TY - JOUR
AU - Fujita, Kento
AB - Abstract We introduce a new effective stability named ‘divisorial stability’ for Fano manifolds that is weaker than K-stability and is stronger than slope stability along divisors. We show that we can test divisorial stability via the volume function. As a corollary, we prove that the first coordinate of the barycenter of the Okounkov body of the anticanonical divisor is not bigger than one for any Kähler–Einstein Fano manifold. In particular, for toric Fano manifolds, the existence of Kähler–Einstein metrics is equivalent to divisorial semistability. Moreover, we find many non-Kähler–Einstein Fano manifolds of dimension 3. 1. Introduction Let $$X$$ be a $${\mathbb {Q}}$$-Fano variety, that is, a projective variety over the complex number field which has at most log-terminal singularities such that the anticanonical divisor $$-K_X$$ of $$X$$ is ample ($$ {\mathbb {Q}}$$-Cartier). If $$X$$ is a Fano manifold (that is, $$X$$ is smooth), then it is known that the existence of Kähler–Einstein metrics is equivalent to K-polystability of the pair $$(X, -K_X)$$ (see [2, 10–13, 16, 37, 38, 49, 53, 54]). The notion of K-polystability is weaker than the notion of K-stability and is stronger than the notion of K-semistability. Our main interest is to test K-(semi)stability of the pair $$(X, -K_X)$$. (In this paper, we do not treat K-polystability.) However, in general, it is hard to test K-(semi)stability of the pair $$(X, -K_X)$$. To overcome the difficulties, Ross and Thomas introduced the notion of slope stability in [47]. This is an epoch-making notion since we can easily calculate. In particular, slope stability of Fano manifolds along divisors is interpreted by the volume function (see [21]). However, unfortunately, slope stability is strictly weaker than K-stability. In fact, as in [46, Example 7.6], if $$X$$ is the blowup of $$ {\mathbb {P}}^2$$ along distinct two points, then $$X$$ is a toric Fano manifold and $$(X, -K_X)$$ is not K-semistable, but $$(X, -K_X)$$ is slope stable. The purpose of this paper is to get a new necessary condition, called divisorial stability, of K-(semi)stability of the pair $$(X, -K_X)$$ which is strictly sharper than slope (semi)stability along divisors (if $$X$$ is smooth; see [21, 47]) and is easy to test. We quickly describe the key idea. We consider the case $$X$$ is smooth for simplicity. For slope stability along a non-zero effective divisor $$D$$, we consider the following flag ideal   \[ {\mathcal{O}}_X(-MD)+ {\mathcal{O}}_X(-(M-1)D)t^1+\cdots+ {\mathcal{O}}_X(-D)t^{M-1}+(t^M)\subset {\mathcal{O}}_{X\times {\mathbb{A}}^1_t} \] for some $$M\in {\mathbb {Z}}_{>0}$$ (for the notion of flag ideals, see [43] or Definition 2). For divisorial stability along $$D$$, we consider the flag ideal   \[ I_M+I_{M-1}t^1+\cdots+I_1t^{M-1}+(t^M)\subset {\mathcal{O}}_{X\times {\mathbb{A}}^1_t}, \] where each $$I_j\subset {\mathcal {O}}_X$$ is the base ideal of the sublinear system of the complete linear system $$|-rK_X|$$ (for some $$r$$) associated to the embedding   \[ H^0(X, {\mathcal{O}}_X(-rK_X-jD))\subset H^0(X, {\mathcal{O}}_X(-rK_X)) \] (see Section 3 in detail). The point is, despite each $$I_j$$ is smaller than $$ {\mathcal {O}}_X(-jD)$$, the associated global sections are same. More precisely, as subspaces of $$H^0(X, {\mathcal {O}}_X(-rK_X))$$, the following equality holds:   \[ H^0(X, {\mathcal{O}}_X(-rK_X-jD))=H^0(X, {\mathcal{O}}_X(-rK_X)\cdot I_j). \] Owing to this property, together with the finite generation of certain section rings (due to [5]), we can easily calculate the Donaldson–Futaki invariant of the semitest configuration (called the basic semitest configuration) obtained by the above flag ideal. The following definition is not the original definition but a consequence of a certain transformation (Theorem 5.2). For the original definition; see Definition 10. Definition 1 (see Definition 10, Proposition 3.3 and Theorem 5.2) Let $$X$$ be a $$ {\mathbb {Q}}$$-Fano variety. Let $$D$$ be a non-zero effective Weil divisor on $$X$$. The pair $$(X, -K_X)$$ is said to be divisorially stable (respectively, divisorially semistable) along$$D$$ if the value   \[ \eta(D):= \operatorname{vol}_X(-K_X)-\int_0^{\infty} \operatorname{vol}_X(-K_X-xD)\,dx \] satisfies that $$\eta (D)>0$$ (respectively, $$\eta (D)\geq 0$$), where $$ \operatorname {vol}_X$$ is the volume function (see Definition 6). The pair $$(X, -K_X)$$ is said to be divisorially stable (respectively, divisorially semistable) if $$(X, -K_X)$$ is divisorially stable (respectively, divisorially semistable) along any non-zero effective Weil divisor. We show that divisorial (semi)stability of Fano manifolds is a necessary condition for K-(semi)stability, and is a sufficient condition for slope stability along divisors. See [21, 47] for the notion of slope stability. Theorem 1.1 (see Remarks 4, 9 and Corollary 8.2) (1) For a$$ {\mathbb {Q}}$$-Fano variety$$X,$$if$$(X, -K_X)$$is K-stable (respectively, K-semistable), then$$(X, -K_X)$$is divisorially stable (respectively, divisorially semistable). For a Fano manifold$$X,$$if$$(X, -K_X)$$is divisorially stable (respectively, divisorially semistable), then$$(X, -K_X)$$is slope stable (respectively, slope semistable) along all divisors. Moreover, we show that divisorial semistability is equivalent to K-semistability if $$X$$ is toric. Theorem 1.2 (see Corollary 6.3) Let$$X$$be a toric$$ {\mathbb {Q}}$$-Fano variety and let$$P\subset M_ {\mathbb {R}}$$be the associated polytope (see Section 6). Then the pair$$(X, -K_X)$$is divisorially semistable if and only if the barycenter of$$P$$is equal to the origin. By the similar argument, we show that divisorial (semi)stability can be interpreted by a certain structure property of the Okounkov body of $$-K_X$$. We remark that the relationship between K-stability and Okounkov bodies has been already pointed out in [57]. Theorem 1.3 (see Theorem 7.1) Let$$X$$be a$$ {\mathbb {Q}}$$-Fano variety of dimension$$n$$and  \[ Z_\bullet\colon X=Z_0\supset Z_1\supset\dots\supset Z_n=\{\text{point}\} \]be an admissible flag in the sense of [34, (1.1)]. Let$$\Delta (-K_X)\subset {\mathbb {R}}^n$$be the Okounkov body of$$-K_X$$with respect to$$Z_\bullet $$in the sense of [34]. If the pair$$(X, -K_X)$$is K-stable (respectively, K-semistable), then the first coordinate$$b_1$$of the barycenter of$$\Delta (-K_X)$$satisfies that$$b_1<1$$(respectively,$$b_1\leq 1)$$. We also see that we can calculate the Donaldson–Futaki invariants of the basic semitest configurations via intersection numbers after we run certain minimal model program (MMP, in short) with scaling (see Section 8). Thus divisorial (semi)stability is easy to test. In fact, we determine divisorial (semi)stability for all smooth $$X$$ of dimension at most 3 (see Proposition 9.6 and Theorem 10.1). As an immediate corollary, we find many (possibly non-toric) non-Kähler–Einstein Fano manifolds of dimension 3. (We heavily depend on the classification result of Mori and Mukai [40].) Theorem 1.4 (see Theorem 10.1 and Table 1) Let$$X$$be a non-toric Fano manifold of dimension 3. Assume that$$X$$belongs to one of the following list: No.$$23,$$No.$$28,$$No. 30 or No. 31 in Table 2$$(\rho (X)=2),$$ No.$$14,$$No.$$16,$$No.$$18,$$No.$$21,$$No.$$22,$$No. 23 or No. 24 in Table 3$$(\rho (X)=3),$$or No. 5 or No. 8 in Table 4$$(\rho (X)=4)$$ in [40]. Then the pair$$(X, -K_X)$$is not K-semistable. In particular,$$X$$does not admit Kähler–Einstein metrics. Remark 1 (see also Remark 12) After the author wrote the article, the author found the results [27, 50]. It has been already known that some (but not all) of $$X$$ in the list of Theorem 1.4 does not admit Kähler–Einstein metrics (see [50, Theorem 1.1, 27, Theorem 6.1]). The article is organized as follows. In Section 2, we recall the notion of K-(semi)stability and geography of models. We also consider the volume function (and the restricted volume functions) of $$X$$ for possibly non-$$ {\mathbb {R}}$$-Cartier divisors. In Section 3, we define the notion of divisorial stability. We construct the basic semitest configuration from a non-zero effective Weil divisor. In Proposition 3.3, we see that we can calculate the Donaldson–Futaki invariant of basic semitest configurations via the growth of the sum of the dimension of certain global sections. In Section 4, we prove a kind of the asymptotic Riemann–Roch theorem in order to calculate the Donaldson–Futaki invariants of basic semitest configurations. Owing to the argument in Section 4, we can rephrase divisorial (semi)stability in Section 5. In Section 6, we consider the case where $$X$$ is toric. We see in Corollary 6.3 that if the barycenter of the associated polytope is not the origin, then $$(X, -K_X)$$ is not divisorially semistable along some torus invariant prime divisor. As a corollary, for a non-K-semistable toric $$ {\mathbb {Q}}$$-Fano variety, we can explicitly construct a flag ideal such that the Donaldson–Futaki invariant of the semitest configuration obtained by the flag ideal is strictly negative (see Examples 1 and 2). In Section 7, by the similar argument in Section 6, we show that divisorial stability along prime divisors can be interpreted by the structure property of Okounkov bodies of $$-K_X$$ (see Theorem 7.1). In Section 8, we rephrase the condition of divisorial (semi)stability via MMP with scaling. As a corollary, we see the relationship between divisorial (semi)stability and slope (semi)stability along divisors. In Section 9, we see some basic properties of divisorial stability. Moreover, we see some examples in order to prove Theorem 10.1. Finally, in Section 10, we determine divisorial (semi)stability for all Fano manifolds of dimension 3. Throughout this paper, we work in the category of algebraic (separated and of finite type) scheme over the complex number field $$ {\mathbb {C}}$$. A variety means a reduced and irreducible algebraic scheme. For the theory of MMP, we refer the readers to [32]. For a complete variety $$X$$, $$\rho (X)$$ denotes the Picard number of $$X$$. For a normal projective variety $$X$$, $$ \operatorname {Nef}(X)$$ (respectively, $$\overline { \operatorname {Eff}}(X)$$) denotes the nef (respectively, pseudo-effective) cone, that is, the closure of the cone in $$ \operatorname {N}^{1}(X)$$ spanned by classes of nef (respectively, effective) divisors on $$X$$, and $$\operatorname {Big}(X)$$ denotes the interior of the cone $$\overline { \operatorname {Eff}}(X)$$. For a Weil divisor $$D$$ on a normal projective variety $$X$$, the divisorial sheaf on $$X$$ is denoted by $$ {\mathcal {O}}_X(D)$$. More precisely, for any open subscheme $$U\subset X$$, the section of $$ {\mathcal {O}}_X(D)$$ on $$U$$ is defined by   \[ \{f\in k(X)\,|\, \operatorname{div}(f)|_U+D|_U\geq 0\}, \] where $$k(X)$$ is the function field of $$X$$. 2. Preliminaries In this section, we fix a $$ {\mathbb {Q}}$$-Fano variety $$X$$ of dimension $$n$$. 2.1. K-stability In this section, we recall the notion of K-stability. Definition 2 ([15, 43, 44, 47, 53]) (1) A flag ideal$$ {\mathcal {I}}$$ is a coherent ideal sheaf $$ {\mathcal {I}}\subset {\mathcal {O}}_{X\times {\mathbb {A}}_t^1}$$ of the form   \[ {\mathcal{I}}=I_M+I_{M-1}t^1+\cdots+I_1t^{M-1}+(t^M)\subset {\mathcal{O}}_{X\times {\mathbb{A}}_t^1}, \] where $$ {\mathcal {O}}_X\supset I_1\supset \cdots \supset I_M$$ is a decreasing sequence of coherent ideal sheaves. Let $$r\in {\mathbb {Z}}_{>0}$$ with $$-rK_X$$ Cartier. A semitest configuration$$( \mathcal {B}, \mathcal {L})/ {\mathbb {A}}^1$$of$$(X, -rK_X)$$obtained by$$ {\mathcal {I}}$$ is defined by the following datum: $$\Pi \colon \mathcal {B}\to X\times {\mathbb {A}}^1$$ is the blowup along $$ {\mathcal {I}}$$, and $$E_ \mathcal {B}\subset \mathcal {B}$$ is the Cartier divisor defined by $$ {\mathcal {O}}_ \mathcal {B}(-E_ \mathcal {B})= {\mathcal {I}}\cdot {\mathcal {O}}_ \mathcal {B}$$; $$ \mathcal {L}:=\Pi ^{\ast }p_1^{\ast } {\mathcal {O}}_X(-rK_X)\otimes {\mathcal {O}}_ \mathcal {B}(-E_ \mathcal {B})$$, where $$p_1\colon X\times {\mathbb {A}}^1\to X$$ is the first projection; and we require the following: $${\mathcal {I}}$$ is not of the form $$(t^M)$$; $$ \mathcal {L}$$ is semiample over $$ {\mathbb {A}}^1$$. Let $$\alpha \colon ( \mathcal {B}, \mathcal {L})\to {\mathbb {A}}^1$$ be the semitest configuration of $$(X, -rK_X)$$ obtained by $$ {\mathcal {I}}$$. Then the multiplicative group $$ \mathbb {G}_m$$ naturally acts on $$( \mathcal {B}, \mathcal {L})$$ and the morphism $$\alpha $$ is $$ \mathbb {G}_m$$-invariant, where the action $$ \mathbb {G}_m\times {\mathbb {A}}^1\to {\mathbb {A}}^1$$ is in a standard way $$(a, t)\mapsto at$$. For $$k\in {\mathbb {Z}}_{>0}$$, $$ \mathbb {G}_m$$ also naturally acts on $$(\alpha _* \mathcal {L}^{\otimes k})|_{\{0\}}$$. Let $$w(k)$$ be the total weight of the action. It is known that $$w(k)$$ is a polynomial of degree at most $$n+1$$ for $$k\gg 0$$. Let $$w_{n+1}$$, $$w_n$$ be the $$(n+1)$$th, $$n$$th coefficient of $$w(k)$$, respectively. We define the Donaldson–Futaki invariant$$ \operatorname {DF}( \mathcal {B}, \mathcal {L})$$ of $$( \mathcal {B}, \mathcal {L})/ {\mathbb {A}}^1$$ such that   \begin{align*} \operatorname{DF}( \mathcal{B}, \mathcal{L}) & := \frac{((-rK_X)^{\cdot n-1}\cdot(-K_X))} {2\cdot(n-1)!}w_{n+1}- \frac{((-rK_X)^{\cdot n})}{n!}w_n\\ & = \frac{((-rK_X)^{\cdot n})}{n!}\left(\frac{n}{2r}w_{n+1}-w_n\right). \end{align*} We say that the pair $$(X, -K_X)$$ is K-stable (respectively, K-semistable) if $$ \operatorname {DF}( \mathcal {B}, \mathcal {L})>0$$ (respectively, $$\geq 0$$) holds for any $$r\in {\mathbb {Z}}_{>0}$$, for any flag ideal $$ {\mathcal {I}}$$ and for any semitest configuration $$( \mathcal {B}, \mathcal {L})/ {\mathbb {A}}^1$$ of $$(X, -rK_X)$$ obtained by $$ {\mathcal {I}}$$. 2.2. On geography of models In this section, we recall the theory of ‘geography of models’ introduced in [48, Section 6]. For the notation in this section, we refer the readers to [28]. Definition 3 ([28, Definition 2.3]) Let $$V$$ be a normal projective variety, $$E_V$$ be an $${\mathbb {R}}$$-Cartier $${\mathbb {R}}$$-divisor on $$V$$ and $$\phi \colon V \dashrightarrow W$$ be a contraction map to a normal projective variety $$W$$ such that $$E_W:=\phi _*E_V$$ is $$ {\mathbb {R}}$$-Cartier. The map $$\phi $$ is said to be $$E_V$$-non-positive if $$\phi $$ is birational, and for a common resolution $$(p, q)\colon \tilde {V}\to V\times W$$, we can write $$p^{\ast }E_V=q^{\ast }E_W+F$$, where $$F$$ is effective and $$q$$-exceptional. The map $$\phi $$ is said to be a semiample model of$$E_V$$ if $$\phi $$ is $$E_V$$-non-positive and $$E_W$$ is semiample. The map $$\phi $$ is said to be the ample model of$$E_V$$ is there exist a birational contraction map $$\phi '\colon V\dashrightarrow W'$$ and a morphism $$\psi \colon W'\to W$$ with connected fibers such that $$\phi =\psi \circ \phi '$$, the map $$\phi '$$ is a semiample model of $$E_V$$ and $$\phi '_*E_V=\psi ^{\ast }A$$, where $$A$$ is an ample $$ {\mathbb {R}}$$-divisor on $$W$$. Throughout the end of the section, we fix a non-zero effective Weil divisor $$D$$ on $$X$$. By [5, Corollary 1.4.3], we can take a projective small $$ {\mathbb {Q}}$$-factorial modification morphism $$\sigma \colon \tilde {X}\to X$$, that is, $$\sigma $$ is a projective birational morphism that is isomorphism in codimension 1 and $$\tilde {X}$$ is $$ {\mathbb {Q}}$$-factorial. We set $$\tilde {D}:=\sigma ^{-1}_*D$$. We note that if $$X$$ is $$ {\mathbb {Q}}$$-factorial (for example, smooth), then $$\sigma \colon \tilde {X}\to X$$ is the identity morphism. Lemma 2.1 The variety$$\tilde {X}$$is a Mori dream space in the sense of [26, Definition 1.10]. Proof Since $$\tilde {X}$$ is projective, having at most log-terminal singularities and $$-K_{\tilde {X}}$$ is nef and big, there exists an effective $$ {\mathbb {Q}}$$-divisor $$\Delta $$ on $$\tilde {X}$$ such that the pair $$(\tilde {X}, \Delta )$$ is klt and $$-(K_{\tilde {X}}+\Delta )$$ is an ample $$ {\mathbb {Q}}$$-divisor. Thus the assertion follows by [5, Corollary 1.3.2]. □ Definition 4 Let $$X$$, $$D$$ and $$\sigma $$ be as above. The pseudo-effective threshold$$\tau (D)$$of$$D$$with respect to$$(X, -K_X)$$ is defined by   \[ \tau(D):=\max\{\tau\in {\mathbb{R}}_{>0}\,|\,-K_{\tilde{X}}-\tau\tilde{D} \in\overline{ \operatorname{Eff}}(\tilde{X})\}. \] The following theorem is important in this paper. Theorem 2.2 ([28, Theorem 4.2]) Let$$X$$,$$D$$and$$\sigma $$be as above. Then there exist an increasing sequence of rational numbers  \[ 0=\tau_0<\tau_1<\cdots<\tau_m=\tau(D), \] normal projective varieties$$X_1,\ldots ,X_m$$, and mutually distinct birational contraction maps$$\phi _i\colon \tilde {X}\dashrightarrow X_i$$$$(1\leq i\leq m)$$ such that the following hold: for any$$x\in [\tau _{i-1}, \tau _i],$$the map$$\phi _i$$is a semiample model of$$-K_{\tilde {X}}-x\tilde {D},$$and if$$x\in (\tau _{i-1}, \tau _i),$$then the map$$\phi _i$$is the ample model of$$-K_{\tilde {X}}-x\tilde {D}$$. Proof The proof follows immediately from Lemma 2.1 and [28, Theorem 4.2]. □ Definition 5 The sequence $$\{(\tau _i, X_i)\}_{1\leq i\leq m}$$ obtained in Theorem 2.2 is called the ample model sequence of$$(X, -K_X; -D)$$. We set $$D_i:=(\phi _i)_*\tilde {D}$$ for $$1\leq i\leq m$$. Remark 2 (1) Since $$\tau _{i-1}<\tau _i$$, both $$K_{X_i}$$ and $$D_i$$ are $$ {\mathbb {Q}}$$-Cartier divisors on $$X_i$$. For any $$x\in (\tau _{i-1}, \tau _i)\cap {\mathbb {Q}}$$ and for any $$k_0\in {\mathbb {Z}}_{>0}$$ with $$-k_0K_{\tilde {X}}-k_0x\tilde {D}$$ Cartier, we have   \[ X_i\simeq \operatorname{Proj}\bigoplus_{k\geq 0}H^0(\tilde{X}, {\mathcal{O}}_{\tilde{X}}(-kk_0K_{\tilde{X}} -kk_0x\tilde{D})) \] by [28, Remark 2.4(i)]. Since $$\sigma \colon \tilde {X}\to X$$ is small, the above is also isomorphic to   \[ \operatorname{Proj}\bigoplus_{k\geq 0}H^0(X, {\mathcal{O}}_X(-kk_0K_X-kk_0xD)). \] Thus the ample model sequence of $$(X, -K_X; -D)$$ does not depend on the choice of $$\sigma $$. In particular, the pseudo-effective threshold $$\tau (D)$$ does not depend on the choice of $$\sigma $$ and $$\tau (D)\in {\mathbb {Q}}_{>0}$$ holds. The map $$\phi _i$$ is $$K_{\tilde {X}}$$-non-positive since $$-K_{\tilde {X}}$$ is nef (see [28, Lemma 2.5]). Hence $$X_i$$ has at most log-terminal singularities. 2.3. On the volume functions In this section, we recall the theory of the volume functions and the restricted volume functions. We refer the readers to [33, 34]. In this section, we fix a projective small $$ {\mathbb {Q}}$$-factorial modification $$\sigma \colon \tilde {X}\to X$$ and we set $$\tilde {D}:=\sigma ^{-1}_*D$$ as in Subsection 2.2. Definition 6 For any $$x\in [0, +\infty )$$, we define   \[ \operatorname{vol}_X(-K_X-xD):= \operatorname{vol}_{\tilde{X}}(-K_{\tilde{X}}-x\tilde{D}), \] where $$ \operatorname {vol}_{\tilde {X}}$$ is the volume function on $$\tilde {X}$$ (see [33, Corollary 2.2.45]). Thus the function $$ \operatorname {vol}_X(-K_X-xD)$$ is continuous over $$[0, +\infty )$$, and, for any $$x\in [0, +\infty )\cap {\mathbb {Q}}$$, $$ \operatorname {vol}_X(-K_X-xD)$$ is equal to   \[ \limsup_{k\to\infty} \frac{h^0(\tilde{X}, {\mathcal{O}}_{\tilde{X}}(-kK_{\tilde{X}}+\lfloor-kx\tilde{D}\rfloor))} {k^n/n!}=\limsup_{k\to\infty} \frac{h^0(X, {\mathcal{O}}_X(-kK_X+\lfloor-kxD\rfloor))}{k^n/n!}, \] where $$\lfloor \bullet \rfloor $$ is the round-down (see [32, Notation 0.4(12)]). In particular, $$ \operatorname {vol}_X(-K_X-xD)$$ does not depend on the choice of $$\sigma $$. Lemma 2.3 Let$$x\in [0, +\infty )$$. The following conditions are satisfied: the equality$$\operatorname {vol}_X(-K_X-xD)=0$$holds if and only if$$x\geq \tau (D);$$ if$$x\in [\tau _{i-1}, \tau _i],$$then$$\operatorname {vol}_X(-K_X-xD)=((-K_{X_i}-xD_i)^{\cdot n})$$. Proof (1) The $$ {\mathbb {R}}$$-divisor $$-K_{\tilde {X}}-x\tilde {D}$$ is big if and only if $$x\in [0, \tau (D))$$. Thus the assertion follows. Both $$ \operatorname {vol}_X(-K_X-xD)$$ and $$((-K_{X_i}-xD_i)^{\cdot n})$$ are continuous functions over $$x\in [\tau _{i-1}, \tau _i]$$. Thus we can assume that $$x\in (\tau _{i-1}, \tau _i)\cap {\mathbb {Q}}$$. We have   \[ \operatorname{vol}_X(-K_X-xD) = \limsup_{k\to\infty} \frac{h^0(X_i, {\mathcal{O}}_{X_i}(-kK_{X_i}+\lfloor-kxD_i\rfloor))}{k^n/n!} = ((-K_{X_i}-xD_i)^{\cdot n}) \] by [28, Remark 2.4 (i)] and the Serre vanishing theorem. □ We define the notion of restricted volume functions. Definition 7 For $$x\in [0, \tau (D))$$, we define the restricted volume$$ \operatorname {vol}_{X|D}(-K_X-xD)$$ such that   \[ \operatorname{vol}_{X|D}(-K_X-xD):=-\frac{1}{n}\frac{d}{dx} \operatorname{vol}_X(-K_X-xD). \] Note that the function $$ \operatorname {vol}_X(-K_X-xD)$$ is $$ {\mathcal {C}}^1$$ over $$x\in [0, \tau (D))$$ by [8, Theorem A]. Thus $$ \operatorname {vol}_{X|D}(-K_X-xD)$$ is well-defined and continuous over $$x\in [0, \tau (D))$$. Proposition 2.4 Assume that$$x\in [\tau _{i-1}, \tau _i]$$. Then  \[ \operatorname{vol}_{X|D}(-K_X-xD)=((-K_{X_i}-xD_i)^{\cdot n-1}\cdot D_i). \]In particular,$$ \operatorname {vol}_{X|D}(-K_X-xD)\geq 0$$for any$$x\in [0, \tau (D))$$. Proof The proof follows from Lemma 2.3(2). □ Definition 8 We define   \[ \operatorname{vol}_{X|D}(-K_X-\tau(D)D):=((-K_{X_m}-\tau(D)D_m)^{\cdot n-1}\cdot D_m) \] for convenience. By Proposition 2.4, $$ \operatorname {vol}_{X|D}(-K_X-xD)$$ is continuous over $$x\in [0, \tau (D)]$$. Proposition 2.5 Assume that$$D$$is a prime divisor and$$x\in [0, \tau (D))\cap {\mathbb {Q}}$$. Then the value$$ \operatorname {vol}_{X|D}(-K_X-xD)$$coincides with the usual restricted volume$$ \operatorname {vol}_{\tilde {X}|\tilde {D}}(-K_{\tilde {X}}-x\tilde {D})$$in [18, 34]. More precisely,  \[ \operatorname{vol}_{X|D}(-K_X-xD)=\limsup_{\substack{k\to\infty\\ -kK_{\tilde{X}}-kx\tilde{D}: \text{ Cartier}}} \frac{\dim V_{(k)}}{k^{n-1}/(n-1)!} \]holds, where$$V_{(k)}$$is the image of the homomorphism  \[ H^0(\tilde{X}, {\mathcal{O}}_{\tilde{X}}(-kK_{\tilde{X}}-kx\tilde{D}))\longrightarrow H^0(\tilde{D}, {\mathcal{O}}_{\tilde{X}}(-kK_{\tilde{X}}-kx\tilde{D})|_{\tilde{D}}). \] Proof We know that $$ \operatorname {Exc}(\sigma )= {\mathbb {B}}_+(-K_{\tilde {X}})$$ by [6, Proposition 2.3], where $$ \operatorname {Exc}(\sigma )$$ is the exceptional locus of $$\sigma $$ and $$ {\mathbb {B}}_+(-K_{\tilde {X}})$$ is the augmented base locus of $$-K_{\tilde {X}}$$ (see [34, Subsection 2.4]). In particular, $$\tilde {D}\not \subset {\mathbb {B}}_+(-K_{\tilde {X}})$$. Thus, by the proof of [34, Corollary 4.25], the assertion follows. □ 3. Divisorial stability We define the notion of divisorial stability for $$ {\mathbb {Q}}$$-Fano varieties. In this section, we fix a $$ {\mathbb {Q}}$$-Fano variety $$X$$ of dimension $$n$$ and a non-zero effective Weil divisor $$D$$ on $$X$$. By Lemma 2.1, the graded $$ {\mathbb {C}}$$-algebra   \[ \bigoplus_{k, j\geq 0}H^0(X, {\mathcal{O}}_X(-kK_X-jD)) \] is finitely generated. We remark that the above algebra is equal to   \[ \bigoplus_{\substack{k\geq 0\\ 0\leq j\leq k\tau(D)}}H^0(X, {\mathcal{O}}_X(-kK_X-jD)), \] since $$H^0(X, {\mathcal {O}}_X(-kK_X-jD))=0$$ for $$j>k\tau (D)$$. We remark that if $$r\in {\mathbb {Z}}_{>0}$$ is sufficiently divisible, then the Veronese-type $$ {\mathbb {C}}$$-subalgebra   \[ \bigoplus_{\substack{k\geq 0\\ 0\leq j\leq kr\tau(D)}} H^0(X, {\mathcal{O}}_X(-krK_X-jD)) \] is generated by   \[ \bigoplus_{0\leq j\leq r\tau(D)}H^0(X, {\mathcal{O}}_X(-rK_X-jD)) \] as a $$ {\mathbb {C}}$$-algebra. Definition 9 We say that a positive integer $$r\in {\mathbb {Z}}_{>0}$$satisfies the generating property with respect to$$(X, -K_X; -D)$$ if $$-rK_X$$ is Cartier, $$r\tau (D)\in {\mathbb {Z}}_{>0}$$ and the $$ {\mathbb {C}}$$-algebra   \[ \bigoplus_{\substack{k\geq 0\\ 0\leq j\leq kr\tau(D)}} H^0(X, {\mathcal{O}}_X(-krK_X-jD)) \] is generated by   \[ \bigoplus_{0\leq j\leq r\tau(D)}H^0(X, {\mathcal{O}}_X(-rK_X-jD)) \] as a $$ {\mathbb {C}}$$-algebra. Remark 3 (1) If $$r\in {\mathbb {Z}}_{>0}$$ is sufficiently divisible, then $$r$$ satisfies the generating property with respect to $$(X, -K_X; -D)$$. We assume that a positive integer $$r\in {\mathbb {Z}}_{>0}$$ satisfies the generating property with respect to $$(X, -K_X; -D)$$. Then the $$ {\mathbb {C}}$$-algebra   \[ \bigoplus_{k\geq 0}H^0(X, {\mathcal{O}}_X(-krK_X)) \] is generated by $$H^0(X, {\mathcal {O}}_X(-rK_X))$$. In particular, the divisor $$-rK_X$$ is very ample. Throughout the end of the section, we fix $$r\in {\mathbb {Z}}_{>0}$$, which satisfies the generating property with respect to $$(X, -K_X; -D)$$. From now on, we construct a semitest configuration of $$(X, -rK_X)$$. For any $$j\geq 0$$, we set the coherent ideal sheaf $$I_j\subset {\mathcal {O}}_X$$ defined by the image of the composition of the homomorphisms   \[ H^0(X, {\mathcal{O}}_X(-rK_X-jD))\otimes_ {\mathbb{C}} {\mathcal{O}}_X(rK_X) \hookrightarrow H^0(X, {\mathcal{O}}_X(-rK_X))\otimes_ {\mathbb{C}} {\mathcal{O}}_X(rK_X)\longrightarrow {\mathcal{O}}_X. \] In other words, $$I_j$$ is the base ideal of the sublinear system of the complete linear system $$|-rK_X|$$ which associates to the embedding   \[ H^0(X, {\mathcal{O}}_X(-rK_X-jD))\subset H^0(X, {\mathcal{O}}_X(-rK_X)). \] Obviously, we have   \[ {\mathcal{O}}_X=I_0\supset I_1\supset\cdots\supset I_{r\tau(D)}\supset I_{r\tau(D)+1}=0. \] For any $$k\in {\mathbb {Z}}_{>0}$$ and $$j\in {\mathbb {Z}}_{\geq 0}$$, we define the coherent ideal sheaf $$J_{(k,j)}\subset {\mathcal {O}}_X$$ such that   \[ J_{(k, j)}:=\sum_{\substack{j_1+\cdots+j_k=j\\ j_1,\ldots,j_k\geq 0}}I_{j_1}\cdots I_{j_k}. \] Lemma 3.1 The above ideal sheaf$$J_{(k, j)}\subset {\mathcal {O}}_X$$is equal to the base ideal of the sublinear system of the complete linear system$$|-krK_X|$$associates to the embedding  \[ H^0(X, {\mathcal{O}}_X(-krK_X-jD))\subset H^0(X, {\mathcal{O}}_X(-krK_X)). \]In other words,$$J_{(k, j)}$$is equal to the image of the composition of the homomorphisms  \[ H^0(X, {\mathcal{O}}_X(-krK_X-jD))\otimes_ {\mathbb{C}} {\mathcal{O}}_X(krK_X) \hookrightarrow H^0(X, {\mathcal{O}}_X(-krK_X))\otimes_ {\mathbb{C}} {\mathcal{O}}_X(krK_X)\to {\mathcal{O}}_X. \]In particular, we have  \[ H^0(X, {\mathcal{O}}_X(-krK_X-jD))=H^0(X, {\mathcal{O}}_X(-krK_X)\cdot J_{(k, j)}) \]as subspaces of$$H^0(X, {\mathcal {O}}_X(-krK_X))$$. Proof We write   \[ V_{(k, j)}:=H^0(X, {\mathcal{O}}_X(-krK_X-jD)) \] for simplicity. By the definition of $$r$$, the homomorphism   \[ \bigoplus_{\substack{j_1+\cdots+j_k=j\\ j_1,\ldots,j_k\geq 0}} V_{(1, j_1)}\otimes_ {\mathbb{C}}\cdots\otimes_ {\mathbb{C}} V_{(1, j_k)} \longrightarrow V_{(k, j)} \] is surjective. For any $$j_i$$, the image of the homomorphism   \[ V_{(1, j_i)}\otimes_ {\mathbb{C}} {\mathcal{O}}_X(rK_X)\longrightarrow {\mathcal{O}}_X \] is equal to $$I_{j_i}$$. Thus the image of the homomorphism   \[ \bigoplus_{\substack{j_1+\cdots+j_k=j\\ j_1,\ldots,j_k\geq 0}} V_{(1, j_1)}\otimes_ {\mathbb{C}}\cdots\otimes_ {\mathbb{C}} V_{(1, j_k)}\otimes_ {\mathbb{C}} {\mathcal{O}}_X(krK_X) \longrightarrow {\mathcal{O}}_X \] is equal to   \[ \sum_{\substack{j_1+\cdots+j_k=j\\ j_1,\ldots,j_k\geq 0}}I_{j_1}\cdots I_{j_k}. \] This is nothing but $$J_{(k, j)}$$. □ We consider the following flag ideal:   \[ {\mathcal{I}}:=I_{r\tau(D)}+I_{r\tau(D)-1}t^1+\cdots+ I_1t^{r\tau(D)-1}+(t^{r\tau(D)}) \subset {\mathcal{O}}_{X\times {\mathbb{A}}^1_t}. \] By construction, for any $$k\in {\mathbb {Z}}_{>0}$$, we have   \[ {\mathcal{I}}^k=J_{(k, kr\tau(D))}+J_{(k, kr\tau(D)-1)}t^1+\cdots+J_{(k, 1)}t^{kr\tau(D)-1} +(t^{kr\tau(D)}). \] Let $$\Pi \colon \mathcal {B}\to X\times {\mathbb {A}}^1$$ be the blowup along $$ {\mathcal {I}}$$ and let $$E_ \mathcal {B}\subset \mathcal {B}$$ be the Cartier divisor on $$ \mathcal {B}$$ defined by $$ {\mathcal {O}}_ \mathcal {B}(-E_ \mathcal {B})= {\mathcal {I}}\cdot {\mathcal {O}}_ \mathcal {B}$$. Moreover, we set $$ \mathcal {L}:=\Pi ^{\ast }p_1^{\ast } {\mathcal {O}}_X(-rK_X)\otimes {\mathcal {O}}_ \mathcal {B}(-E_ \mathcal {B})$$ and $$\alpha :=p_2\circ \Pi \colon \mathcal {B}\to {\mathbb {A}}^1$$, where $$p_1$$ or $$p_2$$ is the first or the second projection morphism, respectively. Lemma 3.2 The above$$\alpha \colon ( \mathcal {B}, \mathcal {L})\to {\mathbb {A}}^1$$is a semitest configuration of$$(X, -rK_X)$$. Proof It is enough to show that $$ \mathcal {L}$$ is semiample over $$ {\mathbb {A}}^1$$. For any $$k\in {\mathbb {Z}}_{>0}$$ and $$j\in {\mathbb {Z}}_{\geq 0}$$, by Lemma 3.1, the homomorphism   \[ H^0(X, {\mathcal{O}}_X(-krK_X)\cdot J_{(k, j)})\otimes_ {\mathbb{C}} {\mathcal{O}}_X\longrightarrow {\mathcal{O}}_X(-krK_X)\cdot J_{(k, j)} \] is surjective. Thus, for any $$k\in {\mathbb {Z}}_{>0}$$, the homomorphism   \[ H^0(X\times {\mathbb{A}}^1, p_1^{\ast} {\mathcal{O}}_X(-krK_X)\cdot {\mathcal{I}}^k)\otimes_{ {\mathbb{C}}[t]} {\mathcal{O}}_{X\times {\mathbb{A}}^1}\longrightarrow p_1^{\ast} {\mathcal{O}}_X(-krK_X)\cdot {\mathcal{I}}^k \] is also surjective. Therefore, by [33, Lemma 5.4.24], we have   \begin{align*} \alpha^{\ast}\alpha_* \mathcal{L}^{\otimes k}&\simeq \alpha^{\ast}(p_2)_*(p_1^{\ast} {\mathcal{O}}_X(-krK_X)\cdot {\mathcal{I}}^k)\\ &=\Pi^{\ast}(H^0(X\times {\mathbb{A}}^1, p_1^{\ast} {\mathcal{O}}_X(-krK_X)\cdot {\mathcal{I}}^k) \otimes_{ {\mathbb{C}}[t]} {\mathcal{O}}_{X\times {\mathbb{A}}^1})\\ &\twoheadrightarrow\Pi^{\ast}(p_1^{\ast} {\mathcal{O}}_X(-krK_X)\cdot {\mathcal{I}}^k)\\ &\twoheadrightarrow \Pi^{\ast}p_1^{\ast} {\mathcal{O}}_X(-krK_X)\otimes {\mathcal{O}}_ \mathcal{B}(-kE_ \mathcal{B})= \mathcal{L}^{\otimes k} \end{align*} for $$k\gg 0$$. This means that $$ \mathcal {L}$$ is semiample over $$ {\mathbb {A}}^1$$. □ Definition 10 (1) The above flag ideal $$ {\mathcal {I}}$$ is called the basic flag ideal with respect to$$(X, -rK_X; -D)$$, and the above semitest configuration $$\alpha \colon ( \mathcal {B}, \mathcal {L})\to {\mathbb {A}}^1$$ is called the basic semitest configuration of$$(X, -rK_X)$$via$$D$$. The pair $$(X, -K_X)$$ is said to be divisorially stable (respectively, divisorially semistable) along$$D$$ if, for any $$r\in {\mathbb {Z}}_{>0}$$ which satisfies the generating property with respect to $$(X, -K_X; -D)$$, the basic semitest configuration $$( \mathcal {B}, \mathcal {L})/ {\mathbb {A}}^1$$ of $$(X, -rK_X)$$ via $$D$$ satisfies that $$ \operatorname {DF}( \mathcal {B}, \mathcal {L})>0$$ (respectively, $$\operatorname {DF}(\mathcal {B}, \mathcal {L}) \geqslant 0$$). (We will see in Theorem 5.1 that the definition does not depend on the choice of $$r$$.) The pair $$(X, -K_X)$$ is said to be divisorially stable (respectively, divisorially semistable) if the pair is divisorially stable (respectively, divisorially semistable) along any non-zero effective Weil divisor. Let $$\alpha \colon ( \mathcal {B}, \mathcal {L})\to {\mathbb {A}}^1$$ be the basic semitest configuration of $$(X, -rK_X)$$ via $$D$$ and let $$w(k)$$ be the total weight of the action of $$ \mathbb {G}_m$$ on $$(\alpha _* \mathcal {L}^{\otimes k})|_{\{0\}}$$. By [43, Lemma 3.3], $$w(k)$$ is equal to   \begin{align*} & -\dim\left(\frac{H^0(X\times {\mathbb{A}}^1, p_1^{\ast} {\mathcal{O}}_X(-krK_X))} {H^0(X\times {\mathbb{A}}^1, p_1^{\ast} {\mathcal{O}}_X(-krK_X)\cdot {\mathcal{I}}^k)}\right)\\ &\quad = -\dim\bigoplus_{j=0}^{kr\tau(D)-1}t^j\cdot\left( \frac{H^0(X, {\mathcal{O}}_X(-krK_X))}{H^0(X, {\mathcal{O}}_X(-krK_X) \cdot J_{(k, kr\tau(D)-j)})}\right)\\ &\quad = -kr\tau(D)h^0(X, {\mathcal{O}}_X(-krK_X))+\sum_{j=1}^{kr\tau(D)} h^0(X, {\mathcal{O}}_X(-krK_X)\cdot J_{(k, j)})\\ &\quad = -kr\tau(D)h^0(X, {\mathcal{O}}_X(-krK_X))+\sum_{j=1}^{kr\tau(D)} h^0(X, {\mathcal{O}}_X(-krK_X-jD)). \end{align*} Thus we have the following proposition. Proposition 3.3 Let$$( \mathcal {B}, \mathcal {L})/ {\mathbb {A}}^1$$be the basic semitest configuration of$$(X, -rK_X)$$via$$D$$. We set  \[ f(k):=\sum_{j=1}^\infty h^0(X, {\mathcal{O}}_X(-krK_X-jD)) =\sum_{j=1}^{kr\tau(D)}h^0(X, {\mathcal{O}}_X(-krK_X-jD)). \]Then$$f(k)$$is a polynomial function of degree at most$$n+1$$for$$k\gg 0$$. Let$$f_{n+1},$$$$f_n$$be the$$(n+1)$$th,$$n$$th coefficient of$$f(k),$$respectively. Then we have  \[ \operatorname{DF}( \mathcal{B}, \mathcal{L})=\frac{r^{2n}((-K_X)^{\cdot n})}{2\cdot(n!)^2}\eta(D), \]where  \[ \eta(D):=\frac{n!}{r^{n+1}}(nf_{n+1}-2rf_n). \] Remark 4 Obviously, if $$(X, -K_X)$$ is K-stable (respectively, K-semistable), then $$(X, -K_X)$$ is divisorially stable (respectively, divisorially semistable). In particular, if $$X$$ admits Kähler–Einstein metrics, then $$(X, -K_X)$$ is divisorially semistable by [2, 15]. Remark 5 The relationship between test configurations and filtered graded linear series is discussed by many authors. See [51, 57] and references therein. Recently, the author generalized the above framework of constructing test configurations. See [22, 23]. 4. On the asymptotic Riemann–Roch theorem In this section, we prove the following proposition. Proposition 4.1 Let$$V$$be a normal projective variety of dimension$$n,$$let$$H_V,$$$$D_V$$be$$ {\mathbb {Q}}$$-Cartier Weil divisors on$$V$$such that$$D_V$$is effective, and let$$a,$$$$b$$be rational numbers with$$a<b$$such that$$H_V-xD_V$$is ample for any$$x\in (a, b)$$. Then, for any sufficiently divisible positive integer$$k$$, the function  \[ v(k):=\sum_{j=ak+1}^{bk}h^0(V, {\mathcal{O}}_V(kH_V-jD_V)) \]satisfies that  \begin{align*} v(k) & = k^{n+1}\frac{1}{n!}\int_a^b((H_V-xD_V)^{\cdot n})\,dx\\ &\quad + k^n\frac{1}{2\cdot (n-1)!}\int_a^b((H_V-xD_V)^{\cdot n-1}\cdot(-K_V-D_V))\,dx + O(k^{n-1}). \end{align*} Proof We set   \[ v_0(k):=\sum_{j=ak+1}^{bk}\chi(V, {\mathcal{O}}_V(kH_V-jD_V)). \] Pick any $$c\in (a, b)\in {\mathbb {Q}}$$ and pick $$k_0\in {\mathbb {Z}}_{>0}$$ such that both $$k_0H_V$$ and $$ck_0D_V$$ are Cartier. By Takao Fujita's vanishing theorem [33, Theorem 1.4.35], there exists a positive integer $$k_1$$ which is divisible by $$k_0$$ such that   \[ H^i(V, {\mathcal{O}}_V((k_1+g_1k_0)H_V-(ck_1+g_2)D_V))=0 \] for any $$i>0$$, $$g_1\in {\mathbb {Z}}_{>0}$$ and $$g_2\in {\mathbb {Z}}$$ with $$ag_1k_0\leq g_2\leq bg_1k_0$$. Thus, for any sufficiently divisible positive integer $$k$$, we have $$H^i(V, {\mathcal {O}}_V(kH_V-jD_V))=0$$ for any $$i>0$$ and any $$j\in {\mathbb {Z}}$$ with $$ak+(c-a)k_1\leq j\leq bk+(c-b)k_1$$. Hence $$v(k)-v_0(k)$$ is equal to   \[ \sum_{\substack{j\in\{ak,\ldots,ak+(c-a)k_1-1, \\ bk+(c-b)k_1+1,\ldots,bk\}}} (h^0(V, {\mathcal{O}}_V(kH_V-jD_V)) -\chi(V, {\mathcal{O}}_V(kH_V-jD_V))). \] Thus $$v(k)-v_0(k)=O(k^{n-1})$$ (see [31, Theorem VI.2.15]). Hence it is enough to show the assertion for $$v_0(k)$$. We fix $$h\in {\mathbb {Z}}_{>0}$$ such that $$hD_V$$ is Cartier. We consider the $$ {\mathbb {Z}}/h {\mathbb {Z}}$$-graded $$ {\mathcal {O}}_V$$-algebra   \[ {\mathcal{A}}:=\bigoplus_{i=0}^{h-1} {\mathcal{O}}_V(-iD_V) \] defined by the effective Cartier divisor $$hD_V$$ ([32, Definition 2.52]). More precisely, we use the multiplication   \[ \begin{cases} {\mathcal{O}}_V(-iD_V)\otimes {\mathcal{O}}_V(-jD_V)\to {\mathcal{O}}_V(-(i+j)D_V), & (i+j< h),\\ {\mathcal{O}}_V(-iD_V)\otimes {\mathcal{O}}_V(-jD_V)\to {\mathcal{O}}_V(-(i+j)D_V)\xrightarrow{s} {\mathcal{O}}_V(-(i+j-h)D_V), & (i+j\geq h), \end{cases} \] where $$s\in H^0(V, {\mathcal {O}}_V(hD_V))$$ corresponds to the effective Cartier divisor $$hD_V$$. We consider the finite morphism   \[ \theta\colon\tilde{V}:= \operatorname{Spec}_{ {\mathcal{O}}_V} {\mathcal{A}}\longrightarrow V. \] Claim 4.2 The algebraic scheme $$\tilde {V}$$ is a reduced scheme which is Gorenstein in codimension 1 and satisfies that Serre's condition $$S_2$$. Moreover, we have $$K_{\tilde {V}}\sim \theta ^{\ast }(K_V+(h-1)D_V)$$, where $$K_{\tilde {V}}$$ is the canonical divisor of $$\tilde {V}$$ (see [25, Definition-Remark 2.7]). Proof of Claim 4.2 It follows from the definition that $$\tilde {V}$$ satisfies Serre's condition $$S_2$$ and is reduced. Pick any irreducible component $$D_0$$ of $$D_V$$ and let $$c_0$$ be the coefficient of $$D_V$$ at $$D_0$$. Let $$p\in D_0$$ be a general point. Then $$p\in V$$ is smooth and we can take an analytic local coordinate $$x_1,\ldots ,x_n$$ around $$p$$ such that $$D_V$$ is defined by the equation $$x_1^{c_0}=0$$. Then $$\tilde {V}$$ around over $$p$$ is defined by the equation $$(t^h-x_1^{c_0h}=0)\subset V\times {\mathbb {A}}^1_t$$. Thus $$\tilde {V}$$ is Gorenstein in codimension 1. Since the canonical sheaf $$\omega _{V\times {\mathbb {A}}^1}$$ of $$V\times {\mathbb {A}}^1$$ is generated by   \[ \frac{d(t^h-x_1^{c_0h})}{t^{h-1}}\wedge dx_1\wedge\cdots\wedge dx_n, \] the canonical sheaf $$\omega _{\tilde {V}}$$ of $$\tilde {V}$$ is generated by   \[ \frac{1}{t^{h-1}}\,dx_1\wedge\cdots\wedge dx_n \] around over $$p$$. Since $$(t^{h-1})^h=(x_1^{c_0h})^{h-1}$$, we get the assertion. □ For any $$k\in {\mathbb {Z}}_{>0}$$ and $$j\in {\mathbb {Z}}$$ such that $$kH_V$$ is Cartier and $$j$$ is divisible by $$h$$, we have   \begin{align*} &\chi(\tilde{V}, \theta^{\ast} {\mathcal{O}}_V(kH_V-jD_V)) =\chi(V, {\mathcal{A}}\otimes {\mathcal{O}}_V(kH_V-jD_V))\\ &\quad =\sum_{i=0}^{h-1}\chi(V, {\mathcal{O}}_V(kH_V-(i+j)D_V)). \end{align*} Therefore, by [43, Lemma 3.5, 47, (4.16)], for a sufficiently divisible positive integer $$k$$, $$v_0(k)$$ is equal to   \begin{align*} &\sum_{j'=ak/h+1}^{bk/h-1}\chi(\tilde{V}, \theta^{\ast} {\mathcal{O}}_V(kH_V-j'hD_V)) +\sum_{j\in\{ak+1,\ldots,ak+h-1\}\cup\{bk\}}\chi(V, {\mathcal{O}}_V(kH_V-jD_V))\\ &\quad =\sum_{j'=ak/h+1}^{bk/h}\chi(\tilde{V}, \theta^{\ast} {\mathcal{O}}_V(kH_V-j'hD_V))\\ &\qquad +k^n\frac{h-1}{n!}(((H_V-aD_V)^{\cdot n})-((H_V-bD_V)^{\cdot n}))+O(k^{n-1})\\ &\quad =\sum_{j'=ak/h+1}^{bk/h}\left\{\frac{k^n}{n!} (\theta^{\ast}(H_V-(j'/k)hD_V)^{\cdot n}) -\frac{k^{n-1}} {2\cdot(n-1)!}(\theta^{\ast}(H_V-(j'/k)hD_V)^{\cdot n-1}\cdot K_{\tilde{V}}) \right\}\\ &\qquad +k^n\frac{h-1}{n!}(((H_V-aD_V)^{\cdot n})-((H_V-bD_V)^{\cdot n}))+O(k^{n-1})\\ &\quad =\frac{k^{n+1}}{n!}\int_{a/h}^{b/h}h((H_V-xhD_V)^{\cdot n})\,dx +k^n\left\{\frac{h-1}{n!} (((H_V-aD_V)^{\cdot n})-((H_V-bD_V)^{\cdot n}))\right.\\ &\qquad +\frac{h}{2\cdot(n-1)!}\int_{a/h}^{b/h}((H_V-xhD_V)^{\cdot n-1}\cdot (-K_V-(h-1)D_V))\,dx\\ &\qquad \left.+\frac{h}{2\cdot(n-1)!}\int_{a/h}^{b/h}((H_V-xhD_V)^{\cdot n-1}\cdot(-hD_V) )\,dx\right\}+O(k^{n-1}). \end{align*} Therefore, we have proved Proposition 4.1. □ 5. Interpretation of $$\eta (D)$$ In this section, we fix a $$ {\mathbb {Q}}$$-Fano variety $$X$$ of dimension $$n$$ and a non-zero effective Weil divisor $$D$$ on $$X$$. In this section, we calculate the value $$\eta (D)$$ in Proposition 3.3 via intersection numbers, via the volume functions and via the restricted volume functions. 5.1. Via intersection numbers Fix $$r\in {\mathbb {Z}}_{>0}$$ which satisfies the generating property with respect to $$(X, -K_X; -D)$$, let $$( \mathcal {B}, \mathcal {L})/ {\mathbb {A}}^1$$ be the basic semitest configuration of $$(X, -rK_X)$$ via $$D$$ and let $$\{(\tau _i, X_i)\}_{1\leq i\leq m}$$ be the ample model sequence of $$(X, -K_X; -D)$$. We set $$f(k):=\sum _{j=1}^{kr\tau (D)}h^0(X, {\mathcal {O}}_X(-krK_X-jD))$$ as in Proposition 3.3. By [28, Remark 2.4(i)], for any sufficiently divisible positive integer $$k$$, we have   \[ f(k)=\sum_{i=1}^m\sum_{j=kr\tau_{i-1}+1}^{kr\tau_i}h^0(X_i, {\mathcal{O}}_{X_i}( -krK_{X_i}-jD_i)). \] By Proposition 4.1, for any $$1\leq i\leq m$$, we have   \begin{align*} &\sum_{j=kr\tau_{i-1}+1}^{kr\tau_i}h^0(X_i, {\mathcal{O}}_{X_i}(-krK_{X_i}-jD_i))\\ &\quad =\frac{k^{n+1}}{n!}\int_{r\tau_{i-1}}^{r\tau_i}((-rK_{X_i}-xD_i)^{\cdot n})\,dx\\ &\qquad +\frac{k^n}{2\cdot(n-1)!}\int_{r\tau_{i-1}}^{r\tau_i}((-rK_{X_i}-xD_i)^{\cdot n-1}\cdot (-K_{X_i}-D_i))dx+O(k^{n-1}). \end{align*} Thus the $$(n+1)$$th and $$n$$th coefficients $$f_{n+1}$$ and $$f_n$$ of $$f(k)$$ are, respectively,   \begin{align*} f_{n+1} &= \sum_{i=1}^m\frac{r^{n+1}}{n!}\int_{\tau_{i-1}}^{\tau_i} ((-K_{X_i}-xD_i)^{\cdot n})\,dx,\\ f_n & = \sum_{i=1}^m\frac{r^n}{2\cdot(n-1)!}\int_{\tau_{i-1}}^{\tau_i} ((-K_{X_i}-xD_i)^{\cdot n-1}\cdot(-K_{X_i}-D_i))\,dx. \end{align*} Therefore, we have proved the following. Theorem 5.1 Let$$X$$be a$$ {\mathbb {Q}}$$-Fano variety of dimension$$n$$and$$D$$be a non-zero effective Weil divisor on$$X$$. Then  \[ \eta(D)=\sum_{i=1}^mn\int_{\tau_{i-1}}^{\tau_i}(1-x)((-K_{X_i}-xD_i)^{\cdot n-1} \cdot D_i)\,dx, \]where$$\eta (D)$$is the value introduced in Proposition 3.3. In particular, divisorial stability and semistability of$$(X, -K_X)$$along$$D$$do not depend on the choice of the value$$r$$in Definition 10. 5.2. Via the volume functions Theorem 5.2 Let$$X,$$$$D$$and$$\eta (D)$$be as above. Then we have  \begin{align*} \eta(D) &= \operatorname{vol}_X(-K_X)-\int_0^{\tau(D)} \operatorname{vol}_X(-K_X-xD)\,dx\\ &= \operatorname{vol}_X(-K_X)-\int_0^\infty \operatorname{vol}_X(-K_X-xD)\,dx\\ &= \operatorname{vol}_X(-K_X)-n\int_0^{\tau(D)}x \operatorname{vol}_{X|D}(-K_X-xD)\,dx. \end{align*} Proof We note that $$\eta (D)$$ is equal to   \[ \sum_{i=1}^m\left\{\left[(x-1)((-K_{X_i}-xD_i)^{\cdot n}) \right]_{\tau_{i-1}}^{\tau_i}-\int_{\tau_{i-1}}^{\tau_i}((-K_{X_i}-xD_i)^{\cdot n})\,dx\right\}. \] By Lemma 2.3, this value is equal to   \[ \operatorname{vol}_X(-K_X)-\int_0^{\tau(D)} \operatorname{vol}_X(-K_X-xD)dx. \] On the other hand, by Definition 7 and Proposition 2.4, this value is also equal to   \[ n\int_0^{\tau(D)}(1-x) \operatorname{vol}_{X|D}(-K_X-xD)\,dx = \operatorname{vol}_X(-K_X)-n\int_0^{\tau(D)}x \operatorname{vol}_{X|D}(-K_X-xD)\,dx. \] Thus the assertion follows. □ 6. Toric case In this section, we see divisorial stability for toric $$ {\mathbb {Q}}$$-Fano varieties. For the theory of toric varieties, we refer the readers to [14]. We fix a lattice $$M:=\bigoplus _{i=1}^n {\mathbb {Z}} e_i$$, the dual lattice $$N:= \operatorname {Hom}_ {\mathbb {Z}}(M, {\mathbb {Z}})=\bigoplus _{i=1}^m {\mathbb {Z}} e_i^{\ast }$$, and we set $$M_ {\mathbb {R}}:=M\otimes _ {\mathbb {Z}} {\mathbb {R}}$$ and $$N_ {\mathbb {R}}:=N\otimes _ {\mathbb {Z}} {\mathbb {R}}$$. We have a natural dual pairing $$\langle , \rangle \colon M_ {\mathbb {R}}\times N_ {\mathbb {R}}\to {\mathbb {R}}$$ with $$\langle e_i, e^{\ast }_j\rangle =\delta _{ij}$$. We fix the canonical Lebesgue measure $$dx$$ on $$M_ {\mathbb {R}}$$, for which $$M_ {\mathbb {R}}/M$$ is of measure 1. Let $$X$$ be a toric $$ {\mathbb {Q}}$$-Fano variety of dimension $$n$$ that corresponds to a fan $$\Sigma $$ in $$N_ {\mathbb {R}}$$. Let $$\{v_\lambda \}_{\lambda \in \Lambda }$$ be the set of the primitive generators of one-dimensional cones in $$\Sigma $$, let $$D_\lambda $$ be the torus invariant prime divisor on $$X$$ associated to the one-dimensional cone $$ {\mathbb {R}}_{\geq 0}v_\lambda \in \Sigma $$. We set the rational polytope $$P\subset M_ {\mathbb {R}}$$ such that   \[ P:=P_{(X, -K_X=\sum_{\lambda\in\Lambda}D_\lambda)}:=\{u\in M_ {\mathbb{R}}\,|\, \langle u, v_\lambda\rangle\geq -1\ (\forall\lambda\in\Lambda)\} \] as in [14, (4.3.2)]. As is known in [4, Proposition 3.2], $$P$$ is a rational polytope that contains the origin in its interior. Let $$b_P\in M_ {\mathbb {R}}$$ be the barycenter of $$P$$, that is,   \[ b_P:=\frac{\int_Px\,dx}{\int_P\,dx}. \] Theorem 6.1 For any$$\lambda \in \Lambda ,$$the signature of$$\eta (D_\lambda )$$is equal to the signature of$$-\langle b_P, v_\lambda \rangle $$. Proof After a certain lattice transform, we can assume that $$v_\lambda =e_1^{\ast }$$. Claim 6.2 For any $$x\in [0, +\infty )$$, we have   \[ \operatorname{vol}_X(-K_X-xD_\lambda)=n!\cdot \operatorname{vol}_{M_ {\mathbb{R}}}(P|_{\nu_1\geq -1+x}), \] where   \begin{align*} P|_{\nu_1\geq -1+x}&:=\{u\in P\,|\,\langle u, e_1^{\ast}\rangle\geq -1+x\},\\ \operatorname{vol}_{M_ {\mathbb{R}}}(P|_{\nu_1\geq -1+x})&:=\int_{P|_{\nu_1\geq -1+x}}\,dx. \end{align*} Proof of Claim 6.2 It is enough to prove Claim 6.2 for the case $$x\in [0, +\infty )\cap {\mathbb {Q}}$$ since both $$ \operatorname {vol}_X(-K_X-xD_\lambda )$$ and $$n!\cdot \operatorname {vol}_{M_ {\mathbb {R}}}(P|_{\nu _1\geq -1+x})$$ are continuous functions over $$x$$. For a sufficiently divisible positive integer $$k$$, we have   \[ H^0(X, {\mathcal{O}}_X(-kK_X))=\bigoplus_{u\in kP\cap M} {\mathbb{C}}\chi^u, \] where $$\chi ^u$$ is the character of the algebraic torus $$( \mathbb {G}_m)^n$$ defined by $$u\in M$$ (see [14, (1.1.1)]). Moreover, $$H^0(X, {\mathcal {O}}_X(-kK_X-kxD_\lambda ))$$ is equal to, as a subspace,   \[ \bigoplus_{\substack{u\in kP\cap M\\ \langle u, e_1^{\ast}\rangle\geq -k+kx}} {\mathbb{C}}\chi^u \] (see [14, Subsection 4.3]). Thus we have   \[ h^0(X, {\mathcal{O}}_X(-kK_X-kxD_\lambda))=\#\{u\in k(P|_{\nu_1\geq -1+x})\cap M\}. \] Hence, by [34, Proposition 2.1], we have the assertion. □ Let $$Q(x)$$ be the (restricted) volume of   \[ P|_{\nu_1=-1+x}:=\{u\in P\,|\,\langle u, e_1^{\ast}\rangle=-1+x\}\subset {\mathbb{R}}^{n-1}. \] By Claim 6.2, we have   \[ Q(x)=-\frac{1}{n!}\frac{d}{dx} \operatorname{vol}_X(-K_X-xD_\lambda) =\frac{1}{(n-1)!} \operatorname{vol}_{X|D_\lambda}(-K_X-xD_\lambda) \] for any $$x\in (0, \tau (D_\lambda ))$$. Thus we get the equation   \[ \eta(D_\lambda)=n!\left( \operatorname{vol}_{M_ {\mathbb{R}}}(P)-\int_0^{\tau(D_\lambda)}xQ(x)\,dx\right) \] from Theorem 5.2. On the other hand, we have   \[ \langle b_P, e_1^{\ast}\rangle=\frac{\int_P\langle x, e_1^{\ast}\rangle dx} { \operatorname{vol}_{M_ {\mathbb{R}}}(P)}=\frac{\int_0^{\tau(D_\lambda)}(-1+x)Q(x)\,dx}{ \operatorname{vol}_{M_ {\mathbb{R}}}(P)} =\frac{\int_0^{\tau(D_\lambda)}xQ(x)\,dx}{ \operatorname{vol}_{M_ {\mathbb{R}}}(P)}-1. \] Thus we get the assertion. □ Corollary 6.3 Let$$X,$$$$P$$and$$b_P$$be as above. The pair$$(X, -K_X)$$is not divisorially stable. Assume that$$b_P\neq 0$$. Then there exists a torus invariant prime divisor$$D$$on$$X$$such that$$(X, -K_X)$$is not divisorially semistable along$$D$$. Proof We consider the case $$b_P\neq 0$$. Let $$c_P\in P$$ be the intersection of the boundary $$\partial P$$ of $$P$$ and the half line   \[ \{(1-t)b_P\,|\,t\in {\mathbb{R}}_{\geq 0}\} \] (cf. [35]). Let $$F_P$$ be a facet (that is, $$(n-1)$$-dimensional face) of $$P$$ with $$c_P\in F_P$$, and let $$ {\mathbb {R}}_{\geq 0}v_\lambda $$ be the one-dimensional cone in $$\Sigma $$ associated to $$F_P$$. By construction, we have $$\langle b_P, v_\lambda \rangle >0$$. Thus $$\eta (D_\lambda )<0$$ by Theorem 6.1. If $$b_P=0$$, then $$\langle b_P, v_\lambda \rangle =0$$ for any $$\lambda \in \Lambda $$. Thus $$\eta (D)=0$$ for any torus invariant prime divisor $$D$$ on $$X$$. □ Remark 6 The converse of Theorem 1.2 follows from [4, Theorem 1.2]. See also [56, 58]. As an immediate consequence of Corollary 6.3, for any non-K-semistable toric $$ {\mathbb {Q}}$$-Fano variety, we can explicitly construct a flag ideal such that the Donaldson–Futaki invariant of the associated semitest configuration is strictly negative. In fact, the basic flag ideal of $$(X, -rK_X; -D)$$ for some $$r\in {\mathbb {Z}}_{>0}$$ and for some torus invariant $$D$$ is a desired flag ideal. We note that, for a toric $$ {\mathbb {Q}}$$-Fano variety $$X$$ and a torus invariant prime divisor $$D_\lambda $$ on $$X$$, a positive integer $$r\in {\mathbb {Z}}_{>0}$$ satisfies the generating property with respect to $$(X, -K_X; -D_\lambda )$$ if and only if the $$ {\mathbb {C}}$$-algebra   \[ \bigoplus_{k\geq 0}H^0(X, {\mathcal{O}}_X(-krK_X)) \] is generated by $$H^0(X, {\mathcal {O}}_X(-rK_X))$$. Indeed, as we have seen in the proof of Theorem 6.1, the space $$H^0(X, {\mathcal {O}}_X(-krK_X-jD_\lambda ))$$ is equal to   \[ \bigoplus_{\substack{u\in krP\cap M\\ \langle u, v_\lambda\rangle\geq -kr+j}} {\mathbb{C}}\chi^u \] (see also [14, Definition 2.2.9]). We see some examples. Example 1 Let $$X$$ be the blowup of $$ {\mathbb {P}}^2$$ along one point and let $$E$$ be the $$(-1)$$-curve on $$X$$. As we have seen in Corollary 6.3, $$(X, -K_X)$$ is not divisorially semistable along $$E$$. In fact, $$\tau (E)=2$$, $$r=1$$ satisfies the generating property with respect to $$(X, -K_X; -E)$$, and the basic flag ideal $$ {\mathcal {I}}=I_2+I_1t+(t^2)$$ with respect to $$(X, -K_X; -E)$$ satisfies that   \begin{align*} I_2&= {\mathcal{O}}_X(-2E),\\ I_1&= {\mathcal{O}}_X(-E). \end{align*} In other words, $$ {\mathcal {I}}$$ is equal to the ideal sheaf $$( {\mathcal {O}}_X(-E)+(t))^2$$. Example 2 Let $$X$$ be the blowup of $$ {\mathbb {P}}^2$$ along distinct two points, let $$E_1$$, $$E_2$$ be the distinct exceptional divisors of $$X\to {\mathbb {P}}^2$$ and let $$E_0$$ be the strict transform of the line passing though the centers of the blowup. Note that $$(X, -K_X)$$ is slope stable by [46, Example 7.6]. As we have seen in Corollary 6.3, $$(X, -K_X)$$ is not divisorially semistable along $$E_0$$. In fact, $$\tau (E_0)=3$$, $$r=1$$ satisfies the generating property with respect to $$(X, -K_X; -E_0)$$, and the basic flag ideal $$ {\mathcal {I}}=I_3+I_2t+I_1t^2+(t^3)$$ with respect to $$(X, -K_X; -E_0)$$ satisfies that   \begin{align*} I_3&= {\mathcal{O}}_X(-3E_0-2E_1-2E_2),\\ I_2&= {\mathcal{O}}_X(-2E_0-E_1-E_2),\\ I_1&= {\mathcal{O}}_X(-E_0). \end{align*} The form looks similar to the one in [24, Corollary 5.3.14]. 7. On the barycenters of Okounkov bodies We see the relation between the value $$\eta (D)$$ and the structure of Okounkov bodies of $$-K_X$$. For the theory of Okounkov bodies, we refer the readers to [34]. (See also [7, 29, 45, 57].) In this section, we fix a $$ {\mathbb {Q}}$$-Fano variety of dimension $$n$$, an admissible flag   \[ Z_\bullet\colon X=Z_0\supset Z_1\supset\cdots\supset Z_n=\{\text{point}\} \] of $$X$$, that is, each $$Z_i\subset X$$ is a subvariety of dimension $$n-i$$ and each $$Z_i$$ is smooth around $$Z_n$$. We set $$\Delta (-K_X):=\Delta _{Z_\bullet }(-K_X) \subset {\mathbb {R}}^n$$ the Okounkov body of $$-K_X$$ with respects to $$Z_\bullet $$ in the sense of [34]. Theorem 7.1 Let$$b_1$$be the first coordinate of the barycenter of the Okounkov body$$\Delta (-K_X)$$. The following are equivalent: $$(X, -K_X)$$is divisorially stable (respectively, divisorially semistable) along$$Z_1;$$ $$b_1<1$$(respectively,$$b_1\leq 1)$$holds. Proof Let $$\sigma \colon \tilde {X}\to X$$ be a projective small $$ {\mathbb {Q}}$$-factorial modification morphism. Since $$Z_n$$ is a smooth point of $$X$$, the morphism $$\sigma $$ is isomorphism around $$Z_n$$ (see [31, Theorem VI.1.5]). Hence we can consider the strict transform $$\tilde {Z}_i$$ of $$Z_i$$, and we get an admissible flag $$\tilde {Z}_\bullet $$ of $$\tilde {X}$$. By the construction of $$\Delta (-K_X)$$, the Okounkov body $$\tilde {\Delta }:=\Delta _{\tilde {Z}_\bullet }(-K_{\tilde {X}})$$ coincides with $$\Delta (-K_X)$$. By the proof of [34, Corollary 4.25] (see also the proof of Proposition 2.5),   \[ \operatorname{vol}_{ {\mathbb{R}}^{n-1}}(\tilde{\Delta}|_{\nu_1=x})=\frac{1}{(n-1)!} \operatorname{vol}_{X|Z_1}(-K_X-xZ_1) \] holds for any $$x\in [0, \tau (Z_1))$$. Hence we have   \[ \eta(Z_1)=n!( \operatorname{vol}_{ {\mathbb{R}}^n}(\tilde{\Delta})-\int_0^{\tau(Z_1)} x \operatorname{vol}_{ {\mathbb{R}}^{n-1}}(\tilde{\Delta}|_{\nu_1=x})\,dx). \] On the other hand, the value $$b_1$$ is equal to   \[ \frac{\int_0^{\tau(Z_1)}x \operatorname{vol}_{ {\mathbb{R}}^{n-1}}(\tilde{\Delta}|_{\nu_1=x})\,dx} { \operatorname{vol}_{ {\mathbb{R}}^n} (\tilde{\Delta})}. \] Thus the assertion follows. □ Remark 7 (cf. [3]) Let $$(b_1,\ldots ,b_n)\in {\mathbb {R}}^n$$ be the barycenter of $$\Delta (-K_X)$$. Assume that $$(X, -K_X)$$ is K-semistable. We may expect that the values $$b_2,\ldots ,b_n$$, in particular $$\sum _{i=1}^nb_i$$, are also small. However, it is not true in general. See the following example. Example 3 Let $$X:= {\mathbb {P}}_{ {\mathbb {P}}^2}(T_{ {\mathbb {P}}^2})$$. We know that $$X$$ is a Fano manifold of dimension 3 and is a rational homogeneous manifold. Thus $$X$$ admits Kähler–Einstein metrics. In particular, $$(X, -K_X)$$ is K-semistable. On the other hand, consider the admissible flag $$Z_\bullet $$ of $$X$$ such that $$Z_1$$ is the inverse image $$\pi ^{-1}(l)$$ of a line $$l\subset {\mathbb {P}}^2$$, where $$\pi \colon X \to {\mathbb {P}}^2$$ is the projection morphism (note that $$Z_1\simeq {\mathbb {P}}_{ {\mathbb {P}}^1} ( {\mathcal {O}}\oplus {\mathcal {O}}(1))$$), $$Z_2$$ is the $$(-1)$$-curve on $$Z_1$$ and $$Z_3$$ is a point on $$Z_2$$. Then the Okounkov body $$\Delta (-K_X)\subset {\mathbb {R}}^3$$ of $$-K_X$$ with respect to $$Z_\bullet $$ is equal to   \[\{(\nu_1,\nu_2,\nu_3)\in {\mathbb{R}}^3\,|\,0\leq \nu_1,\nu_2\leq 2, \ 0\leq \nu_3\leq 2-\nu_1+\nu_2\}. \] The barycenter of $$\Delta (-K_X)$$ is equal to $$(\tfrac {5}{6}, \tfrac {7}{6}, \tfrac {7}{6})$$. 8. MMP with scaling In this section, we fix a $$ {\mathbb {Q}}$$-Fano variety $$X$$ of dimension $$n$$, a non-zero effective Weil divisor $$D$$ on $$X$$, and a projective small $$ {\mathbb {Q}}$$-factorial modification $$\sigma \colon \tilde {X}\to X$$, and we set $$\tilde {D}:=\sigma ^{-1}_*D$$. We show in this section that we can easily calculate the value $$\eta (D)$$ after we run a kind of MMP. More precisely, we run a $$(-\tilde {D})$$-MMP with scaling $$-K_{\tilde {X}}$$ (see [5, Subsection 3.10]). In other words, we consider the following program. Set $$\mu _0:=+\infty $$, $$t_0:=\mu _0^{-1}=0$$, $$\tilde {X}_1:=\tilde {X}$$ and $$\tilde {D}_1:=\tilde {D}$$. Assume that we have constructed $$\mu _{i-1}\in {\mathbb {R}}_{>0}\cup \{+\infty \}$$, a $$ {\mathbb {Q}}$$-factorial projective variety $$\tilde {X}_i$$ and a non-zero effective Weil divisor $$\tilde {D}_i$$ on $$\tilde {X}_i$$ such that $$-\tilde {D}_i+\mu _{i-1}(-K_{\tilde {X}_i})$$ is nef, that is, the value $$t_{i-1}:=\mu _{i-1}^{-1}\in {\mathbb {R}}_{\geq 0}$$ satisfies that $$-K_{\tilde {X}_i}-t_{i-1}\tilde {D}_{i-1}$$ is nef. Let   \[ \mu_i:=\min\{\mu\in {\mathbb{R}}_{>0}\cup\{+\infty\}\,|\,-\tilde{D}_i+\mu(-K_{\tilde{X}_i}) \colon\text{ nef}\}. \] In other words, $$t_i:=\mu _i^{-1}\in {\mathbb {R}}_{\geq 0}$$ satisfies that   \[ t_i=\max\{t\in {\mathbb{R}}_{\geq 0}\,|\,-K_{\tilde{X}_i}-t\tilde{D}_i\colon\text{ nef}\}. \] Since $$\tilde {D}_i$$ is non-zero effective, the values $$\mu _i$$ and $$t_i$$ can be defined and $$t_{i-1}\leq t_i$$. By [26, Proposition 1.11 (1)] and the argument of [5, Lemma 3.10.8], the value $$t_i$$ is a rational number and there exists an extremal ray $$R_i\subset \overline { \operatorname {NE}}(\tilde {X_i})$$ such that $$(-K_{\tilde {X}_i}-t_i\tilde {D}_i\cdot R_i)=0$$ and $$(\tilde {D}_i\cdot R_i)>0$$. Moreover, we get the contraction morphism $$\Phi _i\colon \tilde {X}_i\to \tilde {Y}_i$$ of $$R_i$$. If $$\Phi _i$$ is of fiber type, then we set $$m':=i$$ and we stop the program. If $$\Phi _i$$ is divisorial, then we set $$\tilde {X}_{i+1}:=\tilde {Y}_i$$; if $$\Phi _i$$ is small, then let $$\tilde {X}_{i+1}$$ be the flip of $$\Phi _i$$. Let $$\tilde {D}_{i+1}$$ be the strict transform of $$\tilde {D}_i$$ and we continue the program. Since $$(\tilde {D}_i\cdot R_i)>0$$, the Weil divisor $$\tilde {D}_{i+1}$$ is non-zero effective. By [26, Proposition 1.11(1)], after finitely many steps, we get a contraction morphism $$\Phi _{m'}\colon \tilde {X}_{m'}\to \tilde {Y}_{m'}$$ of fiber type. Thus we get the following data: Proposition 8.1 $$(1)$$Let$$\{(\tau _i, X_i)\}_{1\leq i\leq m}$$be the ample model sequence of$$(X, -K_X; -D)$$. Then we have  \[ \{\tau_0,\ldots,\tau_m\}=\{t_0,\ldots,t_{m'}\}, \]where$$\tau _0:=0$$. In particular, we have$$t_{m'}=\tau (D)$$. We have  \[ \eta(D)= \sum_{i=1}^{m'}n\int_{t_{i-1}}^{t_i}(1-x)((-K_{\tilde{X}_i}-x\tilde{D}_i )^{\cdot n-1}\cdot\tilde{D}_i)\,dx. \] Proof (1) For any $$1\leq i\leq m'$$ with $$t_{i-1}<t_i$$, there exist a $$1\leq j_i\leq m$$ and a birational morphism $$\Psi _i\colon \tilde {X}_i\to X_{j_i}$$ such that $$X_{j_i}$$ is the ample model of $$-K_{\tilde {X}_i}-t\tilde {D}_i$$ (hence also the ample model of $$-K_{\tilde {X}}-t\tilde {D}$$) for any $$t\in (t_{i-1}, t_i)$$. Moreover, the value $$j_i$$ and the morphism $$\Psi _i$$ are uniquely determined. Thus we have $$(t_{i-1}, t_i)\subset (\tau _{j_i-1}, \tau _{j_i})$$. Assume that there exists an $$1\leq i'\leq m'$$ such that $$i'>i$$ and $$j_{i'}=j_i$$. Then $$\Psi _i^{\ast }(-K_{X_{j_i}}-tD_{j_i})=-K_{\tilde {X}_i}-t\tilde {D}_i$$ is nef for any $$t\in (t_{i'-1}, t_{i'})$$. This contradicts to the construction of $$t_i$$. Thus $$j_i\neq j_{i'}$$ if $$i\neq i'$$. Since $$-K_{\tilde {X}_{m'}}-t_{m'}\tilde {D}_{m'}$$ is pseudo-effective but is not big, we have $$t_{m'}=\tau (D)$$. Thus we have proved (1). (2) The right-hand side is equal to   \[ \sum_{\substack{1\leq i\leq m'\\ \text{with }t_{i-1}<t_i}}n\int_{t_{i-1}}^{t_i}(1-x) ((-K_{\tilde{X}_i}-x\tilde{D}_i)^{\cdot n-1}\cdot \tilde{D}_i)\,dx. \] On the other hand, we have   \[ ((-K_{\tilde{X}_i}-x\tilde{D}_i)^{\cdot n-1}\cdot \tilde{D}_i) =(\Psi_i^{\ast}(-K_{X_{j_i}}-xD_{j_i})^{\cdot n-1}\cdot \tilde{D}_i) =((-K_{X_{j_i}}-xD_{j_i})^{\cdot n-1}\cdot D_{j_i}). \] Thus we have proved (2). □ birational contraction maps   \[ \tilde{X}=\tilde{X}_1\dashrightarrow\tilde{X}_2\dashrightarrow\cdots\dashrightarrow \tilde{X}_{m'} \] among $$ {\mathbb {Q}}$$-factorial projective varieties, the strict transform $$\tilde {D}_i$$ of $$\tilde {D}$$ on $$\tilde {X}_i$$ such that all of them are non-zero effective, a non-decreasing sequence   \[ 0=t_0\leq t_1\leq\cdots\leq t_{m'} \] of rational numbers and a contraction morphism $$\Phi _{m'}\colon \tilde {X}_{m'}\to \tilde {Y}_{m'}$$ of fiber type of an extremal ray $$R_{m'}\subset \overline { \operatorname {NE}}(\tilde {X}_{m'})$$ such that $$(-K_{\tilde {X}_{m'}}-t_{m'}\tilde {D}_{m'}\cdot R_{m'})=0$$. From Proposition 8.1, we can see the relationship between divisorial stability and Ross–Thomas' slope stability [47] along divisors (see [21]). Corollary 8.2 $$(1)$$Assume that$$X$$is$$ {\mathbb {Q}}$$-factorial. Then the value$$\tau _1$$is equal to the value$$\varepsilon (D, (X, -K_X)),$$where  \[ \varepsilon(D, (X, -K_X)):=\max\{t\in {\mathbb{R}}_{>0}\,|\,-K_X-tD\,\text{ nef}\}. \] Assume that$$X$$is smooth. If$$m=1,$$then$$\eta (D)=\xi (D),$$where  \[ \xi(D):=n\int_0^{\varepsilon(D, (X, -K_X))}(1-x)((-K_X-xD)^{\cdot n-1}\cdot D )\,dx \] (see [21, Proposition 3.2]). In particular, if$$ \operatorname {Nef}(X)=\overline { \operatorname {Eff}}(X),$$then$$\eta (D)=\xi (D)$$for any non-zero effective divisor$$D$$on$$X$$. If$$(X, -K_X)$$is divisorially stable (respectively, divisorially semistable) along$$D,$$then$$(X, -K_X)$$is slope stable (respectively, slope semistable) along$$D$$. (For the theory of slope stability, we refer the readers to [21, 47].) Proof (1) As we have already seen in the beginning of Section 8, $$t_1=\varepsilon (D, (X, -K_X))$$ holds. In particular, $$t_1$$ is a positive rational number. By Proposition 8.1(1), we have $$t_1=\tau _1$$. (2i) If $$m=1$$, then $$\tau (D)=\tau _1=\varepsilon (D, (X, -K_X))$$. Thus $$\eta (D)=\xi (D)$$. (2ii) Assume that $$\xi (D)\leq 0$$. Then $$t_1>1$$ by [21, Remark 3.4]. Thus   \[ \eta(D)=\xi(D)+\sum_{i=2}^{m'}n\int_{t_{i-1}}^{t_i}(1-x) ((-K_{\tilde{X}_i}-x\tilde{D}_i)^{\cdot n-1}\cdot\tilde{D}_i)\,dx \leq\xi(D) \] holds. □ Remark 8 Take any $$t\in (t_{i-1}, t_i)$$. By Proposition 8.1(1), there exists $$1\leq j\leq m$$ such that $$t_{i-1}=\tau _{j-1}$$ and $$t_i=\tau _j$$. Then the $$ {\mathbb {R}}$$-divisor $$-K_{\tilde {X}_i}-t\tilde {D}_i$$ is semiample and big on $$\tilde {X}_i$$, and the ample model of $$-K_{\tilde {X}_i}-t\tilde {D}_i$$ is $$X_j$$. However, the morphism $$\tilde {X}_i\to X_j$$ is not an isomorphism in general. For example, if $$X$$ is not $$ {\mathbb {Q}}$$-factorial, $$D\sim _ {\mathbb {Q}} -K_X$$ and $$t\in (0, 1)$$, then the morphism $$\tilde {X}_i\to X_j$$ is nothing but the morphism $$\sigma \colon \tilde {X}\to X$$. We note that $$m\neq m'$$ in general. For example, let $$X$$ be the blowup of $$ {\mathbb {P}}^2$$ along distinct two points and let $$D\subset X$$ be the strict transform of the line passing through the centers of the blowup. Then $$X_1=X$$, $$\tau _1=1$$, $$X_2= {\mathbb {P}}^2$$, $$\tau _2=3$$ and $$m=2$$. On the other hand, $$\tilde {X}_1=X$$, $$t_1=1$$, $$\tilde {X}_2$$ is the blowup of $$ {\mathbb {P}}^2$$ along a point, $$t_2=1$$, $$\tilde {X}_3= {\mathbb {P}}^2$$, $$t_3=3$$ and $$m'=3$$. In Corollary 8.2(2ii), we show the inequality $$\xi (D)\geq \eta (D)$$ under the assumption $$\xi (D)\leq 0$$. However, the inequality $$\xi (D)\geq \eta (D)$$ is not true in general. For example, let $$X$$ be the blowup of $$ {\mathbb {P}}^2$$ along a point, let $$\sigma \subset X$$ be the exceptional curve, let $$f\subset X$$ be the strict transform of a line passing through the center of the blowup, and let us consider $$D\subset X$$ with $$D\sim \sigma +3f$$. In this case, $$t_1=\tfrac {1}{2}$$, $$\tilde {X}_2= {\mathbb {P}}^2$$, $$\tilde {D}_2$$ is a cubic curve, $$t_2=1$$ and $$m'=2$$. Thus   \[ \eta(D)-\xi(D)=2\int_{1/2}^1(1-x)\cdot 3\cdot(3-3x)\,dx=\frac{3}{4}>0. \] 9. Basic properties of divisorial stability We see some basic properties of divisorial (semi)stability. In this section, we fix a $$ {\mathbb {Q}}$$-Fano variety $$X$$ of dimension $$n$$ and a non-zero effective Weil divisor $$D$$ on $$X$$. 9.1. Proportional case Lemma 9.1 Assume that a non-zero effective Weil divisor$$D'$$on$$X$$satisfies that$$D'\sim _ {\mathbb {Q}} cD$$for some$$c\in {\mathbb {Q}}_{>0}$$. Assume that$$c=1$$. Then$$\eta (D)=\eta (D')$$. Assume that$$c>1$$. If$$(X, -K_X)$$is divisorially semistable along$$D,$$then$$(X, -K_X)$$is divisorially stable along$$D'$$. Proof This is obvious from the equations   \begin{align*} \eta(D')&= \operatorname{vol}_X(-K_X)-\int_0^{\infty} \operatorname{vol}_X(-K_X-cxD)\,dx\\ &= \operatorname{vol}_X(-K_X)-\frac{1}{c}\int_0^{\infty} \operatorname{vol}_X(-K_X-xD)\,dx. \end{align*} □ Lemma 9.2 Assume that$$-K_X\sim _ {\mathbb {Q}} cD$$for some$$c\in {\mathbb {Q}}_{>0}$$. Then$$\eta (D)>0$$(respectively,$$\eta (D)\geqslant 0)$$holds if and only if$$c<n+1$$(respectively,$$c \leqslant n+1)$$holds. Proof Let $$\{(\tau _i, X_i)\}_{1\leq i\leq m}$$ be the ample model sequence. Then $$m=1$$ and $$\tau _1=c$$. Hence we have $$\eta (D)=((-K_X)^{\cdot n})\cdot (n+1-c)/(n+1)$$. □ Corollary 9.3 Assume that$$X$$is smooth and$$\rho (X)=1$$. If$$X\simeq {\mathbb {P}}^n,$$then$$(X, -K_X)$$is divisorially semistable and is not divisorially stable along a hyperplane. If$$X\not \simeq {\mathbb {P}}^n,$$then$$(X, -K_X)$$is divisorially stable. Proof The proof follows immediately from Lemma 9.2 and [30]. See also [21, Corollary 4.7]. □ Remark 9 There exists a Fano manifold $$X$$ of dimension 3 such that $$\rho (X)=1$$ but $$X$$ does not admit Kähler–Einstein metrics [53]. Thus, from Corollary 9.3, divisorial stability is strictly weaker than K-stability. Recently, the author found Fano manifolds $$X$$ with $$\rho (X)=1$$ such that the pairs $$(X, -K_X)$$ are not K-semistable (see [22]). Thus divisorial semistability is also strictly weaker than K-semistability. 9.2. Some criteria for divisorial stability Lemma 9.4 If$$\eta (D)\leq 0,$$then$$\tau (D)>1$$. In particular, if$$D$$is$$ {\mathbb {Q}}$$-Cartier and$$\eta (D)\leq 0,$$then$$-K_X-D$$is big. Proof The proof follows immediately since $$ \operatorname {vol}_X(-K_X-xD)$$ is a decreasing function. □ Remark 10 Assume that $$X$$ is smooth. It is enough to check the signature of $$\eta (D)$$ for only a finite number of divisors $$D$$ on $$X$$ for testing divisorial (semi)stability. This follows from Lemmas 9.4 and 9.1(1) and the fact   \[ \#\overline{ \operatorname{Eff}}(X)\cap([-K_X]- \operatorname{Big}(X))\cap \operatorname{Pic}(X)<+\infty, \] where $$[-K_X]$$ is the class of $$-K_X$$ in $$ \operatorname {N}^{1}(X)$$. Lemma 9.5 (cf. [21, Proposition 4.1]) Let$$\{(\tau _i, X_i)\}_{1\leq i\leq m}$$be the ample model sequence of$$(X, -K_X; -D)$$and$$D_i\subset X_i$$be the strict transform of$$D$$. Assume the following: for any$$1\leq i\leq m,$$there exists a positive integer$$h_i\in {\mathbb {Z}}_{>0}$$such that$$h_iD_i$$is Cartier and$$H^0(D_i, {\mathcal {O}}_{X_i}(h_iD_i)|_{D_i})\neq 0$$(respectively,$$H^0(D_i, {\mathcal {O}}_{X_i}(-h_iD_i)|_{D_i})\neq 0),$$and $$\tau (D)\leq 2$$(respectively,$$\tau (D)\geqslant 2)$$. Then$$\eta (D)\geqslant 0$$(respectively,$$\eta (D)\leqslant 0)$$. If we further assume one of the conditions: there exists an$$1\leq i\leq m$$such that$$ {\mathcal {O}}_{X_i}(h_iD_i)|_{D_i}$$is not numerically trivial on$$D_i,$$or $$\tau (D)<2$$(respectively,$$\tau (D)>2),$$then$$\eta (D)>0$$(respectively,$$\eta (D)<0)$$. Proof The function $$ \operatorname {vol}_X(-K_X-xD)$$ over $$x\in [0, \tau (D)]$$ is $$ {\mathcal {C}}^1$$ and a monotone decreasing function by Definition 7 and Proposition 2.4. Moreover, by Lemma 2.3,   \[ \frac{d^2}{dx^2} \operatorname{vol}_X(-K_X-xD)=(n-1)((-K_{X_i}-xD_i)^{\cdot n-2} \cdot D_i^{\cdot 2}) \] for any $$x\in (\tau _{i-1}, \tau _i)$$. Thus the assertion follows from the convexity property of the function $$ \operatorname {vol}_X(-K_X-xD)$$ (see also [21, Proof of Proposition 4.1]). For example, assume that $$\tau (D)\leq 2$$ and   \[ \frac{d^2}{dx^2} \operatorname{vol}_X(-K_X-xD)\geq 0 \] for any $$i$$ and $$x\in (\tau _{i-1}, \tau _i)$$. The function $$ \operatorname {vol}_X(-K_X-xD)$$ is $$ {\mathcal {C}}^1$$ and   \[ \frac{d}{dx} \operatorname{vol}_X(-K_X-xD) \] is monotone increasing. Thus   \begin{align*} \operatorname{vol}_X(-K_X-xD)&\leq\left(1-\frac{x}{\tau(D)}\right) \operatorname{vol}_X(-K_X)+\frac{x}{\tau(D)} \operatorname{vol}_X(-K_X-\tau(D)D)\\ &=\left(1-\frac{x}{\tau(D)}\right)((-K_X)^{\cdot n}). \end{align*} Hence   \[ \eta(D)\geq((-K_X)^{\cdot n})-\int_0^{\tau(D)}\left(1-\frac{x}{\tau(D)}\right) ((-K_X)^{\cdot n})dx=\frac{2-\tau(D)}{2}((-K_X)^{\cdot n})\geq 0. \] □ 9.3. Basic examples Proposition 9.6 Assume that$$X$$is smooth and$$n\leq 2$$. Then$$(X, -K_X)$$is K-stable (respectively, K-semistable) if and only if$$(X, -K_X)$$is divisorially stable (respectively, divisorially semistable). Proof The case $$n=1$$ is trivial. We consider the case $$n=2$$. If $$X$$ is toric, then the assertion follows from Corollary 6.3 and [4, Theorem 1.2]. Assume that $$X$$ is not toric. Then $$X$$ is obtained by the blowup of $$ {\mathbb {P}}^2$$ along distinct general points $$p_1,\ldots ,p_k$$ with $$4\leq k\leq 8$$. In this case, $$ \operatorname {Aut}(X)$$ is finite and $$X$$ admits Kähler–Einstein metrics by [52]. Hence $$(X, -K_X)$$ is K-stable by [49, Theorem 1.2]. By Remark 4, $$(X, -K_X)$$ is divisorially stable. □ Lemma 9.7 Let$$Y$$be a Fano manifold of dimension$$n\geq 3$$such that$$ \operatorname {Pic}(Y)= {\mathbb {Z}}[ {\mathcal {O}}_Y(1)]$$. Assume that$$r\in {\mathbb {Z}}_{>0}$$satisfies that$$ {\mathcal {O}}_Y(-K_Y)\simeq {\mathcal {O}}_Y(r)$$. Let$$1\leq d_1<d_2\leq r-1$$and let$$D_1,$$$$D_2\subset Y$$be divisors such that$$ {\mathcal {O}}_Y(D_i)\simeq {\mathcal {O}}_Y(d_i)$$$$(i=1, 2)$$and assume that the scheme theoretic intersection$$C:=D_1\cap D_2\subset Y$$is a smooth codimension 2 subvariety. Let$$X$$be the blowup along$$C$$. Then$$X$$is a Fano manifold. If$$d_2\geq 2d_1,$$then$$(X, -K_X)$$is not divisorially semistable along the strict transform of$$D_1$$. Proof Since $$\overline { \operatorname {NE}}(X)$$ is spanned by the class of a curve contracted by the morphism $$X\to Y$$ and a curve in the strict transform $$\hat {D}_1$$ of $$D_1$$, $$X$$ is a Fano manifold. We consider divisorial stability along $$\hat {D}_1$$. In this case, we have $$\tau _1=1$$, $$X_2=Y$$, $$\tau _2=r/d_2,$$ and $$m=2$$. Hence we get   \begin{align*} \eta(\hat{D}_1) &=\frac{( {\mathcal{O}}_Y(1)^{\cdot n})}{(n+1)d_1(d_2-d_1)^2} \left\{-\left((d_2-d_1)^2-d_1^2\right)(r-d_1)^{n+1}\right.\\ &\quad \left.-\,((n+1)(d_2-d_1)+r-d_2)(r-d_2)^n\right\}. \end{align*} Therefore, if $$d_2-d_1\geq d_1$$, then $$\eta (\hat {D}_1)<0$$. □ Lemma 9.8 Let$$Z$$be a Fano manifold of dimension$$n-1$$with$$n\geq 3$$and$$ \operatorname {Pic}(Z)= {\mathbb {Z}}[ {\mathcal {O}}_Z(1)]$$. Assume that$$r\in {\mathbb {Z}}_{>0}$$satisfies that$$ {\mathcal {O}}_Z(-K_Z)\simeq {\mathcal {O}}_Z(r)$$. We set$$Y:= {\mathbb {P}}_Z( {\mathcal {O}}_Z\oplus {\mathcal {O}}_Z(s)),$$where$$r>s>0$$. Let$$E\subset Y$$be a section of the$$ {\mathbb {P}}^1$$-bundle with$$ {\mathcal {N}}_{E/Y}\simeq {\mathcal {O}}_Z(s)$$and let$$W\subset E$$be a smooth divisor on$$E$$with$$ {\mathcal {O}}_E(W)\simeq {\mathcal {O}}_Z(d)$$such that$$r>d-s$$. Let$$X$$be the blowup of$$Y$$along$$W$$and let$$E'\subset Y$$be the strict transform of the negative section of$$Y\to Z$$. Then$$X$$is a Fano manifold. Moreover, if$$d=2s,$$then$$\eta (E')=0;$$if$$d<2s,$$then$$\eta (E')<0$$. Proof By [20, Lemma 2.5] or [9, Remark 3.6], $$X$$ is a Fano manifold. We consider divisorial stability along $$E'$$. In this case, we have $$\tau _1=1$$, $$X_2= {\mathbb {P}}_Z( {\mathcal {O}}_Z\oplus {\mathcal {O}}_Z(d-s))$$, $$ {\mathcal {N}}_{E'_2/X_2}\simeq {\mathcal {O}}_Z(d-s)$$, where $$E'_2$$ is the image of $$E'$$ on $$X_2$$, and $$\tau _2=2$$, $$m=2$$. If $$d\leq s$$, then $$\eta (E')<0$$ by Lemma 9.5. From now on, we assume that $$s<d\leq 2s$$. Set $$s':=d-s$$. Then $$0<s'\leq s<r$$ holds. Moreover, we have   \[ \eta(E')=\frac{( {\mathcal{O}}_Z(1)^{\cdot n-1})}{n+1}(g_1(s)-g_1(s')), \] where   \[ g_1(x):=\frac{1}{x^2}(r^{n+1}-(nx+r)(r-x)^n). \] Thus it is enough to show that the function $$g_1(x)$$ is a strictly monotone decreasing function over $$x\in (0, r)$$. Since $$g'_1(x)=g_2(x)/x^3$$, where   \[ g_2(x):=(r-x)^{n-1}(n(n-1)x^2+2(n-1)rx+2r^2)-2r^{n+1}, \] and $$g_2(0)=0$$, it is enough to show that the function $$g_2(x)$$ is a strictly monotone decreasing function over $$x\in (0, r)$$. Since   \[ g'_2(x)=-n(n-1)(n+1)(r-x)^{n-2}x^2<0, \] the assertion follows. □ 9.4. On deformations of Fano manifolds We show that the condition of divisorial (semi)stability for Fano manifolds is preserved under deformation. Proposition 9.9 Consider a smooth projective morphism$$X\to T$$with$$-K_X$$ample over$$T$$and$$0\in T$$a pointed smooth (connected) curve. Then$$(X_0, -K_{X_0})$$is divisorially stable (respectively, divisorially semistable) if and only if$$(X_t, -K_{X_t})$$is so for any closed point$$t\in T,$$where$$X_t$$is the fiber at$$t\in T$$. Proof Take any $$L_0^1,\ldots ,L_0^\rho \in \operatorname {Pic}(X_0)$$ which is a minimal generator of the module $$ \operatorname {Pic}(X_0)$$. We remark that $$ \operatorname {Pic}(X_t)=H^2(X_t^{{\rm an}}; {\mathbb {Z}})$$ for any $$t\in T$$ and $$X/T$$ is analytically locally trivial as differentiable manifolds. After replacing $$0\in T$$ with an étale neighborhood, there exist $$L^1,\ldots ,L^\rho \in \operatorname {Pic}(X)$$ such that $$L^i|_{X_0}\simeq L_0^i$$ for any $$1\leq i\leq \rho $$, and if we set $$L_t^i:=L^i|_{X_t}$$ ($$t\in T$$, $$1\leq i\leq \rho $$), then $$L_t^1,\ldots ,L_t^\rho $$ is a minimal generator of $$ \operatorname {Pic}(X_t)$$ for any $$t\in T$$ (see [55, Theorem 6.1]). For any $$a_\bullet =(a_1,\ldots ,a_\rho )\in {\mathbb {Z}}^{\oplus \rho }$$ and $$t\in T$$, we set   \begin{align*} L^{a_\bullet}&:=(L^1)^{\otimes a_1}\otimes\cdots\otimes (L^\rho)^{\otimes a_\rho}, \\ L_t^{a_\bullet}&:=(L_t^1)^{\otimes a_1}\otimes\cdots\otimes (L_t^\rho)^{\otimes a_\rho} =L^{a_\bullet}|_{X_t}. \end{align*} By [55, Theorem 6.1], the natural homomorphism   \[ H^0(X, L^{a_\bullet})\to H^0(X_0, L_0^{a_\bullet}) \] is surjective. Thus, there exists a Zariski open neighborhood $$0\in U^{a_\bullet }\subset T$$ such that $$h^0(X_0, L_0^{a_\bullet })=h^0(X_t, L_t^{a_\bullet })$$ for any $$t\in U^{a_\bullet }$$. Set   \[ U:=\bigcap_{a_\bullet\in {\mathbb{Z}}^{\oplus\rho}}U^{a_\bullet}\subset T. \] Then $$0\in U$$ and $$T\setminus U$$ consists of at most countably many closed points. Take any non-zero effective divisor $$D_0$$ on $$X_0$$. Then there exists $$a_\bullet \in {\mathbb {Z}}^{\oplus \rho }\setminus \{(0,\ldots ,0)\}$$ such that $$ {\mathcal {O}}_{X_0}(D_0)\simeq L_0^{a_\bullet }$$ holds. Pick any $$t\in U$$. Since $$0<h^0(X_0, L_0^{a_\bullet })=h^0(X_t, L_t^{a_\bullet })$$, there exists a non-zero effective divisor $$D_t$$ on $$X_t$$ such that $$ {\mathcal {O}}_{X_t}(D_t)\simeq L_t^{a_\bullet }$$. For any $$x\in {\mathbb {R}}_{\geq 0}$$, we have $$ \operatorname {vol}_{X_0}(-K_{X_0}-xD_0)= \operatorname {vol}_{X_t}(-K_{X_t}-xD_t)$$. Indeed, if we set $$ {\mathcal {O}}_{X_0}(-K_{X_0})\simeq L_0^{c_\bullet }$$ ($$c_\bullet \in {\mathbb {Z}}^{\oplus \rho }$$), then   \begin{align*} \operatorname{vol}_{X_0}(-K_{X_0}-xD_0) &=\limsup_{\substack{k\to\infty\\ kx\in {\mathbb{Z}}}}\frac{h^0(X_0, L_0^{c_\bullet-kxa_\bullet})} {k^n/n!}\\ &=\limsup_{\substack{k\to\infty\\ kx\in {\mathbb{Z}}}}\frac{h^0(X_t, L_t^{c_\bullet-kxa_\bullet})} {k^n/n!} = \operatorname{vol}_{X_t}(-K_{X_t}-xD_t) \end{align*} holds provided that $$x\in {\mathbb {Q}}_{\geq 0}$$. Therefore, $$\eta (D_0)=\eta (D_t)$$ holds. Conversely, take any $$t\in U$$ and take any non-zero effective divisor $$D'_t$$ on $$X_t$$. Then there exists $$a'_\bullet \in {\mathbb {Z}}^{\oplus \rho }\setminus \{(0,\ldots ,0)\}$$ such that $$ {\mathcal {O}}_{X_t}(D'_t)\simeq L_t^{a'_\bullet }$$ holds. Since $$0<h^0(X_t, L_t^{a'_\bullet })=h^0(X_0, L_0^{a'_\bullet })$$, there exists a non-zero effective divisor $$D'_0$$ on $$X_0$$ such that $$ {\mathcal {O}}_{X_0}(D'_0)\simeq L_0^{a'_\bullet }$$. By the same argument as above, we have $$ \operatorname {vol}_{X_0}(-K_{X_0}-xD'_0)= \operatorname {vol}_{X_t}(-K_{X_t}-xD'_t)$$ for any $$x\in {\mathbb {R}}_{\geq 0}$$. Thus $$\eta (D'_0)=\eta (D'_t)$$ holds. Therefore, we have checked that there exists $$0\in U\subset T$$ such that $$T\setminus U$$ consists of at most countably many closed points, and the condition ‘$$(X_0, -K_{X_0})$$ is divisorially stable (respectively, divisorially semistable)’ is equivalent to the condition ‘$$(X_t, -K_{X_t})$$ is divisorially stable (respectively, divisorially semistable) for any $$t\in U$$’. Since $$T$$ consists of uncountably many points, we get the assertion. □ Remark 11 In general, the situation is more complicated. For example, $$ {\mathbb {P}}^1\times {\mathbb {P}}^1$$ degenerates to $$ {\mathbb {P}}(1,1,2)$$, the weighted projective plane of the weights 1, 1, 2. Of course, $$( {\mathbb {P}}^1\times {\mathbb {P}}^1, -K_{ {\mathbb {P}}^1\times {\mathbb {P}}^1})$$ is divisorially semistable by Proposition 9.6 but $$( {\mathbb {P}}(1,1,2), -K_{ {\mathbb {P}}(1,1,2)})$$ is not divisorially semistable by Lemma 9.2. See [42] and [17] for general frameworks. 10. Three-dimensional case In this section, we consider divisorial stability for Fano manifolds of dimension 3. The goal of this section is to prove the following theorem. Theorem 10.1 Let$$X$$be a Fano manifold of dimension 3. The pair$$(X, -K_X)$$is divisorially semistable but not divisorially stable if and only if one of the following is satisfied: $$X\simeq {\mathbb {P}}^3;$$ $$\rho (X)=2$$and$$X$$belongs to either No. 26 or No. 34 in [40, Table 2]; $$\rho (X)=3$$and$$X$$belongs to one of No.$$9,$$No. 19 or No. 25 in [40, Table 3]; $$\rho (X)=4$$and$$X$$belongs to one of No.$$2,$$No. 4 or No. 7 in [40, Table 4]; $$\rho (X)=5$$and$$X$$belongs to No. 3 in [40, Table 5]; $$\rho (X)\geq 6$$. The pair $$(X, -K_X)$$ is not divisorially semistable if and only if one of the following is satisfied: $$\rho (X)=2$$ and $$X$$ belongs to one of No. $$23,$$ No. $$28,$$ No. $$30,$$ No. $$31,$$ No. $$33,$$ No. 35 or No. 36 in [40, Table 2]; $$\rho (X)=3$$ and $$X$$ belongs to one of No. $$14,$$ No. $$16,$$ No. $$18,$$ No. $$21,$$ No. $$22,$$ No. $$23,$$ No. $$24,$$ No. $$26,$$ No. $$28,$$ No. $$29,$$ No. 30 or No. 31 in [40, Table 3]; $$\rho (X)=4$$ and $$X$$ belongs to one of No. $$5,$$ No. $$8,$$ No. $$9,$$ No. $$10,$$ No. 11 or No. 12 in [40, Table 4]; $$\rho (X)=5$$ and $$X$$ belongs to No. 2 in [40, Table 5]. Remark 12 It has been already known in [27, 50] that some of $$X$$ in the list of Theorem 10.1 does not admit Kähler–Einstein metrics. In fact, in [27], Ilten and Süss obtained a concrete criterion for K-polystability of Fano manifolds admitting an algebraic torus action such that general orbits of the action are of codimension 1. We prove Theorem 10.1. We fix the notation. Let $$X$$ be a Fano manifold of dimension 3, let $$D$$ be a non-zero effective divisor on $$X$$ and let $$\{(\tau _i, X_i)\}_{1\leq i\leq m}$$ be the ample model sequence of $$(X, -K_X; -D)$$. We set   \[ \eta_i:=\int_{\tau_{i-1}}^{\tau_i}(1-x) ((-K_{X_i}-xD_i)^{\cdot 2}\cdot D_i)\,dx \] for simplicity. By definition, $$\eta (D)/3=\eta _1+\cdots +\eta _m$$. Let $$V_d$$ be a del Pezzo manifold of degree $$d$$$$(1\leq d\leq 5)$$ and of dimension 3 (see [19, I, Section 8]). Let $$V_7:= {\mathbb {P}}_{ {\mathbb {P}}^2}( {\mathcal {O}}\oplus {\mathcal {O}}(1))$$, $$W_6:= {\mathbb {P}}_{ {\mathbb {P}}^2}(T_{ {\mathbb {P}}^2})$$ and $$Q$$ be a smooth hyperquadric in $$ {\mathbb {P}}^4$$. For $$d\in {\mathbb {Z}}_{>0}$$, we set $$ \mathbb {F}_d:= {\mathbb {P}}_{ {\mathbb {P}}^1}( {\mathcal {O}}\oplus {\mathcal {O}}(d))$$, let $$\sigma _d\subset \mathbb {F}_d$$ be the $$(-d)$$-curve and let $$f_d\subset \mathbb {F}_d$$ be a fiber of the morphism $$ \mathbb {F}_d\to {\mathbb {P}}^1$$. Moreover, on $$ {\mathbb {P}}_{ {\mathbb {P}}^s}( {\mathcal {O}}\oplus {\mathcal {O}}(a_1)\oplus \cdots \oplus {\mathcal {O}}(a_k))$$, let $$\xi _ {\mathbb {P}}$$ be a tautological line bundle and $$H_{ {\mathbb {P}}^s}$$ be a pullback of $$ {\mathcal {O}}_{ {\mathbb {P}}^s}(1)$$ if there is no confusion. Table 1. Numbers of deformation types of Fano threefolds which are divisorially (semi)stable, slope (semi)stable along divisors       div.ss      RT ss        div.  but not  not.  RT  but not  not  $$\rho (X)$$  ALL  stable  div.stable  div.ss  stable  RT Stable  RT ss  1  17  16  1  0  16  1  0  2  36  27  2  7  31  1  4  3  31  16  3  12  26  2  3  4  13  4  3  6  12  1  0  5  3  1  1  1  2  1  0  6  1  0  1  0  0  1  0  7  1  0  1  0  0  1  0  8  1  0  1  0  0  1  0  9  1  0  1  0  0  1  0  10  1  0  1  0  0  1  0        div.ss      RT ss        div.  but not  not.  RT  but not  not  $$\rho (X)$$  ALL  stable  div.stable  div.ss  stable  RT Stable  RT ss  1  17  16  1  0  16  1  0  2  36  27  2  7  31  1  4  3  31  16  3  12  26  2  3  4  13  4  3  6  12  1  0  5  3  1  1  1  2  1  0  6  1  0  1  0  0  1  0  7  1  0  1  0  0  1  0  8  1  0  1  0  0  1  0  9  1  0  1  0  0  1  0  10  1  0  1  0  0  1  0  For the first row, ‘div.stable’, ‘div.ss but not div.stable’, ‘not div.ss’, ‘RT stable’, ‘RT ss but not RT stable’ and ‘not RT ss’ mean ‘divisorially stable’, ‘divisorially semistable but not divisorially stable’, ‘not divisorially semistable’, ‘slope stable along divisors’, ‘slope semistable along divisors but not slope stable along divisors’ and ‘not slope semistable along divisors’, respectively. View Large We can assume that $$\rho (X)\geq 2$$ by Corollary 9.3. Let $$\{l_1,\ldots ,l_k\}$$ be the set of minimal extremal rational curves on $$X$$ as in [39, Section III-3]. We note that the nef cone $$ \operatorname {Nef}(X)$$ of $$X$$ is the dual cone of $$\overline { \operatorname {NE}}(X)$$ and the pseudo-effective cone $$\overline { \operatorname {Eff}}(X)$$ of $$X$$ is equal to   \[ \operatorname{Nef}(X)+\sum_{E\in {\mathcal{E}}} {\mathbb{R}}_{\geq 0}[E], \] where $$ {\mathcal {E}}$$ is the set of the exceptional divisors of all elementary divisorial contractions of $$X$$ (see [1, Proposition 1.2]). Thus we can easily calculate those cones from [39, Section III-3]. We remark that one extremal ray is forgotten in [39, Section III-3, $$B_2=5$$, no. 1]. See Subsection 10.4 for details. To begin with, we see the strategy of the proof of Theorem 10.1. If $$\eta (D)\leq 0$$, then the possibility of the class of $$D$$ is finite by Remark 10. Moreover, we know the structure of the cone $$\overline { \operatorname {Eff}}(X)$$ from [39, Section III-3]. We can calculate those $$\eta (D)$$ by viewing the cone $$\overline { \operatorname {Eff}}(X)$$ carefully. In many cases, we can very easily check the signature of $$\eta (D)$$ from Lemma 9.5. We define the notion of suspicious divisors. The notion is just a quick test for the possibility of destabilizing. Definition 11 The divisor $$D$$ is said to be a suspicious divisor if the following conditions are satisfied: $$-K_X-D$$ is big; the class $$[D]\in \operatorname {Pic}(X)$$ of $$D$$ is primitive in $$ \operatorname {Pic}(X)$$ and is not proportional to $$-K_X$$; $$m\geq 2$$; if $$\rho (X)=2$$ and $$D$$ is nef, then $$\tau (D)>2$$. Assume that $$X$$ is neither a toric, the product of $$ {\mathbb {P}}^1$$ and a del Pezzo surface, nor the blowup of $$Q$$ along a line. Furthermore, if $$\eta (D)\leq 0$$, then $$D$$ must be a suspicious divisor by Corollary 8.2(2ii), [21, Theorem 1.5], Lemmas 9.1, 9.2, 9.4, and 9.5. The following lemma is obvious. Lemma 10.2 Assume that$$m=2,$$$$\tau _1=1,$$$$X_2$$is smooth and$$D_2\simeq D$$. We consider the case$$D\simeq {\mathbb {P}}^1\times {\mathbb {P}}^1$$. If$$\tau _2=\tfrac {3}{2},$$$$ {\mathcal {N}}_{D/X}\simeq {\mathcal {O}}_{ {\mathbb {P}}^1\times {\mathbb {P}}^1}(-1, a)$$and$$ {\mathcal {N}}_{D_2/X_2}\simeq {\mathcal {O}}_{ {\mathbb {P}}^1\times {\mathbb {P}}^1}(0, 4)$$with$$a\geq -1,$$then$$\eta (D)>0$$. If$$\tau _2=2,$$$$ {\mathcal {N}}_{D/X}\simeq {\mathcal {O}}_{ {\mathbb {P}}^1\times {\mathbb {P}}^1}(-1, a)$$and$$ {\mathcal {N}}_{D_2/X_2}\simeq {\mathcal {O}}_{ {\mathbb {P}}^1\times {\mathbb {P}}^1}(2, b)$$with$$a,$$$$b\geq -1,$$then$$\eta (D)>0$$. If$$\tau _2\geq 2,$$$$ {\mathcal {N}}_{D/X}\simeq {\mathcal {O}}_{ {\mathbb {P}}^1\times {\mathbb {P}}^1}(-1, a)$$and$$ {\mathcal {N}}_{D_2/X_2}\simeq {\mathcal {O}}_{ {\mathbb {P}}^1\times {\mathbb {P}}^1}(0, b)$$with$$a\leq 0$$and$$b\leq 1,$$then$$\eta (D)\leq 0$$. Moreover, the equality holds if and only if$$(\tau _2, a, b)=(2, 0, 1)$$. If$$\tau _2=3,$$$$ {\mathcal {N}}_{D/X}\simeq {\mathcal {O}}_{ {\mathbb {P}}^1\times {\mathbb {P}}^1}(-1, 1)$$and$$ {\mathcal {N}}_{D_2/X_2}\simeq {\mathcal {O}}_{ {\mathbb {P}}^1\times {\mathbb {P}}^1}(1, 1),$$then$$\eta (D)>0$$. We consider the case$$D\simeq \mathbb {F}_1$$. If$$\tau _2=2,$$$$ {\mathcal {N}}_{D/X}\simeq {\mathcal {O}}_{ \mathbb {F}_1}(-\sigma _1+af_1)$$and$$ {\mathcal {N}}_{D_2/X_2}\simeq {\mathcal {O}}_{ \mathbb {F}_1}(\sigma _1+2f_1)$$with$$a\geq -1,$$then$$\eta (D)>0$$. If$$\tau _2\geq 2,$$$$ {\mathcal {N}}_{D/X}\simeq {\mathcal {O}}_{ \mathbb {F}_1}(-\sigma _1-f_1)$$and$$ {\mathcal {N}}_{D_2/X_2}\simeq {\mathcal {O}}_{ \mathbb {F}_1}(a\sigma _1+f_1)$$with$$-1\leq a\leq 1,$$then$$\eta (D)\leq 0$$. Moreover, the equality holds if and only if$$(\tau _2, a)=(2, 1)$$. If$$\tau _2=2,$$$$ {\mathcal {N}}_{D/X}\simeq {\mathcal {O}}_{ \mathbb {F}_1}(-\sigma _1)$$and$$ {\mathcal {N}}_{D_2/X_2}\simeq {\mathcal {O}}_{ \mathbb {F}_1}(f_1)$$, then$$\eta (D)>0$$. Proof We only show the proof of (1i) for the readers' convenience. In this case, we have   \begin{align*} \frac{1}{3}\eta(D) & = \int_0^1(1-x)( {\mathcal{O}}_{ {\mathbb{P}}^1\times {\mathbb{P}}^1} (1+x, 2+a-ax)^{\cdot 2})\,dx+\int_1^{3/2}(1-x)( {\mathcal{O}}_{ {\mathbb{P}}^1\times {\mathbb{P}}^1} (2, 6-4x)^{\cdot 2})\,dx\\ &=2\int_0^1(1-x)(1+x)(2+a-ax)\,dx+8\int_1^{3/2}(1-x)(3-2x)\,dx=\frac{1}{6}(5a+14)>0. \end{align*} □ 10.1. The case $$\rho (X)=2$$ We consider the case $$\rho (X)=2$$. We prepare the following lemma. The proof is straightforward. Lemma 10.3 Assume that$$\rho (X)=2,$$$$X_2$$is smooth with$$ \operatorname {Pic}(X_2)= {\mathbb {Z}}[ {\mathcal {O}}_{X_2}(H)],$$$$-K_{X_2}\sim rH$$for some$$r\in {\mathbb {Z}}_{>0},$$and there exists a morphism$$\psi \colon X\to X_2$$which is obtained by the blowup of a smooth curve$$C\subset X_2$$of degree$$d$$and genus$$g$$. Let$$F$$be the exceptional divisor of$$\psi $$and let$$e,$$$$h\in {\mathbb {Z}}_{>0}$$with$$D+hF=\psi ^{\ast }D_2$$and$$D_2\sim eH$$. Then we have the equality  \begin{align*} \frac{1}{3}\eta(D)&=(H^{\cdot 3})\int_0^{\tau_2}e(1-x)(r-ex)^2\,dx\\ &\quad +\int_0^{\tau_1}(1-x)\{-(1-hx)(hr+e+h(hr-3e)x)d+(2g-2)h(1-hx)^2\}\,dx. \end{align*} Proof The proof follows from the fact   \begin{align*} \frac{1}{3}\eta(D)&=\int_0^{\tau_1}(1-x)((r\psi^{\ast}H-F-x(e\psi^{\ast}H-hF) )^{\cdot 2}\cdot e\psi^{\ast}H-hF)\,dx\\ &\quad +(H^{\cdot 3})\int_{\tau_1}^{\tau_2}e(1-x)(r-ex)^2\,dx. \end{align*} □ Assume that $$X$$ belongs to No. 33–36 in [40, Table 2]. Then $$X$$ is toric. Then $$(X, -K_X)$$ is not divisorially stable by Corollary 6.3. Moreover, $$(X, -K_X)$$ is divisorially semistable if and only if $$X$$ belongs to No. 34 in [40, Table 2] by Theorem 1.2 and [36]. Assume that $$X$$ belongs to No. 31 in [40, Table 2]. Then $$(X, -K_X)$$ is not divisorially semistable by Corollary 8.2(2ii) and [21, Theorem 1.5]. Assume that $$X$$ belongs to one of No. 23, No. 28, or No. 30 in [40, Table 2]. Then $$(X, -K_X)$$ is not divisorially semistable by Lemma 9.7. We assume that $$X$$ belongs to neither No. 23, No. 28, No. 30, No. 31, nor No. 33–36. We also assume the existence of a suspicious divisor $$D$$. By [41, Theorem 5.1], the possibility of $$X$$ and $$D$$ is one of the following: $$X$$ belongs to No. 15 in [40, Table 2] and $$D$$ is the strict transform of $$A$$; $$X$$ belongs to No. 19 in [40, Table 2] and $$D$$ is the exceptional divisor of the morphism $$X\to V_4$$; $$X$$ belongs to No. 22 in [40, Table 2] and $$D$$ is the exceptional divisor of the morphism $$X\to V_5$$; $$X$$ belongs to No. 26 in [40, Table 2] and $$D$$ is the exceptional divisor of the morphism $$X\to V_5$$; $$X$$ belongs to No. 26 in [40, Table 2] and $$D$$ is the exceptional divisor of the morphism $$X\to Q$$; $$X$$ belongs to No. 29 in [40, Table 2] and $$ {\mathcal {O}}_X(D)\simeq \operatorname {cont}_{l_2}^{\ast } {\mathcal {O}}_{ {\mathbb {P}}^1}(1)$$, where $$ \operatorname {cont}_{l_2}$$ is the contraction morphism associated to the extremal ray $$ {\mathbb {R}}_{\geq 0}[l_2]$$ [39]. In this case, under the notation in Lemma 10.3, $$\tau _1=1$$, $$\tau _2=2$$, $$X_2= {\mathbb {P}}^3$$, $$r=4$$, $$h=1$$, $$e=2$$, $$d=6$$ and $$g=4$$. Hence $$\eta (D)/3=\tfrac {7}{6}>0$$. In this case, under the notation in Lemma 10.3, $$\tau _1=1$$, $$\tau _2=2$$, $$X_2= {\mathbb {P}}^3$$, $$r=4$$, $$h=1$$, $$e=2$$, $$d=5$$ and $$g=2$$ (see also [41, p. 117]). Hence $$\eta (D)/3=2>0$$. In this case, under the notation in Lemma 10.3, $$\tau _1=1$$, $$\tau _2=2$$, $$X_2= {\mathbb {P}}^3$$, $$r=4$$, $$h=1$$, $$e=2$$, $$d=4$$ and $$g=0$$ (see also [41, p. 117]). Hence $$\eta (D)/3=\tfrac {17}{6}>0$$. In this case, under the notation in Lemma 10.3, $$\tau _1=1$$, $$\tau _2=3$$, $$X_2=Q$$, $$r=3$$, $$h=1$$, $$e=1$$, $$d=3$$ and $$g=0$$ (see also [41, p. 117]). Hence $$\eta (D)/3=0$$. In this case, under the notation in Lemma 10.3, $$\tau _1=\tfrac {1}{2}$$, $$\tau _2=2$$, $$X_2=V_5$$, $$r=2$$, $$h=2$$, $$e=1$$, $$d=1$$ and $$g=0$$. Hence $$\eta (D)/3=\tfrac {239}{48}>0$$. In this case, under the notation in Lemma 10.3, $$\tau _1=1$$, $$\tau _2=3$$, $$X_2=Q$$, $$r=3$$, $$h=1$$, $$e=1$$, $$d=2$$ and $$g=0$$. Hence $$\eta (D)/3=\tfrac {4}{3}>0$$. Therefore, we have proved Theorem 10.1 for the case $$\rho (X)=2$$. 10.2. The case $$\rho (X)=3$$ We consider the case $$\rho (X)=3$$. We assume that $$D$$ is a suspicious divisor. The case No. 1. Assume that $$X$$ belongs to No. 1 in [40, Table 3]. Then $$ \operatorname {Nef}(X)=\overline { \operatorname {Eff}}(X)$$ by [39]. Thus there is no suspicious divisor. Hence $$(X, -K_X)$$ is divisorially stable. The case No. 2. Assume that $$X$$ belongs to No. 2 in [40, Table 3]. Let $$H_1$$, $$H_2$$ be a divisor on $$X$$ that corresponds to the pullback of $$ {\mathcal {O}}_{ {\mathbb {P}}^1\times {\mathbb {P}}^1}(1, 0)$$, $$ {\mathcal {O}}_{ {\mathbb {P}}^1\times {\mathbb {P}}^1}(0, 1)$$, respectively. Let $$F$$ be the exceptional divisor of the morphism $$ \operatorname {cont}_{l_2}$$. Then we have   \begin{align*} \operatorname{Nef}(X)&= {\mathbb{R}}_{\geq 0}[H_1]+ {\mathbb{R}}_{\geq 0}[H_2]+ {\mathbb{R}}_{\geq 0}[2H_1+H_2+2F], \\ \overline{ \operatorname{Eff}}(X)&= {\mathbb{R}}_{\geq 0}[H_1]+ {\mathbb{R}}_{\geq 0}[H_2]+ {\mathbb{R}}_{\geq 0}[F],\\ -K_X&\sim 2H_1+H_2+F,\\ \operatorname{Pic}(X)&= {\mathbb{Z}}[H_1]\oplus {\mathbb{Z}}[H_2]\oplus {\mathbb{Z}}[F]. \end{align*} Hence $$D\sim H_1$$. In this case, $$\tau _1=1$$, $$X_2$$ is the image of the morphism $$ \operatorname {cont}_{l_3}$$, $$ {\mathcal {N}}_{D_2/X_2}$$ is non-zero effective, $$\tau _2=2$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 9.5. Hence $$(X, -K_X)$$ is divisorially stable. The case No. 3. Assume that $$X$$ belongs to No. 3 in [40, Table 3]. Let $$H_1$$, $$H_2$$, $$H_3$$ be a divisor that corresponds to the restriction of $$ {\mathcal {O}}(1, 0, 0)$$, $$ {\mathcal {O}}(0, 1, 0)$$, $$ {\mathcal {O}}(0, 0, 1)$$ on $$X$$, respectively. Let $$E_2$$, $$E_3$$ be the exceptional divisor of $$ \operatorname {cont}_{l_2}$$, $$ \operatorname {cont}_{l_3}$$, respectively. Then $$E_2\sim H_1-H_2+2H_3$$, $$E_3\sim -H_1+H_2+2H_3$$ and   \begin{align*} \operatorname{Nef}(X)&= {\mathbb{R}}_{\geq 0}[H_1]+ {\mathbb{R}}_{\geq 0}[H_2]+ {\mathbb{R}}_{\geq 0}[H_3],\\ \overline{ \operatorname{Eff}}(X)&= {\mathbb{R}}_{\geq 0}[H_1]+ {\mathbb{R}}_{\geq 0}[H_2]+ {\mathbb{R}}_{\geq 0}[H_1-H_2+2H_3] + {\mathbb{R}}_{\geq 0}[-H_1+H_2+2H_3],\\ -K_X&\sim H_1+H_2+H_3,\\ \operatorname{Pic}(X)&= {\mathbb{Z}}[H_1]\oplus {\mathbb{Z}}[H_2]\oplus {\mathbb{Z}}[H_3]. \end{align*} Hence $$D\sim H_1$$ or $$H_2$$. If $$D\sim H_1$$, then $$\tau _1=1$$, $$X_2= {\mathbb {P}}^1\times {\mathbb {P}}^2$$, $$\tau _2=\tfrac {3}{2}$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 9.5. If $$D\sim H_2$$, then we have $$\eta (D)>0$$ in the same way. Hence $$(X, -K_X)$$ is divisorially stable. The case No. 4. Assume that $$X$$ belongs to No. 4 in [40, Table 3]. Let $$H_1$$, $$H_2$$ be a divisor that corresponds to the pullback of $$ {\mathcal {O}}_{ {\mathbb {P}}^1\times {\mathbb {P}}^2}(1, 0)$$, $$ {\mathcal {O}}_{ {\mathbb {P}}^1\times {\mathbb {P}}^2}(0, 1)$$ on $$X$$, respectively. Let $$E$$ be the exceptional divisor of $$ \operatorname {cont}_{l_1}$$. Then we have   \begin{align*} \operatorname{Nef}(X)&= {\mathbb{R}}_{\geq 0}[H_1]+ {\mathbb{R}}_{\geq 0}[H_2]+ {\mathbb{R}}_{\geq 0}[H_2-E], \\ \overline{ \operatorname{Eff}}(X)&= {\mathbb{R}}_{\geq 0}[H_1]+ {\mathbb{R}}_{\geq 0}[H_2-E]+ {\mathbb{R}}_{\geq 0}[E],\\ -K_X&\sim H_1+2H_2-E,\\ \operatorname{Pic}(X)&= {\mathbb{Z}}[H_1]\oplus {\mathbb{Z}}[H_2]\oplus {\mathbb{Z}}[E]. \end{align*} Hence $$D\sim H_2-E$$. In this case, $$D$$ is nef, $$\tau _1=1$$, $$X_2$$ is the image of the morphism $$ \operatorname {cont}_{l_1}$$, $$\tau _2=2$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 9.5. Hence $$(X, -K_X)$$ is divisorially stable. The case No. 5. Assume that $$X$$ belongs to No. 5 in [40, Table 3]. Let $$H_1$$, $$H_2$$ be a divisor that corresponds to the pullback of $$ {\mathcal {O}}_{ {\mathbb {P}}^1\times {\mathbb {P}}^2}(1, 0)$$, $$ {\mathcal {O}}_{ {\mathbb {P}}^1\times {\mathbb {P}}^2}(0, 1)$$ on $$X$$, respectively. Let $$E_1$$, $$E_2$$ be the exceptional divisor of $$ \operatorname {cont}_{l_1}$$, $$ \operatorname {cont}_{l_2}$$, respectively. Then $$E_2\sim 2H_2-E_1$$ and   \begin{align*} \operatorname{Nef}(X)&= {\mathbb{R}}_{\geq 0}[H_1]+ {\mathbb{R}}_{\geq 0}[H_2]+ {\mathbb{R}}_{\geq 0}[2H_1+5H_2-2E_1], \\ \overline{ \operatorname{Eff}}(X)&= {\mathbb{R}}_{\geq 0}[H_1]+ {\mathbb{R}}_{\geq 0}[2H_2-E_1]+ {\mathbb{R}}_{\geq 0}[E_1],\\ -K_X&\sim 2H_1+3H_2-E_1,\\ \operatorname{Pic}(X)&= {\mathbb{Z}}[H_1]\oplus {\mathbb{Z}}[H_2]\oplus {\mathbb{Z}}[E_1]. \end{align*} Hence $$D\sim 2H_2-E_1$$, $$H_1$$ or $$H_1+2H_2-E_1$$. Assume that $$D\sim 2H_2-E_1$$, that is, $$D=E_2$$. Then $$ {\mathcal {N}}_{D/X}\simeq {\mathcal {O}}_{ {\mathbb {P}}^1\times {\mathbb {P}}^1}(-1, -1)$$, $$\tau _1=1$$, $$X_2= {\mathbb {P}}^1\times {\mathbb {P}}^2$$, $$ {\mathcal {N}}_{D_2/X_2}\simeq {\mathcal {O}}_{ {\mathbb {P}}^1\times {\mathbb {P}}^1}(0, 4)$$, $$\tau _2=\tfrac {3}{2}$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 10.2. Assume that $$D\sim H_1$$. In this case, $$\tau _1=1$$, $$X_2$$ is the image of the morphism $$ \operatorname {cont}_{l_2}$$, $$ {\mathcal {N}}_{D_2/X_2}$$ is non-zero effective, $$\tau _2=2$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 9.5. Assume that $$D\sim H_1+2H_2-E_1$$. In this case, $$\tau _1=1$$, $$X_2= {\mathbb {P}}^1\times {\mathbb {P}}^2$$, $$D_2\in | {\mathcal {O}}(1, 2)|$$, $$\tau _2=\tfrac {3}{2}$$ and $$m=2$$. Since $$\eta _1=\tfrac {35}{12}$$ and $$\eta _2=-\tfrac {7}{48}$$, we have $$\eta (D)/3>0$$. Hence $$(X, -K_X)$$ is divisorially stable. The case No. 6. Assume that $$X$$ belongs to No. 6 in [40, Table 3]. Let $$H_3$$ be a divisor that corresponds to the pullback of $$ {\mathcal {O}}_{ {\mathbb {P}}^3}(1)$$, let $$E_1$$, $$E_2$$ be the exceptional divisor of $$ \operatorname {cont}_{l_1}$$, $$ \operatorname {cont}_{l_2}$$, respectively. Let $$H_1:=-E_1+H_3$$ and $$H_2:=-E_2+2H_3$$. Then   \begin{align*} \operatorname{Nef}(X)&= {\mathbb{R}}_{\geq 0}[H_1]+ {\mathbb{R}}_{\geq 0}[H_2]+ {\mathbb{R}}_{\geq 0}[H_3], \\ \overline{ \operatorname{Eff}}(X)&= {\mathbb{R}}_{\geq 0}[H_1]+ {\mathbb{R}}_{\geq 0}[H_2]+ {\mathbb{R}}_{\geq 0}[-H_1+H_3]+ {\mathbb{R}}_{\geq 0}[-H_2+2H_3],\\ -K_X&\sim H_1+H_2+H_3,\\ \operatorname{Pic}(X)&= {\mathbb{Z}}[H_1]\oplus {\mathbb{Z}}[H_2]\oplus {\mathbb{Z}}[H_3]. \end{align*} Hence $$D\sim H_1$$, $$H_2$$ or $$H_1+H_2$$. Assume that $$D\sim H_1$$. In this case, $$\tau _1=1$$, $$X_2$$ is the image of the morphism $$ \operatorname {cont}_{l_1}$$, $$D_2$$ corresponds to the pullback of $$ {\mathcal {O}}_{ {\mathbb {P}}^3}(1)$$, $$\tau _2=2$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 9.5. Assume that $$D\sim H_2$$. In this case, $$\tau _1=1$$, $$X_2$$ is the image of the morphism $$ \operatorname {cont}_{l_2}$$, $$D_2$$ corresponds to the pullback of $$ {\mathcal {O}}_{ {\mathbb {P}}^3}(2)$$, $$\tau _2=\tfrac {3}{2}$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 9.5. Assume that $$D\sim H_1+H_2$$. In this case, $$\tau _1=1$$, $$X_2= {\mathbb {P}}^3$$, $$D_2\in | {\mathcal {O}}_{ {\mathbb {P}}^3}(3)|$$, $$\tau _2=\tfrac {4}{3}$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 9.5. Hence $$(X, -K_X)$$ is divisorially stable. The case No. 7. Assume that $$X$$ belongs to No. 7 in [40, Table 3]. Let $$H_2$$, $$H_3$$ be a divisor that corresponds to the pullback of $$ {\mathcal {O}}_{W_6}(1, 0)$$, $$ {\mathcal {O}}_{W_6}(0, 1)$$, respectively. Let $$E_1$$, $$E_2$$, $$E_3$$ be the exceptional divisor of $$ \operatorname {cont}_{l_1}$$, $$ \operatorname {cont}_{l_2}$$, $$ \operatorname {cont}_{l_3}$$, respectively. Let $$H_1:=-E_1+H_2+H_3$$. Then $$E_2\sim H_1+2H_2-H_3$$, $$E_3\sim H_1-H_2+2H_3$$ and   \begin{align*} \operatorname{Nef}(X)&= {\mathbb{R}}_{\geq 0}[H_1]+ {\mathbb{R}}_{\geq 0}[H_2]+ {\mathbb{R}}_{\geq 0}[H_3], \\ \overline{ \operatorname{Eff}}(X)&= {\mathbb{R}}_{\geq 0}[H_1]+ {\mathbb{R}}_{\geq 0}[H_1+2H_2-H_3] + {\mathbb{R}}_{\geq 0}[-H_1+H_2+H_3]+ {\mathbb{R}}_{\geq 0}[H_1-H_2+2H_3],\\ -K_X&\sim H_1+H_2+H_3,\\ \operatorname{Pic}(X)&= {\mathbb{Z}}[H_1]\oplus {\mathbb{Z}}[H_2]\oplus {\mathbb{Z}}[H_3]. \end{align*} Hence $$D\sim H_1$$, $$H_2$$ or $$H_3$$. Assume that $$D\sim H_1$$. In this case, $$\tau _1=1$$, $$X_2=W_6$$, $$D_2\sim (-\tfrac {1}{2})K_{W_6}$$, $$\tau _2=2$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 9.5. Assume that $$D\sim H_2$$. In this case, $$\tau _1=1$$, $$X_2= {\mathbb {P}}^1\times {\mathbb {P}}^2$$, $$\tau _2=\tfrac {3}{2}$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 9.5. If $$D\sim H_3$$, then we have $$\eta (D)>0$$ in the same way. Hence $$(X, -K_X)$$ is divisorially stable. The case No. 8. Assume that $$X$$ belongs to No. 8 in [40, Table 3]. Let $$H_1$$, $$H_2$$, $$H_3$$ be a divisor that corresponds to the restriction of $$ {\mathcal {O}}_{ \mathbb {F}_1\times {\mathbb {P}}^2}(\sigma _1+f_1, 0)$$, $$ {\mathcal {O}}_{ \mathbb {F}_1\times {\mathbb {P}}^2}(0, 1)$$, $$ {\mathcal {O}}_{ \mathbb {F}_1\times {\mathbb {P}}^2}(f_1, 0)$$, respectively. Let $$E_1$$, $$E_2$$ be the exceptional divisor of $$ \operatorname {cont}_{l_1}$$, $$ \operatorname {cont}_{l_2}$$, respectively. Then $$E_1\sim H_1-H_3$$, $$E_2\sim -H_1+2H_2+H_3$$ and   \begin{align*} \operatorname{Nef}(X)&= {\mathbb{R}}_{\geq 0}[H_1]+ {\mathbb{R}}_{\geq 0}[H_2]+ {\mathbb{R}}_{\geq 0}[H_3], \\ \overline{ \operatorname{Eff}}(X)&= {\mathbb{R}}_{\geq 0}[H_1-H_3]+ {\mathbb{R}}_{\geq 0}[H_3] + {\mathbb{R}}_{\geq 0}[-H_1+2H_2+H_3],\\ -K_X&\sim H_1+H_2+H_3,\\ \operatorname{Pic}(X)&= {\mathbb{Z}}[H_1]\oplus {\mathbb{Z}}[H_2]\oplus {\mathbb{Z}}[H_3]. \end{align*} Hence $$D\sim H_1$$, $$H_3$$ or $$H_1-H_3$$. Assume that $$D\sim H_1$$. In this case, $$\tau _1=1$$, $$X_2= {\mathbb {P}}^1\times {\mathbb {P}}^2$$, $$\tau _2=\tfrac {3}{2}$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 9.5. Assume that $$D\sim H_3$$. In this case, $$\tau _1=1$$, $$X_2$$ is the image of the morphism $$ \operatorname {cont}_{l_1}$$, $$ {\mathcal {N}}_{D_2/X_2}$$ is non-zero effective, $$\tau _2=2$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 9.5. Assume that $$D\sim H_1-H_3$$, that is, $$D=E_1$$. In this case, $$ {\mathcal {N}}_{D/X}\simeq {\mathcal {O}}_{ {\mathbb {P}}^1\times {\mathbb {P}}^1}(-1, 0)$$, $$\tau _1=1$$, $$X_2= {\mathbb {P}}^1\times {\mathbb {P}}^2$$, $$ {\mathcal {N}}_{D_2/X_2}\simeq {\mathcal {O}}_{ {\mathbb {P}}^1\times {\mathbb {P}}^1}(0, 4)$$, $$\tau _2=2$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 9.5. Hence $$(X, -K_X)$$ is divisorially stable. The case No. 9. Assume that $$X$$ belongs to No. 9 in [40, Table 3]. Let $$E_1,\ldots ,E_4$$ be the exceptional divisor of $$ \operatorname {cont}_{l_1},\ldots , \operatorname {cont}_{l_4}$$, respectively. Let $$H:=(E_1+E_3)/4$$. Then $$E_4\sim -2H+E_1+E_2$$ and   \begin{align*} \operatorname{Nef}(X)&= {\mathbb{R}}_{\geq 0}[E_1+E_2]+ {\mathbb{R}}_{\geq 0}[E_1+2E_2]+ {\mathbb{R}}_{\geq 0}[H]+ {\mathbb{R}}_{\geq 0}[E_2+2H], \\ \overline{ \operatorname{Eff}}(X)&= {\mathbb{R}}_{\geq 0}[4H-E_1]+ {\mathbb{R}}_{\geq 0}[-2H+E_1+E_2]+ {\mathbb{R}}_{\geq 0}[E_1] + {\mathbb{R}}_{\geq 0}[E_2],\\ -K_X&\sim H+E_1+2E_2,\\ \operatorname{Pic}(X)&= {\mathbb{Z}}[H]\oplus {\mathbb{Z}}[E_1]\oplus {\mathbb{Z}}[E_2]. \end{align*} Hence $$D\sim 4H-E_1$$, $$2H+E_2$$, $$H$$, $$H+E_2$$, $$E_1$$, $$E_2$$, $$E_1+E_2$$, $$-H+E_1+E_2$$ or $$-2H+E_1+E_2$$. If $$D\sim E_2$$ or $$-2H+E_1+E_2$$, then $$\eta (D)=0$$ by Lemma 9.8. Assume that $$D\sim E_1$$. In this case, $$\tau _1=\tfrac {1}{4}$$, $$X_2$$ is the image of the morphism $$ \operatorname {cont}_{l_2}$$, $$\tau _2=1$$, $$X_3$$ is the projective cone of a Veronese surface, $$\tau _3=\tfrac {5}{4}$$ and $$m=3$$. We have $$\eta _1=\tfrac {1225}{384}$$ and $$\eta _3=-\tfrac {1}{96}$$ since $$((-K_{X_3})^{\cdot 3}) =\tfrac {125}{2}$$. Thus $$\eta (D)/3>\tfrac {1225}{384}-\tfrac {1}{96}>0$$. If $$D\sim 4H-E_1$$, then $$\eta (D)>0$$ in the same way. Assume that $$D\sim E_1+E_2$$. In this case, $$D$$ is nef, $$\tau _1=\tfrac {1}{2}$$, $$X_2$$ is the image of the morphism $$ \operatorname {cont}_{l_2}$$, $$\tau _2=1$$, $$X_3$$ is the projective cone of a Veronese surface, $$\tau _3=\tfrac {5}{4}$$ and $$m=3$$. Thus $$\eta (D)>0$$ by Lemma 9.5. If $$D\sim 2H+E_2$$, then $$\eta (D)>0$$ in the same way. Assume that $$D\sim H_1$$. In this case, $$\tau _1=1$$, $$X_2$$ is the image of the contraction morphism associated to the extremal face spanned by $$ {\mathbb {R}}_{\geq 0}[l_2]$$ and $$ {\mathbb {R}}_{\geq 0}[l_4]$$, $$D_2\sim _ {\mathbb {Q}}(-\tfrac {1}{3})K_{X_2}$$, $$\tau _2=3$$ and $$m=2$$. We note that $$((-K_{X_2})^{\cdot 3})=27$$. Since $$\eta _1=\tfrac {10}{3}$$ and $$\eta _2=\tfrac {4}{3}$$, we have $$\eta (D)/3>0$$. Assume that $$D\sim H+E_2$$. In this case, $$\tau _1=1$$, $$X_2$$ is the projective cone of a Veronese surface, $$D_2\sim _ {\mathbb {Q}}(-\tfrac {3}{5})K_{X_2}$$, $$\tau _2=\tfrac {5}{3}$$ and $$m=2$$. Since $$\eta _1=\tfrac {19}{6}$$ and $$\eta _2=-\tfrac {2}{9}$$, we have $$\eta (D)/3>0$$. If $$D\sim -H+E_1+E_2$$, then $$\eta (D)>0$$ in the same way. Hence $$(X, -K_X)$$ is divisorially semistable but not divisorially stable. The case No. 10. Assume that $$X$$ belongs to No. 10 in [40, Table 3]. Let $$H_3$$ be a divisor that corresponds to the pullback of $$ {\mathcal {O}}_Q(1)$$, let $$E_1$$, $$E_2$$ be the exceptional divisor of $$ \operatorname {cont}_{l_1}$$, $$ \operatorname {cont}_{l_2}$$, respectively. Let $$H_1:=H_3-E_2$$ and $$H_2:=H_3-E_1$$. Then we have   \begin{align*} \operatorname{Nef}(X)&= {\mathbb{R}}_{\geq 0}[H_1]+ {\mathbb{R}}_{\geq 0}[H_2]+ {\mathbb{R}}_{\geq 0}[H_3], \\ \overline{ \operatorname{Eff}}(X)&= {\mathbb{R}}_{\geq 0}[H_1]+ {\mathbb{R}}_{\geq 0}[H_2]+ {\mathbb{R}}_{\geq 0}[-H_1+H_3] + {\mathbb{R}}_{\geq 0}[-H_2+H_3],\\ -K_X&\sim H_1+H_2+H_3,\\ \operatorname{Pic}(X)&= {\mathbb{Z}}[H_1]\oplus {\mathbb{Z}}[H_2]\oplus {\mathbb{Z}}[H_3]. \end{align*} Hence $$D\sim H_1$$, $$H_2$$ or $$H_1+H_2$$. Assume that $$D\sim H_1$$. In this case, $$\tau _1=1$$, $$X_2$$ is the blowup of $$Q$$ along a conic, $$D_2$$ corresponds to a pullback of $$ {\mathcal {O}}_Q(1)$$, $$\tau _2=2$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 9.5. If $$D\sim H_2$$, then $$\eta (D)>0$$ in the same way. Assume that $$D\sim H_1+H_2$$. In this case, $$\tau _1=1$$, $$X_2=Q$$, $$D_2\in | {\mathcal {O}}_Q(2)|$$, $$\tau _2=\tfrac {3}{2}$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 9.5. Hence $$(X, -K_X)$$ is divisorially stable. The case No. 11. Assume that $$X$$ belongs to No. 11 in [40, Table 3]. Let $$H_2$$, $$H_3$$ be a divisor that corresponds to the pullback of $$H_{ {\mathbb {P}}^2}$$, $$\xi _ {\mathbb {P}}$$ on $$V_7$$, respectively. Let $$E_1,\ldots ,E_3$$ be the exceptional divisor of $$ \operatorname {cont}_{l_1},\ldots , \operatorname {cont}_{l_3}$$, respectively. Let $$H_1:=-E_1+H_2+H_3$$. Then $$E_2\sim -H_2+H_3$$, $$E_3\sim H_1+2H_2-H_3$$ and   \begin{align*} \operatorname{Nef}(X)&= {\mathbb{R}}_{\geq 0}[H_1]+ {\mathbb{R}}_{\geq 0}[H_2]+ {\mathbb{R}}_{\geq 0}[H_3], \\ \overline{ \operatorname{Eff}}(X)&= {\mathbb{R}}_{\geq 0}[H_1]+ {\mathbb{R}}_{\geq 0}[H_1+2H_2-H_3] + {\mathbb{R}}_{\geq 0}[-H_2+H_3]+ {\mathbb{R}}_{\geq 0}[-H_1+H_2+H_3],\\ -K_X&\sim H_1+H_2+H_3,\\ \operatorname{Pic}(X)&= {\mathbb{Z}}[H_1]\oplus {\mathbb{Z}}[H_2]\oplus {\mathbb{Z}}[H_3]. \end{align*} Hence $$D\sim H_3$$, $$-H_2+H_3$$, $$H_1$$, $$H_2$$, $$H_1+H_2$$ or $$H_1+2H_2-H_3$$. Assume that $$D\sim H_3$$. In this case, $$\tau _1=1$$, $$X_2= {\mathbb {P}}^1\times {\mathbb {P}}^2$$, $$\tau _2=\tfrac {3}{2}$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 9.5. Assume that $$D\sim -H_2+H_3$$, that is, $$D=E_2$$. In this case, $$ {\mathcal {N}}_{D/X}\simeq {\mathcal {O}}_{ \mathbb {F}_1}(-\sigma _1-f_1)$$, $$\tau _1=1$$, $$X_2= {\mathbb {P}}^1\times {\mathbb {P}}^2$$, $$D_2\in | {\mathcal {O}}(1, 1)|$$, $$\tau _2=2$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 10.2. Assume that $$D\sim H_1$$. In this case, $$\tau _1=1$$, $$X_2=V_7$$, $$D_2\sim (-\tfrac {1}{2})K_{V_7}$$, $$\tau _2=2$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 9.5. Assume that $$D\sim H_2$$. In this case, $$\tau _1=1$$, $$X_2$$ is the blowup of $$ {\mathbb {P}}^3$$ along a quartic which is an intersection of two quadrics, $$D_2$$ corresponds to the pullback of $$ {\mathcal {O}}_{ {\mathbb {P}}^3}(1)$$, $$\tau _2=2$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 9.5. Assume that $$D\sim H_1+H_2$$. In this case, $$\tau _1=1$$, $$X_2= {\mathbb {P}}^3$$, $$\tau _2=\tfrac {4}{3}$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 9.5. Assume that $$D\sim H_1+2H_2-H_3$$, that is, $$D=E_3$$. In this case, $$\tau _1=\tfrac {1}{2}$$, $$X_2$$ is the blowup of $$ {\mathbb {P}}^3$$ along a quartic which is an intersection of two quadrics, $$D_2$$ corresponds to the sum of the pullback of $$ {\mathcal {O}}_{ {\mathbb {P}}^3}(1)$$ and the pullback of $$ {\mathcal {O}}_{ {\mathbb {P}}^1}(1)$$, $$\tau _2=1$$, $$X_3= {\mathbb {P}}^3$$, $$D_3\in | {\mathcal {O}}_{ {\mathbb {P}}^3}(3)|$$, $$\tau _3=\tfrac {4}{3}$$ and $$m=3$$. Since $$\tau _2=\tfrac {173}{192}$$ and $$\tau _3=-\tfrac {1}{36}$$, we have $$\eta (D)/3>\tfrac {173}{192}-\tfrac {1}{36}>0$$. Hence $$(X, -K_X)$$ is divisorially stable. The case No. 12. Assume that $$X$$ belongs to No. 12 in [40, Table 3]. Let $$H_3$$ be a divisor that corresponds to the pullback of $$ {\mathcal {O}}_{ {\mathbb {P}}^3}(1)$$. Let $$E_1,\ldots ,E_3$$ be the exceptional divisor of $$ \operatorname {cont}_{l_1},\ldots , \operatorname {cont}_{l_3}$$, respectively. Let $$H_1:=-E_1+H_3$$ and $$H_2:=-E_2+2H_3$$. Then $$E_3\sim H_1+2H_2-H_3$$ and   \begin{align*} \operatorname{Nef}(X)&= {\mathbb{R}}_{\geq 0}[H_1]+ {\mathbb{R}}_{\geq 0}[H_2]+ {\mathbb{R}}_{\geq 0}[H_3], \\ \overline{ \operatorname{Eff}}(X)&= {\mathbb{R}}_{\geq 0}[H_1]+ {\mathbb{R}}_{\geq 0}[H_1+2H_2-H_3] + {\mathbb{R}}_{\geq 0}[-H_1+H_3]+ {\mathbb{R}}_{\geq 0}[-H_2+2H_3],\\ -K_X&\sim H_1+H_2+H_3,\\ \operatorname{Pic}(X)&= {\mathbb{Z}}[H_1]\oplus {\mathbb{Z}}[H_2]\oplus {\mathbb{Z}}[H_3]. \end{align*} Hence $$D\sim -H_1+H_3$$, $$H_3$$, $$H_1$$, $$H_2$$ or $$H_1+H_2$$. Assume that $$D\sim -H_1+H_3$$, that is, $$D=E_1$$. In this case, $$ {\mathcal {N}}_{D/X}\simeq {\mathcal {O}}_{ {\mathbb {P}}^1\times {\mathbb {P}}^1}(-1, 1)$$, $$\tau _1=1$$, $$X_2= {\mathbb {P}}^1\times {\mathbb {P}}^2$$, $$ {\mathcal {N}}_{D_2/X_2}\simeq {\mathcal {O}}_{ {\mathbb {P}}^1\times {\mathbb {P}}^1}(0, 4)$$, $$\tau _2=\tfrac {3}{2}$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 10.2. Assume that $$D\sim H_3$$. In this case, $$\tau _1=1$$, $$X_2= {\mathbb {P}}^1\times {\mathbb {P}}^2$$, $$\tau _2=\tfrac {3}{2}$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 9.5. Assume that $$D\sim H_1$$. In this case, $$\tau _1=1$$, $$X_2$$ is the blowup of $$ {\mathbb {P}}^3$$ along a twisted cubic, $$D_2$$ corresponds to the pullback of $$ {\mathcal {O}}_{ {\mathbb {P}}^3}(1)$$, $$\tau _2=2$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 9.5. Assume that $$D\sim H_2$$. In this case, $$\tau _1=1$$, $$X_2= {\mathbb {P}}_{ {\mathbb {P}}^1}( {\mathcal {O}}^{\oplus 2}\oplus {\mathcal {O}}(1))$$, $$D_2\in |\xi _ {\mathbb {P}}^{\otimes 2}|$$, $$\tau _2=\tfrac {3}{2}$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 9.5. Assume that $$D\sim H_1+H_2$$. In this case, $$\tau _1=1$$, $$X_2= {\mathbb {P}}^3$$, $$\tau _2=\tfrac {4}{3}$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 9.5. Hence $$(X, -K_X)$$ is divisorially stable. The case No. 13. Assume that $$X$$ belongs to No. 13 in [40, Table 3]. Let $$H_2$$, $$H_3$$ be a divisor that corresponds to the pullback of $$ {\mathcal {O}}_{W_6}(1, 0)$$, $$ {\mathcal {O}}_{W_6}(0, 1)$$, respectively. Let $$E_1,\ldots ,E_3$$ be the exceptional divisor of $$ \operatorname {cont}_{l_1},\ldots , \operatorname {cont}_{l_3}$$, respectively. Let $$H_1:=-E_1+H_2+H_3$$. Then $$E_2\sim H_1-H_2+H_3$$, $$E_3\sim H_1+H_2-H_3$$ and   \begin{align*} \operatorname{Nef}(X)&= {\mathbb{R}}_{\geq 0}[H_1]+ {\mathbb{R}}_{\geq 0}[H_2]+ {\mathbb{R}}_{\geq 0}[H_3], \\ \overline{ \operatorname{Eff}}(X)&= {\mathbb{R}}_{\geq 0}[H_1+H_2-H_3]+ {\mathbb{R}}_{\geq 0}[H_1-H_2+H_3] + {\mathbb{R}}_{\geq 0}[-H_1+H_2+H_3],\\ -K_X&\sim H_1+H_2+H_3,\\ \operatorname{Pic}(X)&= {\mathbb{Z}}[H_1]\oplus {\mathbb{Z}}[H_2]\oplus {\mathbb{Z}}[H_3]. \end{align*} Hence $$D\sim H_1$$, $$H_2$$ or $$H_3$$. Assume that $$D\sim H_1$$. In this case, $$\tau _1=1$$, $$X_2=W_6$$, $$D_2\sim (-\tfrac {1}{2})K_{W_6}$$, $$\tau _2=2$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 9.5. If $$D\sim H_2$$ or $$H_3$$, then $$\eta (D)>0$$ in the same way. Hence $$(X, -K_X)$$ is divisorially stable. The case No. 14. Assume that $$X$$ belongs to No. 14 in [40, Table 3]. By Lemma 9.8, $$(X, -K_X)$$ is not divisorially semistable. The case No. 15. Assume that $$X$$ belongs to No. 15 in [40, Table 3]. Let $$H_3$$ be a divisor that corresponds to the pullback of $$ {\mathcal {O}}_Q(1)$$. Let $$E_1,\ldots ,E_3$$ be the exceptional divisor of $$ \operatorname {cont}_{l_1},\ldots , \operatorname {cont}_{l_3}$$, respectively. Let $$H_1:=-E_2+H_3$$ and $$H_2:=-E_1+H_3$$. Then $$E_3\sim H_1+2H_2-H_3$$ and   \begin{align*} \operatorname{Nef}(X)&= {\mathbb{R}}_{\geq 0}[H_1]+ {\mathbb{R}}_{\geq 0}[H_2]+ {\mathbb{R}}_{\geq 0}[H_3], \\ \overline{ \operatorname{Eff}}(X)&= {\mathbb{R}}_{\geq 0}[H_1]+ {\mathbb{R}}_{\geq 0}[H_1+2H_2-H_3] + {\mathbb{R}}_{\geq 0}[-H_1+H_3]+ {\mathbb{R}}_{\geq 0}[-H_2+H_3],\\ -K_X&\sim H_1+H_2+H_3,\\ \operatorname{Pic}(X)&= {\mathbb{Z}}[H_1]\oplus {\mathbb{Z}}[H_2]\oplus {\mathbb{Z}}[H_3]. \end{align*} Hence $$D\sim H_3$$, $$-H_1+H_3$$, $$-H_2+H_3$$, $$H_1$$, $$H_2$$, $$H_1+H_2$$ or $$H_1+2H_2-H_3$$. Assume that $$D\sim H_3$$. In this case, $$\tau _1=1$$, $$X_2= {\mathbb {P}}^1\times {\mathbb {P}}^2$$, $$D_2\in | {\mathcal {O}}(1, 2)|$$, $$\tau _2=\tfrac {3}{2}$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 9.5. Assume that $$D\sim -H_1+H_3$$, that is, $$D=E_2$$. In this case, $$ {\mathcal {N}}_{D/X}\simeq {\mathcal {O}}_{ {\mathbb {P}}^1\times {\mathbb {P}}^1}(-1, 2)$$, $$\tau _1=1$$, $$X_2= {\mathbb {P}}^1\times {\mathbb {P}}^2$$, $$ {\mathcal {N}}_{D_2/X_2}\simeq {\mathcal {O}}_{ {\mathbb {P}}^1\times {\mathbb {P}}^1}(0, 2)$$, $$\tau _2=\tfrac {3}{2}$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 10.2. Assume that $$D\sim -H_2+H_3$$, that is, $$D=E_1$$. In this case, $$ {\mathcal {N}}_{D/X}\simeq {\mathcal {O}}_{ \mathbb {F}_1}(-\sigma _1)$$, $$\tau _1=1$$, $$X_2= {\mathbb {P}}^1\times {\mathbb {P}}^2$$, $$ {\mathcal {N}}_{D_2/X_2}\simeq {\mathcal {O}}_{ \mathbb {F}_1}(\sigma _1+2f_1)$$, $$\tau _2=2$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 10.2. Assume that $$D\sim H_1$$. In this case, $$\tau _1=1$$, $$X_2$$ is the blowup of $$Q$$ along a line, $$D_2$$ corresponds to the pullback of $$ {\mathcal {O}}_Q(1)$$, $$\tau _2=2$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 9.5. Assume that $$D\sim H_2$$. In this case, $$\tau _1=1$$, $$X_2$$ is the blowup of $$Q$$ along a conic, $$D_2$$ corresponds to the pullback of $$ {\mathcal {O}}_Q(1)$$, $$\tau _2=2$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 9.5. Assume that $$D\sim H_1+H_2$$. In this case, $$\tau _1=1$$, $$X_2=Q$$, $$\tau _2=\tfrac {3}{2}$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 9.5. Assume that $$D\sim H_1+2H_2-H_3$$, that is, $$D=E_3$$. In this case, $$\tau _1=\tfrac {1}{2}$$, $$X_2$$ is the blowup of $$Q$$ along a conic, $$D_2$$ corresponds to the sum of the pullback of $$ {\mathcal {O}}_Q(1)$$ and the pullback of $$ {\mathcal {O}}_{ {\mathbb {P}}^1}(1)$$, $$\tau _2=1$$, $$X_3=Q$$, $$D_2\in | {\mathcal {O}}_Q(2)|$$, $$\tau _3=\tfrac {3}{2}$$ and $$m=3$$. Since $$\eta _2=\tfrac {9}{8}$$ and $$\eta _3=-\tfrac {1}{12}$$, we have $$\eta (D)/3> \tfrac {9}{8}-\tfrac {1}{12}>0$$. Hence $$(X, -K_X)$$ is divisorially stable. The case No. 16. Assume that $$X$$ belongs to No. 16 in [40, Table 3]. We consider the case that $$D$$ is the strict transform of the negative section of the morphism $$V_7\to {\mathbb {P}}^2$$. Then $$ {\mathcal {N}}_{D/X}\simeq {\mathcal {O}}_{ \mathbb {F}_1}(-\sigma _1-f_1)$$, $$\tau _1=1$$, $$X_2=W_6$$, $$D_2\in | {\mathcal {O}}_{W_6}(1, 0)|$$, $$\tau _2=2$$ and $$m=2$$. Thus $$\eta (D)<0$$ by Lemma 10.2. Hence $$(X, -K_X)$$ is not divisorially semistable. The case No. 17. Assume that $$X$$ belongs to No. 17 in [40, Table 3]. Let $$H_1$$, $$H_2$$, $$H_3$$ be a divisor that corresponds to the restriction of $$ {\mathcal {O}}_{ {\mathbb {P}}^1\times {\mathbb {P}}^1\times {\mathbb {P}}^2}(1, 0, 0)$$, $$ {\mathcal {O}}_{ {\mathbb {P}}^1\times {\mathbb {P}}^1\times {\mathbb {P}}^2}(0, 1, 0)$$, $$ {\mathcal {O}}_{ {\mathbb {P}}^1\times {\mathbb {P}}^1\times {\mathbb {P}}^2}(0, 0, 1)$$, respectively. Let $$E_1$$, $$E_2$$ be the exceptional divisor of $$ \operatorname {cont}_{l_1}$$, $$ \operatorname {cont}_{l_2}$$, respectively. Then $$E_1\sim H_1-H_2+H_3$$, $$E_2\sim -H_1+H_2+H_3$$ and   \begin{align*} \operatorname{Nef}(X)&= {\mathbb{R}}_{\geq 0}[H_1]+ {\mathbb{R}}_{\geq 0}[H_2]+ {\mathbb{R}}_{\geq 0}[H_3], \\ \overline{ \operatorname{Eff}}(X)&= {\mathbb{R}}_{\geq 0}[H_1]+ {\mathbb{R}}_{\geq 0}[H_2] + {\mathbb{R}}_{\geq 0}[H_1-H_2+H_3]+ {\mathbb{R}}_{\geq 0}[-H_1+H_2+H_3],\\ -K_X&\sim H_1+H_2+2H_3,\\ \operatorname{Pic}(X)&= {\mathbb{Z}}[H_1]\oplus {\mathbb{Z}}[H_2]\oplus {\mathbb{Z}}[H_3]. \end{align*} Hence $$D\sim H_1+H_3$$, $$H_2+H_3$$, $$H_1-H_2+H_3$$, $$-H_1+H_2+H_3$$, $$H_1$$ or $$H_2$$. Assume that $$D\sim H_1+H_3$$. In this case, $$\tau _1=1$$, $$X_2= {\mathbb {P}}^1\times {\mathbb {P}}^2$$, $$\tau _2=\tfrac {3}{2}$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 9.5. If $$D\sim H_2+H_3$$, then $$\eta (D)>0$$ in the same way. Assume that $$D\sim H_1-H_2+H_3$$. In this case, $$\tau _1=1$$, $$X_2= {\mathbb {P}}^1\times {\mathbb {P}}^2$$, $$D_2\in | {\mathcal {O}}(0, 2)|$$, $$\tau _2=\tfrac {3}{2}$$ and $$m=2$$. Since $$\eta _1=\tfrac {31}{6}$$ and $$\eta _2=-\tfrac {1}{3}$$, we have $$\eta (D)/3>0$$. If $$D\sim -H_1+H_2+H_3$$, then $$\eta (D)>0$$ in the same way. Assume that $$D\sim H_1$$. In this case, $$\tau _1=1$$, $$X_2= {\mathbb {P}}^1\times {\mathbb {P}}^2$$, $$D_2\in | {\mathcal {O}}(1, 1)|$$, $$\tau _2=2$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 9.5. If $$D\sim H_2$$, then $$\eta (D)>0$$ in the same way. Hence $$(X, -K_X)$$ is divisorially stable. The case No. 18. Assume that $$X$$ belongs to No. 18 in [40, Table 3]. We consider the case that $$D$$ is the strict transform of the plane in $$ {\mathbb {P}}^3$$ passing through the conic which is the center of the blowup. Then $$ {\mathcal {N}}_{D/X}\simeq {\mathcal {O}}_{ \mathbb {F}_1}(-\sigma _1-f_1)$$, $$\tau _1=1$$, $$X_2= {\mathbb {P}}_{ {\mathbb {P}}^1}( {\mathcal {O}}^{\oplus 2}\oplus {\mathcal {O}}(1))$$, $$D_2\in |\xi _ {\mathbb {P}}|$$, $$\tau _2=3$$ and $$m=2$$. Thus $$\eta (D)<0$$ by Lemma 10.2. Hence $$(X, -K_X)$$ is not divisorially semistable. The case No. 19. Assume that $$X$$ belongs to No. 19 in [40, Table 3]. Let $$H$$ be a divisor that corresponds to the pullback of $$ {\mathcal {O}}_Q(1)$$. Let $$E_1,\ldots ,E_4$$ be the exceptional divisor of $$ \operatorname {cont}_{l_1},\ldots , \operatorname {cont}_{l_4}$$, respectively. Then $$E_3\sim -2E_2+H$$, $$E_4\sim -2E_1+H$$ and   \begin{align*} \operatorname{Nef}(X)&= {\mathbb{R}}_{\geq 0}[H]+ {\mathbb{R}}_{\geq 0}[-E_1+H]+ {\mathbb{R}}_{\geq 0}[-E_2+H] + {\mathbb{R}}_{\geq 0}[-E_1-E_2+H], \\ \overline{ \operatorname{Eff}}(X)&= {\mathbb{R}}_{\geq 0}[E_1]+ {\mathbb{R}}_{\geq 0}[E_2] + {\mathbb{R}}_{\geq 0}[-2E_1+H]+ {\mathbb{R}}_{\geq 0}[-2E_2+H],\\ -K_X&\sim -2E_1-2E_2+3H,\\ \operatorname{Pic}(X)&= {\mathbb{Z}}[E_1]\oplus {\mathbb{Z}}[E_2]\oplus {\mathbb{Z}}[H]. \end{align*} Hence $$D\sim E_1$$, $$E_2$$, $$-E_1-E_2+H$$, $$-E_1+H$$, $$-E_2+H$$, $$-2E_1+H$$, $$-2E_2+H$$, $$-2E_1+E_2+H$$, $$E_1-2E_2+H$$, $$-2E_1-E_2+2H$$, $$-E_1-2E_2+2H$$, $$-3E_1-E_2+2H$$ or $$-E_1-3E_2+2H$$. Assume that $$D\sim E_1$$ or $$E_2$$. In this case, $$\eta (D)=0$$ by Lemma 10.2. Assume that $$D\sim -E_1-E_2+H$$. In this case, $$\tau _1=1$$, $$X_2=Q$$, $$D_2\in | {\mathcal {O}}_Q(1)|$$, $$\tau _2=3$$ and $$m=2$$. Since $$\eta _1=\tfrac {8}{3}$$ and $$\eta _2=-\tfrac {5}{6}$$, we have $$\eta (D)/3>0$$. Assume that $$D\sim -E_1+H$$. In this case, $$D$$ is nef, $$\tau _1=1$$, $$X_2=V_7$$, $$D_2\sim (-\tfrac {1}{2})K_{V_7}$$, $$\tau _2=2$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 9.5. If $$D\sim -E_2+H$$, then $$\eta (D)>0$$ in the same way. Assume that $$D\sim -2E_1+H$$, that is, $$D=E_4$$. In this case, $$\tau _1=1$$, $$X_2= {\mathbb {P}}^3$$, $$D_2\in | {\mathcal {O}}(2)|$$, $$\tau _2=2$$ and $$m=2$$. Since $$\eta _1=\tfrac {31}{6}$$ and $$\eta _2=-\tfrac {2}{3}$$, we have $$\eta (D)/3>0$$. If $$D\sim -2E_2+H$$, then $$\eta (D)>0$$ in the same way. Assume that $$D\sim -2E_1+E_2+H$$. In this case, $$\tau _1=\tfrac {1}{2}$$, $$X_2=V_7$$, $$D_2\in |\xi _ {\mathbb {P}}\otimes H_{ {\mathbb {P}}^2}^{\otimes 2}|$$, $$\tau _2=1$$, $$X_3= {\mathbb {P}}^3$$, $$D_3\in | {\mathcal {O}}(3)|$$, $$\tau _3=\tfrac {4}{3}$$ and $$m=3$$. Since $$\eta _2=\tfrac {91}{64}$$ and $$\eta _3=-\tfrac {1}{36}$$, we have $$\eta (D)/3>\tfrac {91}{64}-\tfrac {1}{36}>0$$. If $$D\sim E_1-2E_2+H$$, then $$\eta (D)>0$$ in the same way. Assume that $$D\sim -2E_1-E_2+2H$$. In this case, $$D$$ is nef, $$\tau _1=1$$, $$X_2= {\mathbb {P}}^3$$, $$D_2\in | {\mathcal {O}}(3)|$$, $$\tau _2=\tfrac {4}{3}$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 9.5. If $$D\sim -E_1-2E_2+2H$$, then $$\eta (D)>0$$ in the same way. Assume that $$D\sim -3E_1-E_2+2H$$. In this case, $$\tau _1=\tfrac {2}{3}$$, $$X_2$$ is the blowup of $$Q$$ along a point, $$D_2\sim -E+2H$$, where $$E$$ is the exceptional divisor of the morphism $$X_2\to Q$$ and $$H$$ corresponds to the pullback of $$ {\mathcal {O}}_Q(1)$$, $$\tau _2=1$$, $$X_3= {\mathbb {P}}^3$$, $$D_3\in | {\mathcal {O}}(3)|$$, $$\tau _3=\tfrac {4}{3}$$ and $$m=3$$. Since $$\eta _2=\tfrac {125}{324}$$ and $$\eta _3=-\tfrac {1}{36}$$, we have $$\eta (D)/3>\tfrac {125}{324}-\tfrac {1}{36}>0$$. If $$D\sim -E_1-3E_2+2H$$, then $$\eta (D)>0$$ in the same way. Hence $$(X, -K_X)$$ is divisorially semistable but not divisorially stable. The case No. 20. Assume that $$X$$ belongs to No. 20 in [40, Table 3]. Let $$H$$ be a divisor that corresponds to the pullback of $$ {\mathcal {O}}_Q(1)$$. Let $$E_1,\ldots ,E_3$$ be the exceptional divisor of $$ \operatorname {cont}_{l_1},\ldots , \operatorname {cont}_{l_3}$$, respectively. Then $$E_3\sim -E_1-E_2+H$$ and   \begin{align*} \operatorname{Nef}(X)&= {\mathbb{R}}_{\geq 0}[H]+ {\mathbb{R}}_{\geq 0}[-E_1+H]+ {\mathbb{R}}_{\geq 0}[-E_2+H], \\ \overline{ \operatorname{Eff}}(X)&= {\mathbb{R}}_{\geq 0}[E_1]+ {\mathbb{R}}_{\geq 0}[E_2] + {\mathbb{R}}_{\geq 0}[-E_1-E_2+H],\\ -K_X&\sim -E_1-E_2+3H,\\ \operatorname{Pic}(X)&= {\mathbb{Z}}[E_1]\oplus {\mathbb{Z}}[E_2]\oplus {\mathbb{Z}}[H]. \end{align*} Hence $$D\sim E_1$$, $$E_2$$, $$E_1+E_2$$, $$H$$, $$-E_1+H$$, $$-E_2+H$$, $$-E_1-E_2+H$$, $$-E_1-E_2+2H$$, $$-2E_1-E_2+2H$$ or $$-E_1-2E_2+2H$$. Assume that $$D\sim E_1$$. In this case, $$ {\mathcal {N}}_{D/X}\simeq {\mathcal {O}}_{ \mathbb {F}_1}(-\sigma _1)$$, $$\tau _1=1$$, $$X_2=W_6$$, $$D_2\in | {\mathcal {O}}_{W_6}(1, 0)|$$, $$\tau _2=2$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 10.2. If $$D\sim E_2$$, then $$\eta (D)>0$$ in the same way. Assume that $$D\sim E_1+E_2$$. In this case, $$\tau _1=\tfrac {1}{2}$$, $$X_2=W_6$$, $$D_2\sim (-\tfrac {1}{2})K_{W_6}$$, $$\tau _2=2$$ and $$m=2$$. Since $$\eta _2=\tfrac {27}{32}$$, we have $$\eta (D)>\tfrac {27}{32}>0$$. Assume that $$D\sim -E_1+H$$. In this case, $$D$$ is nef, $$\tau _1=1$$, $$X_2$$ is the blowup of $$Q$$ along a line, $$D_2$$ corresponds to the pullback of $$ {\mathcal {O}}_Q(1)$$, $$\tau _2=2$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 9.5. If $$D\sim -E_2+H$$, then $$\eta (D)>0$$ in the same way. Assume that $$D\sim -E_1-E_2+H$$, that is, $$D=E_3$$. In this case, $$ {\mathcal {N}}_{D/X}\simeq {\mathcal {O}}_{ {\mathbb {P}}^1\times {\mathbb {P}}^1}(-1, 1)$$, $$\tau _1=1$$, $$X_2=Q$$, $$D_2\in | {\mathcal {O}}_Q(1)|$$, $$\tau _2=3$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 10.2. Assume that $$D\sim -E_1-E_2+2H$$. In this case, $$D$$ is nef, $$\tau _1=1$$, $$X_2=Q$$, $$\tau _2=\tfrac {3}{2}$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 9.5. Assume that $$D\sim -2E_1-E_2+2H$$. In this case, $$\tau _1=\tfrac {1}{2}$$, $$X_2$$ is the blowup of $$Q$$ along a line, $$D_2\sim -E+2H$$, where $$H$$ corresponds to the pullback of $$ {\mathcal {O}}_Q(1)$$ and $$E$$ is the exceptional divisor of the morphism $$X_2\to Q$$, $$\tau _2=1$$, $$X_3=Q$$, $$D_3\in | {\mathcal {O}}_Q(2)|$$, $$\tau _3=\tfrac {3}{2}$$ and $$m=2$$. Since $$\eta _2=\tfrac {241}{192}$$ and $$\eta _3=-\tfrac {1}{12}$$, we have $$\eta (D)/3>\tfrac {241}{192}-\tfrac {1}{12}>0$$. If $$D\sim -E_1-2E_2+2H$$, then $$\eta (D)>0$$ in the same way. Hence $$(X, -K_X)$$ is divisorially stable. The case No. 21. Assume that $$X$$ belongs to No. 21 in [40, Table 3]. We consider the case that $$D$$ is the strict transform of the divisor in $$| {\mathcal {O}}_{ {\mathbb {P}}^1\times {\mathbb {P}}^2}(0, 1)|$$ passing through the conic which is the center of the blowup. Then $$ {\mathcal {N}}_{D/X}\simeq {\mathcal {O}}_{ {\mathbb {P}}^1\times {\mathbb {P}}^1}(-1, -1)$$, $$\tau _1=1$$, $$X_2= {\mathbb {P}}^1\times {\mathbb {P}}^2$$, $$D_2\in | {\mathcal {O}}(0, 1)|$$, $$\tau _2=3$$ and $$m=2$$. Thus $$\eta (D)<0$$ by Lemma 10.2. Hence $$(X, -K_X)$$ is not divisorially semistable. The case No. 22. Assume that $$X$$ belongs to No. 22 in [40, Table 3]. By Lemma 9.8, $$(X, -K_X)$$ is not divisorially semistable. The case No. 23. Assume that $$X$$ belongs to No. 23 in [40, Table 3]. We consider the case that $$D$$ is the strict transform of the hyperplane in $$ {\mathbb {P}}^3$$ passing through the conic which is the center of the blowup. Then $$ {\mathcal {N}}_{D/X}\simeq {\mathcal {O}}_{ \mathbb {F}_1}(-\sigma _1-f_1)$$, $$\tau _1=1$$, $$X_2=V_7$$, $$ {\mathcal {N}}_{D_2/X_2}\simeq {\mathcal {O}}_{ \mathbb {F}_1}(f_1)$$, $$\tau _2=2$$, $$X_3= {\mathbb {P}}^3$$, $$D_3\in | {\mathcal {O}}(1)|$$, $$\tau _3=4$$ and $$m=3$$. Thus $$\eta (D)/3<\eta _1+\eta _2<0$$ by Lemma 10.2. Hence $$(X, -K_X)$$ is not divisorially semistable. The case No. 24. Assume that $$X$$ belongs to No. 24 in [40, Table 3]. We consider the case that $$D$$ is the exceptional divisor of the morphism $$X\to W_6$$. Then $$ {\mathcal {N}}_{D/X}\simeq {\mathcal {O}}_{ {\mathbb {P}}^1\times {\mathbb {P}}^1}(-1, 0)$$, $$\tau _1=1$$, $$X_2= {\mathbb {P}}^1\times {\mathbb {P}}^2$$, $$D_2\in | {\mathcal {O}}(0, 1)|$$, $$\tau _2=3$$ and $$m=2$$. Thus $$\eta (D)<0$$ by Lemma 10.2. Hence $$(X, -K_X)$$ is not divisorially semistable. The case No. 25–31. Assume that $$X$$ belongs to one of No. 25–31 in [40, Table 3]. Then $$X$$ is toric. Then $$(X, -K_X)$$ is not divisorially stable by Corollary 6.3. Moreover, $$(X, -K_X)$$ is divisorially semistable if and only if $$X$$ belongs to either No. 25 or No. 27 in [40, Table 3] by Theorem 1.2 and [36]. Therefore, we have proved Theorem 10.1 for the case $$\rho (X)=3$$. 10.3. The case $$\rho (X)=4$$ We consider the case $$\rho (X)=4$$. We assume that $$D$$ is a suspicious divisor. The case No. 1. Assume that $$X$$ belongs to No. 1 in [40, Table 4]. Let $$H_1,\ldots ,H_4$$ be a divisor that corresponds to the restriction of $$ {\mathcal {O}}_{ {\mathbb {P}}^1\times {\mathbb {P}}^1\times {\mathbb {P}}^1\times {\mathbb {P}}^1}(1, 0, 0, 0),\ldots , {\mathcal {O}}_{ {\mathbb {P}}^1\times {\mathbb {P}}^1\times {\mathbb {P}}^1\times {\mathbb {P}}^1}(0, 0, 0, 1)$$, respectively. Let $$E_1,\ldots ,E_4$$ be the exceptional divisor of $$ \operatorname {cont}_{l_1},\ldots , \operatorname {cont}_{l_4}$$, respectively. Then $$E_1\sim -H_1+H_2+H_3+H_4$$, $$E_2\sim H_1-H_2+H_3+H_4$$, $$E_3\sim H_1+H_2-H_3+H_4$$, $$E_4\sim H_1+H_2+H_3-H_4$$ and   \begin{align*} \operatorname{Nef}(X)&= {\mathbb{R}}_{\geq 0}[H_1]+ {\mathbb{R}}_{\geq 0}[H_2]+ {\mathbb{R}}_{\geq 0}[H_3]+ {\mathbb{R}}_{\geq 0}[H_4], \\ \overline{ \operatorname{Eff}}(X)&= {\mathbb{R}}_{\geq 0}[H_1]+ {\mathbb{R}}_{\geq 0}[H_2] + {\mathbb{R}}_{\geq 0}[H_3]+ {\mathbb{R}}_{\geq 0}[H_4]\\ &\quad + {\mathbb{R}}_{\geq 0}[-H_1+H_2+H_3+H_4]+ {\mathbb{R}}_{\geq 0}[H_1-H_2+H_3+H_4]\\ &\quad + {\mathbb{R}}_{\geq 0}[H_1+H_2-H_3+H_4]+ {\mathbb{R}}_{\geq 0}[H_1+H_2+H_3-H_4],\\ -K_X&\sim H_1+H_2+H_3+H_4,\\ \operatorname{Pic}(X)&= {\mathbb{Z}}[H_1]\oplus {\mathbb{Z}}[H_2]\oplus {\mathbb{Z}}[H_3]\oplus {\mathbb{Z}}[H_4]. \end{align*} Hence $$D\sim H_1$$, $$H_2$$, $$H_3$$ or $$H_4$$. Assume that $$D\sim H_1$$. In this case, $$\tau _1=1$$, $$X_2= {\mathbb {P}}^1\times {\mathbb {P}}^1\times {\mathbb {P}}^1$$, $$D_2\in | {\mathcal {O}}(1, 1, 1)|$$, $$\tau _2=2$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 9.5. If $$D\sim H_2$$, $$H_3$$ or $$H_4$$, then $$\eta (D)>0$$ in the same way. Hence $$(X, -K_X)$$ is divisorially stable. The case No. 2. Assume that $$X$$ belongs to No. 2 in [40, Table 4]. Let $$H_1$$, $$H_2$$ be a divisor that corresponds to the pullback of $$ {\mathcal {O}}_{ {\mathbb {P}}^1\times {\mathbb {P}}^1}(1, 0)$$, $$ {\mathcal {O}}_{ {\mathbb {P}}^1\times {\mathbb {P}}^1}(0, 1)$$, respectively. Let $$E_1$$, $$E_2$$, $$E_3$$, $$E_5$$ be the exceptional divisor of $$ \operatorname {cont}_{l_1}$$, $$ \operatorname {cont}_{l_2}$$, $$ \operatorname {cont}_{l_3}$$, $$ \operatorname {cont}_{l_5}$$, respectively. Then $$E_1\sim H_1+H_2-E_3+E_5$$, $$E_2\sim H_1+H_2+E_3-E_5$$ and   \begin{align*} \operatorname{Nef}(X)&= {\mathbb{R}}_{\geq 0}[H_1]+ {\mathbb{R}}_{\geq 0}[H_2]+ {\mathbb{R}}_{\geq 0}[H_1+H_2+E_3] + {\mathbb{R}}_{\geq 0}[H_1+H_2+E_5]\\ &\quad + {\mathbb{R}}_{\geq 0}[H_1+H_2+E_3+E_5], \\ \overline{ \operatorname{Eff}}(X)&= {\mathbb{R}}_{\geq 0}[H_1]+ {\mathbb{R}}_{\geq 0}[H_2] + {\mathbb{R}}_{\geq 0}[E_3]+ {\mathbb{R}}_{\geq 0}[E_5]\\ &\quad + {\mathbb{R}}_{\geq 0}[H_1+H_2+E_3-E_5]+ {\mathbb{R}}_{\geq 0}[H_1+H_2-E_3+E_5],\\ -K_X&\sim 2H_1+2H_2+E_3+E_5,\\ \operatorname{Pic}(X)&= {\mathbb{Z}}[H_1]\oplus {\mathbb{Z}}[H_2]\oplus {\mathbb{Z}}[E_3]\oplus {\mathbb{Z}}[E_5]. \end{align*} Hence $$D\sim H_1$$, $$H_2$$, $$H_1+H_2$$, $$E_3$$, $$E_5$$, $$H_1+E_3$$, $$H_2+E_3$$, $$H_1+E_5$$, $$H_2+E_5$$, $$H_1+H_2+E_3$$, $$H_1+H_2+E_5$$, $$H_1+H_2+E_3-E_5$$ or $$H_1+H_2-E_3+E_5$$. Assume that $$D\sim H_1$$. In this case, $$\tau _1=1$$, $$X_2$$ is the image of the morphism associated to the extremal face spanned by $$ {\mathbb {R}}_{\geq 0}[l_3]$$ and $$ {\mathbb {R}}_{\geq 0}[l_5]$$, $$ {\mathcal {N}}_{D_2/X_2}$$ is non-zero effective, $$\tau _2=2$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 9.5. If $$D\sim H_2$$, then $$\eta (D)>0$$ in the same way. Assume that $$D\sim H_1+H_2$$. In this case, $$\tau _1=1$$, $$X_2$$ is the image of the morphism associated to the extremal face spanned by $$ {\mathbb {R}}_{\geq 0}[l_3],\ldots , {\mathbb {R}}_{\geq 0}[l_6]$$, $$\tau _2=2$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 9.5. Assume that $$D\sim E_3$$. In this case, $$ {\mathcal {N}}_{D/X}\simeq {\mathcal {O}}_{ {\mathbb {P}}^1\times {\mathbb {P}}^1}(-1, -1)$$, $$\tau _1=1$$, $$ {\mathcal {N}}_{D_2/X_2}\simeq {\mathcal {O}}_{ {\mathbb {P}}^1\times {\mathbb {P}}^1}(1, 1)$$, $$\tau _2=2$$ and $$m=2$$. Thus $$\eta (D)=0$$ by the same calculation in Lemma 9.8. If $$D\sim E_5$$, then $$\eta (D)=0$$ in the same way. Assume that $$D\sim H_1+E_3$$. In this case, $$\tau _1=1$$, $$X_2= {\mathbb {P}}_{ {\mathbb {P}}^1}( {\mathcal {O}}\oplus {\mathcal {O}}(1)^{\oplus 2})$$, $$D_2\in |\xi _ {\mathbb {P}}^{\otimes 2}\otimes H_{ {\mathbb {P}}^1}^{\otimes (-1)}|$$, $$\tau _2=\tfrac {3}{2}$$ and $$m=2$$. Since $$\eta _1=\tfrac {47}{12}$$ and $$\eta _2=-\tfrac {13}{48}$$, we have $$\eta (D)/3>0$$. If $$D\sim H_1+E_5$$, $$H_2+E_3$$ or $$H_2+E_5$$, then $$\eta (D)>0$$ in the same way. Assume that $$D\sim H_1+H_2+E_3$$. In this case, $$D$$ is nef, $$\tau _1=1$$, $$X_2$$ is the projective cone of a quadric hypersurface in $$ {\mathbb {P}}^3$$, $$\tau _2=\tfrac {3}{2}$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 9.5. If $$D\sim H_1+H_2+E_5$$, then $$\eta (D)>0$$ in the same way. Assume that $$D\sim H_1+H_2+E_3-E_5$$, that is, $$D=E_2$$. In this case, $$\tau _1=\tfrac {1}{2}$$, $$\tau _2=1$$, $$X_3$$ is the projective cone of a quadric hypersurface in $$ {\mathbb {P}}^3$$, $$D_3\sim (-2/3)K_{X_3}$$, $$\tau _3=\tfrac {3}{2}$$ and $$m=3$$. Since $$\eta _1=\tfrac {67}{16}$$ and $$\eta _3=-\tfrac {1}{12}$$, we have $$\eta (D)/3>\tfrac {67}{16}-\tfrac {1}{12}>0$$. If $$D\sim H_1+H_2-E_3+E_5$$, then $$\eta (D)>0$$ in the same way. Hence $$(X, -K_X)$$ is divisorially semistable but not divisorially stable. The case No. 3. Assume that $$X$$ belongs to No. 3 in [40, Table 4]. Let $$H_1,\ldots ,H_3$$ be a divisor that corresponds to the pullback of $$ {\mathcal {O}}_{ {\mathbb {P}}^1\times {\mathbb {P}}^1\times {\mathbb {P}}^1}(1, 0, 0),\ldots , {\mathcal {O}}_{ {\mathbb {P}}^1\times {\mathbb {P}}^1\times {\mathbb {P}}^1}(0, 0, 1)$$, respectively. Let $$E_1,\ldots ,E_4$$ be the exceptional divisor of $$ \operatorname {cont}_{l_1},\ldots , \operatorname {cont}_{l_4}$$, respectively. Then $$E_1\sim 2H_2+H_3-E_4$$, $$E_2\sim 2H_1+H_3-E_4$$, $$E_3\sim H_1+H_2-E_4$$ and   \begin{align*} \operatorname{Nef}(X)&= {\mathbb{R}}_{\geq 0}[H_1]+ {\mathbb{R}}_{\geq 0}[H_2]+ {\mathbb{R}}_{\geq 0}[H_3] + {\mathbb{R}}_{\geq 0}[H_1+H_2+H_3-E_4], \\ \overline{ \operatorname{Eff}}(X)&= {\mathbb{R}}_{\geq 0}[H_1]+ {\mathbb{R}}_{\geq 0}[H_2] + {\mathbb{R}}_{\geq 0}[H_3]+ {\mathbb{R}}_{\geq 0}[H_1+H_2-E_4]\\ &\quad + {\mathbb{R}}_{\geq 0}[2H_1+H_3-E_4]+ {\mathbb{R}}_{\geq 0}[2H_2+H_3-E_4]+ {\mathbb{R}}_{\geq 0}[E_4],\\ -K_X&\sim 2H_1+2H_2+2H_3-E_4,\\ \operatorname{Pic}(X)&= {\mathbb{Z}}[H_1]\oplus {\mathbb{Z}}[H_2]\oplus {\mathbb{Z}}[H_3]\oplus {\mathbb{Z}}[E_4]. \end{align*} Hence $$D\sim H_1$$, $$H_2$$, $$H_3$$, $$H_1+H_3$$, $$H_2+H_3$$, $$H_1+H_2-E_4$$ or $$H_1+H_2+H_3-E_4$$. Assume that $$D\sim H_1$$. In this case, $$\tau _1=1$$, $$X_2= {\mathbb {P}}^1\times \mathbb {F}_1$$, $$D_2\in | {\mathcal {O}}_{ {\mathbb {P}}^1\times \mathbb {F}_1}(1, \sigma _1+f_1)|$$, $$\tau _2=2$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 9.5. If $$D\sim H_2$$, then $$\eta (D)>0$$ in the same way. Assume that $$D\sim H_3$$. In this case, $$\tau _1=1$$, $$X_2\in | {\mathcal {O}}_{ {\mathbb {P}}^1\times {\mathbb {P}}^1\times {\mathbb {P}}^2}(1, 1, 2)|$$, $$D_2\in | {\mathcal {O}}_{X_2}(0, 0, 1)|$$, $$\tau _2=2$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 9.5. Assume that $$D\sim H_1+H_3$$. In this case, $$\tau _1=1$$, $$X_2= {\mathbb {P}}^1\times {\mathbb {P}}^2$$, $$D_2\in | {\mathcal {O}}(1, 2)|$$, $$\tau _2=\tfrac {3}{2}$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 9.5. If $$D\sim H_2+H_3$$, then $$\eta (D)>0$$ in the same way. Assume that $$D\sim H_1+H_2-E_4$$, that is, $$D=E_3$$. In this case, $$ {\mathcal {N}}_{D/X}\simeq {\mathcal {O}}_{ {\mathbb {P}}^1\times {\mathbb {P}}^1}(-1, 0)$$, $$\tau _1=1$$, $$X_2= {\mathbb {P}}^1\times {\mathbb {P}}^1\times {\mathbb {P}}^1$$, $$D_2\in | {\mathcal {O}}(1, 1, 0)|$$, $$\tau _2=2$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 10.2. Assume that $$D\sim H_1+H_2+H_3-E_4$$. In this case, $$D$$ is nef, $$\tau _1=1$$, $$X_2= {\mathbb {P}}^1\times {\mathbb {P}}^1\times {\mathbb {P}}^1$$, $$D_2\in | {\mathcal {O}}(1, 1, 1)|$$, $$\tau _2=2$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 9.5. Hence $$(X, -K_X)$$ is divisorially stable. The case No. 4. Assume that $$X$$ belongs to No. 4 in [40, Table 4]. Let $$H$$ be a divisor that corresponds to the pullback of $$ {\mathcal {O}}_Q(1)$$. Let $$E_1,\ldots ,E_5$$ be the exceptional divisor of $$ \operatorname {cont}_{l_1},\ldots , \operatorname {cont}_{l_5}$$, respectively. Then $$E_3\sim -2E_1+H$$, $$E_4\sim -2E_2+H$$ and   \begin{align*} \operatorname{Nef}(X)&= {\mathbb{R}}_{\geq 0}[H]+ {\mathbb{R}}_{\geq 0}[-E_1+H]+ {\mathbb{R}}_{\geq 0}[-E_2+H]\\ &\quad + {\mathbb{R}}_{\geq 0}[-E_1-E_2+H]+ {\mathbb{R}}_{\geq 0}[-E_1-E_2+H-E_5], \\ \overline{ \operatorname{Eff}}(X)&= {\mathbb{R}}_{\geq 0}[E_1]+ {\mathbb{R}}_{\geq 0}[E_2] + {\mathbb{R}}_{\geq 0}[-2E_1+H]+ {\mathbb{R}}_{\geq 0}[-2E_2+H]\\ &\quad + {\mathbb{R}}_{\geq 0}[-E_1-E_2+H-E_5]+ {\mathbb{R}}_{\geq 0}[E_5],\\ -K_X&\sim -2E_1-2E_2+3H-E_5,\\ \operatorname{Pic}(X)&= {\mathbb{Z}}[E_1]\oplus {\mathbb{Z}}[E_2]\oplus {\mathbb{Z}}[H]\oplus {\mathbb{Z}}[E_5]. \end{align*} Hence $$D\sim -2E_1-E_2+2H-2E_5$$, $$-E_1-2E_2+2H-2E_5$$, $$-E_1+H-E_5$$, $$-E_2+H-E_5$$, $$-E_1-E_2+H-E_5$$, $$-2E_1-E_2+2H-E_5$$, $$-E_1-2E_2+2H-E_5$$, $$-3E_1-E_2+2H-E_5$$, $$-E_1-3E_2+2H-E_5$$, $$-2E_1-2E_2+2H-E_5$$, $$E_1$$, $$E_2$$, $$-E_1+H$$, $$-E_2+H$$, $$-2E_1+H$$, $$-2E_2+H$$, $$-E_1-E_2+H$$, $$E_5$$, $$E_1+E_5$$ or $$E_2+E_5$$. Assume that $$D\sim -2E_1-E_2+2H-2E_5$$. In this case, $$\tau _1=\tfrac {1}{2}$$, $$X_2$$ is the blowup of $$Q$$ along general two points, $$D_2\sim -2E_1-E_2+2H$$, where $$E_1$$, $$E_2$$ are the exceptional divisors of the morphism $$X_2\to Q$$, $$H$$ corresponds to the pullback of $$ {\mathcal {O}}_Q(1)$$, $$\tau _2=1$$, $$X_3= {\mathbb {P}}^3$$, $$D_3\in | {\mathcal {O}}(3)|$$, $$\tau _3=\tfrac {4}{3}$$ and $$m=3$$. Since $$\eta _1=\tfrac {991}{192}$$ and $$\eta _3=-\tfrac {1}{36}$$, we have $$\eta (D)/3>\tfrac {991}{192}-\tfrac {1}{36}>0$$. If $$D\sim -E_1-2E_2+2H-2E_5$$, then $$\eta (D)>0$$ in the same way. Assume that $$D\sim -E_1+H-E_5$$. In this case, $$\tau _1=1$$, $$X_2=V_7$$, $$D_2\sim (-\tfrac {1}{2})K_{V_7}$$, $$\tau _2=2$$ and $$m=2$$. Since $$\eta _1=\tfrac {13}{3}$$ and $$\eta _2=-\tfrac {7}{12}$$, we have $$\eta (D)/3>0$$. If $$D\sim -E_2+H-E_5$$, then $$\eta (D)>0$$ in the same way. Assume that $$D\sim -E_1-E_2+H-E_5$$. In this case, $$\tau _1=1$$, $$X_2$$ is the blowup of $$Q$$ along general two points, $$D_2\sim -E_1-E_2+H$$, where $$E_1$$, $$E_2$$ are the exceptional divisors of the morphism $$X_2\to Q$$, $$H$$ corresponds to the pullback of $$ {\mathcal {O}}_Q(1)$$, $$\tau _2=2$$, $$X_3=Q$$, $$D_3\in | {\mathcal {O}}(1)|$$, $$\tau _3=3$$ and $$m=3$$. Since $$\eta _1=3$$, $$\eta _2=-\tfrac {5}{3}$$ and $$\eta _3=-\tfrac {5}{6}$$, we have $$\eta (D)/3>0$$. Assume that $$D\sim -2E_1-E_2+2H-E_5$$. In this case, $$D$$ is nef, $$\tau _1=1$$, $$X_2= {\mathbb {P}}^3$$, $$D_2\in | {\mathcal {O}}(3)|$$, $$\tau _2=\frac {4}{3}$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 9.5. If $$D\sim -E_1-2E_2+2H-E_5$$, then $$\eta (D)>0$$ in the same way. Assume that $$D\sim -3E_1-E_2+2H-E_5$$. In this case, $$\tau _1=\tfrac {1}{2}$$, $$X_2$$ is the blowup of $$ {\mathbb {P}}^3$$ along a line and a conic, $$\tau _2=1$$, $$X_3= {\mathbb {P}}^3$$, $$D_3\in | {\mathcal {O}}(3)|$$, $$\tau _3=\tfrac {4}{3}$$ and $$m=3$$. Since $$\eta _1=\tfrac {257}{48}$$ and $$\eta _3=-\tfrac {1}{36}$$, we have $$\eta (D)/3> \tfrac {257}{48}-\tfrac {1}{36}>0$$. If $$D\sim -E_1-3E_2+2H-E_5$$, then $$\eta (D)>0$$ in the same way. Assume that $$D\sim -2E_1-2E_2+2H-E_5$$. In this case, $$D$$ is nef, $$\tau _1=1$$, $$X_2=Q$$, $$D_2\in | {\mathcal {O}}(2)|$$, $$\tau _2=\tfrac {3}{2}$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 9.5. Assume that $$D\sim E_1$$. In this case, $$ {\mathcal {N}}_{D/X}\simeq {\mathcal {O}}_{ \mathbb {F}_1}(-\sigma _1-f_1)$$, $$\tau _1=1$$, $$X_2$$ belongs to No. 30 in [40, Table 3], $$ {\mathcal {N}}_{D_2/X_2}\simeq {\mathcal {O}}_{ \mathbb {F}_1}(\sigma _1+f_1)$$, $$\tau _2=2$$ and $$m=2$$. Thus $$\eta (D)=0$$ by Lemma 10.2. If $$D\sim E_2$$, then $$\eta (D)=0$$ in the same way. Assume that $$D\sim -E_1+H$$. In this case, $$D$$ is nef, $$\tau _1=1$$, $$X_2= {\mathbb {P}}_{ {\mathbb {P}}^1}( {\mathcal {O}}^{\oplus 2}\oplus {\mathcal {O}}(1))$$, $$D_2\in |\xi _ {\mathbb {P}}^{\otimes 2}|$$, $$\tau _2=\tfrac {3}{2}$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 9.5. If $$D\sim -E_2+H$$, then $$\eta (D)>0$$ in the same way. Assume that $$D\sim -2E_1+H$$. In this case, $$\tau _1=\tfrac {1}{2}$$, $$X_2$$ is the blowup of $$ {\mathbb {P}}^3$$ along a line and a conic, $$\tau _2=1$$, $$X_3= {\mathbb {P}}_{ {\mathbb {P}}^1}( {\mathcal {O}}^{\oplus 2}\oplus {\mathcal {O}}(1))$$, $$D_3\in |\xi _ {\mathbb {P}}^{\otimes 2}|$$, $$\tau _3=\tfrac {3}{2}$$ and $$m=3$$. Since $$\eta _1=\tfrac {377}{96}$$ and $$\eta _3=-\tfrac {5}{24}$$, we have $$\eta (D)/3> \tfrac {377}{96}-\tfrac {5}{24}>0$$. If $$D\sim -2E_2+H$$, then $$\eta (D)>0$$ in the same way. Assume that $$D\sim -E_1-E_2+H$$. In this case, $$D$$ is nef, $$\tau _1=1$$, $$X_2$$ is the blowup of $$Q$$ along a conic, $$D_2$$ corresponds to the pullback of $$ {\mathcal {O}}_Q(1)$$, $$\tau _2=2$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 9.5. Assume that $$D\sim E_1+E_5$$. In this case, $$\tau _1=1$$, $$X_2= {\mathbb {P}}_{ {\mathbb {P}}^1}( {\mathcal {O}}^{\oplus 2}\oplus {\mathcal {O}}(1))$$, $$D_2\in |\xi _ {\mathbb {P}}^{\otimes 2}\otimes H_{ {\mathbb {P}}^1}^{\otimes (-1)}|$$, $$\tau _2=\tfrac {3}{2}$$ and $$m=2$$. Since $$\eta _1=4$$ and $$\eta _2=-\tfrac {19}{48}$$, we have $$\eta (D)/3>0$$. If $$D\sim E_2+E_5$$, then $$\eta (D)>0$$ in the same way. Assume that $$D\sim E_5$$. In this case, $$ {\mathcal {N}}_{D/X}\simeq {\mathcal {O}}_{ {\mathbb {P}}^1\times {\mathbb {P}}^1}(-1, 0)$$, $$\tau _1=1$$, $$X_2$$ is the blowup of $$Q$$ along a conic, $$D_2$$ is the exceptional divisor of the morphism $$X_2\to Q$$, $$ {\mathcal {N}}_{D_2/X_2}\simeq {\mathcal {O}}_{ {\mathbb {P}}^1\times {\mathbb {P}}^1}(-1, 2)$$, $$\tau _2=2$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 10.2. Hence $$(X, -K_X)$$ is divisorially semistable but not divisorially stable. The case No. 5. Assume that $$X$$ belongs to No. 5 in [40, Table 4]. We consider the case that $$D$$ is the strict transform of the divisor in $$| {\mathcal {O}}_{ {\mathbb {P}}^1\times {\mathbb {P}}^2}(0, 1)|$$ passing through the conic which is one of the centers of the blowup $$X\to {\mathbb {P}}^1\times {\mathbb {P}}^2$$. Then $$ {\mathcal {N}}_{D/X}\simeq {\mathcal {O}}_{ {\mathbb {P}}^1\times {\mathbb {P}}^1}(-1, -1)$$, $$\tau _1=1$$, $$X_2= {\mathbb {P}}^1\times {\mathbb {P}}^2$$, $$D_2\in | {\mathcal {O}}(0, 1)|$$, $$\tau _2=3$$ and $$m=2$$. Thus $$\eta (D)<0$$ by Lemma 10.2. Hence $$(X, -K_X)$$ is not divisorially semistable. The case No. 6. Assume that $$X$$ belongs to No. 6 in [40, Table 4]. Let $$H_1,\ldots ,H_3$$ be a divisor that corresponds to the pullback of $$ {\mathcal {O}}_{ {\mathbb {P}}^1\times {\mathbb {P}}^1\times {\mathbb {P}}^1}(1, 0, 0),\ldots , {\mathcal {O}}_{ {\mathbb {P}}^1\times {\mathbb {P}}^1\times {\mathbb {P}}^1}(0, 0, 1)$$, respectively. Let $$E_1,\ldots ,E_4$$ be the exceptional divisor of $$ \operatorname {cont}_{l_1},\ldots , \operatorname {cont}_{l_4}$$, respectively. Then $$E_1\sim H_2+H_3-E_4$$, $$E_2\sim H_1+H_3-E_4$$, $$E_3\sim H_1+H_2-E_4$$ and   \begin{align*} \operatorname{Nef}(X)&= {\mathbb{R}}_{\geq 0}[H_1]+ {\mathbb{R}}_{\geq 0}[H_2]+ {\mathbb{R}}_{\geq 0}[H_3] + {\mathbb{R}}_{\geq 0}[H_1+H_2+H_3-E_4], \\ \overline{ \operatorname{Eff}}(X)&= {\mathbb{R}}_{\geq 0}[H_1]+ {\mathbb{R}}_{\geq 0}[H_2] + {\mathbb{R}}_{\geq 0}[H_3]+ {\mathbb{R}}_{\geq 0}[H_1+H_2-E_4]+ {\mathbb{R}}_{\geq 0}[H_1+H_3-E_4]\\ &\quad + {\mathbb{R}}_{\geq 0}[H_2+H_3-E_4]+ {\mathbb{R}}_{\geq 0}[E_4],\\ -K_X&\sim 2H_1+2H_2+2H_3-E_4,\\ \operatorname{Pic}(X)&= {\mathbb{Z}}[H_1]\oplus {\mathbb{Z}}[H_2]\oplus {\mathbb{Z}}[H_3]\oplus {\mathbb{Z}}[E_4]. \end{align*} Hence $$D\sim H_1+H_2-E_4$$, $$H_1+H_3-E_4$$, $$H_2+H_3-E_4$$, $$H_1+H_2+H_3-E_4$$, $$H_1$$, $$H_2$$, $$H_3$$, $$H_1+H_2$$, $$H_1+H_3$$, $$H_2+H_3$$, $$H_1+H_2+H_3$$, $$E_4$$, $$H_1+E_4$$, $$H_2+E_4$$ or $$H_3+E_4$$. Assume that $$D\sim H_1+H_2-E_4$$, that is, $$D=E_3$$. In this case, $$ {\mathcal {N}}_{D/X}\simeq {\mathcal {O}}_{ {\mathbb {P}}^1\times {\mathbb {P}}^1}(-1, 1)$$, $$\tau _1=1$$, $$X_2= {\mathbb {P}}^1\times {\mathbb {P}}^1\times {\mathbb {P}}^1$$, $$D_2\in | {\mathcal {O}}(1, 1, 0)|$$, $$\tau _2=2$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 10.2. If $$D\sim H_1+H_3-E_4$$ or $$H_2+H_3-E_4$$, then $$\eta (D)>0$$ in the same way. Assume that $$D\sim H_1+H_2+H_3-E_4$$. In this case, $$D$$ is nef, $$\tau _1=1$$, $$X_2= {\mathbb {P}}^1\times {\mathbb {P}}^1\times {\mathbb {P}}^1$$, $$D_2\in | {\mathcal {O}}(1, 1, 1)|$$, $$\tau _2=2$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 9.5. Assume that $$D\sim H_1$$. In this case, $$\tau _1=1$$, $$X_2= {\mathbb {P}}_{ {\mathbb {P}}^1\times {\mathbb {P}}^1}( {\mathcal {O}}(1, 0)\oplus {\mathcal {O}}(0, 1))$$, $$D_2$$ corresponds to the pullback of $$ {\mathcal {O}}_{ {\mathbb {P}}^3}(1)$$, $$\tau _2=2$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 9.5. If $$D\sim H_2$$ or $$H_3$$, then $$\eta (D)>0$$ in the same way. Assume that $$D\sim H_1+H_2$$. In this case, $$\tau _1=1$$, $$X_2= {\mathbb {P}}_{ {\mathbb {P}}^1}( {\mathcal {O}}^{\oplus 2}\oplus {\mathcal {O}}(1))$$, $$D_2\in |\xi _ {\mathbb {P}}^{\otimes 2}|$$, $$\tau _2=\tfrac {3}{2}$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 9.5. If $$D\sim H_1+H_3$$ or $$H_2+H_3$$, then $$\eta (D)>0$$ in the same way. Assume that $$D\sim H_1+H_2+H_3$$. In this case, $$\tau _1=1$$, $$X_2= {\mathbb {P}}^3$$, $$D_2\in | {\mathcal {O}}(3)|$$, $$\tau _2=\tfrac {4}{3}$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 9.5. Assume that $$D\sim E_4$$. In this case, $$ {\mathcal {N}}_{D/X}\simeq {\mathcal {O}}_{ {\mathbb {P}}^1\times {\mathbb {P}}^1}(-1, 2)$$, $$\tau _1=1$$, $$X_2= {\mathbb {P}}^3$$, $$D_2\in | {\mathcal {O}}(2)|$$, $$\tau _2=2$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 10.2. Assume that $$D\sim H_1+E_4$$. In this case, $$\tau _1=\tfrac {1}{2}$$, $$X_2= {\mathbb {P}}_{ {\mathbb {P}}^1\times {\mathbb {P}}^1}( {\mathcal {O}}(1, 0)\oplus {\mathcal {O}}(0, 1))$$, $$\tau _2=1$$, $$X_3= {\mathbb {P}}^3$$, $$D_3\in | {\mathcal {O}}(3)|$$, $$\tau _3=\tfrac {4}{3}$$ and $$m=3$$. Since $$\eta _1=\tfrac {1045}{192}$$ and $$\eta _3=-\tfrac {1}{36}$$, we have $$\eta (D)/3>\tfrac {1045}{192}-\tfrac {1}{36}>0$$. If $$D\sim H_2+E_4$$ or $$H_3+E_4$$, then $$\eta (D)>0$$ in the same way. Hence $$(X, -K_X)$$ is divisorially stable. The case No. 7. Assume that $$X$$ belongs to No. 7 in [40, Table 4]. Let $$H_1$$, $$H_2$$ be a divisor that corresponds to the pullback of $$ {\mathcal {O}}_{W_6}(1, 0)$$, $$ {\mathcal {O}}_{W_6}(0, 1)$$, respectively. Let $$E_1,\ldots ,E_4$$ be the exceptional divisor of $$ \operatorname {cont}_{l_1},\ldots , \operatorname {cont}_{l_4}$$, respectively. Then $$E_3\sim -E_1+H_2$$, $$E_4\sim -E_2+H_1$$ and   \begin{align*} \operatorname{Nef}(X)&= {\mathbb{R}}_{\geq 0}[H_1]+ {\mathbb{R}}_{\geq 0}[H_2]+ {\mathbb{R}}_{\geq 0}[-E_1+H_1] + {\mathbb{R}}_{\geq 0}[-E_2+H_2],\\ \overline{ \operatorname{Eff}}(X)&= {\mathbb{R}}_{\geq 0}[E_1]+ {\mathbb{R}}_{\geq 0}[E_2] + {\mathbb{R}}_{\geq 0}[-E_1+H_1]+ {\mathbb{R}}_{\geq 0}[-E_2+H_1]\\ &\quad + {\mathbb{R}}_{\geq 0}[-E_1+H_2]+ {\mathbb{R}}_{\geq 0}[-E_2+H_2],\\ -K_X&\sim -E_1-E_2+2H_1+2H_2,\\ \operatorname{Pic}(X)&= {\mathbb{Z}}[E_1]\oplus {\mathbb{Z}}[E_2]\oplus {\mathbb{Z}}[H_1]\oplus {\mathbb{Z}}[H_2]. \end{align*} Hence $$D\sim E_1$$, $$E_2$$, $$H_1$$, $$H_2$$, $$-E_1+H_1$$, $$-E_2+H_2$$, $$-E_1+H_2$$, $$-E_2+H_1$$, $$E_1-E_2+H_1$$, $$-E_1+E_2+H_2$$, $$-E_1+E_2+H_1$$, $$E_1-E_2+H_2$$, $$-E_1+H_1+H_2$$, $$-E_2+H_1+H_2$$, $$-2E_1+H_1+H_2$$, $$-2E_2+H_1+H_2$$ or $$-E_1-E_2+H_1+H_2$$. Assume that $$D\sim E_1$$. In this case, $$ {\mathcal {N}}_{D/X}\simeq {\mathcal {O}}_{ {\mathbb {P}}^1\times {\mathbb {P}}^1}(-1, 0)$$, $$\tau _1=1$$, $$X_2= {\mathbb {P}}^1\times \mathbb {F}_1$$, $$D_2\in | {\mathcal {O}}_{ {\mathbb {P}}^1\times \mathbb {F}_1}(0, \sigma _1+f_1)|$$, $$\tau _2=2$$ and $$m=2$$. Thus $$\eta (D)=0$$ by Lemma 10.2. If $$D\sim E_2$$, then $$\eta (D)=0$$ in the same way. Assume that $$D\sim H_1$$. In this case, $$\tau _1=1$$, $$X_2= {\mathbb {P}}^1\times \mathbb {F}_1$$, $$D_2\in | {\mathcal {O}}_{ {\mathbb {P}}^1\times \mathbb {F}_1}(1, \sigma _1+f_1)|$$, $$\tau _2=2$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 9.5. If $$D\sim H_2$$, then $$\eta (D)>0$$ in the same way. Assume that $$D\sim -E_1+H_1$$. In this case, $$D$$ is nef, $$\tau _1=1$$, $$X_2=W_6\times _{ {\mathbb {P}}^2} \mathbb {F}_1$$, $$D_2$$ corresponds to the pullback of $$ {\mathcal {O}}_{W_6}(1, 0)$$, $$\tau _2=2$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 9.5. If $$D\sim -E_2+H_2$$, then $$\eta (D)>0$$ in the same way. Assume that $$D\sim -E_1+H_2$$, that is, $$D=E_3$$. In this case, $$ {\mathcal {N}}_{D/X}\simeq {\mathcal {O}}_{ \mathbb {F}_1}(-\sigma _1+f_1)$$, $$\tau _1=1$$, $$X_2= {\mathbb {P}}^1\times {\mathbb {P}}^2$$, $$D_2\in | {\mathcal {O}}(1, 1)|$$, $$\tau _2=2$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 10.2. If $$D\sim -E_2+H_1$$, then $$\eta (D)>0$$ in the same way. Assume that $$D\sim E_1-E_2+H_1$$. In this case, $$\tau _1=\tfrac {1}{2}$$, $$X_2= {\mathbb {P}}^1\times \mathbb {F}_1$$, $$D_2\in | {\mathcal {O}}_{ {\mathbb {P}}^1\times \mathbb {F}_1}(1, \sigma _1+2f_1)|$$, $$\tau _2=1$$, $$X_3= {\mathbb {P}}^1\times {\mathbb {P}}^2$$, $$D_3\in | {\mathcal {O}}(1, 2)|$$, $$\tau _3=\tfrac {3}{2}$$ and $$m=3$$. Since $$\eta _2=\tfrac {259}{192}$$ and $$\eta _3=-\tfrac {7}{48}$$, we have $$\eta (D)/3>\tfrac {259}{192}-\tfrac {7}{48}>0$$. If $$D\sim -E_1+E_2+H_2$$, then $$\eta (D)>0$$ in the same way. Assume that $$D\sim -E_1+E_2+H_1$$. In this case, $$\tau _1=1$$, $$X_2= {\mathbb {P}}^1\times {\mathbb {P}}^2$$, $$D_2\in | {\mathcal {O}}(0, 2)|$$, $$\tau _2=\tfrac {3}{2}$$ and $$m=2$$. Since $$\eta _1=\tfrac {21}{4}$$ and $$\eta _2=-\tfrac {1}{3}$$, we have $$\eta (D)/3>0$$. If $$D\sim E_1-E_2+H_2$$, then $$\eta (D)>0$$ in the same way. Assume that $$D\sim -E_1+H_1+H_2$$. In this case, $$D$$ is nef, $$\tau _1=1$$, $$X_2= {\mathbb {P}}^1\times {\mathbb {P}}^2$$, $$D_2\in | {\mathcal {O}}(1, 2)|$$, $$\tau _2=\tfrac {3}{2}$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 9.5. If $$D\sim -E_2+H_1+H_2$$, then $$\eta (D)>0$$ in the same way. Assume that $$D\sim -2E_1+H_1+H_2$$. In this case, $$\tau _1=1$$, $$X_2=W_6\times _{ {\mathbb {P}}^2} \mathbb {F}_1$$, $$\tau _2=1$$, $$X_3= {\mathbb {P}}^1\times {\mathbb {P}}^2$$, $$D_3\in | {\mathcal {O}}(1, 2)|$$, $$\tau _3=\tfrac {3}{2}$$ and $$m=3$$. Since $$\eta _1=\tfrac {163}{32}$$ and $$\eta _3=-\tfrac {7}{48}$$, we have $$\eta (D)/3> \tfrac {163}{32}-\frac {7}{48}>0$$. If $$D\sim -2E_2+H_1+H_2$$, then $$\eta (D)>0$$ in the same way. Assume that $$D\sim -E_1-E_2+H_1+H_2$$. In this case, $$D$$ is nef, $$\tau _1=1$$, $$X_2=W_6$$, $$D_2\sim (-\tfrac {1}{2})K_{W_6}$$, $$\tau _2=2$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 9.5. Hence $$(X, -K_X)$$ is divisorially semistable but not divisorially stable. The case No. 8. Assume that $$X$$ belongs to No. 8 in [40, Table 4]. We consider the case that $$D$$ is the strict transform of the divisor in $$| {\mathcal {O}}_{ {\mathbb {P}}^1\times {\mathbb {P}}^1\times {\mathbb {P}}^1}(1, 0, 0)|$$ passing through the center of the blowup $$X\to {\mathbb {P}}^1\times {\mathbb {P}}^1\times {\mathbb {P}}^1$$. Then $$ {\mathcal {N}}_{D/X}\simeq {\mathcal {O}}_{ {\mathbb {P}}^1\times {\mathbb {P}}^1}(-1, -1)$$, $$\tau _1=1$$, $$X_2= {\mathbb {P}}^1\times {\mathbb {P}}^1\times {\mathbb {P}}^1$$, $$D_2\in | {\mathcal {O}}(1, 0, 0)|$$, $$\tau _2=2$$ and $$m=2$$. Thus $$\eta (D)<0$$ by Lemma 9.5. Hence $$(X, -K_X)$$ is not divisorially semistable. The case No. 9–12. Assume that $$X$$ belongs to one of No. 9–12 in [40, Table 4]. Then $$X$$ is toric. Then $$(X, -K_X)$$ is not divisorially semistable by Theorem 1.2 and [36]. The case No. 13. Assume that $$X$$ belongs to No. 13 in [40, Table 4]. Let $$l_1$$, $$l_2$$, $$l_3\subset X$$ be the strict transform of a curve on $$ {\mathbb {P}}^1\times {\mathbb {P}}^1\times {\mathbb {P}}^1$$ of tridegree $$(1, 0, 0)$$, $$(0, 1, 0)$$, $$(0, 0, 1)$$, passing through the center of the blowup $$X\to {\mathbb {P}}^1\times {\mathbb {P}}^1\times {\mathbb {P}}^1$$, respectively. Let $$l_4\subset X$$ be the strict transform of a curve on $$ {\mathbb {P}}^1\times {\mathbb {P}}^1\times {\mathbb {P}}^1$$ of tridegree $$(1, 1, 0)$$ which is contained in the divisor in $$| {\mathcal {O}}_{ {\mathbb {P}}^1\times {\mathbb {P}}^1\times {\mathbb {P}}^1}(1, 1, 0)|$$ which contains the center of the blowup $$X\to {\mathbb {P}}^1\times {\mathbb {P}}^1\times {\mathbb {P}}^1$$. Let $$l_5\subset X$$ be an exceptional curve of the blowup $$X\to {\mathbb {P}}^1\times {\mathbb {P}}^1\times {\mathbb {P}}^1$$. Then $$\overline { \operatorname {NE}}(X)$$ is spanned by the classes of $$l_1,\ldots ,l_5$$. Let $$H_1,\ldots ,H_3$$ be a divisor that corresponds to the pullback of $$ {\mathcal {O}}_{ {\mathbb {P}}^1\times {\mathbb {P}}^1\times {\mathbb {P}}^1}(1, 0, 0),\ldots , {\mathcal {O}}_{ {\mathbb {P}}^1\times {\mathbb {P}}^1\times {\mathbb {P}}^1} (0, 0, 1)$$, respectively. Let $$E_1$$, $$E_2$$, $$E_3$$, $$E_5$$ be the exceptional divisor of $$ \operatorname {cont}_{l_1}$$, $$ \operatorname {cont}_{l_2}$$, $$ \operatorname {cont}_{l_3}$$, $$ \operatorname {cont}_{l_5}$$, respectively. Then $$E_1\sim 3H_2+H_3-E_5$$, $$E_2\sim 3H_1+H_3-E_5$$, $$E_3\sim H_1+H_2-E_5$$ and   \begin{align*} \operatorname{Nef}(X)&= {\mathbb{R}}_{\geq 0}[H_1]+ {\mathbb{R}}_{\geq 0}[H_2]+ {\mathbb{R}}_{\geq 0}[H_3]\\ &\quad + {\mathbb{R}}_{\geq 0}[2H_1+H_2+H_3-E_5]+ {\mathbb{R}}_{\geq 0}[H_1+2H_2+H_3-E_5], \\ \overline{ \operatorname{Eff}}(X)&= {\mathbb{R}}_{\geq 0}[H_1]+ {\mathbb{R}}_{\geq 0}[H_2]+ {\mathbb{R}}_{\geq 0}[H_3] + {\mathbb{R}}_{\geq 0}[H_1+H_2-E_5]\\ &\quad + {\mathbb{R}}_{\geq 0}[3H_1+H_3-E_5]+ {\mathbb{R}}_{\geq 0}[3H_2+H_3-E_5]+ {\mathbb{R}}_{\geq 0}[E_5],\\ -K_X&\sim 2H_1+2H_2+2H_3-E_5,\\ \operatorname{Pic}(X)&= {\mathbb{Z}}[H_1]\oplus {\mathbb{Z}}[H_2]\oplus {\mathbb{Z}}[H_3]\oplus {\mathbb{Z}}[E_5]. \end{align*} Hence $$D\sim H_1$$, $$H_2$$, $$H_3$$, $$H_1+H_3$$, $$H_2+H_3$$, $$H_1+H_2-E_5$$ or $$H_1+H_2+H_3-E_5$$. Assume that $$D\sim H_1$$. In this case, $$\tau _1=1$$, $$X_2= {\mathbb {P}}_{ {\mathbb {P}}^1}( {\mathcal {O}}\oplus {\mathcal {O}}(1)^{\oplus 2})$$, $$D_2\in |\xi _ {\mathbb {P}}^{\otimes 2}|$$, $$\tau _2=\tfrac {3}{2}$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 9.5. If $$D\sim H_2$$, then $$\eta (D)>0$$ in the same way. Assume that $$D\sim H_3$$. In this case, $$\tau _1=1$$ and $$X_2$$ is the image of the morphism $$ \operatorname {cont}_{l_3}$$. If we see $$ \operatorname {N}^{1}(X_2)$$ as a subspace of $$ \operatorname {N}^{1}(X)$$, then $$-K_{X_2}\sim 3H_1+3H_2+2H_3-2E_5$$ and $$D_2\sim H_1+H_2+H_3-E_5$$. Moreover, $$ \operatorname {Nef}(X_2)$$ is spanned by the classes of $$H_1$$, $$H_2$$, $$2H_1+H_2+H_3-E_5$$ and $$H_1+2H_2+H_3-E_5$$. Since $$\tau (D)=2$$ and $$-K_{X_2}-2D_2\in \operatorname {Nef}(X_2)$$, we have $$\tau _2=2$$ and $$m=2$$. Since $$ {\mathcal {N}}_{D_2/X_2}$$ is non-zero effective, we have $$\eta (D)>0$$ by Lemma 9.5. Assume that $$D\sim H_1+H_3$$. In this case, $$\tau _1=1$$, $$X_2$$ is the image of the contraction of the negative section of $$ {\mathbb {P}}_{ {\mathbb {P}}^1}( {\mathcal {O}}\oplus {\mathcal {O}}(1)^{\oplus 2})$$, $$D_2\sim (-\tfrac {2}{3})K_{X_2}$$, $$\tau _2=\tfrac {3}{2}$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 9.5. If $$D\sim H_2+H_3$$, then $$\eta (D)>0$$ in the same way. Assume that $$D\sim H_1+H_2-E_5$$, that is, $$D=E_3$$. In this case, $$ {\mathcal {N}}_{D/X}\simeq {\mathcal {O}}_{ {\mathbb {P}}^1\times {\mathbb {P}}^1}(-1, -1)$$, $$\tau _1=1$$, $$X_2= {\mathbb {P}}^1\times {\mathbb {P}}^1\times {\mathbb {P}}^1$$, $$D_2\in | {\mathcal {O}}(1, 1, 0)|$$, $$\tau _2=2$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 10.2. Assume that $$D\sim H_1+H_2+H_3-E_5$$. In this case, $$\tau _1=1$$, $$X_2= {\mathbb {P}}^1\times {\mathbb {P}}^1\times {\mathbb {P}}^1$$, $$D_2\in | {\mathcal {O}}(1, 1, 1)|$$, $$\tau _2=2$$ and $$m=2$$. Since $$\eta _1=\tfrac {41}{12}$$ and $$\eta _2=-\tfrac {1}{2}$$, we have $$\eta (D)/3>0$$. Hence $$(X, -K_X)$$ is divisorially stable. Therefore, we have proved Theorem 10.1 for the case $$\rho (X)=4$$. 10.4. The case $$\rho (X)=5$$ We consider the case $$\rho (X)=5$$. We assume that $$D$$ is a suspicious divisor. The case No. 1. Assume that $$X$$ belongs to No. 1 in [40, Table 5]. Let $$E_7\subset X$$ be the prime divisor such that the center on $$Q$$ is a conic. In [39], $$l_7$$ is a fiber of the ruling $$E_7\simeq {\mathbb {P}}^1\times {\mathbb {P}}^1\to {\mathbb {P}}^1$$. Let $$l_8\subset X$$ be a fiber of the other ruling $$E_7\simeq {\mathbb {P}}^1\times {\mathbb {P}}^1\to {\mathbb {P}}^1$$. Then $$\overline { \operatorname {NE}}(X)$$ is spanned by the classes of $$l_1\dots , l_8$$ (in [39], the ray $$ {\mathbb {R}}_{\geq 0}[l_8]$$ is forgotten). Let $$H$$ be a divisor that corresponds to the pullback of $$ {\mathcal {O}}_Q(1)$$. Let $$E_1,\ldots ,E_6$$ be the exceptional divisor of $$ \operatorname {cont}_{l_1},\ldots , \operatorname {cont}_{l_6}$$, respectively. Then $$E_4\sim -2E_1+H$$, $$E_5\sim -2E_2+H$$, $$E_6\sim -2E_3+H$$ and   \begin{align*} \operatorname{Nef}(X)&= {\mathbb{R}}_{\geq 0}[H]+ {\mathbb{R}}_{\geq 0}[-E_1+H]+ {\mathbb{R}}_{\geq 0}[-E_2+H] + {\mathbb{R}}_{\geq 0}[-E_1-E_2+H]\\ &\quad + {\mathbb{R}}_{\geq 0}[-E_3+H]+ {\mathbb{R}}_{\geq 0}[-E_1-E_3+H]+ {\mathbb{R}}_{\geq 0}[-E_2-E_3+H]\\ &\quad + {\mathbb{R}}_{\geq 0}[-E_1-E_2-E_3-E_7+H], \\ \overline{ \operatorname{Eff}}(X)&= {\mathbb{R}}_{\geq 0}[E_1]+ {\mathbb{R}}_{\geq 0}[E_2] + {\mathbb{R}}_{\geq 0}[E_3]+ {\mathbb{R}}_{\geq 0}[E_7]+ {\mathbb{R}}_{\geq 0}[-2E_1+H]\\ &\quad + {\mathbb{R}}_{\geq 0}[-2E_2+H]+ {\mathbb{R}}_{\geq 0}[-2E_3+H]+ {\mathbb{R}}_{\geq 0}[-E_1-E_2-E_3-E_7+H],\\ -K_X&\sim -2E_1-2E_2-2E_3-E_7+3H,\\ \operatorname{Pic}(X)&= {\mathbb{Z}}[E_1]\oplus {\mathbb{Z}}[E_2]\oplus {\mathbb{Z}}[E_3]\oplus {\mathbb{Z}}[E_7]\oplus {\mathbb{Z}}[H]. \end{align*} Hence $$D\sim E_1$$, $$E_2$$, $$E_3$$, $$E_7$$, $$E_1+E_7$$, $$E_2+E_7$$, $$E_3+E_7$$, $$-E_2-E_3-E_7+H$$, $$-E_1-E_3-E_7+H$$, $$-E_1-E_2-E_7+H$$, $$-E_1-E_2-E_3-E_7+H$$, $$-E_2-E_3+H$$, $$-E_1-E_3+H$$, $$-E_1-E_2+H$$, $$-E_1-E_2-E_3+H$$, $$-E_1-2E_2-2E_3-2E_7+2H$$, $$-2E_1-E_2-2E_3-2E_7+2H$$, $$-2E_1-2E_2-E_3-2E_7+2H$$, $$-E_1-2E_2-2E_3-E_7+2H$$, $$-2E_1-E_2-2E_3-E_7+2H$$, $$-2E_1-2E_2-E_3-E_7+2H$$ or $$-2E_1-2E_2-2E_3-E_7+2H$$. Assume that $$D\sim E_1$$. In this case, $$ {\mathcal {N}}_{D/X}\simeq {\mathcal {O}}_{ \mathbb {F}_1}(-\sigma _1-f_1)$$, $$\tau _1=1$$, $$\tau (D)=2$$ and $$X_2$$ is the image of the morphism associated to the extremal face spanned by $$ {\mathbb {R}}_{\geq 0}[l_4]$$ and $$ {\mathbb {R}}_{\geq 0}[l_8]$$. If we see $$ \operatorname {N}^{1}(X_2)$$ as a subspace of $$ \operatorname {N}^{1}(X)$$, then $$-K_{X_2}\sim -4E_1-2E_2-2E_3+4H$$ and $$D_2\sim -E_1+E_7+H$$. Moreover, $$ \operatorname {Nef}(X_2)$$ is spanned by the classes of $$-E_1-E_2+H$$, $$-E_1-E_3+H$$ and $$-E_1-E_2-E_3-E_7+H$$. Since $$-K_{X_2}-2D_2$$ is nef, we have $$\tau _2=2$$ and $$m=2$$. Since $$\eta _1=\tfrac {9}{4}$$ and $$\eta _2=-\tfrac {4}{3}$$, we have $$\eta (D)/3>0$$. If $$D\sim E_2$$ or $$E_3$$, then $$\eta (D)>0$$ in the same way. Assume that $$D\sim E_7$$. In this case, $$ {\mathcal {N}}_{D/X}\simeq {\mathcal {O}}_{ {\mathbb {P}}^1\times {\mathbb {P}}^1}(-1, -1)$$, $$\tau _1=1$$, $$X_2$$ is the blowup of $$Q$$ along a conic, $$D_2$$ is the exceptional divisor of the morphism $$X_2\to Q$$, $$ {\mathcal {N}}_{D_2/X_2}\simeq {\mathcal {O}}_{ {\mathbb {P}}^1\times {\mathbb {P}}^1}(-1, 2)$$, $$\tau _2=2$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 10.2. Assume that $$D\sim E_1+E_7$$. In this case, $$\tau _1=1$$, $$X_2= {\mathbb {P}}_{ {\mathbb {P}}^1}( {\mathcal {O}}^{\oplus 2}\oplus {\mathcal {O}}(1))$$, $$D_2\in |\xi _ {\mathbb {P}}^{\otimes 2}\otimes H_{ {\mathbb {P}}^1}^{\otimes (-1)}|$$, $$\tau _2=\tfrac {3}{2}$$ and $$m=2$$. Since $$\eta _1=\tfrac {19}{6}$$ and $$\eta _2=-\tfrac {5}{24}$$, we have $$\eta (D)/3>0$$. If $$D\sim E_2+E_7$$ or $$E_3+E_7$$, then $$\eta (D)>0$$ in the same way. Assume that $$D\sim -E_2-E_3-E_7+H$$. In this case, $$\tau _1=1$$ and $$X_2$$ is the image of the morphism associated to the extremal face spanned by $$ {\mathbb {R}}_{\geq 0}[l_4]$$, $$ {\mathbb {R}}_{\geq 0}[l_7]$$ and $$ {\mathbb {R}}_{\geq 0}[l_8]$$. If we see $$ \operatorname {N}^{1}(X_2)$$ as a subspace of $$ \operatorname {N}^{1}(X)$$, then $$-K_{X_2}\sim -4E_1-2E_2-2E_3+4H$$ and $$D_2\sim -2E_1-E_2-E_3+2H$$. Thus $$D_2\sim (-\tfrac {1}{2})K_{X_2}$$. Hence $$\tau _2=2$$ and $$m=2$$. Since $$\eta _1=\tfrac {23}{6}$$ and $$\eta _2=-\tfrac {1}{2}$$, we have $$\eta (D)/3>0$$. If $$D\sim -E_1-E_3-E_7+H$$ or $$-E_1-E_2-E_7+H$$, then $$\eta (D)>0$$ in the same way. Assume that $$D\sim -E_1-E_2-E_3-E_7+H$$. In this case, $$\tau _1=1$$, $$X_2$$ is the blowup of $$Q$$ along general three points, $$D_2\sim -E_1-E_2-E_3+H$$, where $$E_1,\ldots ,E_3$$ are the exceptional divisors of the morphism $$X_2\to Q$$ and $$H$$ corresponds to the pullback of $$ {\mathcal {O}}_Q(1)$$, $$\tau _2=2$$, $$X_3=Q$$, $$D_3\in | {\mathcal {O}}_Q(1)|$$, $$\tau _3=3$$ and $$m=3$$. Since $$\eta _1=\tfrac {5}{2}$$, $$\eta _2=-\tfrac {19}{12}$$ and $$\eta _3=-\tfrac {5}{6}$$, we have $$\eta (D)/3=\tfrac {1}{12}>0$$. Assume that $$D\sim -E_2-E_3+H$$. In this case, $$D$$ is nef, $$\tau _1=1$$, $$X_2= {\mathbb {P}}_{ {\mathbb {P}}^1}( {\mathcal {O}}^{\oplus 2}\oplus {\mathcal {O}}(1))$$, $$D_2\in |\xi _ {\mathbb {P}}^{\otimes 2}|$$, $$\tau _2=\tfrac {3}{2}$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 9.5. If $$D\sim -E_1-E_3+H$$ or $$-E_1-E_2+H$$, then $$\eta (D)>0$$ in the same way. Assume that $$D\sim -E_1-E_2-E_3+H$$. In this case, $$\tau _1=1$$, $$X_2$$ is the blowup of $$Q$$ along a conic, $$D_2$$ corresponds to the pullback of $$ {\mathcal {O}}_Q(1)$$, $$\tau _2=2$$ and $$m=2$$. Since $$\eta _1=\tfrac {41}{12}$$ and $$\eta _2=-\tfrac {5}{6}$$, we have $$\eta (D)/3>0$$. Assume that $$D\sim -E_1-2E_2-2E_3-2E_7+2H$$. In this case, $$\tau _1=\tfrac {1}{2}$$, $$X_2$$ is the blowup of $$Q$$ along general three points, $$D_2\sim -E_1-2E_2-2E_3+2H$$, where $$E_1,\ldots ,E_3$$ are the exceptional divisors of the morphism $$X_2\to Q$$ and $$H$$ corresponds to the pullback of $$ {\mathcal {O}}_Q(1)$$, $$\tau _2=1$$, $$X_3= {\mathbb {P}}^3$$, $$D_3\in | {\mathcal {O}}(3)|$$, $$\tau _3=\tfrac {4}{3}$$ and $$m=3$$. Since $$\eta _1=\tfrac {23}{4}$$ and $$\eta _3=-\tfrac {1}{36}$$, we have $$\eta (D)/3>\tfrac {23}{4}-\tfrac {1}{36}>0$$. If $$D\sim -2E_1-E_2-2E_3-2E_7+2H$$ or $$-2E_1-2E_2-E_3-2E_7+2H$$, then $$\eta (D)>0$$ in the same way. Assume that $$D\sim -E_1-2E_2-2E_3-E_7+2H$$. In this case, $$D$$ is nef, $$\tau _1=1$$, $$X_2= {\mathbb {P}}^3$$, $$D_2\in | {\mathcal {O}}(3)|$$, $$\tau _2=\tfrac {4}{3}$$ and $$m=2$$. Thus $$\eta (D)>0$$ by Lemma 9.5. If $$D\sim -2E_1-E_2-2E_3-E_7+2H$$ or $$-2E_1-2E_2-E_3-E_7+2H$$, then $$\eta (D)>0$$ in the same way. Assume that $$D\sim -2E_1-2E_2-2E_3-E_7+2H$$. In this case, $$\tau _1=1$$, $$X_2=Q$$, $$D_2\in | {\mathcal {O}}_Q(2)|$$, $$\tau _2=\tfrac {3}{2}$$ and $$m=2$$. Since $$\eta _1=5$$ and $$\eta _2=-\tfrac {1}{12}$$, we have $$\eta (D)/3>0$$. Hence $$(X, -K_X)$$ is divisorially stable. The case No. 2–3. Assume that $$X$$ belongs to either No. 2 or No. 3 in [40, Table 5]. Then $$X$$ is toric. Then $$(X, -K_X)$$ is not divisorially stable by Corollary 6.3. Moreover, $$(X, -K_X)$$ is divisorially semistable if and only if $$X$$ belongs to No. 3 in [40, Table 5] by Theorem 1.2 and [36]. Therefore, we have proved Theorem 10.1 for the case $$\rho (X)=5$$. 10.5. 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TI - On K-stability and the volume functions of ℚ-Fano varieties
JO - Proceedings of the London Mathematical Society
DO - 10.1112/plms/pdw037
DA - 2016-09-12
UR - https://www.deepdyve.com/lp/oxford-university-press/on-k-stability-and-the-volume-functions-of-fano-varieties-PMfe2cCeld
SP - 541
EP - 582
VL - 113
IS - 5
DP - DeepDyve
ER -