Proof. Assume the set of the images of the $g_i$ is not minimal as a generating set. Then w.l.o.g. $g_1$ can be expressed as $g_1 = \sum _{j \geq 2} b^0_j g_j + f$ , where $f \in I_+^2$ and $b^0_j \in A_0$ .

Since the $g_1, \ldots , g_m$ generate A as a $A_0$ -algebra we may write f as a polynomial in $g_1, \ldots , g_m$ with coefficients in $A_0$ . Now for $\alpha \in \mathbb {Z}_{\geq 0}^m$ we set $g^\alpha =g_1^{\alpha _1} \cdots g_m^{\alpha _m}$ and $|\alpha |=\sum _j \alpha _j$ and we write

$$\begin{align*}f= \sum_\alpha b^0_\alpha g^{\alpha}.\end{align*}$$

After restricting to the homogeneous part of degree $\deg (g_1)$ of the equation $g_1 = \sum _{j \geq 2}b^0_j g_j + \sum _\alpha b^0_\alpha g^\alpha $ , we may assume that $b_j^0 = 0$ for $\deg (g_j) \neq \deg (g_1)$ and $b^0_\alpha = 0$ for $\deg (g^\alpha ) \neq \deg (g_1)$ . From the fact that $f \in I_+^2$ it follows that $|\alpha |> 1$ for $b^0_\alpha \neq 0$ . Now, if $\alpha _1 \neq 0$ and $|\alpha |> 1$ , then $\deg (g^{\alpha })> \deg (g_1)$ , since the degrees of the $g_i$ are all positive. Therefore, the coefficient $b^0_\alpha $ must be $0$ in this case. Hence,

$$\begin{align*}g_1 = \sum_{j\geq 2} b^0_jg_j + \sum_\alpha b^0_\alpha g^{\alpha} \end{align*}$$

expresses $g_1$ as a polynomial in $g_2,\ldots ,g_m$ with coefficients in $A_0$ , but this contradicts the minimality of $\{g_1,\ldots , g_m\}$ as a set of $A_0$ -algebra generators for A.

Proof. Assume $\mathcal {X}=\operatorname {\mathrm {Spec}} A \to \operatorname {\mathrm {Spec}} B$ . Then for an appropriate homomorphism $M \to \mathbb {Z}$ the B-algebra A is positively graded with $A_0=B$ and there is a minimal set of homogeneous generators $g_1 \ldots g_m$ . By Lemma 3.1 their images give a minimal generating set of $I_+/I^2_+$ . Now, it follows from Nakayama’s Lemma (or its graded version, respectively) that the images of the generating set form a basis $\bar {g}_1, \ldots ,\bar g_m$ of $T^*=(I_+/I^2_+) \otimes _B (B/\mathfrak {m})$ , the Zariski cotangent space of C at the vertex $0$ . Hence, we have an equivariant embedding $\mathcal {X} \hookrightarrow T \times \operatorname {\mathrm {Spec}} B$ , induced by the B-algebra homomorphism

$$\begin{align*}B \otimes_{\mathbb{C}} \operatorname{Sym}(T^*) \to A;\quad \bar g_i \mapsto g_i\\[-24pt]\end{align*}$$

Proof. This follows from Lemma 3.2 when taking B to be the graded local ring $\mathbb {C}[t]$ .

Proof. Recall that $X_0$ was a $\hat {\mathbb {T}}$ -fixed point of $\mathcal {H}$ . We may choose a $\hat {\mathbb {T}}$ -invariant open affine neighbourhood $U=\operatorname {\mathrm {Spec}} B$ of $X_0$ in $\mathcal {H}$ . Then $\psi =\psi _{\mathcal {X}}$ corresponds to a $\mathbb {Z}$ -graded homomorphism $\psi ^\# \colon B \to \mathbb {C}[t]$ . The scheme-theoretic image of $\psi $ in U is given by $C=\operatorname {\mathrm {Spec}}(B/\ker \psi ^\#)$ . By the definition of a special degeneration, $\mathbb {C}^*$ acts with weight $1$ on the base of $\mathcal {X}$ . It will therefore act with positive weight on $T_{X_0}C$ . Indeed, on the level of coordinate rings we have $\deg (t)=-1$ . Hence, $\mathbb {C}[t]$ is negatively graded with respect to $\hat v$ . Then $B/\ker \psi ^\#$ will be negatively graded as well. Hence, $\mathbb {C}^*$ also acts via $\hat v$ with negative weights on the cotangent space and therefore with positive weights on the tangent space. From this we deduce that $\langle u, \hat v \rangle> 0$ for $u \in \operatorname {\mathrm {supp}} T_{X_0}C$ and therefore $\hat v \in \operatorname {\mathrm {int}} \Sigma _{\mathcal {X}}$ as claimed in (ii).

Then applying Lemma 3.2 to the trivial family $C \to \operatorname {\mathrm {Spec}} \mathbb {C}$ gives an equivariant embedding

$$\begin{align*}C \hookrightarrow T_{X_0}C \subset T_{X_0}\mathcal{H}.\end{align*}$$

We have $\operatorname {\mathrm {supp}} T_{X_0}C \subset \operatorname {\mathrm {supp}} T_{X_0}\mathcal {H} \subset \ker (\hat M_{\mathbb {R}} \to M_{\mathbb {R}})$ . Consider now the image $\psi (1) \in C \hookrightarrow T_{X_0}C \subset T_{X,0}\mathcal {H}$ . Then we have $\operatorname {\mathrm {supp}}(\psi (1)) \subset \operatorname {\mathrm {supp}} T_{X_0}C$ . Now, $\psi (1)$ is fixed under the subtorus $\mathbb {T}'$ corresponding to the surjection $\hat M \to \hat M/\operatorname {\mathrm {span}}_{\mathbb {R}}(\operatorname {\mathrm {supp}}\psi (1))$ . Since $\psi (1)$ parametrizes $\mathcal {X}_1\cong X$ in the Hilbert scheme $\mathcal {H}$ it follows that X admits an action of this subtorus. Because X was assumed to be a $\mathbb {T}$ -variety in our strict sense, it follows that $\mathbb {T}=\mathbb {T}'$ and $\operatorname {\mathrm {span}}_{\mathbb {R}}(\operatorname {\mathrm {supp}}\psi (1)) = \ker (\hat M_{\mathbb {R}} \to M_{\mathbb {R}})$ . But then also $\operatorname {\mathrm {span}}_{\mathbb {R}}(\operatorname {\mathrm {supp}} T_{X_0}C) = \ker (\hat M_{\mathbb {R}} \to M_{\mathbb {R}})$ holds. This shows (i).

To prove (iii) we need to refine the ideas from above a little bit. We consider the $\hat T$ -invariant subspace

$$\begin{align*}W:=(\ker \psi^\#) \cap B_0 \oplus \bigoplus_{\langle u, \hat v \rangle>0} B_u \subset B.\end{align*}$$

Then $W \subset \ker \psi ^\#$ holds, since $\mathbb {C}[t]$ was negatively graded. Let $I \subset B$ be the $\hat M$ -graded ideal generated by W. Then $\psi \colon \mathbb A^1 \to \operatorname {\mathrm {Spec}} B$ factors through a $\hat T$ -equivariant closed embedding $S:=\operatorname {\mathrm {Spec}} B/I \hookrightarrow \operatorname {\mathrm {Spec}} B$ . Note that $B/I$ is negatively graded by construction and $X_0 \in S$ is the unique fixed point of the corresponding $\mathbb {C}^*$ -action. Hence, by Lemma 3.1 we have a $\hat T$ -equivariant embedding

$$\begin{align*}S\hookrightarrow T_{X_0}S \subset T_{X_0}\mathcal{H}\end{align*}$$

which maps $X_0 \in S$ to $0 \in T_{X_0}S$ . Now, we have

$$\begin{align*}\psi(1) \in S \subset T_{X_0}S \subset T_{X_0}\mathcal{H}\end{align*}$$

and for every $\hat w \in \operatorname {\mathrm {int}} \Sigma _{\mathcal {X}}$ the corresponding $\mathbb {C}^*$ -action has positive weights along the homogeneous components of $\psi (1)$ in $T_{X_0}S$ . Hence, in $T_{X_0}S$ we have

$$\begin{align*}\lim_{t\to 0} \gamma_{\hat w}(t).\psi(1) = 0,\end{align*}$$

which corresponded to $X_0 \in S \subset \mathcal {H}$ . In this way we obtain a special degeneration of X to $X_0$ and inducing the $\mathbb {C}^*$ -action given by $\hat w$ .

Proof. We may assume wlog that $S=\operatorname {\mathrm {Spec}} B$ is affine and $\mathcal {X}=\operatorname {\mathrm {Spec}} \mathcal A$ with $\mathcal {A}=(B \otimes R)/\mathcal {I}$ and the ideal $\mathcal {I}=(f_1,\ldots ,f_r)$ being generated by homogeneous elements of weight $\deg f_i=: w_i \in \hat M$ . The fibre over $s \in S$ is then given by $\mathcal {X}_s = \operatorname {\mathrm {Spec}} \mathcal {A}_s$ with $\mathcal {A}_s = \mathcal {A} \otimes \mathbb {C}(s) = R/\mathcal {I}_s$ . Note that $\mathcal {I}_s = \mathcal {I} \otimes \mathbb {C}(s)$ is generated by elements in degrees $w_1, \ldots , w_r$ , as well.

We consider the non-trivial homogeneous components of $T^1_{\mathcal {X}_s}=\operatorname {\mathrm {Hom}}_R(\mathcal {I}_s, R/\mathcal {I}_s)$ . Let $\varphi \in \operatorname {\mathrm {Hom}}_R(\mathcal {I}_s, R/\mathcal {I}_s)$ be a non-trivial element. Then we have $0 \neq \varphi (f_i) \in (R/\mathcal {I}_s)_w$ for at least one $i \in \{1,\ldots ,r\}$ and some $w \in \hat \sigma ^\vee $ . In this case we have $\deg \varphi =w-w_i$ . Hence, we get $T^1_{\mathcal {X}_s}(u) \neq 0$ only for $u \in \left (G + \hat \sigma ^\vee \right )$ , where $G=\{-w_1, \ldots ,-w_r\}$ .

Consider a sequence of pairwise distinct elements $(u_i)_{i \in \mathbb {N}}\in \hat M \setminus \hat \sigma ^\vee $ with $T^1_{\mathcal {X}_s}(u) \neq 0$ for some $s \in S$ . Then we have ${u_i}/{|u_i|} \in ({G}/{|u_i|} + \hat \sigma ^\vee ) \setminus \hat \sigma ^\vee $ . Since the $u_i$ are lattice points we necessarily have $|u_i| \to +\infty $ . We see that the distance between ${u_i}/{|u_i|}$ and $\hat \sigma ^\vee $ goes to $0$ . Hence, every accumulation point has to lie on the boundary of $\hat \sigma ^\vee $ .

Now, we show (ii). Consider a sequence $u_i \to u \in \operatorname {\mathrm {int}} \tau \prec \hat \sigma ^\vee $ . Assuming $\langle \xi _{u_i}, u_i \rangle \leq 0$ we obtain $\lim _{i\to \infty } \langle {\xi _{u_i}}/{|\xi _{u_i}|} , u \rangle = \lim _{i\to \infty } \langle {\xi _{u_i}}/{|\xi _{u_i}|} , u_i \rangle \leq 0.$ On the other hand, it follows from $u \in \hat \sigma ^\vee $ that $\langle {\xi _{u_i}}/{|\xi _{u_i}|} , u \rangle \geq 0$ and therefore $\lim _{i\to \infty } \langle {\xi _{u_i}}/{|\xi _{u_i}|} , u \rangle = 0$ . In other words, the $\xi _{u_i}$ accumulate at $\tau ^*=u^\perp \cap \hat \sigma $ .

The statement (i) follows from (ii), since the finite set $\{\xi _0\} \subset \operatorname {\mathrm {int}} \hat \sigma $ cannot have accumulation points at the boundary of $\hat \sigma $ .

Proof. Let $\mathcal {Y}\rightarrow T$ be the bounding family for the projective varieties $Y_i$ and $\mathcal {A}$ be the bounding divisor for $A_i$ . We may assume that $\mathcal {A}$ is ample over T (see, e.g., [Reference Lazarsfeld29, Theorem 1.2.13]). Then, $\mathcal {A}^{\otimes m}$ is T-very ample for some m only depending on the pair $(\mathcal {Y},\mathcal {A})$ . We may further assume that if $\mathcal {Y}_t$ admits a $\mathbb {T}$ -action, then $\mathcal {A}_t^{\otimes m}$ admits a $\mathbb {T}$ -linearization. Hence, we may embed all fibres of $\mathcal {Y}\rightarrow T$ to $\mathbb {P}^N$ as varieties of degree at most d. We call this embedding $\phi _t$ . Here, both N and d only depend on the bounding family. Furthermore, if $\mathcal {Y}_t$ admits a $\mathbb {T}$ -action, then this embedding is $\mathbb {T}$ -equivariant. Let $\mathbb {T}_0\leqslant \mathrm {PGL}_N(\mathbb {K})$ be a maximal torus. For every $\mathcal {Y}_t$ that admits a torus action, we can find $g\in \mathrm {PGL}_N(\mathbb {K})$ such that $\mathcal {Y}_t$ is equivariantly isomorphic to $g^{-1}\phi _t(\mathcal {Y}_t)g$ with the action endowed by a subtorus of $\mathbb {T}_0$ . Since the degree of $g^{-1}\phi _t(\mathcal {Y}_t)g$ is d and it is a non-degenerate variety, then there are only finitely many tori $\mathbb {T}_1,\dots ,\mathbb {T}_k \leqslant \mathbb {T}_0$ that can act on it. Then, the disjoint union $\bigcup _{i=1}^k \mathrm {Hilb}_{d,N}^{\mathbb {T}_i}$ of $\mathbb {T}_i$ -invariant points of the Hilbert scheme gives an equivariant bounding family for the pairs $(Y_i,A_i)$ . By taking the cone of the $Y_i$ with respect to the $\mathbb {Q}$ -polarization $A_i$ on this family, we obtain the desired result.

Proof. After passing to an open affine subset of S we may assume that S and $\mathcal {X}$ are affine. Since $\mathcal {X} \to S$ is a $\mathbb {T}$ -invariant family of $\mathbb {T}$ -cone singularities, we have $\mathbb {C}[\mathcal {X}]^{\mathbb {T}} = \mathbb {C}[S]$ and the structure morphism of the family is induced by the inclusion of algebras.

It follows from Theorem 2.24 that there exists a semi-projective variety $\mathcal {Y}$ , a proper $\mathbb {T}$ -equivariant morphism $r \colon \widetilde {\mathcal {X}} \to \mathcal {X}$ and good quotient morphisms $\pi \colon \widetilde {\mathcal {X}} \to \mathcal {Y}$ . We obtain $H^0(\mathcal {O}_{\mathcal {Y}})=H^0(\mathcal {O}_{\widetilde {\mathcal {X}}})^{\mathbb {T}}=H^0(\mathcal {O}_{\mathcal {X}})^{\mathbb {T}}$ . In particular, $\operatorname {\mathrm {Spec}}(H^0(\mathcal {O}_{\mathcal {Y}}))=S$ holds and $\mathcal {Y}$ is projective over S. Altogether we have a commutative diagram

Let $X_\sigma $ be the toric variety corresponding to the Reeb cone $\sigma $ . By Remark 2.23 there is an open subset $\mathcal {Y}^\circ \subset \mathcal {Y}$ , such that $X_\sigma \times \mathcal {Y}^\circ $ is $\mathbb {T}$ -equivariantly isomorphic to an open subset $V \subset \widetilde {X}$ . Let $U=\mathcal {Y}^\circ \setminus \overline {\pi (E^{\text {vert}})}$ . Here, $E^{\text {vert}}$ denotes the union of the vertical components of the exceptional locus of $r \colon \widetilde {\mathcal {X}} \to \mathcal {X}$ . By vertical components we mean those whose image under $\pi $ has positive codimension. We then set $S'=p(U)$ .

Let $X_\Sigma $ be the toric variety corresponding to the fan $\Sigma $ obtained by star subdividing $\sigma $ with respect to $\xi $ . Let $\bar X_\Sigma $ be some $\mathbb {T}$ -equivariant completion of $X_\Sigma $ . There is a $\mathbb {T}$ -equivariant birational map $\varphi \colon X_\Sigma \times \mathcal {Y} \dashrightarrow \widetilde {\mathcal {X}}$ and we obtain $\mathcal {X}'$ as a resolution of indetermacy by taking the normalization of the closure of the graph of $\varphi $ in $(X_\Sigma \times \mathcal {Y}) \times _S \widetilde {\mathcal {X}}$ . This results in the following commutative diagram.

Here, $p_1$ and $p_2$ are the projections to the first and second factor, respectively. Notice that $p_2$ is proper because $\bar X_\Sigma \times \mathcal {Y}$ is proper over S and properness is preserved under base extension. Therefore, the induced map $\mathcal {X}' \to \widetilde {\mathcal {X}}$ is proper because it is the composition of the normalization with a closed embedding and $p_2$ . Hence, the composition $\psi \colon \mathcal {X}' \to \widetilde {X} \to \mathcal {X}$ is proper as well.

The rational map $\bar X_\Sigma \times \mathcal {Y} \dashrightarrow \widetilde {\mathcal {X}}$ is defined on the open subset $X_\Sigma \times U$ . Indeed, here, it is induced simply by the toric morphism $X_\Sigma \to X_\sigma $ . Therefore, over $X_\Sigma \times U$ the resolution of indeterminacy $\mathcal {X}' \to \bar X_\Sigma \times \mathcal {Y}$ is an isomorphism and we may identify $X_\Sigma \times U$ with a $\mathbb {T}$ -invariant open subset of $\mathcal {X}'$ . We now set

$$\begin{align*}V' \;:=\; X_\xi \times U \;\subset\; \mathcal{X}'.\end{align*}$$

Then $D_\xi \times U$ is the only component of the exceptional locus of $\psi \colon \mathcal {X}' \to \mathcal {X}$ which intersects $V'$ . Indeed, $D_\xi \times U$ is the only horizontal closed $\mathbb {T}$ -invariant subset in $X_\xi \times U$ , i.e. the only one which projects onto U. On the other hand, we had chosen U, such that $X_\Sigma \times U$ cannot contain any vertical component.

Proof. First, we may reduce to the case of a normal base S. This can be done by passing to the normalization of the underlying reduced scheme and considering the pullback of the family along the normalization. Then we may treat the resulting finitely many connected components separately. By induction on the dimension of S it is sufficient to show that the maps are constant on a Zariski open subset of S. Hence, we may additionally assume that S is affine, $\mathcal {X} \to S$ is flat and the total space is $\mathbb {Q}$ -Gorenstein.

We may consider $\xi $ as a torus invariant valuation on $\mathcal {X}$ . Moreover, using an approximation argument we may assume that $\xi $ is a primitive lattice element.

We now consider the proper birational morphism $f \colon \mathcal {X}' \to \mathcal {X}$ and the open subset $V' \cong X_\xi \times U$ from Lemma 4.2. We have $V^{\prime }_s= X_\xi \times U_s$ and $(D_\xi \times U)|_{V^{\prime }_s} = D_\xi \times U_s$ The product structure of $V^{\prime }_s$ lets us conclude that, as in the toric setting, the divisorial valuation corresponding to $D_\xi \times U_s$ coincides with the one induced by $\xi $ . We are now comparing the log discrepancy of $D_\xi \times U$ and $D_\xi \times U_s$ on the open subsets $V'\subset \mathcal {X}'$ and $V^{\prime }_s\subset \mathcal {X}^{\prime }_s$ , respectively. We are interested in the coefficient of the divisor $D_s:=K_{X^{\prime }_s}+\psi _*^{-1}\Delta _s - \psi ^*(K_{\mathcal {X}_s}+\Delta _s)$ at $\overline {D_\xi \times U_s}$ . Let us set $D:=K_{X'/S'}+\psi _*^{-1}\Delta - \psi ^*(K_{\mathcal {X}/S'}+\Delta )$ . Then we have $D_s =D|_{\mathcal {X}_s'}.$ Since D is only supported in exceptional divisors we have $D|_{V'}=a_{(\mathcal {X},\Delta )}(\overline {D_\xi \times U})\cdot (D_\xi \times U)$ . It follows that

$$\begin{align*}a_{(\mathcal{X}_s,\Delta_s)}(\overline{D_\xi \times U_s})\cdot(D_\xi \times U_s)= D|_{V^{\prime}_s}= a_{(\mathcal{X},\Delta)}(\overline{D_\xi \times U})\cdot(D_\xi \times U_s)\end{align*}$$

Now, comparing coefficients gives

$$\begin{align*}a_{(\mathcal{X}_s,\Delta_s)}(\xi)=a_{(\mathcal{X}_s,\Delta_s)}(\overline{D_\xi \times U_s})=a_{(\mathcal{X},\Delta)}(\overline{D_\xi \times U}). \end{align*}$$

In particular, $a_{(\mathcal {X}_s,\Delta _s)}(\xi )$ does not depend on s.

It now follows from flatness that $\operatorname {\mathrm {vol}}_{\mathcal {X}_s}(\xi )$ and therefore also $\operatorname {\mathrm {\widehat {vol}}}_{(\mathcal X_s,\Delta _s)}$ is constant along $S'$ .

Proof. By Lemma 3.6 and using the approximation result of [Reference Li and Xu35, Lemma 2.10] we can find a valuation w on X, inducing a degeneration to $X_0$ and the polarization $\xi '$ . We then have $\operatorname {\mathrm {\widehat {vol}}}_{(X_0,\Delta _0)}(\xi ') = \operatorname {\mathrm {\widehat {vol}}}_{(X,\Delta )}(w) \geq \operatorname {\mathrm {\widehat {vol}}}(X,\Delta )$ by definition of the normalized volume. Moreover, we obtain $\operatorname {\mathrm {\widehat {vol}}}(X,\Delta )=\operatorname {\mathrm {\widehat {vol}}}_{(X,\Delta )}(\xi )$ by the K-semistability of $(X,\Delta ;\xi )$ and $\operatorname {\mathrm {\widehat {vol}}}_{(X_0,\Delta _0)}(\xi ) = \operatorname {\mathrm {\widehat {vol}}}_{(X,\Delta )}(\xi )$ by [Reference Li and Xu35, Lem. 3.2.].

Let now $\xi _0$ be the unique minimizer of $\operatorname {\mathrm {\widehat {vol}}}_{(X_0,\Delta _0)}$ on $\hat \sigma \cap V$ (existence and uniqueness follows from the convexity properties of the volume functional). Since $\xi _0$ was a minimizer we have $\operatorname {\mathrm {\widehat {vol}}}_{(X_0,\Delta _0)}(\xi ) \geq \operatorname {\mathrm {\widehat {vol}}}_{(X_0,\Delta _0)}(\xi _0)$ . If we assume $\xi _0 \in \Sigma _{\mathcal {X}}\setminus N_{\mathbb {R}}$ , then as we have just seen, $\operatorname {\mathrm {\widehat {vol}}}_{(X_0,\Delta _0)}(\xi ) \leq \operatorname {\mathrm {\widehat {vol}}}_{(X_0,\Delta _0)}(\xi _0)$ holds. Hence, we obtain $\operatorname {\mathrm {\widehat {vol}}}_{(X_0,\Delta _0)}(\xi ) = \operatorname {\mathrm {\widehat {vol}}}_{(X_0,\Delta _0)}(\xi _0)$ . By the uniqueness of the minimizer this implies $\xi _0=\xi $ , but this is a contradiction to $\xi \in N_{\mathbb {R}}$ .

Proof. After passing to one of the strata in Proposition 4.3 we may assume that the minimizer $\xi _0 \in \hat \sigma $ of the normalized volume on $\mathcal {X}_s$ is the same for every $s \in S$ .

Recall that the subtori $\mathbb {T} \subset \hat {\mathbb {T}}$ are in one-to-one correspondence with surjective lattice homomorphisms $\hat M \twoheadrightarrow M$ or equivalently to rational subspaces $K=\ker (\hat M_{\mathbb {R}} \twoheadrightarrow M_{\mathbb {R}})$ .

Assume there is an infinite sequence of subtori $\mathbb {T}_i$ corresponding to rational subspaces $K_i\subset \hat M$ , with non-product $\mathbb {T}$ -invariant special degenerations $\mathcal {Y}_i$ of a K-semistable cone $\mathbb {T}_i$ -singularity to an element $\mathcal {X}_{s_i}$ of our family and with $G_i:=\operatorname {\mathrm {supp}} T\mathcal {Y}_i$ . According to Lemma 3.6 we have $K_i = \operatorname {\mathrm {span}} G_i$ and by definition we have $\Sigma _{\mathcal {Y}_i} = (\sum _{u\in G_i}\mathbb {R}_{\geq 0}\cdot u)^\vee $ .

Let K be a subspace of maximal dimension (possibly $0$ ), which is contained in infinitely many of the $K_i$ . We may pass to an infinite subsequence such that always $K \subset K_i$ holds. We then consider $G_i' = G_i \setminus K$ . By the maximality of K, multiples of an element $u \in \hat M$ can only appear in finitely many of the $G_i'$ . By applying Proposition 4.4 for $V=K^\perp $ we see that $\langle u, \xi _0 \rangle \leq 0$ holds for at least one u in each $G_i'$ , but this leads to a contradiction, since by Lemma 3.8 this inequality can only hold for finitely many such u.

Notation 5.2. Given a polyhedron $P\subset N_{\mathbb {Q}}$ , we denote by $P^{(k)}$ the set of k-dimensional faces of P. Given an element $v\in N_{\mathbb {Q}}$ , we denote by $\mu (v)$ the smallest positive integer for which $\mu (v)v\in N$ .

Proof. By [Reference Petersen and Süss44, Prop. 3.1] every Cartier divisor on $X=X(\mathcal {D})$ is in fact principal. Hence, it has the form as in (6). Assume now $\operatorname {div}(f\cdot \chi ^u)=m(K_X+\Delta )$ with $K_X$ as in (7). If we set $\sum _y a_y \cdot \{y\} = {1}/{m}\cdot \operatorname {div}(f)$ , then comparing the coefficients of $ {1}/{m}\operatorname {div}(f\cdot \chi ^u)$ in front of $D_\rho $ and $D_{y,v}$ , respectively, in (6) with those of $K+\Delta $ leads to the equations (8) and (9), respectively.

Now, comparing the coefficient of $D_{y,w} \subset X(\mathcal {S})$ and $D_\tau \subset X(\mathcal {S})$ in equation (6) for $\operatorname {div}(f \cdot \chi ^u)$ with that of $-K_{\widetilde {X}}$ as in (7) leads to the log discrepancies (10) and (11), respectively.