see, for example, [Reference Kollár26, Theorem 3.32] or [Reference Barth, Peters and Van de Ven2, III. 2.(ii) and Theorem III.5.1]. Note that this description is formulated for chains with no $(-1)$ -curves – that is, for minimal resolutions of cyclic singularities. Starting from an arbitrary negative definite chain of ${\mathbb {P}}^1$ ’s, contract all $(-1)$ -curves in this chain and its images. Then the image of the chain is either a smooth point or a chain as in the cited theorems.

(b) This is proved in [Reference Kollár27, 3.1.10]. One applies the formula in the cited theorem to the three possible cases concerning the ordering of the $T_i$ mentioned in Definition 6.8. For the two cases when T is a connected component, the outcome reads the same.

Proof. Write $\phi =q\circ \sigma \circ p$ , where $p\colon (Y,\Delta )\to (\bar {Y},\bar {B})$ is a contraction of some admissible twigs of $\Delta $ over D, q is a birational morphism, and $\sigma $ is an extremal ray contraction obtained by contracting a prime divisor $\bar {A}$ . Thus, $\bar {A}^2<0$ and $\bar {A}\cdot (K_{\bar {Y}}+\bar {B})<0$ . Write $a(p)=\sum _{E} a(E,\bar Y,\bar B) E,$ where the sum runs over all irreducible components E of $\mathrm {Ex}(p)$ , and $a(E,\bar Y,\bar B)$ is the log discrepancy. Put $B=p^{-1}_{*}\bar {B}$ , $A=p^{-1}_{*}\bar {A}$ . We have $0>\bar {A}^{2}=A\cdot p^{*}\bar {A}\geq A^{2}$ and

$$ \begin{align*}0>\bar{A}\cdot (K_{\bar{Y}}+\bar{B})=A\cdot (K_{Y}+\Delta-a(p))=-2+2p_{a}(A)+\beta_{B}(A)+A\cdot (\mathrm{Ex} (p)-a(p)).\end{align*} $$

Note that $p_{a}(A)\geq 0$ , and $\beta _{B}(A)\geq 1$ since the connected component of B containing A also contains a component of $\mu ^{-1}_{*}D$ . Moreover, by Proposition 6.9, we have ${A\cdot (\mathrm {Ex}(p)-a(p))=\sum _{i=1}^{n}\left (1-{1}/{d(T_{i})}\right )\geq 0,}$ where $T_1,\dots , T_n$ are the twigs of $\Delta $ meeting A and contracted by p. Note that $T_i$ are also the connected components of $\mathrm {Ex}(p)$ meeting A. It follows that $p_{a}(A)=0$ and $\beta _{B}(A)=1$ , so A is an admissible tip of B. Also, $\sum _{i=1}^{n} (1-1/d(T_{i}))<1$ . So we can assume that $d(T_{i})=1$ for $i>1$ – that is, $T_{2},\dots , T_{n}$ get contracted to smooth points. The image of $T_{1}+A$ after this contraction must then be an admissible twig of the image of $\Delta $ .

It follows that $\alpha \circ \upsilon $ factors through $p\circ \sigma $ . Applying this fact inductively, starting from $p=\mathrm {id}$ , we see that $\alpha \circ \upsilon $ factors through $\phi $ . Conversely, if A is an irreducible component of $\mathrm {Ex}(\alpha \circ \upsilon )-\mathrm {Ex}(\phi )$ , then reversing the above computation, we see that $\bar {A}=\phi _{*}A$ satisfies $\bar {A}^{2}<0$ , $\bar {A}\cdot (K_{Y'}+\Delta ')<0$ , which is impossible. Thus, $\phi =\alpha \circ \upsilon $ . This proves (a). Then (b) and (d) are easy consequences.

For (c), replace $(Y,\Delta )$ by $(\upsilon (Y),\upsilon _{*}\Delta )$ , and assume $\upsilon =\mathrm {id}$ . Let R be the sum of all components of $\Delta $ satisfying (1)–(3). We need to prove that $\Delta -R=B$ , where B is the sum of all admissible twigs of $\Delta $ over D. Let E be a component of $\Delta $ . Clearly, if E satisfies (1) or (2), then $E\subseteq \Delta -B$ . Otherwise, let T be a maximal subchain in the sense of inclusion of $\Delta $ containing E whose irreducible components do not satisfy (1) or (2). Then either T is as in (3), or one of the tips of T is a tip of $\Delta $ , and $T\subseteq \Delta -\mu ^{-1}_{*}D$ . Note that all irreducible components C of $\Delta -\mu ^{-1}_{*}D$ satisfy $C^2<0$ , $C\simeq \mathbb {P}^1$ . Since $\Delta -\mu ^{-1}_{*}D$ is negative definite, T is an admissible twig over D, so $E\subseteq B$ .

Proof. (a) Suppose that $\mu _1:Y_1\to X$ is another m-separating log resolution of $(X,D,\Sigma )$ and that $\mu _1=\mu \circ \pi $ for a morphism $\pi :Y_1\to Y$ . Suppose F is an irreducible component of $\Delta _1=(\mu _1^*D)_{red}$ such that $\mu _1(F)\subset \Sigma $ , $\mathrm {ord}_F(D)$ divides m, and F does not get contracted on the minimal model of $(Y_1,\Delta _1)$ over X. We can assume that $\pi $ is a composition of blowups of singular points of preimages of D since blowing up smooth points of preimages of D does not introduce exceptional divisors which survive on the relative minimal model by Proposition 6.10. If F is $\pi $ -exceptional, then there must exist two distinct irreducible components $E_1$ and $E_2$ of $\Delta $ such that $\pi (F)=E_1\cap E_2$ . But this implies that $\mathrm {ord}_F(D)\ge N_1+N_2>m$ cannot divide m, which is a contradiction. Thus, F is not $\pi $ -exceptional. Hence, F gives an essential m-valuation of $(X,D,\Sigma )$ .

(b) If $E\subset \Delta $ corresponds to a dlt m-valuation of $(X,D,\Sigma )$ , then $E=\pi (F)$ with $\pi $ and F as above. By Proposition 6.10, F must satisfy one of the conditions (1)–(3) with $(\mu _1,Y_1,\Delta _1)$ replacing $(\mu , Y,\Delta )$ . It is easy to see that this is equivalent to E satisfying one of the conditions (1)–(3) for $(\mu , Y,\Delta )$ . Thus, $E=\pi (F)$ does not get contracted on the minimal model of $(Y,\Delta )$ and the conditions (1)–(3) characterize such E. Alternatively, one also sees that E does not get contracted on the minimal model of $(Y,\Delta )$ by applying the more general Lemma 3.5.

Proof. It only remains to show that each of the sets in the decomposition – that is, $Z_j$ with $j \in \{1,\ldots ,g\}$ and ${\mathscr X}_{m,i}$ with $i \in S^{\prime }_m$ – is closed in the Zariski topology. By [Reference Ein, Lazarsfeld and Mustaţă15, Theorem A], each of these subsets is a cylinder in ${\mathscr L}({\mathbb {C}}^2,0)$ . Thus, for $l \gg 0$ , and are constructible subsets in ${\mathscr L}_l({\mathbb {C}}^2,0)$ . Moreover, $Z_j$ (resp. ${\mathscr X}_{m,i}$ ) is closed in ${\mathscr L}({\mathbb {C}}^2,0)$ if and only if $Z^l_j$ (resp. ${\mathscr X}^l_{m,i}$ ) is closed in ${\mathscr L}_l({\mathbb {C}}^2,0)$ . Therefore, it is enough to prove that each $Z^l_j$ and ${\mathscr X}^l_{m,i}$ is closed in the analytic topology of ${\mathscr L}_l({\mathbb {C}}^2,0)$ .

Arguing by contradiction, we suppose there are divisors $E_i$ and $E_{i'}$ such that ${\mathscr X}^l_{m,i}$ and ${\mathscr X}^l_{m,i'}$ are contained in different sets of the decomposition (7.1) and ${\mathscr X}^l_{m,i} \cap \overline {{\mathscr X}^l_{m,i'}}\neq \varnothing $ . Since ${\mathscr X}^l_{m,i}$ and ${\mathscr X}^l_{m,i'}$ are constructible, we may apply the Curve Selection Lemma to obtain a ${\mathbb {C}}$ -analytic map germ $\alpha : ({\mathbb {C}},0) \to {\mathscr X}^l_m$ such that $\alpha (0) \in {\mathscr X}^l_{m,i}$ and $\alpha (s) \in {\mathscr X}^l_{m,i'}$ for $s \neq 0$ . Note that since the l-jet space is finite-dimensional, the classical version of the Curve Selection Lemma is used here, not the more delicate version that appears often in the context of the infinite-dimensional arc spaces. Using the zero section of the truncation map ${\mathscr L}({\mathbb {C}}^2,0) \to {\mathscr L}_l({\mathbb {C}}^2,0)$ , we may regard $\alpha $ as a ${\mathbb {C}}$ -analytic map germ to ${\mathscr L}({\mathbb {C}}^2,0)$ such that for every s, the entries of the arc $\alpha (s)$ are all polynomials in t.

Let f be an equation for C. Note that $f(\alpha (s))$ is a ${\mathbb {C}}$ -analytic map germ of polynomials of constant order m. Therefore, we can find a ${\mathbb {C}}$ -analytic map germ $u: ({\mathbb {C}},0) \to {\mathscr L}({\mathbb {C}},0)$ such that for every s, $u(s)$ is convergent and $f(\alpha (s)) = u(s)^m$ . To see this, we can write $u(s)$ explicitly by picking a holomorphic m-th root of the coefficient of $t^m$ in $f(\alpha (s))$ and using the general formulas relating the coefficients of both sides of $ (\sum _{k\ge 0}b_kt^k )^m= \sum _{k\ge 0}c_kt^k$ with $c_0\neq 0, $ – namely, $c_0=b_0^m$ , $c_k=\frac {1}{b_0}\sum _{i=1}^k(im-k+i)b_ic_{k-i}$ for $k\ge 1$ . This shows that u is indeed analytic. Also note that $\mathrm {ord}_t(u(s)) = 1$ , so we can make the change of variables $(\tilde {s}, \tilde {t}) = (s, u(s)(t))$ , and then $f(\alpha (\tilde {s}, t(\tilde {s},\tilde {t}))) = \tilde {t}^m$ . In summary, after a reparametrization we can suppose that for every s, $\alpha (s)$ is a convergent arc with $f(\alpha (s)) = t^m$ and such that $\alpha (0) \in {\mathscr X}^l_{m,i}$ and $\alpha (s) \in {\mathscr X}^l_{m,i'}$ for $s \neq 0$ .

We can assume that ${\alpha }(s)$ is defined for $s\in [0,1]$ . For any non-constant arc $\gamma \in {\mathscr L}({\mathbb {C}}^2,0)$ , denote by $\tilde {\gamma }$ the lift of $\gamma $ to the resolution. Let $p_0 \in E_i^{\circ }$ and $p_1 \in E_{i'}^{\circ }$ be the centers of $\tilde {{\alpha }}(0)$ and $\tilde {{\alpha }}(1)$ , respectively, where . There exist coordinate charts $(U_0,\psi _0)=(z_1^{(0)},z_2^{(0)})$ and $(U_1,\psi _1)=(z_1^{(1)},z_2^{(1)})$ at $p_0$ and $p_1$ , respectively, such that $\psi _0(p_0) = (0,0)$ and $f \circ \mu |_{U_0}(z_1^{(0)}, z_2^{(0)}) = (z_1^{(0)})^{N_i}$ – and analogously for $(U_1,\psi _1)$ . After possibly reducing the radius of convergence of the arcs in the family $\alpha $ , we may assume that $U_0$ and $U_1$ contain the images of $\tilde {{\alpha }}(0)$ and $\tilde {{\alpha }}(1)$ , respectively.

We write $\tilde {{\alpha }}(0)$ in the local coordinates of $(U_0,\psi _0)$ as $\tilde {{\alpha }}(0)(t)=(a_1(t), a_2(t))$ . Note that $a_1(t)^{N_i} = f(\alpha (0)(t)) = t^m$ , and therefore $a_1(t) = t^{m/N_i}$ , after possibly multiplying the coordinate $z_1^{(0)}$ by a root of unity, which does not affect the properties of $\psi _0$ . Hence, $(a_1(t), \lambda a_2(t))$ with $\lambda \in [0,1]$ defines a continuous path in ${\mathscr X}_m$ joining $\tilde {\alpha }(0)$ and the arc which in the coordinates of $(U_0,\psi _0)$ is given by $(t^{m/N_i}, 0)$ . After a similar construction for $\alpha (1)$ and pushing the paths to ${\mathbb {C}}^2$ via $\mu $ , we can concatenate and obtain a path $\beta : [0,1] \to {\mathscr X}_m$ such that for every s, $\beta (s)$ is convergent and $f(\beta (s)) = t^m$ , and in the local coordinates of $(U_0,\psi _0)$ and $(U_1,\psi _1)$ , we have $\tilde {\beta }(0) = (t^{m/N_i},0)$ and $\tilde {\beta }(1)=(t^{m/N_{i'}},0)$ , respectively. From now on, we write $\beta (s,t)$ for $\beta (s)(t)$ .

Let $0 < \delta \ll \varepsilon \ll 1$ be Milnor radii for f. Denote by $B_\varepsilon $ the open ball in ${\mathbb {C}}^2$ centered at the origin of radius $\varepsilon $ and by $D_\delta $ the open disk in ${\mathbb {C}}$ centered at the origin of radius $\delta $ . We pull-back the Milnor fibration to Y via $\mu $ – that is, we put and consider the map $f \circ \mu \colon Y'\to D_\delta $ . Its restriction over $S^1_\delta $ is a locally trivial fibration isomorphic to the Milnor fibration (in the tube).

We now recall a construction of a topological model for this fibration, given by A’Campo in [Reference A’Campo1, §2]. Let $\tau \colon Y_{\log }\to Y'$ be a real oriented blowup along $(f\circ \mu )^{-1}(0)$ . Let $\eta \colon A\to Y_{\log }$ be a continuous map which is an identity over $\bigsqcup _{k}\tau ^{-1}(E_{k}^{\circ })$ , and for each pair $E_{k},E_{k'}$ it replaces $\tau ^{-1}(E_{k}\cap E_{k'})$ by $\tau ^{-1}(E_{k}\cap E_{k'})\times [0,1]$ ; see [Reference A’Campo1, p. 239]. The resulting topological space A is a manifold with boundary and corners, equipped with a continuous map and a locally trivial fibration $f_{A}\colon \partial A\to S^{1}$ which is topologically equivalent to the Milnor fibration in the tube (i.e., admits a homeomorphism $h\colon \mu ^{-1}(B_\varepsilon \cap f^{-1}(S^1_\delta ))\to \partial A$ such that $f_{A}\circ h=\frac {1}{\delta }\cdot f\circ \mu $ ). Let $(\phi _\theta \colon {\mathbb {F}}_1 \to {\mathbb {F}}_{e^{2\pi \imath \theta }})_{\theta \in {\mathbb {R}}}$ , where , be the monodromy trivialization of $f_{A}$ . We now list some useful properties of the decomposition $\partial A=\bigsqcup _{k} \pi _{A}^{-1}(E_{k}^{\circ })\sqcup \bigsqcup _{k,k'} \pi _{A}^{-1}(E_{k}\cap E_{k'})$ ; see Figure 2.