Proof Since X is not K-stable, by Theorem 2.3, there exists a dreamy prime divisor F over X such that $\beta (F)\leq 0$ . Meanwhile, [Reference Fujita and OdakaFO18, Lemma 3.3] implies that $A_X(F)\geq \alpha (X)\tau (F)={n}{/(n+1)}\tau (F)$ . Then by [Reference FujitaFuj19a, Lemma 3.1, Proposition 3.2 and 3.3] we know that there exists a proper birational morphism $\sigma :Y\to X$ with Y normal such that $F=\mathrm {Exc}(\sigma )$ is a prime divisor on Y, $-F$ is $\sigma $ -ample, $\sigma (F)$ is a closed point $x\in X$ , and parts (1) and (2) hold. Moreover, part (3) on $\alpha (F,\Delta _F)\geq 1$ implies that $(F, \Delta _F)$ is klt, that is, F is a Kollár component. Thus it suffices to show that part (3) holds.

Finally we prove part (3). From part (2), we know that $F\cong Z$ are both normal. Denote by $\Delta _{F}$ , the different divisor of F in Y. Let $B\sim _{\mathbb {Q}} -K_{F}-\Delta _{F}$ be any effective $\mathbb {Q}$ -divisor on F. We denote by $B_Z:=(\pi |_{F})_*B$ . From parts (1) and (2), we know that

$$ \begin{align*} (-K_Y-2F)|_{F}=(\sigma^*(-K_X)-(n+1)F)|_{F}=-(n+1)F|_{F}. \end{align*} $$

Hence $-K_{F}-\Delta _{F}=(-K_Y-F)|_{F}=-n F|_{F}$ . Since $\pi |_F:F\to Z$ is an isomorphism, we know that

$$ \begin{align*} \sigma^*(-K_X)-(n+1)F\sim_{\mathbb{Q}}\frac{n+1}{n}\pi^*(\pi|_{F})_*(-K_{F}-\Delta_{F})\sim_{\mathbb{Q}}\frac{n+1}{n}\pi^*B_Z. \end{align*} $$

It is clear that $\pi ^*B_Z$ does not contain F as a component. Thus $B_X:=\sigma _*\pi ^*B_Z$ satisfies that $B_X\sim _{\mathbb {Q}}-{n}{/(n+1)}K_X$ and $\mathrm {ord}_F(B_X)=n$ . Since $\alpha (X)={n}{/(n+1)}$ , the pair $(X,B_X)$ is log canonical. We know

$$ \begin{align*} \sigma^*(K_X+B_X)=K_Y-(n-1)F+\pi^*B_Z+nF=K_Y+F+\pi^*B_Z. \end{align*} $$

Hence $(Y,F+\pi ^*B_Z)$ is log canonical. By adjunction, $(F,\Delta _{F}+B)$ is log canonical since $B=(\pi ^*B_Z)|_{F}$ . By choosing B whose support contains a log canonical center of $(F, \Delta _{F})$ , we see that $(F, \Delta _{F})$ is klt. Hence inversion of adjunction implies that F is a Kollár component over $x\in X$ and $\alpha (F,\Delta _F)\geq 1$ .

By part (2) we know that

$$ \begin{align*} (\sigma^*(-K_X)-(n+1)F)^n=(-K_X)^n-(n+1)^n (-1)^{n-1}(F^n)=0. \end{align*} $$

It is clear that $\mathrm {vol}_{X,x}(\mathrm {ord}_F)=(-1)^{n-1}(F^n)$ . From part (1), we know that $A_X(F)=n$ . Thus we get $(-K_X)^n=(1+\frac {1}{n})^n\widehat {\mathrm {vol}}_{X,x}(\mathrm {ord}_F)$ . Since $\alpha (X)={n}{/(n+1)}$ , Theorem 1.2 implies that X is K-semistable. Hence [Reference LiuLiu18, Theorem 21] implies that $\mathrm {ord}_F$ is a divisorial minimizer of the normalized volume function $\widehat {\mathrm {vol}}$ on $\mathrm {Val}_{X,x}$ . Thus the proof of part (3) is finished.

Proof Denote by $\mathfrak {a}_j:=\{f\in \mathcal {O}_{X,x}\mid \mathrm {ord}_F(f)\geq j\}$ for $j\in \mathbb {Z}$ . From the proof of [Reference LiuLiu18, Lemma 33], we know that there exists a special test configuration $(\mathcal {X};\mathcal {L})$ such that $\mathrm {Fut}(\mathcal {X};\mathcal {L})=0$ and $\mathcal {X}_0\cong \mathrm {Proj} (\oplus _{j=0}^\infty \mathfrak {a}_j/\mathfrak {a}_{j+1})[s]$ . Since F is a Kollár component over $x\in X$ , by adjunction of plt pairs we know that $F|_F$ is a $\mathbb {Q}$ -divisor on F that is well-defined up to $\mathbb {Z}$ -linear equivalence. Moreover, if $\{F|_F\}=\sum _i\frac {a_i}{b_i}B_i$ for $a_i< b_i$ coprime positive integers and $B_i$ prime divisors on F, then (using the same notation as in Theorem 3.1) $\Delta _F=\frac {b_i-1}{b_i}B_i$ (see, e.g., [Reference KollárK⁺ 92, Section 16]). By [Reference Li and XuLX20, Section 2.4] or Proposition 2.10, we know that $\oplus _{j=0}^\infty \mathfrak {a}_j/\mathfrak {a}_{j+1}\cong \oplus _{m=0}^\infty H^0(F,\mathcal {O}_F(\lfloor -mF|_F\rfloor ))$ as graded rings which implies $\mathcal {X}_0\cong C_p(F,(-F)|_F)$ . Since $\alpha (F,\Delta _F)\geq 1$ , Theorem 1.2 implies that $(F,\Delta _F)$ is K-stable. Hence Proposition 2.11 implies that $\mathcal {X}_0$ is K-polystable since $-K_F-\Delta _F\sim _{\mathbb {Q}}-n F|_F$ . To show $\alpha (\mathcal {X}_0)={1}{/(n+1)}$ , we notice that $\alpha (\mathcal {X}_0)\geq {1}{/(n+1)}$ by [Reference Fujita and OdakaFO18, Theorem 3.5], and $\alpha (\mathcal {X}_0)\leq {1}{/(n+1)}$ since $-K_{\mathcal {X}_0}\sim _{\mathbb {Q}} (n+1)F_{\infty }$ . Hence the proof is finished.

Proof By assumption X is birational to Y over C. Let m be a sufficiently large and divisible integer and let $D_1\in |-mK_{X_0}|$ , $D_2\in |-mK_{Y_0}|$ be general divisors in the corresponding linear systems. Choose effective divisors $D_{X,1}\sim _{\mathbb {Q}} -mK_X$ , $D_{Y,2}\sim _{\mathbb {Q}} -mK_Y$ not containing $X_0$ or $Y_0$ such that $D_{X,1}|_{X_0}=D_1$ and $D_{Y,2}|_{Y_0}=D_2$ . Let $D_{Y,1}$ and $D_{X,2}$ be their strict transforms to the other family. Let $D_X=\frac {\alpha (Y_0)}{\alpha (X_0)+\alpha (Y_0)}D_{X,1}+\frac {\alpha (X_0)}{\alpha (X_0)+\alpha (Y_0)}D_{X,2}$ and similar $D_Y=\frac {\alpha (Y_0)}{\alpha (X_0)+\alpha (Y_0)}D_{Y,1}+\frac {\alpha (X_0)}{\alpha (X_0)+\alpha (Y_0)}D_{Y,2}$ . Apparently $D_Y$ is the strict transform of $D_X$ . As $D_1$ is a general member of a very ample linear system and $D_{X,2}|_{X_0}\sim _{\mathbb {Q}} -mK_{X_0}$ , by the definition of alpha invariant we see that $(X_0,\frac {1}{m}D_X|_{X_0})$ is klt. Similarly $(Y_0,\frac {1}{m}D_Y|_{Y_0})$ is klt. The result then follows from the next lemma.

Proof This follows from the exact same proof of [Reference Li, Wang and XuLWX19, Theorem 5.2] (see also [Reference Blum and XuBX19, Proposition 3.2]).

Proof Proof of Theorem 1.5

Boundedness follows from [Reference JiangJia17a]; openness follows from [Reference Blum and LiuBL18b, Theorem B]; separatedness follows from Proposition 3.3; the existence of coarse moduli space is due to Keel and Mori [Reference Keel and MoriKM97].

Proof Assume that $\mathrm {Aut}(X)$ is not finite, then it contains $\mathbb {G}_m$ or $\mathbb {G}_a$ . Let $\mathcal {X}_1=\mathcal {X}_2=X\times \mathbb {A}^1$ . If $\mathbb {G}_m\subseteq \mathrm {Aut}(X)$ , then it induces an isomorphism $\mathcal {X}_1\times _{\mathbb {A}^1} (\mathbb {A}^{1}\setminus \{0\}) \stackrel {\sim }{\longrightarrow } \mathcal {X}_2\times _{\mathbb {A}^1} (\mathbb {A}^{1}\setminus \{0\})=X\times (\mathbb {A}^{1}\setminus \{0\})$ through the diagonal action of $\mathbb {G}_m$ . Since $\alpha (X)>\frac {1}{2}$ , this map extends to an isomorphism of $\mathcal {X}_1$ and $\mathcal {X}_2$ over $\mathbb {A}^1$ by Proposition 3.3. Thus the map $\mathbb {G}_m\rightarrow \mathrm {Aut}(X)$ extends to a map $\mathbb {A}^1\rightarrow \mathrm {Aut}(X)$ , a contradiction. Similarly, if $\mathbb {G}_a\subseteq \mathrm {Aut}(X)$ , then the inclusion $\mathbb {A}^{1}\setminus \{0\}\subseteq \mathbb {G}_a$ given by $t\mapsto t^{-1}$ induces an automorphism of $X\times (\mathbb {A}^{1}\setminus \{0\})$ over $\mathbb {A}^{1}\setminus \{0\}$ which extends to an automorphism over $\mathbb {A}^1$ , hence $\mathbb {G}_a\rightarrow \mathrm {Aut}(X)$ extends to a map $\mathbb {P}^1\rightarrow \mathrm {Aut}(X)$ , again a contradiction.

Proof By Theorem 3.1, it suffices to show that under the given assumptions, X cannot have a weakly exceptional singularity whose corresponding Kollár component has log discrepancy n. For 3-fold terminal singularities, this follows from [Reference ProkhorovPro00, Corollary 4.8]. In the surface case, it is well known (and not hard to verify) that a klt surface singularity is weakly exceptional if and only if the dual graph of the exceptional divisors on the minimal resolution has a fork, in which case the Kollár component lives on the minimal resolution and corresponds to the central fork. In particular, the log discrepancy of the Kollár component is at most $1$ and does not satisfy our requirement.

Proof Let $X^{\prime }$ be the unique K-polystable special degeneration of X whose existence is proved in [Reference Li, Wang and XuLWX18, Theorem 1.3]. Then we have $\alpha (X^{\prime })\leq \alpha (X)={1}{/(n+1)}$ by [Reference Blum and LiuBL18b, Theorem B] and $\alpha (X^{\prime })\geq {1}{/(n+1)}$ by [Reference Fujita and OdakaFO18, Theorem 3.5]. Thus $\alpha (X^{\prime })={1}{/(n+1)}$ . Then by [Reference JiangJia17b, Proposition 3.1], there exists a prime divisor V on $X^{\prime }$ such that $(X^{\prime },V)$ is plt and $-K_{X^{\prime }}\sim _{\mathbb {Q}}(n+1)V$ . Let $M\sim V|_V$ be a $\mathbb {Q}$ -Cartier $\mathbb {Q}$ -divisor on V. Thus Lemma 2.12 implies that there exists a special test configuration $(\mathcal {X}^{\prime };\mathcal {L}^{\prime })$ of $X^{\prime }$ such that the central fiber $\mathcal {X}_0^{\prime }$ is isomorphic to $C_p(V,M)$ and $\mathrm {Fut}(\mathcal {X}^{\prime };\mathcal {L}^{\prime })=0$ . Since $X^{\prime }$ is K-polystable, we have $X^{\prime }\cong \mathcal {X}_0^{\prime }\cong C_p(V,M)$ . Denote $\Delta :=\mathrm {Diff}_V(0)$ , then it is clear that $\Delta $ has standard coefficients, $\Delta =\Delta _M$ , and

$$ \begin{align*} -K_V-\Delta=(-K_X-V)|_V\sim_{\mathbb{Q}} nV|_V=nM. \end{align*} $$

Since $C_p(V,M)\cong X^{\prime }$ is K-polystable, the log Fano pair $(V,\Delta )$ is also K-polystable by Proposition 2.11. Hence we finish the proof.

Proof Assume to the contrary that $\Delta _F=0$ . Then by Theorem 3.1 and Lemma 3.10, we know that $X\cong C_p(F,\mathcal {O}_F(-F))$ and $\alpha (X)\leq {1}{/(n+1)}$ which contradicts to $\alpha (X)={n}{/(n+1)}$ .

Proof Let $D:=\sigma ^*(-K_X)-\epsilon (F)\cdot F$ . By definition we know that D is a nef $\mathbb {R}$ -divisor that is not big. Since $\epsilon (F)>A_X(F)>0$ , $-K_Y-F=\sigma ^*(-K_X)-A_X(F)\cdot F$ is ample. Hence the cone theorem implies that $\overline {NE}(Y)$ is polyhedral, so $\epsilon (F)\in \mathbb {Q}$ . By Shokurov’s basepoint-free theorem and Kawamata–Viehweg vanishing theorem, the $\mathbb {Q}$ -divisor D is semiample and defines an algebraic fiber space $\pi : Y\to Z$ . Since $-K_Y-F=D+(\epsilon (F)-A_X(F))F$ is ample, F is a $\pi $ -ample $\mathbb {Q}$ -Cartier divisor. Following the same argument as in [Reference FujitaFuj19a, Proposition 3.3], the fibers of $\pi $ are one-dimensional, $\pi $ is a generic $\mathbb {P}^1$ -fibration, and $\pi |_F:F\to Z$ is an isomorphism. In particular, $-K_Y-2F$ has zero intersection number with a generic fiber of $\pi $ , hence $-K_Y-2F=D$ which yields $\epsilon (F)=A_X(F)+1$ .

Let $F^0$ be the biggest open subset of the smooth locus $F_{\mathrm {reg}}$ of F such that $\mathcal {O}_Y(F)$ is locally free along $F^0$ . By assumption we have $\mathrm {codim}_F F\setminus F^0\geq 2$ . Denote by $Y^0:=Y\setminus (F\setminus F^0)$ with $i: Y^0\hookrightarrow Y$ and $j:F^0\hookrightarrow F$ the open immersions, hence $\mathrm {codim}_Y Y\setminus Y^0\geq 3$ . Denote by $Z^0:=\pi (F^0)$ . Since $\mathcal {O}_Y(F)$ is locally free along $F^0$ , we have the short exact sequence

(3.1) $$ \begin{align} 0\to \mathcal{O}_{Y^0}((m-1)F^0)\to\mathcal{O}_{Y^0}(mF^0)\to \mathcal{O}_{F^0}(mF^0|_{F^0})\to 0. \end{align} $$

By taking $i_*$ , we get an exact sequence

(3.2) $$ \begin{align} 0\to i_*\mathcal{O}_{Y^0}((m-1)F^0)\to i_*\mathcal{O}_{Y^0}(mF^0)\to j_*\mathcal{O}_{F^0}(mF^0|_{F^0}) \to R^1 i_*\mathcal{O}_{Y^0}((m-1)F^0). \end{align} $$

Since $Y^0$ and $F^0$ are open subsets of Y and $F,$ respectively, whose complements have codimension at least $2$ , we have these natural isomorphisms

$$ \begin{align*} i_*\mathcal{O}_{Y^0}(mF^0)\cong \mathcal{O}_Y(mF),\quad j_*\mathcal{O}_{F^0}(mF^0|_{F^0})\cong \mathcal{O}_F(mF|_F). \end{align*} $$

From the assumption that F is Cartier in codimension $2$ , we have that $\mathcal {O}_F(mF|_F)$ is a well-defined $\mathbb {Q}$ -Cartier Weil divisorial sheaf on F satisfying $\mathcal {O}_F(mF|_F)\cong \mathcal {O}_F(F|_F)^{[m]}$ . By [Reference Kollár and MoriKM98, Corollary 5.25] we have that $\mathcal {O}_Y(mF)$ is Cohen–Macaulay for any $m\in \mathbb {Z}$ . Since $\mathrm {codim}_Y Y\setminus Y^0\geq 3$ , the local cohomology long exact sequence implies that

$$ \begin{align*} R^1 i_* \mathcal{O}_{Y^0}((m-1)F^0)\cong \mathcal{H}^2_{Y\setminus Y^0}(Y,\mathcal{O}_Y((m-1)F))=0. \end{align*} $$

Hence the exact sequence (3.2) becomes

(3.3) $$ \begin{align} 0\to \mathcal{O}_Y((m-1)F)\to\mathcal{O}_Y(mF)\to \mathcal{O}_F(mF|_F)\to 0. \end{align} $$

Let M be a $\mathbb {Q}$ -Cartier Weil divisor on F representing $\mathcal {O}_F(-F)$ , then $M|_{F^0}$ is Cartier on $F^0$ . Denote by $M_Z:=(\pi |_F)_*M$ . After tensoring (3.3) by $\mathcal {O}_Y(m\cdot \pi ^*M_Z)$ and taking the reflexive hull, we get an exact sequence

(3.4) $$ \begin{align} 0\to \mathcal{O}_Y((m-1)F+m\cdot\pi^*M_Z))\to\mathcal{O}_Y(m(F+\pi^*M_Z))\to \mathcal{O}_F\to 0. \end{align} $$

The reason that (3.4) is exact can be deduced as follows: we first tensor (3.1) by $\mathcal {O}_{Y^0}(m\cdot \pi ^*M_Z)$ (since $\mathcal {O}_{Y^0}(m\cdot \pi ^*M_Z)$ is locally free along $F^0$ , the exactness remains), then taking $i_*$ (since $\pi ^*M_Z$ is a $\mathbb {Q}$ -Cartier Weil divisor on Y, we again use [Reference Kollár and MoriKM98, Corollary 5.25] to show that $\mathcal {O}_Y((m-1)F+m\cdot \pi ^*M_Z))$ is Cohen–Macaulay); then the argument above works in a similar way.

Now applying $\pi _*$ to the exact sequence (3.4) when $m=1$ , we get an exact sequence

(3.5) $$ \begin{align} 0\to \pi_*\mathcal{O}_Y(\pi^*M_Z)\to\pi_*\mathcal{O}_Y(F+\pi^*M_Z)\to \mathcal{O}_Z\to R^1\pi_*\mathcal{O}_Y(\pi^*M_Z). \end{align} $$

It is clear that $K_Y\sim _{\mathbb {Q},\pi }-2F$ is $\pi $ -anti-ample, hence Kawamata–Viehweg vanishing implies that $R^1\pi _*\mathcal {O}_Y(\pi ^*M_Z)=0$ . The projection formula yields $(\pi _*\mathcal {O}_{Y}(\pi ^*M_Z))|_{Z^0}\cong \mathcal {O}_{Z^0}(M_Z)$ . As abuse of notation we also denote the open immersion $Z^0\hookrightarrow Z$ by j. Hence

$$ \begin{align*} \pi_*\mathcal{O}_Y(\pi^*M_Z)\cong j_*\left((\pi_*\mathcal{O}_{Y}(\pi^*M_Z))|_{Z^0}\right) \cong j_* \mathcal{O}_{Z^0}(M_Z)\cong \mathcal{O}_{Z}(M_Z), \end{align*} $$

where the first equality follows from the fact that $\mathcal {O}_Y(\pi ^*M_Z)$ satisfies $S_2$ -condition and $\pi $ is flat. As a result, (3.5) becomes

(3.6) $$ \begin{align} 0\to \mathcal{O}_Z(M_Z)\to\pi_*\mathcal{O}_Y(F+\pi^*M_Z)\to \mathcal{O}_Z\to 0. \end{align} $$

Hence $\pi _*\mathcal {O}_Y(F+\pi ^*M_Z)$ is an extension of $\mathcal {O}_Z$ by $\mathcal {O}_Z(M_Z)$ . It is clear that

$$ \begin{align*} \mathrm{Ext}^1(\mathcal{O}_Z,\mathcal{O}_Z(M_Z))\cong H^1(Z, \mathcal{O}_Z(M_Z))\cong H^1(F,\mathcal{O}_F(M)). \end{align*} $$

By assumption we have that

$$ \begin{align*} K_F=(K_Y+F)|_F\sim_{\mathbb{Q}}(\sigma^*K_X+A_X(F)\cdot F)|_F =-A_X(F)\cdot M. \end{align*} $$

Hence $M-K_F\sim _{\mathbb {Q}}(A_X(F)+1)M$ is ample. Then Kawamata–Viehweg vanishing yields $H^1(F,\mathcal {O}_F(M))=0$ . So the exact sequence (3.6) splits, that is, $\pi _*\mathcal {O}_Y(F+\pi ^*M_Z)\cong \mathcal {O}_Z(M_Z)\oplus \mathcal {O}_Z$ . By restricting to $Z^0$ , tensoring with $\mathcal {O}_{Z^0}(-M_Z)$ and taking $i_*$ , we have

$$ \begin{align*} \pi_*\mathcal{O}_Y(F)\cong\mathcal{O}_Z\oplus\mathcal{O}_Z(-M_Z). \end{align*} $$

Denote by $Y^{\prime }:=\pi ^{-1}(Z^0)$ . From [Reference FujitaFuj19a, Proposition 3.3] and [Reference Liu and ZhuangLZ18, Lemma 5] we know that $\pi ^0=\pi |_{Y^{\prime }}:Y^{\prime }\to Z^0$ is a smooth $\mathbb {P}^1$ -fibration. Since $\pi ^0$ admits a section, it is a $\mathbb {P}^1$ -bundle. As a consequence,

(3.7) $$ \begin{align} \bigoplus_{m=0}^\infty(\pi^0)_*\mathcal{O}_{Y^{\prime}}(mF^0)\cong \bigoplus_{m=0}^\infty\mathrm{Sym}^m \big((\pi^0)_*\mathcal{O}_{Y^{\prime}} (F^0)\big). \end{align} $$

Applying $j_*$ to (3.7) yields

$$ \begin{align*} \bigoplus_{m=0}^\infty\pi_*\mathcal{O}_{Y}(mF)&\cong \bigoplus_{m=0}^\infty j_*~\mathrm{Sym}^m \big((\pi^0)_*\mathcal{O}_{Y^{\prime}} (F^0)\big)\\ &\cong \bigoplus_{m=0}^\infty j_*\left(\bigoplus_{k=0}^m\mathcal{O}_{Z^0}(-kM_Z)\right)\\ &\cong \bigoplus_{m=0}^\infty \bigoplus_{k=0}^m\mathcal{O}_{Z}(-kM_Z). \end{align*} $$

Since F is $\pi $ -ample, we have

$$ \begin{align*} Y\cong \mathrm{Proj}_{Z}\bigoplus_{m=0}^\infty\pi_*\mathcal{O}_{Y}(mF)\cong \mathrm{Proj}_Z\bigoplus_{m=0}^\infty\bigoplus_{k=0}^m\mathcal{O}_{Z}(-kM_Z). \end{align*} $$

Hence $(Y,F)$ is isomorphic to the canonical blow-up of the projective orbifold cone $C_p(F,\mathcal {O}_F(M))$ at the vertex, which implies $X\cong C_p(F,\mathcal {O}_F(M))$ .

Since $-K_F\sim _{\mathbb {Q}} A_X(F)\cdot M$ , we have that $-K_X\sim _{\mathbb {Q}} (A_X(F)+1) F_{\infty }$ where $F_\infty $ is the section of the projective orbifold cone at infinity. Hence $\alpha (X)\leq \frac {1}{A_X(F)+1}$ .

Proof The proof is similar to the hypersurface case in [Reference CheltsovChe01]. Let $D\sim _{\mathbb {Q}} H$ be an effective divisor on X. By [Reference SuzukiSuz17, Proposition 2.1], there exists a closed subset $Z\subseteq X$ of dimension at most $r-1$ such that $\mathrm {mult}_x D\le 1$ for all $x \in X\backslash Z$ . In particular, $(X,D)$ is lc outside Z. Let $W\subseteq X$ be a linear space section of X of codimension r that is disjoint from Z. Let $H_1,\ldots ,H_{r+1}\subseteq X$ be general hyperplane sections containing W and let $\Delta =(1-\epsilon )D+\frac {r}{r+1}(H_1+\cdots +H_{r+1})$ where $0<\epsilon \ll 1$ . Then $(X,\Delta )$ is klt outside $Z\cup W$ by construction. As $-(K_X+\Delta )\sim _{\mathbb {Q}} (s-r-1+\epsilon )H$ is ample, $\mathrm {Nklt}(X,\Delta )$ is connected by Kollár–Shokurov’s connectedness lemma. Since $\mathrm {mult}_W \Delta \ge r$ , W is already a non-klt center of $(X,\Delta )$ ; hence since Z is disjoint from W, we deduce that $(X,\Delta )$ is klt along Z and it follows that $(X,(1-\epsilon )D)$ is also klt along Z. Hence $(X,D)$ is lc and $\mathrm {lct}(X;|H|_{\mathbb {Q}})\ge 1$ . The other direction of the inequality is obvious.

Proof The proof is similar to that of [Reference PukhlikovPuk05, Theorem 2], based on the technique of hypertangent divisors. Denote by c the right hand side of the above inequality. Let $D\sim _{\mathbb {Q}} H$ be an effective divisor on X. It suffices to show that $(X,cD)$ is lc. Since $c\le 1$ , the non-lc center of the pair $(X,cD)$ has codimension at least $2$ . Let $x\in X$ be a point in the non-lc center of $(X,cD)$ . Let $\pi :\tilde {X}\rightarrow X$ be the blowup of x, let E be the exceptional divisor and let $\tilde {D}$ be the strict transform of D. Then by (the proof of) [Reference PukhlikovPuk13, Section 7, Proposition 2.3] (applied to a general surface section of $(X,cD)$ ), there exists a hyperplane $\Pi \subseteq E$ such that

(4.2) $$ \begin{align} \mathrm{mult}_x (cD) + \mathrm{mult}_\Pi (c\tilde{D})>2. \end{align} $$

Let $\mathcal {L}$ be the linear system of hyperplane sections of X whose strict transform contains $\Pi $ and let $L\in \mathcal {L}$ be a general member. Note that $\mathcal {L}$ is spanned by the tangent hyperplanes of X at x and another hyperplane. Thus by the P-regularity condition, $\mathrm {Bs}(\mathcal {L})$ has codimension at least $r+1\ge 2$ in X and the divisor $L\cdot D \subseteq L$ is well defined. By (4.2), we have $\mathrm {mult}_x(cD\cdot L)>2$ or

(4.3) $$ \begin{align} \frac{\mathrm{mult}_x}{\deg}(D\cdot L)>\frac{2}{c\cdot \deg X}. \end{align} $$

Let $\Lambda _{L,\ell }$ be the $\ell $ th hypertangent linear system of L at x. Since X is $(m-1)$ -strongly P-regular, the cycle $Z_\ell =\mathrm {Bs}(\Lambda _{L,\ell })$ is irreducible and reduced when $1\le \ell \le m-1$ .

By (4.3), there exists an irreducible component $W_1$ of $D\cdot L$ such that

$$ \begin{align*}\frac{\mathrm{mult}_x}{\deg} W_1 \ge \frac{\mathrm{mult}_x}{\deg}(D\cdot L)>\frac{2}{c\cdot \deg X}\ge \frac{2}{\deg X}\end{align*} $$

and hence by (4.1) with $\ell =1,$ we have $W_1\neq Z_1$ (recall that $\beta _1=2$ and indeed we have $\beta _\ell \le 2$ for all $\ell $ ). It follows that $W_1$ is not contained in the divisor $H_1 = \Lambda _{L,1}$ (assuming $m-1\ge 1$ ) and we have a well-defined codimension $2$ cycle $W_1\cdot H_1$ in L with

$$ \begin{align*}\frac{\mathrm{mult}_x}{\deg}(W_1\cdot H_1)\ge \frac{2}{1}\cdot \frac{\mathrm{mult}_x}{\deg} W_1=\beta_1 \cdot \frac{\mathrm{mult}_x}{\deg} W_1\end{align*} $$

as $\mathrm {mult}_x H_1 = 2$ and $H_1\sim H$ . Let $W_2$ be an irreducible component of $W_1\cdot H_1$ such that

$$ \begin{align*}\frac{\mathrm{mult}_x}{\deg} W_2\ge\frac{\mathrm{mult}_x}{\deg}(W_1\cdot H_1)>\frac{2\beta_1}{c\cdot \deg X} \ge \frac{2\beta_1}{\deg X}.\end{align*} $$

Assume that $\beta _2>1$ , then by (4.1) again we have $W_2\neq Z_2$ (assuming $m-1\ge 2$ ). Since $W_2$ is already contained in $H_1$ and L we deduce that $W_2$ is not contained in a general divisor $H_2\in \Lambda _{L,2}$ , and in this case we get a codimension $3$ cycle $W_3$ in L as an irreducible component of $W_2\cdot H_2$ with

(4.4) $$ \begin{align} \frac{\mathrm{mult}_x}{\deg} W_3 \ge \frac{\mathrm{mult}_x}{\deg}(W_2\cdot H_2)\ge \beta_2 \cdot \frac{\mathrm{mult}_x}{\deg} W_2> \frac{2\beta_1 \beta_2}{c\cdot \deg X}. \end{align} $$

If $\beta _2=1$ , we simply set $W_3=W_2$ and the inequality (4.4) clearly still holds. Iterating this process, we find an irreducible cycle $W_m$ of codimension $\lambda _m=\lambda _{m-1}+1$ (where $\lambda _\ell :=\#\{i\le \ell \,|\,\beta _i>1\}$ ) in L with

$$ \begin{align*}\frac{\mathrm{mult}_x}{\deg} W_m> \frac{2\beta_1 \beta_2 \cdots \beta_{m-1}}{c\cdot \deg X}.\end{align*} $$

By the P-regularity condition, $Z_\ell $ has codimension $\lambda _\ell $ in a neighbourhood of x, hence if $\beta _{m+1}>1$ (so that $\lambda _{m+1}=\lambda _m +1$ ) then $W_m$ is not contained in $\mathrm {Bs}(\Lambda _{L,m+1})$ and we may repeat the construction above to get an irreducible cycle $W_{m+1}$ (of codimension $\lambda _{m+1}$ in L) in the support of $W_m\cdot H_{m+1}$ (where $H_{m+1}$ is a general divisor in $\Lambda _{L,m+1}$ ) such that

$$ \begin{align*}\frac{\mathrm{mult}_x}{\deg} W_{m+1}> \frac{2\beta_1 \beta_2 \cdots \beta_{m-1} \beta_{m+1}}{c\cdot \deg X}.\end{align*} $$

If $\beta _{m+1}=1$ then we simply set $W_{m+1}=W_m$ and the above inequality still holds for obvious reason. Iterate this process again and eventually we obtain an irreducible cycle $W_k$ (where $k = \min \{d,n+r-2\}$ ) such that

$$ \begin{align*}\frac{\mathrm{mult}_x}{\deg} W_k> \frac{2\beta_1 \beta_2 \cdots \beta_{m-1} \beta_{m+1}\cdots \beta_k}{c\cdot \deg X}.\end{align*} $$

But we always have $\frac {\mathrm {mult}_x}{\deg } W \le 1$ for any subvarieties $Z\subseteq X$ , hence

$$ \begin{align*}c>\frac{2}{\deg X}\prod_{i\ge 1, i\neq m} \beta_i\end{align*} $$

(recall that $\beta _i=1$ when $i>k$ ), a contradiction. Therefore, $(X,cD)$ is lc at every $x\in X$ .

Proof This should be well known to experts and is essentially a direct consequence of the work of Pukhlikov [Reference PukhlikovPuk18, Section 4]. For reader’s convenience we sketch the proof. Let $\mathbb {P}=\mathbb {P}^{n+r}$ . Let $U\rightarrow B\subseteq \prod _{i=1}^r \mathbb {P} H^0(\mathbb {P},\mathcal {O}_{\mathbb {P}}(d_i))$ be the universal family of smooth complete intersections of given type and $U\rightarrow \mathbb {P}$ be the natural projection. It suffices to show that for a smooth complete intersection containing x to violate the P-regularity at x imposes at least $n+1$ conditions on the coefficients of $q_1,\ldots ,q_d$ for any $x\in \mathbb {P}$ , since then the locus of irregular points has codimension at least $n+1$ in U and cannot dominate B. For this we may fix the coefficients of $q_1,\ldots ,q_r$ and further assume that $d\ge n+r-2$ by introducing auxiliary terms in the sequence $q_1,\ldots ,q_d$ .

Let $p_1,\ldots ,p_{n-2}$ be the sequence obtained by restricting $q_{r+1},\ldots ,q_{n+r-2}$ to $T_x X=(q_1=\cdots =q_r=0)\cap T_x \mathbb {P}$ and let $m_i=\deg p_i\ge 2$ . Let $\Pi =\mathbb {P}(T_x X)$ . Then the P-regularity of X at x is equivalent to saying that every hyperplane section of $W=(p_1=\cdots =p_{n-2}=0)\subseteq \Pi $ is zero dimensional. There are now two cases to consider. If all the $p_i$ vanish on a line, it would require at least

(4.5) $$ \begin{align} \sum_{i=1}^{n-2} (m_i+1) - 2(n-2) = \sum_{i=1}^r \frac{a_i(a_i+1)}{2} - (n+r-2) \end{align} $$

conditions on the coefficients of $p_i$ (where $a_i$ is the largest subscript j such that $q_{i,j}$ appears in $q_1,\ldots ,q_{n+r-2}$ ). As $\sum _{i=1}^r a_i=n+r-2$ , an elementary argument shows that if $n\ge 2r+3$ then this number is $\ge n+1$ . On the other hand, for a fixed hyperplane $\Sigma \subseteq \Pi $ we claim that for $W\cap \Sigma $ to contain a component of positive dimension that is not a line would require at least $2n$ conditions on the coefficients of $p_i$ . As the hyperplane $\Sigma $ varies in an $(n-1)$ -dimensional family, this will prove the statement we want.

Let $p^{\prime }_i=p_i|_\Sigma $ and let $1\le \ell \le n-2$ . To verify the claim, it suffices to consider the case where $p^{\prime }_1,\ldots ,p^{\prime }_{\ell -1}$ form a regular sequence and $p^{\prime }_\ell |_B=0$ for some component B of $(p^{\prime }_1=\cdots =p^{\prime }_{\ell -1}=0)\subseteq \Sigma $ whose linear span $\left \langle B \right \rangle $ is different from a line. If $m_\ell =2,$ then $\ell \le r$ and as $\dim B=n-\ell -1,$ we see that (see, e.g., Lemma 4.13) this already imposes at least

(4.6) $$ \begin{align} \binom{n-\ell+1}{2}\ge \binom{n-r+1}{2}\ge 2n \end{align} $$

conditions on $p^{\prime }_\ell $ (given that $n\ge 2r+3$ ). Similarly if $m_\ell =3$ and $\ell \le r+1$ , then the vanishing of $p^{\prime }_\ell $ on B imposes at least

(4.7) $$ \begin{align} \binom{n-\ell+2}{3}\ge \binom{n-r+1}{3}\ge 2n \end{align} $$

conditions on $p^{\prime }_\ell $ . Thus we may assume that either $m_\ell \ge 4$ or $\ell \ge r+2$ ; in particular, this implies $\sum _{i=1}^\ell m_i\ge 2\ell +2$ . Since any nonzero product $\prod _{j=1}^{m_i} \ell _j$ (where each $\ell _j$ is a linear form on $\left \langle B \right \rangle $ ) does not vanish on B and the set of such products has dimension $m_i(n-2-b)+1$ (where $b=\mathrm {codim}_\Sigma \left \langle B \right \rangle $ ) in the space of degree $m_i$ homogeneous forms on $\left \langle B \right \rangle $ , we deduce that our assumption on $p^{\prime }_1,\ldots ,p^{\prime }_\ell $ imposes at least

(4.8) $$ \begin{align} \left(\sum_{i=1}^\ell m_i\right)(n-2-b)+\ell-\dim \mathrm{Gr}(b,n-1) \ge (2\ell+2)(n-2-b)+\ell-b(n-1-b) \end{align} $$

conditions on their coefficients. Since $\mathrm {codim} B\le \ell -1$ and $\left \langle B \right \rangle $ is not a line, we have $b\le \ell -1$ and $b\le n-4$ when $\ell =n-2$ . Therefore, it is straightforward to check that the right hand side of (4.8), as a quadratic function in b, in minimized at $b=\ell -1$ when $\ell \le n-3$ or at $b=n-4$ when $\ell =n-2$ . It then follows from another elementary argument that the right hand side of (4.8) is $\ge 2n$ .

Proof As before we assume $d=\sum _{i=1}^r d_i \ge n+r-2$ . Since $s<r$ , a general such complete intersection is smooth and does not contain y. For each $x\neq y\in \mathbb {P}^{n+r}$ , we may choose affine linear coordinates $z_*=(z_1,\ldots ,z_{n+r})$ with origin at x such that a hypersurface has a cone singularity at y if and only if its equation only involves the variables $z_2,\ldots ,z_{n+r}$ . After a further change of variable, we may also assume that $T_x X$ (throughout the proof we use the same notations as in the proof of Lemma 4.5) is defined by the vanishing of some of the $z_i$ ’s. As before, it suffices to show that violation of P-regularity condition imposes at least $n+1$ conditions on the coefficients of the $p_i$ . If $z_1$ vanishes on $T_x X$ , then the restricted expression $p_i$ can be an arbitrary degree $m_i$ homogeneous polynomial and the same calculation in Lemma 4.5 applies. If $z_1$ does not vanish on $T_x X$ , the only restriction is that when $p_i$ comes from some $Q_j$ with $j\le s$ , it can only vary among the equations of (arbitrary) degree $m_i$ hypersurfaces with a cone singularity at $w=[1:0:\cdots :0]\in \Pi $ .

The proof then proceeds as before. For instance (this is the most complicated case), if $\Sigma \subseteq \Pi $ is a hyperplane containing w and $B\subseteq \Sigma $ is a positive dimensional subvariety whose linear span $\left \langle B \right \rangle $ also contains w but is different from a line, then requiring $p^{\prime }_1,\ldots ,p^{\prime }_\ell $ ( $m_\ell \ge 3$ ) to vanish on B imposes at least

$$ \begin{align*}\left(\sum_{i=1}^\ell m_i\right)(n-2-b)+\ell-\dim \mathrm{Gr}(b,n-2)- \sum\nolimits' m_i \ge 3n-5-\sum_{i=1}^s \frac{1}{2}d_i(d_i+1)\ge 2n\end{align*} $$

conditions on the coefficients of $p^{\prime }_1,\ldots ,p^{\prime }_\ell $ (where b is the codimension of $\left \langle B \right \rangle $ in $\Sigma $ and $\sum ^{\prime }$ sums over all $1\le i\le \ell $ such that $p_i$ comes from some $Q_j$ with $1\le j\le s$ ; clearly $\sum ^{\prime } m_i\le \sum _{i=1}^s \frac {1}{2}d_i(d_i+1)$ ). On the other hand, requiring all $p_i$ to contain a line passing through w imposes at least $m_i+1$ (resp. $1$ ) on coefficients of $p_i$ if $p_i$ comes from the equation of $Q_j$ for some $j>s$ (resp. $j\le s$ ), and we get a total of at least

$$ \begin{align*} \sum_{i=s+1}^r \frac{1}{2}a_i (a_i+1) - (r-s) & \ge \sum_{i=s+1}^r \frac{1}{2}a_i (a_i+1) - r \\ & \ge 2\sum_{i=s+1}^r a_i -r \ge 2 \left(n+r-2-\sum_{i=1}^s d_i\right) - r \ge n+1 \end{align*} $$

conditions (to see the second inequality, note that $\sum _{i=s+1}^r a_i \ge n+r-2-\sum _{i=1}^s d_i\ge 3r$ and the quadratic form $\sum a_i (a_i+1)$ is minimized when the differences between the $a_i$ ’s are at most $1$ , hence we may assume $a_i\ge 3$ for all $s+1\le i\le r$ ). The remaining cases can be treated in a similar and easier fashion as in Lemma 4.5.

Proof Since X is CY, $d=\sum _{i=1}^r d_i = n+r+1$ , hence by the definition of $\beta _i$ , the product $\prod _{i\ge 1} \beta _i = \beta _1\beta _2\cdots \beta _{n+r-2}$ is obtained by removing three smallest factors (and then adding r factors of $1$ ) from the following expression

$$ \begin{align*}\frac{2}{1}\cdot \frac{3}{2}\cdot \ldots \cdot \frac{d_1}{d_1-1}\cdot \frac{2}{1}\cdot \ldots \cdot \frac{d_2}{d_2-1}\cdot \ldots \cdot \frac{2}{1}\cdot \ldots \cdot \frac{d_r}{d_r-1},\end{align*} $$

which equals $\deg X = d_1 d_2\cdots d_r$ . As $d_r\ge 12$ , we get

(4.9) $$ \begin{align} \prod_{i\ge1} \beta_i \ge \frac{d_r-3}{d_r-2}\cdot \frac{d_r-2}{d_r-1}\cdot \frac{d_r-1}{d_r} \cdot \deg X \ge \frac{3}{4} \deg X. \end{align} $$

By rearranging the sequence $q_1,\ldots ,q_d$ we may assume that the first quadratic term $q_{r+1}$ comes from the degree $d_r$ equation and hence $\beta _{r+1}=\frac {3}{2}>1$ . As X is r-strongly P-regular under our assumptions, we have

$$ \begin{align*}\mathrm{lct}(X;|H|_{\mathbb{Q}})\ge \min\left\{1,\frac{2}{\beta_{r+1}\deg X}\prod_{i\ge1} \beta_i\right\} = \min\left\{1,\frac{4}{3\deg X}\prod_{i\ge1} \beta_i\right\}\ge 1\end{align*} $$

by Lemma 4.4 and (4.9).

Proof By Lemma 4.6, X is P-regular. By assumption, $n+2-e\ge 12$ and X is Calabi–Yau. Hence the result follows directly from Corollary 4.7.

Proof Consider the following variant of the m-strongly P-regular condition on complete intersections (we use the same notation as in Definition 4.3):

1. (I _m ) For any linear form $h=h(z_*)$ not in the span of $q_1,\ldots ,q_r$ , the algebraic set $Z=(h=q_1=\cdots =q_m=0)\cap X$ is equidimensional and for any irreducible component $Z_i$ of Z, we have

   $$ \begin{align*}\frac{\mathrm{mult}_x}{\deg}(Z)=\frac{\mathrm{mult}_x}{\deg}(Z_i).\end{align*} $$

When $m=2$ and X is a hypersurface, this condition is implied by the conditions (R1.2)–(R1.3) of [Reference PukhlikovPuk05, Section §2.1], thus by Lemma 4.5 and [Reference PukhlikovPuk05, Proposition 4], a general hypersurface of degree $d\ge n+1$ is P-regular and satisfies the condition $(I_2)$ when $n\ge 5$ (although [Reference PukhlikovPuk05, Proposition 4] is only stated for degree $n+1$ hypersurfaces, the same proof works in general). It is also not hard to see that the proof of Lemma 4.4 applies without change to P-regular complete intersections that satisfy $(I_{m-1})$ . In particular, taking $m=3$ in Lemma 4.4, we have $\mathrm {lct}(X;|H|_{\mathbb {Q}})\ge {3(n-1)}{/2d} \ge {(n+1)}{/d}$ as desired.

Proof When $n=4$ , the result follows from [Reference Ahmadinezhad, Cheltsov and SchichoACS18, Theorem 1.2 and Lemma 3.2]. If $n\ge 7$ , this is a special case of Corollary 4.9.

Proof Denote the given singularity by $(0\in X)$ . The weighted blow-up of $\mathbb {A}^{n+1}$ with weights $(n+1,\ldots ,n+1,n)$ induces a plt blow-up $\pi :(E\subseteq Y)\rightarrow (0\in X)$ with Kollár component E. The corresponding log Fano pair $(E,\mathrm {Diff}_E(0))$ is isomorphic to (c.f. [Reference KudryavtsevKud01, Example 2.4])

$$ \begin{align*}\left( (f(x_1,\ldots,x_n)+x_{n+1}=0)\subseteq \mathbb{P}(1,\ldots,1,n), \frac{n}{n+1}(x_{n+1}=0)\right) = \left( \mathbb{P}^{n-1},\frac{n}{n+1}S \right),\end{align*} $$

where $S\subseteq \mathbb {P}^{n-1}$ is the hypersurface defined by $f(x_1,\ldots ,x_n)$ . By Theorem 2.7, our statement is equivalent to

(4.10) $$ \begin{align} \alpha\left( \mathbb{P}^{n-1},\frac{n}{n+1}S \right)\ge 1. \end{align} $$

To see this, let $D\subseteq \mathbb {P}^{n-1}$ be an effective divisor such that $D\sim _{\mathbb {Q}} H \sim _{\mathbb {Q}} -\frac {n+1}{n}(K_{\mathbb {P}^{n-1}}+{n}{/(n+1)}S)$ , we need to show that $(\mathbb {P}^{n-1},{n}{/(n+1)}(S+D))$ is lc. We may assume that D is irreducible since being lc is preserved under convex linear combinations. If D is supported on S then $D=\frac {1}{n}S$ and the result is clear. If D is not supported on S then by Corollary 4.10, the pair $(S,{(n-1)}{/n}D|_{S})$ is lc, hence by inversion of adjunction, $(\mathbb {P}^{n-1},S+{(n-1)}{/n}D)$ is lc. Since $\mathrm {lct}(\mathbb {P}^{n-1};|H|_{\mathbb {Q}})=1$ , the pair $(\mathbb {P}^{n-1},D)$ is also lc. By taking convex linear combinations, we see that $(\mathbb {P}^{n-1},{n}{/(n+1)}(S+D))$ is lc as well since

$$ \begin{align*}\frac{n}{n+1}(S+D)=\frac{n}{n+1}\left( S+\frac{n-1}{n}D \right) + \frac{1}{n+1} D. \end{align*} $$

This finishes the proof.

Proof Clearly $\mathcal {S}_X$ is a linear subspace of $\mathcal {S}$ . Let $\phi :\mathbb {P}^n\dashrightarrow \mathbb {P}^m$ be a general linear projection whose restriction to X has dense image, and let $\mathcal {N}=\phi ^{-1} |\mathbb {P}^m,\mathcal {O}_{\mathbb {P}^m}(d)|\subseteq \mathcal {S}$ , then by construction none of the hypersurfaces in $\mathcal {N}$ contain X, thus $\mathcal {N}\cap \mathcal {S}_X=\emptyset $ and we have $\mathrm {codim}_{\mathcal {S}} \mathcal {S}_X\ge \dim \mathcal {N}+1= \binom {m+d}{d}$ .

Proof Let $\mathcal {T}_j \subseteq \mathcal {S}$ be the set of $(g_i)_{1\le i \le q}$ such that Z has codimension exactly j. By induction on c it suffices to show that

(4.11) $$ \begin{align} \mathrm{codim}_{\mathcal{S}}\ \mathcal{T}_c\ge \sum_{i=1}^{q-c} \binom{m-c+d_i}{d_i}. \end{align} $$

Let $\sigma =(i_1,\ldots ,i_c)\subseteq \{1,2,\ldots ,q\}$ be an index set and let $\mathcal {T}_{c,\sigma }\subseteq \mathcal {S}$ be the set of $(g_i)$ such that $Z_\sigma =\cap _{i\in \sigma } Z(f_i+g_i)$ has codimension c in X, then clearly

(4.12) $$ \begin{align} \mathcal{T}_c\subseteq \bigcup_{|\sigma|=c} \mathcal{T}_{c,\sigma}. \end{align} $$

Fix an index set $\sigma =(i_1,\ldots ,i_c)$ and a subsequence $(g_j)_{j\in \sigma }$ such that $Z_\sigma =\cap _{i\in \sigma } Z(f_i+g_i)$ has codimension c in X, then $(g_1,\ldots ,g_q)\in \mathcal {T}_c$ if and only if $f_i+g_i$ vanishes on some irreducible component of $Z_\sigma $ for each $i\not \in \sigma $ . Then set of such $(g_i)_{i\not \in \sigma }$ is either empty or (after translating by an element in this set) isomorphic to the set of $g_i$ that vanishes on some irreducible component of $Z_\sigma $ . As $Z_\sigma $ is a complete intersection of dimension $m-c$ , this imposes at least $\binom {m-c+d_i}{d_i}$ conditions on $g_i$ by Lemma 4.13 (we identify $k[x_1,\ldots ,x_n]_{\le d}$ with $H^0(\mathbb {P}^n,\mathcal {O}_{\mathbb {P}^n}(d))$ ). Since this holds for every $(g_j)_{j\in \sigma }$ , we get