2.1.a Birational (super)rigidity of Fano varieties

Let X be a normal $\mathbb {Q}$ -factorial variety, D a $\mathbb {Q}$ -divisor on X and $\mathcal {M}$ a movable linear system on X. For a prime divisor E over X, we define $\operatorname {ord}_E (D)$ to be the coefficient of E in $\varphi ^*D$ , where $\varphi \colon Y \to X$ is a birational morphism such that $E \subset Y$ , and we set $m_E (\mathcal {M}) := \operatorname {ord}_E (M)$ , where M is a general member of $\mathcal {M}$ . For a positive rational number $\lambda $ , we say that a pair $(X, \lambda \mathcal {M})$ is canonical if

$$\begin{align*}a_E (K_X) \ge \lambda m_E (\mathcal{M}) \end{align*}$$

for any exceptional prime divisor E over X.

Let X be a Fano variety of Picard number one. Note that we can view X (or more precisely the structure morphism $X \to \operatorname {Spec} \mathbb {C}$ ) as a Mori fiber space.

2.1.b Log canonical thresholds and alpha invariants

Definition 2.5. Let $(X, \Delta )$ be a pair, D an effective $\mathbb {Q}$ -divisor on X, and let $\mathsf {p} \in X$ be a point. Assume that $(X, \Delta )$ has at most log canonical singularities. We define the log canonical threshold (abbreviated as LCT) of $(X, \Delta; D)$ at $\mathsf {p}$ and the log canonical threshold of $(X, \Delta; D)$ to be the numbers

$$ \begin{align*} \operatorname{lct}_{\mathsf{p}} (X, \Delta; D) &= \sup \{\, c \in \mathbb{Q}_{\ge 0} \mid (X, \Delta + c D)\ \text{is log canonical at}\ \mathsf{p} \,\}, \\ \operatorname{lct} (X, \Delta; D) &= \sup \{\, c \in \mathbb{Q}_{\ge 0} \mid (X, \Delta + c D)\ \text{is log canonical} \,\}, \end{align*} $$

respectively. We set $\operatorname {lct}_{\mathsf {p}} (X; D) = \operatorname {lct}_{\mathsf {p}} (X, 0; D)$ and $\operatorname {lct}_{\mathsf {p}} (X; D) = \operatorname {lct} (X, \Delta; D)$ when $\Delta = 0$ . Assume that $\left | -K_X \right |{}_{\mathbb {Q}} \ne \emptyset $ . Then we define the alpha invariant of X at $\mathsf {p}$ and the alpha invariant of X to be the numbers

$$ \begin{align*} \alpha_{\mathsf{p}} (X) &= \inf \{\, \operatorname{lct}_{\mathsf{p}} (X, D) \mid D \in \left| -K_X \right|{}_{\mathbb{Q}} \,\}, \\ \alpha (X) &= \inf \{\, \alpha_{\mathsf{p}} (X) \mid \mathsf{p} \in X \, \}, \end{align*} $$

respectively.

The following fact is frequently used.

Remark 2.6. Let $\mathsf {p}$ be a point on X, and let $D_1, D_2$ be effective $\mathbb {Q}$ -divisors on X. If both $(X, D_1)$ and $(X, D_2)$ are log canonical at $\mathsf {p}$ , then the pair

$$\begin{align*}(X, \lambda D_1 + (1-\lambda) D_2) \end{align*}$$

is log canonical at $\mathsf {p}$ for any $\lambda \in \mathbb {Q}$ such that $0 \le \lambda \le 1$ . In particular, if $\alpha _{\mathsf {p}} (X) < c$ for some number $c> 0$ , then there exists an irreducible $\mathbb {Q}$ -divisor $D \in \left |-K_X \right |{}_{\mathbb {Q}}$ such that $(X, c D)$ is not log canonical at $\mathsf {p}$ . Here, a $\mathbb {Q}$ -divisor is irreducible if its support $\operatorname {Supp} (D)$ is irreducible.

2.1.c Cyclic quotient singularities and orbifold multiplicities

Definition 2.7. Let $r> 0$ and $a_1, \dots , a_n$ be integers. Suppose that the cyclic group ${\boldsymbol {\mu }}_r$ of rth roots of unity in $\mathbb {C}$ acts on the affine n-space $\mathbb {A}^n$ with affine coordinates $x_1, \dots , x_n$ via

$$\begin{align*}(x_1, \dots, x_n) \mapsto (\zeta^{a_1} x_1, \dots, \zeta^{a_n} x_n), \end{align*}$$

where $\zeta \in {\boldsymbol {\mu }}_r$ is a fixed primitive rth root of unity. We denote by $\bar {o} \in \mathbb {A}^n/{\boldsymbol {\mu }}_r$ the image of the origin $o \in \mathbb {A}^n$ under the quotient morphism $\mathbb {A}^n \to \mathbb {A}^n/{\boldsymbol {\mu }}_r$ . A singularity $\mathsf {p} \in X$ is a cyclic quotient singularity of type $\frac {1}{r} (a_1, \dots , a_n)$ if $\mathsf {p} \in X$ is analytically isomorphic to an analytic germ $\bar {o} \in \mathbb {A}^n/{\boldsymbol {\mu }}_r$ . In this case, r is called the index of the cyclic quotient singularity $\mathsf {p} \in X$ .

Note that, by convention, the case $r = 1$ is allowed in the definition of cyclic quotient singularity. A cyclic quotient singularity $\mathsf {p} \in X$ of index $1$ is nothing but a smooth point $\mathsf {p} \in X$ and in that case the quotient morphism $q_{\mathsf {p}} \colon \check {X} \to X$ is simply an isomorphism.

Definition 2.9. Let $\mathsf {p} \in X$ be a cyclic quotient singularity, and let $q_{\mathsf {p}} \colon \check {X} \to X$ be its quotient morphism with $\check {\mathsf {p}} \in \check {X}$ the preimage of $\mathsf {p}$ . For an effective $\mathbb {Q}$ -divisor D on X, we define

$$\begin{align*}\operatorname{omult}_{\mathsf{p}} (D) := \operatorname{mult}_{\check{\mathsf{p}}} (q_{\mathsf{p}}^*D) \end{align*}$$

and call it the orbifold multiplicity of D at $\mathsf {p}$ . By convention, we set $\operatorname {omult}_{\mathsf {p}} (D) = \operatorname {mult}_{\mathsf {p}} (D)$ when $\mathsf {p} \in X$ is a smooth point.

2.1.d Kawamata blowup

Let $\mathsf {p} \in V$ be a three-dimensional terminal quotient singularity. Then it is of type $\frac {1}{r} (1, a, r-a)$ , where r and a are coprime positive integers with $r> a$ (see [Reference Morrison and StevensMS84]). Let $\varphi \colon W \to V$ be the weighted blowup of V at $\mathsf {p}$ with weight $\frac {1}{r} (1,a,r-a)$ . By [Reference KawamataKaw96], $\varphi $ is the unique divisorial contraction centered at $\mathsf {p}$ and we call $\varphi $ the Kawamata blowup of V at $\mathsf {p}$ . If we denote by E the $\varphi $ -exceptional divisor, then $E \cong \mathbb {P} (1,a,r-a)$ and we have

$$\begin{align*}K_W = \varphi^*K_V + \frac{1}{r} E, \end{align*}$$

and

$$\begin{align*}(E^3) = \frac{r^2}{a (r-a)}. \end{align*}$$

2.2.a Weighted projective space

Let N be a positive integer. For positive integers $a_0, \dots , a_N$ , let

$$\begin{align*}R (a_0, \dots, a_N) := \mathbb{C} [x_0, \dots, x_N] \end{align*}$$

be the graded ring whose grading is given by $\deg x_i = a_i$ . We define

$$\begin{align*}\mathbb{P} (a_0,\dots, a_N) := \operatorname{Proj} R (a_0, \dots, a_N) \end{align*}$$

and call it the weighted projective space with homogeneous coordinates $x_0, \dots , x_N$ (of degree $\deg x_i = a_i$ ). We sometimes denote

$$\begin{align*}\mathbb{P} (a_0, \dots, a_N)_{x_0, \dots, x_N} \end{align*}$$

in order to make it clear the homogeneous coordinates $x_0, \dots , x_N$ . For $i = 0, \dots , N$ , we denote by

(2.1) $$ \begin{align} \mathsf{p}_{x_i} = (0\!:\!\cdots\!:\!1\!:\!\cdots\!:\!0) \in \mathbb{P} (a_0, \dots, a_N) \end{align} $$

the coordinate point at which only the coordinate $x_i$ does not vanish. Let $f \in R := R (a_0, \dots , a_N) = \mathbb {C} [x_0, \dots , x_N]$ be a polynomial. We say that f is quasi-homogeneous (resp. homogeneous) if it is homogeneous with respect to the grading $\deg x_i = a_i$ (resp. $\deg x_i = 1$ ) for $i = 0, 1, \dots , N$ . For a polynomial $f \in \mathbb {C} [x_0, \dots , x_N]$ and a monomial $M = x_0^{m_0} \cdots x_N^{m_N}$ , we denote by

$$\begin{align*}\operatorname{coeff}_f (M) \in \mathbb{C} \end{align*}$$

the coefficient of M in f, and, by a slight abuse of notation, we write $M \in f$ if $\operatorname {coeff}_f (M) \ne 0$ . For quasi-homogeneous polynomials $f_1, \dots , f_k \in R$ , we denote by

$$\begin{align*}(f_1 = \cdots = f_k = 0) \subset \mathbb{P} (a_0, \dots, a_N) \end{align*}$$

the closed subscheme defined by the quasi-homogeneous ideal $(f_1, \dots , f_k) \subset R$ . Moreover, for a closed subscheme $X \subset \mathbb {P} (a_0, \dots , a_N)$ and quasi-homogeneous polynomials $g_1, \dots , g_l \in R$ , we define

$$\begin{align*}(g_1 = \cdots = g_l = 0)_X := (g_1 = \cdots = g_l = 0) \cap X, \end{align*}$$

which is a closed subscheme of X. For $i = 0, \dots , N$ , we define

(2.2) $$ \begin{align} \mathcal{H}_{x_i} &:= (x_i = 0) \subset \mathbb{P} (a_0, \dots, a_N), \nonumber\\ \mathcal{U}_{x_i} &:= \mathbb{P} (a_0, \dots, a_N) \setminus \mathcal{H}_{x_i}. \end{align} $$

Remark 2.10. The weighted projective space $\mathbb {P} (a, b, c, d, e)$ with homogeneous coordinates x, y, z, t, w of degrees a, b, c, d, e, respectively, is sometimes denoted by

$$\begin{align*}\mathbb{P} (a, b, c, d, e)_{x, y, z, t, w} \end{align*}$$

in order to emphasize the homogeneous coordinates. For a coordinate $v \in \{x, \dots , w\}$ , the point $\mathsf {p}_v \in \mathbb {P} (a, b, c, d, e)$ , the quasi-hyperplane $\mathcal {H}_v = (v = 0) \subset \mathbb {P} (a, b, c, d, e)$ and the open set $\mathcal {U}_v = \mathbb {P} (a, b, c, d, e) \setminus \mathcal {H}_v$ are similarly defined as in equations (2.1) and (2.2).

2.2.b Well-formedness and quasi-smoothness

Definition 2.11. We say that a weighted projective space $\mathbb {P} (a_0, \dots , a_N)$ is well-formed if

$$\begin{align*}\gcd \{a_0, \dots, \hat{a}_i, \dots, a_N\} = 1 \end{align*}$$

for any $i = 0, 1, \dots , N$ .

Definition 2.12. Let $\mathbb {P} (a_0, \dots , a_N)$ be a weighted projective space such that $\gcd \{a_0, \dots , a_N\} = 1$ . For $j = 0, 1, \dots , N$ , we set

$$ \begin{align*} l_j &:= \gcd \{ a_0, a_1, \dots, \hat{a}_j, \dots, a_N\}, \\ m_j &:= l_0 l_1 \cdots \hat{l}_j \cdots l_N, \\ b_j &:= \frac{a_j}{m_j}. \end{align*} $$

We then define

$$\begin{align*}\mathbb{P} (a_0, \dots, a_N)^{\operatorname{wf}} := \mathbb{P} (b_0, \dots, b_N) \end{align*}$$

and call it the well-formed model of $\mathbb {P} (a_0, \dots , a_N)$ .

Remark 2.13. Any weighted projective space is isomorphic to a well-formed one (see, e.g., [Reference Iano-FletcherIF00, Lemma 5.7]). More precisely, for a weighted projective space $\mathbb {P} = \mathbb {P} (a_0, \dots , a_N)$ with $\gcd \{a_0, \dots , a_N\} = 1$ , there exists an isomorphism

$$\begin{align*}\phi \colon \mathbb{P} (a_0, \dots, a_N)_{x_0, \dots, x_N} \to \mathbb{P}^{\operatorname{wf}} = \mathbb{P} (b_0, \dots, b_N)_{y_0, \dots, y_N} \end{align*}$$

such that $\phi ^* \mathcal {H}_{y_i} = m_i \mathcal {H}_{x_i}$ for $i = 0, 1, \dots , N$ , where $\mathcal {H}_{x_i} = (x_i = 0) \subset \mathbb {P}$ and $\mathcal {H}_{y_i} = (y_i = 0) \subset \mathbb {P}^{\operatorname {wf}}$ .

In the following, we set $\mathbb {P} := \mathbb {P} (a_0, \dots , a_N)$ and we denote by

$$\begin{align*}\Pi \colon \mathbb{A}^{N+1} \setminus \{o\} \to \mathbb{P}, \quad (\alpha_0, \dots, \alpha_N) \mapsto (\alpha_0\!:\!\cdots\!:\!\alpha_N), \end{align*}$$

the canonical projection. Let $X \subset \mathbb {P}$ be a closed subscheme. We set $C_X^* := \Pi ^{-1} (X)$ and call it the punctured affine quasi-cone over X. The affine quasi-cone $C_X$ over X is the closure of $C_X^*$ in $\mathbb {A}^{N+1}$ . We set $\pi := \Pi |_{C_X^*} \colon C_X^* \to X$ .

Definition 2.15. Let $X \subset \mathbb {P}$ be a closed subscheme as above. We define the quasi-smooth locus of X as

$$\begin{align*}\operatorname{QSm} (X) := \pi (\operatorname{Sm} (C^*_X)) \subset X. \end{align*}$$

Let S be a subset of X. We say that X is quasi-smooth along S if $S \subset \operatorname {QSm} (X)$ . We simply say that X is quasi-smooth when $X = \operatorname {QSm} (X)$ .

2.2.c Orbifold charts

Let $\mathcal {U}_{x_i}$ be the open subset of $\mathbb {P} = \mathbb {P} (a_0, \dots , a_N)_{x_0, \dots , x_N}$ as in equation (2.2), where $i \in \{0, 1, \dots , N\}$ . We call $\mathcal {U}_{x_i}$ the standard affine open subset of $\mathbb {P}$ containing $\mathsf {p}_{x_i}$ . We denote by $\breve {\mathcal {U}}_{x_i}$ the affine N-space $\mathbb {A}^N$ with affine coordinates $\breve {x}_0, \dots , \widehat {\breve {x}_i}, \dots , \breve {x}_N$ . Consider the ${\boldsymbol {\mu }}_{a_i}$ -action on $\breve {\mathcal {U}}_i$ defined by

$$\begin{align*}\breve{x}_j \mapsto \zeta^{a_j} \breve{x}_j, \quad \text{for}\ j = 0, \dots, \hat{i}, \dots, N, \end{align*}$$

where $\zeta \in {\boldsymbol {\mu }}_{a_i}$ is a primitive $a_i$ th root of unity. Then the open set $\mathcal {U}_i$ can be naturally identified with the quotient $\breve {\mathcal {U}}_{x_i}/{\boldsymbol {\mu }}_{a_i}$ . In fact, this can be seen by the identification

$$\begin{align*}\breve{x}_j = \frac{x_j}{x_i^{a_j/a_i}}, \quad \text{for}\ j = 0, \dots, \hat{i}, \dots, N. \end{align*}$$

The quotient morphism $\breve {\mathcal {U}}_{x_i} \to \breve {\mathcal {U}}_{x_i}/{\boldsymbol {\mu }}_{a_i} = \mathcal {U}_{x_i}$ is denoted by

$$\begin{align*}\rho_{x_i} \colon \breve{\mathcal{U}}_{x_i} \to \mathcal{U}_{x_i} \end{align*}$$

and is called the orbifold chart of $\mathbb {P}$ containing $\mathsf {p}_{x_i}$ .

Let $X \subset \mathbb {P}$ be a subscheme. Usually, we denote by $U_{x_i} \subset X$ the open set $\mathcal {U}_{x_i} \cap X$ , and we call $U_{x_i}$ the standard affine open subset of X containing $\mathsf {p}_{x_i}$ . In this case, we set $\breve {U}_{x_i} = \rho _{x_i}^{-1} (U_{x_i}) \subset \breve {\mathcal {U}}_{x_i}$ . By a slight abuse of notation, the morphism $\rho _{x_i}|_{U_i} \colon \breve {U}_{x_i} \to U_{x_i}$ is also denote by

$$\begin{align*}\rho_{x_i} \colon \breve{U}_{x_i} \to U_{x_i} \end{align*}$$

and is called the orbifold chart of X containing $\mathsf {p}_{x_i}$ . When we are using the notation $\mathsf {p} = \mathsf {p}_{x_i}$ , the morphism $\rho _{x_i}$ is sometimes denoted by $\rho _{\mathsf {p}}$ . Note that $\breve {U}_{x_i}$ is not necessary smooth in general.

Suppose that $X \subset \mathbb {P}$ is a closed subvariety containing the point $\mathsf {p} = \mathsf {p}_{x_i}$ . The preimage $\breve {\mathsf {p}}$ of $\mathsf {p}$ is the origin of $\breve {U}_{x_i} \subset \breve {\mathcal {U}}_{x_i} = \mathbb {A}^N$ . It is straightforward to see that X is quasi-smooth at $\mathsf {p}$ if and only if $\breve {U}_{x_i}$ is smooth at $\breve {\mathsf {p}}$ . Suppose that X is quasi-smooth at $\mathsf {p}$ . A system of local coordinates of $U_{x_i}$ at $\breve {\mathsf {p}}$ is called a system of local orbifold coordinates of X at $\mathsf {p}$ . In this case, $\mathsf {p} \in X$ is a cyclic quotient singularity of index $a_i$ and $\rho _{x_i} \colon \breve {U}_{x_i} \to U_{x_i}$ can be identified with (or analytically equivalent to) the quotient morphism $q_{\mathsf {p}}$ of $\mathsf {p} \in X$ after shrinking $U_{x_i}$ and then $\breve {U}_{x_i}$ . Moreover, if X is quasi-smooth, then $\breve {U}_i$ is smooth for any i.

2.2.d Weighted hypersurfaces and quasi-tangent divisors

As in the previous subsections, we work with $\mathbb {P} = \mathbb {P} (a_0, \dots , a_N)_{x_0, \dots , x_N}$ .

Definition 2.18. We say that a subvariety $S \subset \mathbb {P}$ is a quasi-linear subspace of $\mathbb {P}$ if it is a complete intersection in $\mathbb {P}$ defined by quasi-linear equations of the form

$$\begin{align*}\ell_1 + f_1 = \ell_2 + f_2 = \cdots = \ell_k + f_k = 0, \end{align*}$$

where $\ell _1, \ell _2, \dots , \ell _k$ are linearly independent linear forms in variables $x_0, \dots , x_{n+1}$ and $f_1, \dots , f_k \in \mathbb {C} [x_0, \dots , x_{n+1}]$ are quasi-homogeneous polynomials which are not quasi-linear. A quasi-linear subspace of $\mathbb {P}$ of codimension $1$ (resp. dimension $1$ ) is called a quasi-hyperplane (resp. quasi-line) of $\mathbb {P}$ .

It is clear that a quasi-linear subspace of $\mathbb {P}$ is isomorphic to a weighted projective space. In particular, a quasi-line is isomorphic to $\mathbb {P}^1$ .

Let X be a hypersurface in $\mathbb {P} = \mathbb {P} (a_0, \dots , a_N)$ defined by a quasi-homogeneous polynomial of degree d. We often denote it as $X = X_d \subset \mathbb {P} (a_0, \dots , a_N)$ . Suppose that X is quasi-smooth at a point $\mathsf {p} = \mathsf {p}_{x_i}$ . Then the defining polynomial $F = F (x_0, \dots , x_N)$ of X can be written as

(2.3) $$ \begin{align} F = x_i^m f + x_i^{m-1} g_{m-1} + \cdots + x_i g_1 + g_0, \end{align} $$

where $m \ge 0$ , $f = f (x_0, \dots , x_N)$ is a quasi-homogeneous polynomial of degree $d - m a_i$ which is quasi-linear and $g_k = g_k (x_0, \dots ,\hat {x}_i, \dots , x_N)$ is a quasi-homogeneous polynomial of degree $d - k a_i$ which is not quasi-linear for $0 \le k \le m - 1$ . Note that the expression (2.3) is uniquely determined once the homogeneous coordinates of $\mathbb {P}$ are fixed.

Remark 2.20. Let $X = X_7 \subset \mathbb {P} (1, 1, 1, 2, 3)_{x, y, z, t, w}$ be a weighted hypersurface of degree $7$ . Suppose that its defining polynomial is of the form

$$\begin{align*}F = t^3 x + t^2 w + t g_5 + g_7, \end{align*}$$

where $g_5, g_7 \in \mathbb {C} [x, y, z, w]$ are quasi-homogeneous polynomials of degree $5, 7$ , respectively. In this case X is quasi-smooth at $\mathsf {p} = \mathsf {p}_t$ . The quasi-tangent polynomial of X at $\mathsf {p}$ is $t x + w$ . Note that x is not a quasi-tangent coordinate of X at $\mathsf {p}$ because of the presence of $t^2 w \in F$ .

Lemma 2.21. Let $X \subset \mathbb {P}$ be a weighted hypersurface of degree d. Assume that X is quasi-smooth at a point $\mathsf {p} = \mathsf {p}_{x_i}$ for some $i = 0, 1, \dots , N$ , and let $x_j$ be a homogeneous coordinate such that $x_j \in f$ , where f is the quasi-tangent polynomial of X at $\mathsf {p}$ . Then, after a suitable choice of homogeneous coordinates $x_0, \dots , x_N$ , the defining polynomial F of X can be written as

$$\begin{align*}F = x_i^m x_j + x_i^{m-1} g_{m - 1} + \cdots + x_i g_1 + g_0, \end{align*}$$

where $g_k = g_k (x_0, \dots , \hat {x}_i, \dots , x_N)$ is a quasi-homogeneous polynomial of degree $d - k a_i$ which is not quasi-linear.

Proof. We can write $F = x_i^m f + g$ , where $m \ge 0$ , $f = f (x_0, \dots , x_N) \ni x_j$ is the quasi-tangent polynomial and g is a quasi-homogeneous polynomial of degree d which does not involve a monomial divisible by $x_i^m$ and which is contained in the ideal $(x_0, \dots , \hat {x}_i, \dots , x_N)^2$ . We write $g = x_j^e h_{e} + \cdots + x_j h_1 + h_0$ , where $e \ge 0$ and $h_k$ is a quasi-homogeneous polynomial of degree $d - k a_j$ which does not involve the variables $x_j$ . By rescaling $x_j$ , we may assume $f = x_j - \tilde {f}$ , where $x_j \notin \tilde {f}$ , and we write $\tilde {f} = x_i^n \tilde {f}_n + \cdots + x_i \tilde {f}_1 + \tilde {f}_0$ , where $\tilde {f}_k$ is a quasi-homogeneous polynomial of degree $d - (m + k) a_i$ which does not involve the variable $x_i$ . We consider the coordinate change $x_j \mapsto x_j + \tilde {f}$ . Then the new defining polynomial can be written as

$$ \begin{align*} F &= x_i^m x_j + (x_j + x_i^n \tilde{f}_n + \cdots)^e h_{e} + \cdots + (x_j + x_i^n \tilde{f}_n + \cdots) h_1 + h_0 \\ &= (x_i^{n e - m} \tilde{f}_n^e + \cdots + x_j) x_i^m + \cdots \end{align*} $$

It follows that the new quasi-tangent polynomial is $x_i^{n e - m} \tilde {f}_n^e + \cdots + x_j$ . We claim that $n e - m< n$ . We have $d = m a_i + a_j$ , $d = e a_j + \deg h_e \ge e a_j$ and $a_j = n a_i + \deg \tilde {f}_n> n a_i$ , which implies $n e - m < n$ . Thus, repeating the above coordinate change, we can drop the degree of the quasi-tangent coordinate with respect to $x_i$ , and we may assume $F = x_i^m f + x_i^{m-1} g_{m-1} + \cdots + x_i g + g_0$ , where $f \ni x_j$ and $g_k$ are quasi-homogeneous polynomials of degree $a_j$ and $d - k a_i$ , respectively, which do not involve the variable $x_i$ . Moreover, $g_k$ is not quasi-linear for $0 \le k \le m-1$ . Finally, replacing $x_j$ , we may assume $f = x_j$ and this completes the proof.

2.3.a Definition of the families

As it is explained in Section 1.2.b, quasi-smooth Fano 3-fold weighted hypersurfaces of index 1 are classified and they form 95 families. According to the classification, the minimum of the weights of an ambient space is $1$ . Hence a family is determined by a quadruple $(a_1, a_2, a_3, a_4)$ , which means that the family corresponding to a quadruple $(a_1, a_2, a_3, a_4)$ is the family of weighted hypersurfaces of degree $d = a_1 + a_2 + a_3 + a_4$ in $\mathbb {P} (1, a_1, a_2, a_3, a_4)$ . The 95 families are numbered in the lexicographical order on $(d, a_1, a_2, a_3, a_4)$ , and each family is referred to as family No. $\mathsf {i}$ for $\mathsf {i} \in \{1, 2, \dots , 95\}$ . Families No. $1$ and $3$ are the families consisting of quartic 3-folds and degree $6$ hypersurfaces in $\mathbb {P} (1, 1, 1, 1, 3)$ , respectively, and for any smooth member of these two families, K-stability (and hence the existence of KE metrics) is known.

Definition 2.23. We set

$$\begin{align*}\mathsf{I} := \{1, 2, \dots, 95\} \setminus \{1, 3\}, \end{align*}$$

and, for $\mathsf {i} \in \mathsf {I}$ , we denote by $\mathcal {F}_{\mathsf {i}}$ the family consisting of the quasi-smooth members of family No. $\mathsf {i}$ .

The main objects of this article is thus the members of $\mathcal {F}_{\mathsf {i}}$ for $\mathsf {i} \in \mathsf {I}$ .

We set

$$\begin{align*}\mathsf{I}_1 := \{2, 4, 5, 6, 8, 10, 14\}. \end{align*}$$

The set $\mathsf {I}_1$ is characterized as follows: Let $X = X_d \subset \mathbb {P} (1, a_1, a_2, a_3, a_4)$ , $a_1 \le a_2 \le a_3 \le a_4$ , be a member of a family $\mathcal {F}_{\mathsf {i}}$ with $\mathsf {i} \in \mathsf {I}$ . Then $\mathsf {i} \in \mathsf {I}_1$ if and only if $a_2 = 1$ . The computations of alpha invariants will be done in a relatively systematic way for families $\mathcal {F}_{\mathsf {i}}$ with $\mathsf {i} \in \mathsf {I} \setminus \mathsf {I}_1$ (see Sections 4 and 5), while the computations will be done separately for families $\mathcal {F}_{\mathsf {i}}$ with $\mathsf {i} \in \mathsf {I}_1$ (see Section 6).

We explain notation and conventions concerning the main objects of this article. Let $X = X_d \subset \mathbb {P} (1, a_1, a_2, a_3, a_4) =: \mathbb {P}$ be a member of a family $\mathcal {F}_{\mathsf {i}}$ with $\mathsf {i} \in \mathsf {I}$ .

• • Unless otherwise specified, we assume $a_1 \le a_2 \le a_3 \le a_4$ .

• • In many situations (especially when we treat a specific family), we denote by $x, y, z, t, w$ the homogeneous coordinates of $\mathbb {P}$ of degree, respectively, $1, a_1, a_2, a_3, a_4$ .

• • We denote by F the polynomial defining X in $\mathbb {P}$ , which is quasi-homogeneous of degree $d = a_1 + a_2 + a_3 + a_4$ .

• • We set $A = -K_X$ , which is the positive generator of of $\operatorname {Cl} (X) \cong \mathbb {Z}$ . Note that we have

  $$\begin{align*}(-K_X)^3 = (A^3) = \frac{d}{a_1 a_2 a_3 a_4} = \frac{a_1 + a_2 + a_3 + a_4}{a_1 a_2 a_3 a_4}. \end{align*}$$

2.3.b Definitions of QI and EI centers and birational (super)rigidity

In this subsection, let

$$\begin{align*}X = X_d \subset \mathbb{P} (1, a_1, a_2, a_3, a_4)_{x, y, z, t, w} \end{align*}$$

be a member of a family $\mathcal {F}_{\mathsf {i}}$ with $\mathsf {i} \in \mathsf {I}$ , where $a_1 \le a_2 \le a_3 \le a_4$ . We give definitions of QI and EI centers, which are particular singular points on X and are important for understanding birational (super)rigidity of X. For EI centers, we only give an ad hoc definition (see [Reference Corti, Pukhlikhov and ReidCPR00, Section 4.10] and [Reference Cheltsov and ParkCP17, Section 4.2] for more detailed treatments).

Note that $|\mathsf {I}_{\mathrm {BSR}}| = 48$ and $|\mathsf {I}_{\mathrm {BR}}| = 45$ . The following is a more precise version of Theorem 1.2.

We emphasize that a family $\mathcal {F}_{\mathsf {i}}$ , where $\mathsf {i} \in \mathsf {I}_{\mathrm {BR}}$ , can contain (in fact does contain for most of $\mathsf {i} \in \mathsf {I}_{\mathrm {BR}}$ ) birationally superrigid Fano 3-folds as special members.

2.3.c Numerics on weights and degrees

Let

$$\begin{align*}X = X_d \subset \mathbb{P} (1, a_1, a_2, a_3, a_4)_{x, y, z, t, w} \end{align*}$$

be a member of $\mathcal {F}_{\mathsf {i}}$ . Throughout the subsection, we assume that $\mathsf {i} \in \mathsf {I} \setminus \mathsf {I}_1$ and that $a_1 \le a_2 \le a_3 \le a_4$ . We collect some elementary numerical results on weights $a_1, \dots , a_4$ , the degree $d = a_1 + a_2 + a_3 + a_4$ of the defining polynomial $F = F (x, y, z, t, w)$ of X, and the anticanonical degree $(A^3)$ of X which will be repeatedly used in the rest of this article.

Proof. We see that either $w^n \in F$ for some $n \ge 2$ or $x^n v \in F$ for some $n \ge 1$ and $v \in \{x, y, z, t\}$ by the quasi-smoothness of X.

Suppose $w^n \in F$ for some $n \ge 2$ . Then we have

$$\begin{align*}d = n a_4 = a_1 + a_2 + a_3 + a_4 < 4 a_4. \end{align*}$$

Hence, $n = 2, 3$ and we are in case (1) or (2). Suppose $w^n v \in F$ for some $n \ge 1$ and $v \in \{y, z, t\}$ . Then we have $d = n a_4 + a_j$ and moreover we have

$$\begin{align*}a_4 + a_j < d = a_1 + a_2 + a_3 + a_4 < 3 a_4 + a_j. \end{align*}$$

This shows $n = 2$ , that is, $d = 2 a_4 + a_j$ .

If $a_1 = 1$ , then the proof is completed. It remains to show that the case $d = 2 a_4 + 1$ does not take place assuming $a_1 \ge 2$ . Suppose $d = 2 a_4 + 1$ and $a_1 \ge 2$ . Then $w^2 x \in F$ and the singularity of $\mathsf {p}_w \in X$ is of type $\frac {1}{a_4} (a_1, a_2, a_3)$ . There exist distinct $i, j \in \{1,2,3\}$ such that $a_i + a_j$ is divisible by $a_4$ since $\mathsf {p}_w \in X$ is terminal. We have $a_i + a_j = a_4$ since $0 < a_i + a_j < 2 a_4$ . Let $k \in \{1,2,3\}$ be such that $\{i, j, k\} = \{1,2,3\}$ . Then

$$\begin{align*}d = a_1 + a_2 + a_3 + a_4 = a_k + 2 a_4. \end{align*}$$

Combining this with $d = 2 a_4 + 1$ , we have $a_k = 1$ . This is a contradiction since $a_k \ge a_1 \ge 2$ .

Proof. We prove (1). The ‘only if’ part is obvious. Suppose $d = 3 a_4$ and $a_1 = 1$ . Then we have $2 a_4 = 1 + a_2 + a_3$ . This implies $a_2 = a_4 - 1$ and $a_3 = a_4$ since $a_2 \le a_3 \le a_4$ . Then, by setting $a = a_2 \ge 2$ , X is a weighted hypersurface in $\mathbb {P} (1, 1, a, a+1, a+1)$ of degree $3 (a+1)$ . Suppose $\mathsf {p}_z \notin X$ . Then some power of z is contained in F and this implies that $3 (a+1)$ is divisible by a. In particular, we have $a = 3$ and this case corresponds to $\mathsf {i} = 17$ . Suppose $\mathsf {p}_z \notin X$ , then either $3 (a+1) \equiv 1 \pmod {a}$ or $3 (a+1) \equiv a+1 \pmod {a}$ by the quasi-smoothness of X. In both cases, we have $a = 2$ , ad hence $\mathsf {i} = 9$ . Thus, (1) is proved.

The assertion (2) follows immediately since we have

$$\begin{align*}a_1 a_2 a_3 (A^3) = \frac{d}{a_4} \le 3 \end{align*}$$

and $d \le 3 a_4$ by Lemma 2.28.

We prove (3). Note that $2 \le a_2 \le a_3 \le a_4$ . Note also that $a_4> a_2$ because otherwise X has nonisolated singularity along $L_{xy}$ which is impossible. In particular, we have $a_1 + \cdots + a_4 < 4 a_4$ and $a_2 a_3 \ge 4$ and we have

$$\begin{align*}a_1 (A^3) = \frac{a_1 (a_1 + a_2 + a_3 + a_4)}{a_1 a_2 a_3 a_4} < \frac{4}{a_2 a_3} \le 1, \end{align*}$$

which proves (3).

We prove (4). We have $a_2 \ge 3$ since $a_2> a_1 > 1$ and thus

$$\begin{align*}a_1 a_3 (A^3) = \frac{d}{a_2 a_4} \le \frac{3}{a_2} \le 1. \end{align*}$$

We prove (5). We have $d> 2 a_4$ by assumption. Then, by Lemma 2.28, we have $d = 2 a_4 + a_j$ for some $j \in \{1,2,3,4\}$ , and combining this with $d = a_1 + a_2 + a_3 + a_4$ , we have

$$\begin{align*}a_4 = a_1 + a_2 + a_3 - a_j \le a_2 + a_3. \end{align*}$$

If $a_1> 1$ , then we have $a_2, a_3 \ge 3$ and thus

$$\begin{align*}a_1 a_4 (A^3) = \frac{a_1 + a_2 + a_3 + a_4}{a_2 a_3} < \frac{3 a_2 + 2 a_3}{a_2 a_3} = \frac{3}{a_3} + \frac{2}{a_2} \le \frac{5}{3}. \end{align*}$$

Suppose $a_1 = 1$ . In this case $2 \le a_2 \le a_3$ . If $a_3 \ge 3$ , then

$$\begin{align*}a_4 (A^3) = \frac{1 + a_2 + a_3 + a_4}{a_2 a_3} \le \frac{1 + 2 a_2 + 2 a_3}{a_2 a_3} = \frac{1}{a_2 a_3} + \frac{2}{a_3} + \frac{2}{a_2} \le \frac{11}{6}. \end{align*}$$

Suppose $a_3 = 2$ , that is, $a_2 = a_3 = 2$ . Then we have $a_4 = 3$ and $d = 8$ since $d = 5 + a_4> 2 a_4$ and $a_4$ is odd. In this case, we have $a_4 (A^3) = 2$ . This proves (5).

We prove (6). By Lemma 2.28 and (1), either $d = 2 a_4$ or $d = 3 a_4$ and $a_1 \ge 2$ . If $d = 2 a_4$ (resp. $d = 3 a_4$ and $a_1 \ge 2$ ), then

$$\begin{align*}a_2 a_3 (A^3) = \frac{2}{a_1} \le 2 \quad (\text{resp.\ } a_2 a_3 (A^3) = \frac{3}{a_1} \le 2). \end{align*}$$

This proves (6).

We prove (7). By Lemma 2.28, we have $d = 2 a_4 + a_j$ for some $j \in \{1, 2, 3\}$ . Then we have $a_4 = a_1 + a_2 + a_3 - a_j \le a_2 + a_3$ . If $a_1 \ge 3$ , then

$$\begin{align*}a_2 a_4 (A^3) = \frac{a_1 + a_2 + a_3 + a_4}{a_1 a_3} \le \frac{a_1 + 4 a_3}{a_1 a_3} = \frac{1}{a_3} + \frac{4}{a_1} \le \frac{5}{3}. \end{align*}$$

We continue the proof assuming $a_1 = 2$ . If in addition $a_2 < a_3$ , then

$$\begin{align*}a_2 a_4 (A^3) = \frac{2 + a_2 + a_3 + a_4}{2 a_3} \le \frac{2 + 2 a_2 + 2 a_3}{2 a_3} \le \frac{4 a_3}{2 a_3} = 2. \end{align*}$$

We continue the proof assuming $a_1 = 2$ and $a_2 = a_3$ . In this case, by setting $a = a_2 = a_3$ and $b = a_4$ , X is a weighted hypersurface of degree d in $\mathbb {P} (1, 2, a, a, b)$ and either $d = 2 b + 2$ or $d = 2 b + a$ . If $d = 2 b + 2$ , then $b = 2 a$ but this is impossible since X has only terminal singularities. Hence, $d = 2 b + a$ . In this case $b = a + 2$ and $d = 3 a + 4$ . By the quasi-smoothness of X, we see that $d = 3 a + 4$ is divisible by a. This implies that $a \in \{2, 4\}$ . This is impossible since X has only terminal singularities. Therefore, (7) is proved.

2.3.d How to compute alpha invariants?

Let X be a member of a family $\mathcal {F}_{\mathsf {i}}$ with $\mathsf {i} \in \mathsf {I}$ . For the proof of Theorem 1.8, it is necessary to show $\alpha _{\mathsf {p}} (X) \ge 1/2$ for any point $\mathsf {p} \in X$ . Let $\mathsf {p} \in X$ be a point. We briefly explain the most typical method of bounding $\alpha _{\mathsf {p}} (X)$ from below, which goes as follows.

1. 1. Choose and fix a divisor S on X which vanishes at $\mathsf {p}$ to a relatively large (orbifold) multiplicity $m = \operatorname {omult}_{\mathsf {p}} (S)> 0$ . In some cases, $S = H_x$ (when $\mathsf {p} \in H_x$ ), and in other cases, S is the quasi-tangent divisor of X at $\mathsf {p}$ . Let a be the positive integer such that $S \sim a A$ .

2. 2. Let $D \in |A|_{\mathbb {Q}}$ be an irreducible $\mathbb {Q}$ -divisor other than $\frac {1}{a} S$ . Then $D \cdot S$ is an effective $1$ -cycle on X.

3. 3. Find a $\mathbb {Q}$ -divisor $T \in |e A|_{\mathbb {Q}}$ for some $e \in \mathbb {Z}_{> 0}$ such that $\operatorname {mult}_{\mathsf {p}} (T) \ge 1$ and $\operatorname {Supp} (T)$ does not contain any component of $D \cdot S$ . We will find such a $\mathbb {Q}$ -divisor T by considering $\mathsf {p}$ -isolating set or class which will be explained in Section 3.1.c.

4. 4. Let $q = q_{\mathsf {p}}$ be the quotient morphism of $\mathsf {p} \in X$ and $\check {\mathsf {p}}$ be the preimage of $\mathsf {p}$ via q. By the above choices, $\operatorname {Supp} (q^*D) \cap \operatorname {Supp} (q^*S) \cap \operatorname {Supp} (q^*T)$ is a finite set of points including $\check {\mathsf {p}}$ , and hence the local intersection number $(q^*D \cdot q^*S \cdot q^*T)_{\check {\mathsf {p}}}$ is defined (see Section 3.1.a). Then we have the inequalities

   $$\begin{align*}m \operatorname{omult}_{\mathsf{p}} (D) \le (q_{\mathsf{p}}^*D \cdot q_{\mathsf{p}}^*S \cdot q_{\mathsf{p}}^*T)_{\check{\mathsf{p}}} \le r (D \cdot S \cdot T) = r a e (A^3), \end{align*}$$
   where r is the index of the cyclic quotient singularity $\mathsf {p} \in X$ . Note that q is the identity morphism and $r = 1$ when $\mathsf {p} \in X$ is a smooth point. By Lemma 3.2 which will be explained below, we have
   $$\begin{align*}\operatorname{lct}_{\mathsf{p}} (X;D) \ge \frac{r a e (A^3)}{m} \end{align*}$$
   for any D as in (2).

5. 5. As a conclusion, we have

   $$\begin{align*}\alpha_{\mathsf{p}} (X) \ge \min \left\{ \operatorname{lct}_{\mathsf{p}} (X; S), \ \frac{r a e (A^3)}{m} \right\}. \end{align*}$$

6. 6. It remains to bound $\operatorname {lct}_{\mathsf {p}} (X;S)$ from below. This is easy when S is quasi-smooth at $\mathsf {p}$ because in that case we have $\operatorname {lct}_{\mathsf {p}} (X;S) = 1$ . The computation gets involved when S is the quasi-tangent divisor but will be done by considering suitable weighted blowups which will be explained in Section 3.2.b.

We need to consider variants of the above explained method or other methods especially for points in special positions. These will be explained in Section 3.

3.1.a Some results on multiplicities and log canonicity

Let V be an n-dimensional variety. For effective Cartier divisors $D_1, \dots , D_n$ on V and a point $\mathsf {p} \in V$ which is an isolated component of $\operatorname {Supp} (D_1) \cap \cdots \cap \operatorname {Supp} (D_n)$ , the intersection multiplicity

$$\begin{align*}i (\mathsf{p}, D_1 \cdots D_n;V) \end{align*}$$

is defined (see [Reference FultonFul98, Example 7.1.10]). Suppose that V is $\mathbb {Q}$ -factorial. Then this definition is naturally generalized to effective $\mathbb {Q}$ -divisors $D_1, \dots , D_n$ as follows:

$$\begin{align*}i (\mathsf{p}, D_1, \cdots, D_n;V) := \frac{1}{d^n} i (\mathsf{p}, d D_1, \cdots, d D_n;V), \end{align*}$$

where d is a positive integer such that $d D_i$ is a Cartier divisor for any i. In this paper, we set

$$\begin{align*}(D_1 \cdots D_n)_{\mathsf{p}} := i (\mathsf{p}, D_1 \cdots D_n;V) \end{align*}$$

and call it the local intersection number of $D_1, \dots , D_n$ at $\mathsf {p}$ .

Remark 3.1. If $\mathsf {p} \in V$ is a smooth point, $D_1, \dots , D_n$ are effective divisors defined by $f_1, \dots , f_n \in \mathcal {O}_{V, \mathsf {p}}$ around $\mathsf {p}$ , and $\mathsf {p}$ is an isolated component of $\operatorname {Supp} (D_1) \cap \cdots \cap \operatorname {Supp} (D_n)$ , then

$$\begin{align*}(D_1 \cdots D_n)_{\mathsf{p}} = \dim_{\mathbb{C}} \mathcal{O}_{V, P}/(f_1, \dots, f_n). \end{align*}$$

If $X \subset \mathbb {P} (a_0, \dots , a_N)$ is an n-dimensional subvariety which is quasi-smooth at $\mathsf {p} = \mathsf {p}_{x_i} \in V$ , $D_1 = (G_1 = 0)_X, \dots , D_n = (G_n = 0)_X$ are effective Weil divisors such that $\mathsf {p}$ is an isolated component of $D_1 \cap \cdots \cap D_n$ , where $G_i = G_i (x_0, \dots , x_N)$ is a quasi-homogeneous polynomial of degree $d_i$ , then

$$\begin{align*}(D_1 \cdots D_n)_{\mathsf{p}} = \frac{1}{a_i} (\rho^*D_1 \cdots \rho^* D_n)_{\breve{\mathsf{p}}} = \frac{1}{a_i} \dim_{\mathbb{C}} \mathcal{O}_{\breve{U}_{\mathsf{p}}, \breve{\mathsf{p}}}/(g_1, \dots, g_n), \end{align*}$$

where $\rho = \rho _{\mathsf {p}} \colon \breve {U}_{\mathsf {p}} \to U_{\mathsf {p}} := X \cap \mathcal {U}_{\mathsf {p}}$ is the orbifold chart with $\breve {\mathsf {p}} \in \breve {U}_{\mathsf {p}}$ the preimage of $\mathsf {p}$ and $g_i = G (\breve {x}_0, \dots , 1, \dots , \breve {x}_N)$ with $\breve {x}_j = x_j/x_i^{a_j/a_i}$ for $j \ne i$ .

We will frequently use the following property of local intersection numbers. Let $D_1, \dots , D_n$ be effective $\mathbb {Q}$ -divisors on X and $\mathsf {p} \in X$ be a smooth point. If $\mathsf {p}$ is an isolated component of $\operatorname {Supp} (D_1) \cap \cdots \cap \operatorname {Supp} (D_n)$ , then

$$\begin{align*}(D_1 \cdot \ldots \cdot D_n)_{\mathsf{p}} \ge \prod_{i=1}^n \operatorname{mult}_{\mathsf{p}} (D_i). \end{align*}$$

We refer readers to [Reference FultonFul98, Corollary 12.4] for a proof. Although the following results are well-known to experts, we include their proofs for readers’ convenience.

Lemma 3.2. Let $\mathsf {p} \in X$ be either a germ of a smooth variety or a germ of a cyclic quotient singular point, and let D be an effective $\mathbb {Q}$ -divisor on X. Then the inequality

$$\begin{align*}\frac{1}{\operatorname{omult}_{\mathsf{p}} (D)} \le \operatorname{lct}_{\mathsf{p}} (X,D) \end{align*}$$

holds.

Proof. Let $q = q_{\mathsf {p}} \colon \check {X} \to X$ be the quotient morphism of $\mathsf {p} \in X$ , which is étale in codimension $1$ , and let $\check {\mathsf {p}} \in \check {X}$ be the preimage of $\mathsf {p}$ . By [Reference KollárKol+92, 20.4 Corollary], we have $\operatorname {lct}_{\mathsf {p}} (X;D) = \operatorname {lct}_{\check {\mathsf {p}}} (\check {X};q^*D)$ . Note that $\check {\mathsf {p}} \in \check {X}$ is smooth. Hence, by [Reference KollárKol97, 8.10 Lemma], we have

$$\begin{align*}\frac{1}{\operatorname{omult}_{\mathsf{p}} (D)} = \frac{1}{\operatorname{mult}_{\check{\mathsf{p}}} (q^*D)} \le \operatorname{lct}_{\check{\mathsf{p}}} (\check{X};q^*D), \end{align*}$$

and the proof is completed.

Proof. We set $m = \operatorname {mult}_{\mathsf {p}} (D)$ . By [Reference CortiCor00, Corollary 3.5], one of the following holds.

1. 1. $m> 2 n$ .

2. 2. There is a line $L \subset E$ such that the pair

   $$\begin{align*}\left(Y, \left(\frac{m}{n} - 1\right)E + \frac{1}{n} \tilde{D} \right) \end{align*}$$
   is not log canonical at the generic point of L.

Note that in [Reference CortiCor00, Corollary 3.5] the boundary is a movable linear system $\mathcal {H}$ , but the same argument applies if we replace $\mathcal {H}$ by an effective $\mathbb {Q}$ -divisor D. We may assume $m \le 2n$ because otherwise $\operatorname {mult}_{\mathsf {p}} (D|_T)> 2 n$ for any prime divisor T which is smooth at $\mathsf {p}$ and the assertion follows by choosing any line on E. Thus, the option (2) takes place. Let T be a prime divisor on X such that T is smooth at $\mathsf {p}$ and $\tilde {T} \supset L$ . We have

$$\begin{align*}K_Y + \left( \frac{m}{n} - 1 \right) E + \frac{1}{n} \tilde{D} + \tilde{T} = \varphi^* \left( K_X + \frac{1}{n} D + T \right). \end{align*}$$

Note that $E|_{\tilde {T}} = L$ , and we can write $\tilde {D}|_{\tilde {T}} = \alpha L + G$ , where $\alpha \ge 0$ is a rational number and G is an effective $\mathbb {Q}$ -divisor on $\tilde {T}$ . Thus, by restricting the above equation to $\tilde {T}$ , we have

$$\begin{align*}K_{\tilde{T}} + \left( \frac{m}{n} - 1 + \alpha \right) L + G = \varphi^* \left(K_T + \frac{1}{n} D|_T \right), \end{align*}$$

and the pair

$$\begin{align*}\left( \tilde{T}, \left( \frac{m}{n} - 1 + \alpha \right) L + G \right) \end{align*}$$

is not log canonical at the generic point of L. This implies $\frac {m}{n} - 1 + \alpha> 1$ , and we have

$$\begin{align*}\frac{1}{n} \operatorname{mult}_{\mathsf{p}} (D|_T) = \left( \frac{m}{n} - 1 + \alpha \right) + 1> 2. \end{align*}$$

Thus, $\operatorname {mult}_{\mathsf {p}} (D|_T)> 2 n$ and the proof is completed.

Lemma 3.6. Let

$$\begin{align*}X = X_d \subset \mathbb{P} (1, b_1, b_2, b_3, b_4)_{x, y_1, y_2, y_3, y_4} \end{align*}$$

be a member of a family $\mathcal {F}_{\mathsf {i}}$ with $\mathsf {i} \in \mathsf {I}$ . Let $i \in \{1, 2, 3, 4\}$ be such that $b_i> 1$ and $\mathsf {p} := \mathsf {p}_{y_i} \in X$ . If $H_x$ is the quasi-tangent divisor of X at $\mathsf {p}$ , then the pair $(X, H_x)$ is not canonical at $\mathsf {p}$ .

Proof. Note that the point $\mathsf {p} \in X$ is of type $\frac {1}{b_i} (b_j, b_k, b_l)$ , where $\{i, j, k, l\} = \{1, 2, 3, 4\}$ , and it is a terminal singularity. Let $\varphi \colon Y \to X$ be the Kawamata blowup with exceptional divisor E. Since $H_x$ is the quasi-tangent divisor of X at $\mathsf {p}$ , we have

$$\begin{align*}\operatorname{ord}_E (H_x)> \frac{1}{b_i}. \end{align*}$$

Combining this with

$$\begin{align*}K_Y = \varphi^*K_X + \frac{1}{b_i} E, \end{align*}$$

we see that the discrepancy of the pair $(X, H_x)$ along E is negative. This completes the proof.

3.1.b Some results on singularities of weighted hypersurfaces

Proof. Let $x_0, \dots , x_4$ be the homogeneous coordinates of $\mathbb {P} = \mathbb {P} (b_0, \dots , b_4)$ of degree $b_0, \dots , b_4$ , respectively. Let $F = F (x_0,\dots ,x_4)$ be the defining polynomial of X, and let S be a quasi-hyperplane section on X. After replacing homogeneous coordinates, we may assume $S = (x_4 = 0)_X = (x_4 = F = 0) \subset \mathbb {P}$ . It is enough to show that the singular locus $\operatorname {Sing} (S)$ of S is a finite set of points.

We write $F = x_4 G + \bar {F}$ , where $G = G (x_0, \dots , x_4)$ and $\bar {F} = \bar {F} (x_0, \dots , x_3)$ are quasi-homogeneous polynomials. We set $\bar {\mathbb {P}} = \mathbb {P} (b_0, \dots , b_3)$ . We claim that $\bar {\mathbb {P}}$ is well-formed. Suppose it is not. Then $\operatorname {Sing} (\mathbb {P})$ contains a two-dimensional stratum. We have $\operatorname {Sing} (X) = \operatorname {Sing} (\mathbb {P}) \cap X$ since a quasi-smooth weighted hypersurface is well-formed ([Reference Iano-FletcherIF00, Theorem 6.17]). It follows that $\operatorname {Sing} (X)$ cannot be a finite set of points. This is a contradiction, and the claim is proved. The surface S is identified with the hypersurface $(\bar {F} = 0) \subset \bar {\mathbb {P}}$ , and we have

$$\begin{align*}\operatorname{Sing} (S) = (S \setminus \operatorname{QSm} (S)) \cup (\operatorname{Sing} (\bar{\mathbb{P}}) \cap S). \end{align*}$$

We claim that $\operatorname {Sing} (\bar {\mathbb {P}}) \cap S$ is a finite set of points. Suppose not. Then $\operatorname {Sing} (\bar {\mathbb {P}}) \cap S$ contains a curve and so does $\operatorname {Sing} (\mathbb {P}) \cap S$ . In particular, $\operatorname {Sing} (X) = \operatorname {Sing} (\mathbb {P}) \cap X$ contains a curve. This is impossible since $\operatorname {Sing} (X)$ is a finite set of points, and the claim is proved.

It remains to show that the closed subset $\Sigma := S \setminus \operatorname {QSm} (S) \subset \mathbb {P}$ is a finite set of points. Let $\Pi \colon \mathbb {A}^5 \setminus \{o\} \to \mathbb {P}$ be the natural quotient morphism. Then $\Sigma = \Pi (\operatorname {Sing} (C^*_S))$ , where $C^*_S \subset \mathbb {A}^5 \setminus \{o\}$ is the punctured affine quasi-cone of S. We have $\operatorname {Sing} (C_X) \cap C_S = \operatorname {Sing} (C_S) \cap (G = 0)$ . By the quasi-smoothness of X, we have $\operatorname {Sing} (C_X) = \{o\} \subset \mathbb {A}^5$ . This implies $\Sigma \cap (G = 0) = \emptyset $ . Since $(G = 0)$ is an ample divisor on $\mathbb {P}$ , we see that $\Sigma $ is a finite set of points. This completes the proof.

Proof. Let $x_0, x_1, x_2, x_3$ be the homogeneous coordinates of $\mathbb {P} = \mathbb {P} (b_0, b_1, b_2, b_3)$ of degree $b_0, b_1, b_2, b_3$ , respectively, and let $F= F (x_0, x_1, x_2, x_3)$ be the defining polynomial of S. We may assume $T = H_{x_3} \subset \mathbb {P}$ , and we write $F = x_3 G + \bar {F}$ , where $G = G (x_0, x_1, x_2, x_3)$ and $\bar {F} = \bar {F} (x_0, x_1, x_2)$ are quasi-homogeneous polynomials. Then $S \cap T$ is the closed subscheme in $\mathbb {P} (b_0,b_1,b_2,b_3)$ defined by $x_3 = \bar {F} = 0$ . By the quasi-smoothness of $S \cap T$ at $\mathsf {p}$ , there exists $i \in \{0, 1, 2\}$ such that

$$\begin{align*}\frac{\partial \bar{F}}{\partial x_i} (\mathsf{p}) \ne 0. \end{align*}$$

It follows that

$$\begin{align*}\frac{\partial F}{\partial x_i} (\mathsf{p}) = \frac{\partial \bar{F}}{\partial x_i} (\mathsf{p}) \ne 0 \end{align*}$$

since $\mathsf {p} \in H_{x_3}$ . Thus, S is quasi-smooth at $\mathsf {p}$ .

Lemma 3.9. Let S be a normal weighted hypersurface in a well-formed weighted projective $3$ -space $\mathbb {P} (b_0,\dots , b_3)$ and $T \subset \mathbb {P} (b_0,\dots , b_3)$ a quasi-hyperplane such that $T \ne S$ . Let $\Gamma $ be an irreducible component of $S \cap T$ , and we assume that

$$\begin{align*}T|_S = \Gamma + \Delta, \end{align*}$$

where $\Delta $ is an effective divisor on S such that $\Gamma \not \subset \operatorname {Supp} (\Delta )$ . If $\Gamma $ is a smooth weighted complete intersection curve and S is quasi-smooth at each point of $\Gamma \cap \operatorname {Supp} (\Delta )$ , then S is quasi-smooth along $\Gamma $ and the pair $(S, \Gamma )$ is purely log terminal (plt) along $\Gamma $ .

Remark 3.10. Let S, T and $\Gamma $ be as in Lemma 3.9. We assume in addition that $\Gamma $ is rational, that is, $\Gamma \cong \mathbb {P}^1$ . Let $\operatorname {Sing}_{\Gamma } (S) = \{\mathsf {p}_1,\dots ,\mathsf {p}_n\}$ be the set of singular points of S along $\Gamma $ , and let $m_i$ be the index of the quotient singular point $\mathsf {p}_i \in S$ . Then, since the pair $(S, \Gamma )$ is plt along $\Gamma $ , we can apply [Reference KollárKol+92, Proposition 16.6] and we have

$$\begin{align*}(K_S + \Gamma)|_{\Gamma} = K_{\Gamma} + \sum_{i=1}^n \frac{m_i-1}{m_i} \mathsf{p}_i. \end{align*}$$

Thus, we have

$$\begin{align*}(\Gamma^2)_S = - (K_S \cdot \Gamma)_S -2 + \sum_{i=1}^n \frac{m_i-1}{m_i}. \end{align*}$$

3.1.c Isolating set and class

We recall the definitions of isolating set and class which are introduced by Corti, Pukhlikov and Reid [Reference Corti, Pukhlikhov and ReidCPR00] as well as their basic properties.

Let V be a normal projective variety embedded in a weighted projective space $\mathbb {P} = \mathbb {P} (a_0, \dots , a_N)$ with homogeneous coordinates $x_0, \dots , x_N$ with $\deg x_i = a_i$ , and let A be a Weil divisor on V such that $\mathcal {O}_V (A) \cong \mathcal {O}_V (1)$ . We do not assume that $a_0 \le \cdots \le a_N$ .

Definition 3.11. Let $\mathsf {p} \in V$ be a point. We say that a set $\{g_1, \dots , g_m\}$ of quasi-homogeneous polynomials $g_1, \dots , g_m \in \mathbb {C} [x_0, \dots , x_N]$ isolates $\mathsf {p}$ or is a $\mathsf {p}$ -isolating set if $\mathsf {p}$ is an isolated component of the set

$$\begin{align*}(g_1 = \cdots = g_m = 0) \cap V. \end{align*}$$

Lemma 3.13 [Reference Corti, Pukhlikhov and ReidCPR00, Lemma 5.6.4].

Let $\mathsf {p} \in V$ be a smooth point. If $\{g_1, \dots , g_m\}$ is a $\mathsf {p}$ -isolating class, then $l A$ is a $\mathsf {p}$ -isolating class, where

$$\begin{align*}l = \max \{\, \deg g_i \mid i = 1, 2, \dots, m \,\}. \end{align*}$$

Lemma 3.14. Let $\mathsf {p} \in V$ be a point, $Z_1, \dots , Z_k$ irreducible closed subsets of V such that $\dim Z_i> 0$ for any i, and let $g_1, \dots , g_n \in \mathbb {C} [x_0, \dots , x_N]$ be quasi-homogeneous polynomials. Suppose that V is quasi-smooth at $\mathsf {p}$ and that $\{g_1, \dots , g_n\}$ isolates $\mathsf {p}$ . We set $G_i = (g_i = 0)_V$ , and we set

$$\begin{align*}\mu := \min \left\{\, \frac{\operatorname{omult}_{\mathsf{p}} (G_i)}{\deg g_i} \; \middle| \; i = 1, \dots, n \,\right\}. \end{align*}$$

Then there exists an effective $\mathbb {Q}$ -divisor $T \sim _{\mathbb {Q}} A$ such that $\operatorname {omult}_{\mathsf {p}} (T) \ge \mu $ and $\operatorname {Supp} (T)$ does not contain any $Z_i$ .

Proof. Let d be the least common multiple of $\deg g_1, \dots , \deg g_n$ , and we set $e_i = d/\deg g_i$ . Consider the linear system $\Lambda \subset |d A|$ on V generated by $g_1^{e_1}, \dots , g_n^{e_n}$ . We see that $\mathsf {p}$ is an isolating component of $\operatorname {Bs} \Lambda $ since $\{g_1, \dots , g_n\}$ isolates $\mathsf {p}$ . Hence, a general $D \in \Lambda $ does not contain any $Z_i$ in its support. Moreover, for any $D \in \Lambda $ , we have

$$\begin{align*}\operatorname{omult}_{\mathsf{p}} (D) \ge \min \{ \, e_i \operatorname{omult}_{\mathsf{p}} (D_i) \mid i = 1, \dots, n \,\} = d \mu. \end{align*}$$

Thus, the assertion follows by setting $T = \frac {1}{d} D \sim _{\mathbb {Q}} A$ for a general $D \in \Lambda $ .

Remark 3.15. Lemma 3.14 will be frequently applied in the following way: under the same notation and assumptions as in Lemma 3.14, there exists an effective $\mathbb {Q}$ -divisor $T \sim _{\mathbb {Q}} e A$ , where

$$\begin{align*}e = \max \{ \, \deg g_i \mid i = 1, \dots, n \,\} \end{align*}$$

such that $\operatorname {omult}_{\mathsf {p}} (T) \ge 1$ and $\operatorname {Supp} (T)$ does not contain any $Z_i$ .

Proof. We can write $\mathsf {p} = (0\!:\!1\!:\!\alpha _2\!:\!\alpha _3\!:\!\alpha _4)$ for some $\alpha _2, \alpha _3, \alpha _4 \in \mathbb {C}$ . Then it is easy to see that the set

$$\begin{align*}\{ x, z^{a_1} - \alpha_2^{a_1} y^{a_2}, t^{a_1} - \alpha_3^{a_3} y^{a_3}, w^{a_1} - \alpha_4^{a_4} y^{a_4} \} \end{align*}$$

isolates $\mathsf {p}$ , and thus $a_1 a_4 A$ isolates $\mathsf {p}$ .

Suppose that $w^k$ appears in the defining polynomial of X. Then the natural projection $\mathbb {P} (1, a_1, \dots , a_4) \dashrightarrow \mathbb {P} (1, a_1, a_2, a_3)$ restricts to a finite morphism $\pi \colon X \to \mathbb {P} (1, a_1, a_2, a_3)$ . The common zero locus (in X) of the sections contained in the set

$$\begin{align*}\{ x, z^{a_1} - \alpha_2^{a_1} y^{a_2}, t^{a_1} - \alpha_3^{a_3} y^{a_3} \} \end{align*}$$

coincides with the set $\pi ^{-1} (\mathsf {q})$ , where $\mathsf {q} = (0\!:\!1\!:\!\alpha _2\!:\!\alpha _3) \in \mathbb {P} (1, a_1, a_2, a_3)$ . It follows that the above set isolates $\mathsf {p}$ since $\pi ^{-1} (\mathsf {q})$ is a finite set containing $\mathsf {p}$ . Thus, $a_1 a_3 A$ isolates $\mathsf {p}$ .