A gap theorem for minimal log discrepancies of non-canonical singularities in dimension three

TL;DR

This paper proves a uniform gap in minimal log discrepancies for non-canonical singularities in three-dimensional varieties, with implications for the boundedness of certain Calabi-Yau threefolds.

Contribution

It establishes a positive lower bound gap for minimal log discrepancies in threefolds with non-canonical singularities, advancing the understanding of their structure.

Findings

Existence of a positive gap elta in minimal log discrepancies.

Boundedness of non-canonical klt Calabi-Yau 3-folds modulo flops.

Upper bounds on global indices of klt Calabi-Yau 3-folds.

Abstract

We show that there exists a positive real number $\delta>0$ such that for any normal quasi-projective $\mathbb{Q}$-Gorenstein $3$-fold $X$, if $X$ has worse than canonical singularities, that is, the minimal log discrepancy of $X$ is less than $1$, then the minimal log discrepancy of $X$ is not greater than $1-\delta$. As applications, we show that the set of all non-canonical klt Calabi-Yau $3$-folds are bounded modulo flops, and the global indices of all klt Calabi-Yau $3$-folds are bounded from above.

Equations

$$\{{\mathrm{mld}}_{{\eta }_Z}(X,\Delta )\mid (X,\Delta )\text{ is lc},\dim X\le d,Z\subset X,\mathrm{coeff}(\Delta )\in I\}$$

$$K_Y=f^{\ast }(K_X+B)+\sum a_i{F}_i,$$

$$a(F_i;X,B)={\text{mult}}_{F_i}(K_Y-f^{\ast }(K_X+B))+1=a_i+1.$$

$${\mathrm{mld}}_Z(X,B)=\underset{E}{\inf }\{a(E;X,B)\mid {\text{center}}_X(E)\subset Z\},$$

$${\mathrm{mld}}_{{\eta }_Z}(X,B)=\underset{E}{\inf }\{a(E;X,B)\mid {\text{center}}_X(E)=Z\}.$$

$${\mathbf{\mu }}_r\ni \xi :(x_0,\dots ,x_n)\mapsto ({\xi }^{a_0}{x}_0,\dots ,{\xi }^{a_n}{x}_n)$$

$${\mathbb{A}}^{n+1}=\text{Spec}\mathbb{C}[\overline{M}\cap {\sigma }^{\vee }]$$

$${\mathbb{A}}^{n+1}/{\mathbf{\mu }}_r=\text{Spec}\mathbb{C}[M\cap {\sigma }^{\vee }]=T_N(\Delta ),$$

$${\mathbf{\mu }}_r\ni \xi :(x_0,\dots ,x_n)\mapsto ({\xi }^{a_0}{x}_0,\dots ,{\xi }^{a_n}{x}_n).$$

$${\mathbf{\mu }}_r\ni \xi :f\mapsto {\xi }^{e}f$$

$$s={\text{Res}}_Y\frac{d{x}_0\wedge \cdots \wedge d{x}_n}{f}=\frac{d{x}_1\wedge \cdots \wedge d{x}_n}{\partial f/\partial x_0}\in {\omega }_Y$$

$${\mathbf{\mu }}_r\ni \xi :s\mapsto {\xi }^{a_0+\cdots +a_n-e}s.$$

$$\alpha (f):=\min \{\alpha ({\mathbf{x}}^{\mathbf{m}})\mid {\mathbf{x}}^{\mathbf{m}}\in f\}.$$

$$1-\delta <\beta (x_0\cdots x_n)-\beta (f)<1;$$

$$\alpha (x_0\cdots x_n)-\alpha (f)>1.$$

$${\alpha }^{\prime }(x_0\cdots x_n)-{\alpha }^{\prime }(f)>1.$$

$$1-\delta <\beta (x_0\cdots x_n)-\beta (f)<1.$$

$$K_{Z_{\beta }}+X_{\beta }={\varphi }_{\beta }^{\ast }(K_Z+X)+(\beta (x_0\cdots x_n)-\beta (f)-1)E_{\beta },$$

$$K_{Z_{\beta }}+X_{\beta }+t{E}_{\beta }={\varphi }_{\beta }^{\ast }(K_Z+X),$$

$$K_{X_{\beta }}+B_{\beta }=(K_{Z_{\beta }}+X_{\beta }+t{E}_{\beta }){|}_{X_{\beta }}={\varphi }_{\beta }{|}_{X_{\beta }}^{\ast }(K_X),$$

$$1-\delta <\beta (x_0\cdots x_n)-\beta (f)<1.$$

$${\alpha }^{\prime }(x_0\cdots x_n)-{\alpha }^{\prime }(f)=m(\alpha (x_0\cdots x_n)-\alpha (f))>2(1-\delta )\ge 1.$$

$$\sum _{i=1}^4\overline{a_{i}k}=\overline{ek}+k+r.$$

$$\sum _{i=1}^4\overline{a_i{k}_0}=\overline{e{k}_0}+k_0;$$

$$\sum _{i=1}^4\overline{a_{i}k}\ge \overline{ek}+r.$$

$${\text{mld}}_0(Z)=\min \{\alpha (x_0\cdots x_4)\mid \alpha \in N\cap \text{relin}(\sigma )\},$$

$$\beta =\frac{1}{r}(\overline{a_1{k}_0},\dots ,\overline{a_4{k}_0},\overline{-e{k}_0})\in N\cap \text{relin}(\sigma ).$$

$$\beta (x_0\cdots x_4)=\frac{1}{r}(\sum _{i=1}^4\overline{a_i{k}_0}+r-\overline{e{k}_0})=1+\frac{k_0}{r}<2.$$

$$\alpha (x_0\cdots x_4)\ge \frac{1}{r}(\sum _{i=1}^4\overline{a_{i}j}+r-\overline{ej})\ge 2.$$

$$\sum _{i=1}^3(\overline{a_i{k}^{\prime }}{)}_q=\frac{1}{p}\sum _{i=1}^3\overline{a_{i}p{k}^{\prime }}=\frac{1}{p}(\sum _{i=1}^4\overline{a_{i}p{k}^{\prime }}-\overline{ep{k}^{\prime }})\ge q;$$

Full text

1. 1 Introduction

2. 2 Preliminaries

3. 2.1 Residues of integers

4. 2.2 Pairs, singularities, and minimal log discrepancies

5. 2.3 Log Calabi–Yau pairs

6. 2.4 Bounded pairs

7. 2.5 Extremely non-canonical singularities

8. 2.6 Cyclic quotient singularities and hyperquotient singularities

9. 2.7 ACC for minimal log discrepancies of cyclic quotient singularities

10. 2.8 The terminal lemma and the non-canonical lemma

11. 3 Reduction to extremely non-canonical singularities

12. 4 The $1$-gap theorem for $3$-dimensional extremely non-canonical singularities: the hyperquotient case

13. 4.1 Settings and rules

14. 4.2 Reduction to the terminal lemma

15. 4.3 The $cA$ case

16. 4.4 The odd case

17. 4.5 The $cD$-$E$ case

18. 5 The $1$-gap theorem for $3$-dimensional non-canonical singularities: the general case

19. 6 Boundedness of global indices of klt Calabi–Yau $3$-folds

1. Introduction

Throughout this paper, we work over the complex number field ${\mathbb{C}}$.

Canonical and terminal singularities, introduced by Reid, appear naturally in the minimal model program and play important roles in the birational classification of higher dimensional algebraic varieties. Such singularities are well-understood in dimension $3$, while the property of non-canonical singularities is still mysterious. In this paper, we investigate the difference between canonical and non-canonical singularities via minimal log discrepancies.

The minimal log discrepancy (mld) of a normal quasi-projective ${\mathbb{Q}}$-Gorenstein variety $X$, introduced by Shokurov, is defined to be the infimum of log discrepancies of all prime divisors on all birational models of $X$. It is an important invariant for singularities in the minimal model program, and is known to be related to the termination of flips and other topics of interest, see [46, 9]. Here we recall the following deep conjecture regarding the behavior of minimal log discrepancies proposed by Shokurov.

Conjecture 1.1** (ACC for minimal log discrepancies, cf. [42, Problem 5], [45, Conjecture 4.2]).**

Fix a positive integer $d$ and a DCC set $I\subset[0,1]$. Then the set

[TABLE]

satisfies the ACC.

Here ACC stands for the ascending chain condition whilst DCC stands for the descending chain condition.

Conjecture 1.1 is proved in dimension 2 by Alexeev [1] and Shokurov [43], and for toric pairs by Borisov [11] and Ambro [4]. Although some partial results are known [24, 39, 38, 25, 32, 34, 22], Conjecture 1.1 still remains open in its full generality in dimensions $3$ and higher.

Recall that for a normal quasi-projective ${\mathbb{Q}}$-Gorenstein variety $X$, ${\text{\rm mld}}(X)\geq 1$ if and only if $X$ has canonical singularities. Hence in this paper, we are only interested in the following special case of Conjecture 1.1.

Conjecture 1.2** (1-gap conjecture for minimal log discrepancies).**

Fix a positive integer $d$. Then $1$ is not an accumulation point from below for the set of minimal log discrepancies of all normal quasi-projective ${\mathbb{Q}}$-Gorenstein varieties of dimension $d$.

Conjecture 1.2 asserts that there is a gap for minimal log discrepancies between canonical and non-canonical singularities, and it already has interesting applications related to the boundedness of Calabi–Yau varieties (see [13]). Note that in Conjecture 1.2, we are interested in the global minimal log discrepancies rather than the local ones at closed points. Although it is much weaker than Conjecture 1.1, Conjecture 1.2 was still open even in dimension $3$.

As the main result of this paper, we give an affirmative answer to Conjecture 1.2 in dimension $3$.

Theorem 1.3**.**

There exists a positive real number $\delta>0$ with the following property: if $X$ is a normal quasi-projective ${\mathbb{Q}}$-Gorenstein $3$-fold with ${\text{\rm mld}}(X)<1$, then ${\text{\rm mld}}(X)\leq 1-\delta$.

Remark 1.4*.*

We explain the strategy of proving Theorem 1.3 briefly. The goal is to show that there is no $3$-fold $X$ with $1-\delta<{\text{\rm mld}}(X)<1$ for a sufficiently small $\delta>0$. The first step is to reduce to the case that all but one exceptional divisors over $X$ have log discrepancies greater than $1$, in which case $X$ is called extremely non-canonical (see Section 3). Also it is easy to reduce to the case that $X$ is an isolated singularity which is a hyperquotient of an isolated cDV singularity in ${\mathbb{A}}^{4}$. To deal with this case, we replay the game for the classification of $3$-dimensional terminal singularities by Mori [35] as explained by Reid [41], and show that such a singularity does not exist. Of course in our situation rules are changed which makes the game more complicated, but it will be in control after some essential modifications (see Section 4 for more explanations).

Remark 1.5*.*

In many applications, it suffices to know the existence of such a positive number $\delta$. But by our method, it is possible to determine the number $\delta$ in Theorem 1.3 effectively. In fact, by the proof of Theorem 1.3, we can take $\delta=\delta_{0}$, where $\delta_{0}$ is a positive constant given in Lemma 2.12 which is related to the gap of minimal log discrepancies of isolated cyclic quotient singularities in dimensions $3$ and $5$. After the first version of this paper appeared on arXiv, the author was informed by Liu and Xiao [33] that they computed that $\delta_{0}=\frac{1}{19}$ in Lemma 2.12, which then gives an optimal value $\delta=\frac{1}{13}$ for Theorem 1.3 after some extra effort.

Next we explain the applications of Theorem 1.3 to boundedness problem for singular Calabi–Yau $3$-folds.

A normal projective variety $X$ is a Calabi–Yau variety if $K_{X}\equiv 0$. According to the minimal model program, Calabi–Yau varieties form a fundamental class in birational geometry as building blocks of algebraic varieties. Calabi–Yau varieties are also interesting objects in differential geometry and mathematical physics. Hence, it is interesting to ask whether such kind of varieties satisfies any finiteness properties, namely, whether some invariants of them are in a finite set, or they can be parametrized by finitely many families. For recent developments on this direction in birational geometry, see [2, 3, 14, 13, 7]. We recall that Alexeev [2, Theorem 6.9] showed that all Calabi–Yau varieties in dimension $2$ with worse than du Val singularities form a bounded family. Motivated by Alexeev’s result, [13] considers rationally connected klt Calabi–Yau $3$-folds and showed their boundedness modulo flops assuming Theorem 1.3.

As an application of Theorem 1.3, we show that the set of all non-canonical klt Calabi–Yau $3$-folds are bounded modulo flops, which is a weak version of the analogue of Alexeev’s result in dimension $3$.

Theorem 1.6** (=Theorem 6.1).**

The set of non-canonical klt Calabi–Yau $3$-folds forms a bounded family modulo flops.

Note that [13] only considers Theorem 1.6 for rationally connected klt Calabi–Yau $3$-folds, but we are able to remove the rational connectedness condition in this paper.

As a consequence, the global indices of all klt Calabi–Yau $3$-folds are bounded from above.

Corollary 1.7** (=Corollary 6.2).**

There exists a positive integer $m$ such that for any klt Calabi–Yau $3$-fold $X$, $mK_{X}\sim 0$.

Here we remark that Theorem 1.7 was known for canonical Calabi–Yau $3$-folds by Kawamata [26] and Morrison [36]. So we only need to deal with the case of non-canonical klt Calabi–Yau $3$-folds, which follows from Theorem 1.6. Also we recall that Blache and Zhang [10, 47, 48] studied klt Calabi–Yau surfaces (also known as log Enriques surfaces) and showed that for any such surface $S$, $mK_{S}\sim 0$ for some $m\leq 21$. So Corollary 1.7 is a generalization of their results in dimension $3$. Of course it is very interesting to ask for an effective bound of the global indices, but our method can not give an effective bound.

It is worthwhile to mention that Jingjun Han brought our attention to another application of Theorem 1.3, which is the termination of log twists (introduced by Birkar and Shokurov) in dimension $3$. See [9, Proposition 3.4] for details.

This paper is organized as follows. In Section 2, we recall basic definitions and make preparation for the proof. In Section 3, we reduce Conjecture 1.2 to the case of extremely non-canonical singularities. In Section 4, we prove Theorem 1.3 for $3$-dimensional isolated hyperquotient extremely non-canonical singularities, using the method from classification of $3$-dimensional terminal singularities. In Section 5, we prove Theorem 1.3 for the general case. In Section 6, we prove Theorem 1.6 and Corollary 1.7.

2. Preliminaries

We adopt the standard notation and definitions in [28] and [30], and will freely use them. We work over ${\mathbb{C}}$.

2.1. Residues of integers

For a positive integer $r$, $(\overline{n})_{r}$ denotes the smallest non-negative residue modulo $r$, i.e., the number $m$ such that $0\leq m<r$ and $n\equiv m\bmod r$. Usually $r$ is clear in the context, so we simply write $\overline{n}$ instead of $(\overline{n})_{r}$. We will often use the following easy fact: $\overline{n}+\overline{-n}=\begin{cases}r&\text{ if }n\not\equiv 0\bmod r;\\ 0&\text{ if }n\equiv 0\bmod r.\end{cases}$

2.2. Pairs, singularities, and minimal log discrepancies

A log pair $(X,B)$ consists of a normal quasi-projective variety $X$ and an effective ${\mathbb{R}}$-divisor $B$ on $X$ such that $K_{X}+B$ is ${\mathbb{R}}$-Cartier.

Let $f\colon Y\rightarrow X$ be a log resolution of the log pair $(X,B)$. Write

[TABLE]

where $\{F_{i}\}$ are distinct prime divisors. For a non-negative real number $\epsilon$, the log pair $(X,B)$ is called

• (a)

kawamata log terminal (klt for short) if $a_{i}>-1$ for all $i$;

• (b)

$\epsilon$-log canonical ($\epsilon$-lc for short) if $a_{i}\geq-1+\epsilon$ for all $i$;

• (c)

terminal if $a_{i}>0$ for all $f$-exceptional divisors $F_{i}$ and all $f$;

• (d)

canonical if $a_{i}\geq 0$ for all $f$-exceptional divisors $F_{i}$ and all $f$;

• (e)

purely log terminal (plt for short) if $a_{i}\geq 0$ for all $f$-exceptional divisors $F_{i}$ and all $f$.

Usually we write $X$ instead of $(X,0)$ in the case when $B=0$. In this case, when we talk about singularities as above, we automatically assume that $X$ is ${\mathbb{Q}}$-Gorenstein, that is, $K_{X}$ is ${\mathbb{Q}}$-Cartier. Note that we usually use lc instead of [math]-lc. Also note that $\epsilon$-lc singularities only make sense if $\epsilon\in[0,1]$.

The log discrepancy of the divisor $F_{i}$ is defined to be

[TABLE]

It does not depend on the choice of the log resolution $f$. Here we identify divisors on different birational models by its divisorial valuation. When $B=0$, we simply write $a(F_{i};X)$ instead of $a(F_{i};X,B)$.

Let $(X,B)$ be a log pair and $Z\subset X$ an irreducible closed subset with $\eta_{Z}$ the generic point of $Z$. The minimal log discrepancy of $(X,B)$ over $Z$ is defined as

[TABLE]

and the minimal log discrepancy of $(X,B)$ at $\eta_{Z}$ is defined as

[TABLE]

Moreover, we write $\mathrm{ld}(X,B)$ instead of $\mathrm{mld}_{X}(X,B)$, and call it the total log discrepancy of $(X,B)$. We define the the minimal log discrepancy of $(X,B)$ to be $\mathrm{mld}(X,B)=\inf_{Z}\mathrm{mld}_{Z}(X,B)$ where $Z$ runs over all subvarieties of codimension $2$. Note that the difference between total log discrepancy and minimal log discrepancy is just whether codimension $1$ points (or prime divisors) on $X$ are considered or not. If $B=0$, we simply write $\mathrm{ld}(X)$ and $\mathrm{mld}(X)$. Note that if $\mathrm{mld}(X)\leq 1$, then $\mathrm{ld}(X)=\mathrm{mld}(X)$.

Note that $\mathrm{ld}(X,B)\geq\epsilon$ (resp. $>0$) if and only if $(X,B)$ is $\epsilon$-lc (resp. klt), and $\mathrm{mld}(X,B)\geq 1$ (resp. $>0$) if and only if $(X,B)$ is canonical (resp. terminal). So $X$ is non-canonical if and only if $\mathrm{mld}(X)<1$.

2.3. Log Calabi–Yau pairs

A normal projective variety $X$ is a Calabi–Yau variety if $K_{X}\equiv 0$. If $K_{X}\sim_{\mathbb{Q}}0$, then the global index of $X$ is the minimal positive integer $m$ such that $mK_{X}\sim 0$.

A log pair $(X,B)$ is called a log Calabi–Yau pair if $X$ is projective and $K_{X}+B\equiv 0$. Recall that if $(X,B)$ is lc, this is equivalent to $K_{X}+B\sim_{\mathbb{R}}0$ by [17].

2.4. Bounded pairs

A collection of projective varieties $\mathcal{D}$ is said to be bounded (resp., bounded in codimension one) if there exists a projective morphism $h\colon\mathcal{Z}\rightarrow S$ between schemes of finite type such that each $X\in\mathcal{D}$ is isomorphic (resp., isomorphic in codimension one) to $\mathcal{Z}_{s}$ for some closed point $s\in S$ where $\mathcal{Z}_{s}=h^{-1}(s)$.

We say that a collection of projective log pairs $\mathcal{D}$ is log bounded (resp., log bounded in codimension one) if there is a quasi-projective scheme $\mathcal{Z}$, a reduced divisor $\mathcal{E}$ on $\mathcal{Z}$, and a projective morphism $h\colon\mathcal{Z}\to S$, where $S$ is of finite type and $\mathcal{E}$ does not contain any fiber, such that for every $(X,B)\in\mathcal{D}$, there is a closed point $s\in S$ and a birational map $f\colon\mathcal{Z}_{s}\dashrightarrow X$ which is isomorphic (resp., isomorphic in codimension one) such that $\mathcal{E}_{s}:=\mathcal{E}|_{\mathcal{Z}_{s}}$ coincides with the support of $f_{*}^{-1}B$.

Moreover, if $\mathcal{D}$ is a set of klt Calabi–Yau varieties (resp., klt log Calabi–Yau pairs), then it is said to be bounded modulo flops (resp., log bounded modulo flops) if it is (log) bounded in codimension one, each fiber $\mathcal{Z}_{s}$ corresponding to $X$ in the definition is normal projective, and $K_{\mathcal{Z}_{s}}$ is ${\mathbb{Q}}$-Cartier (resp., $K_{\mathcal{Z}_{s}}+f_{*}^{-1}B$ is ${\mathbb{R}}$-Cartier).

Note that if $\mathcal{D}$ is a set of klt log Calabi–Yau pairs which is log bounded modulo flops, and $(X,B)\in\mathcal{D}$ with a birational map $f\colon\mathcal{Z}_{s}\dashrightarrow X$ isomorphic in codimension one as in the definition, then $(\mathcal{Z}_{s},f_{*}^{-1}B)$ is also a klt log Calabi–Yau pair by the negativity lemma. Moreover, $(X,B)$ is $\epsilon$-lc if and only if $(\mathcal{Z}_{s},f_{*}^{-1}B)$ is $\epsilon$-lc. A similar statement holds when $\mathcal{D}$ is a set of klt Calabi–Yau varieties.

Here the name “modulo flops” comes from the fact that, if we assume that $X$ and $\mathcal{Z}_{s}$ are both ${\mathbb{Q}}$-factorial, then they are connected by flops by running a $(K_{X}+B+\delta f_{*}H)$-MMP where $H$ is an ample divisor on $\mathcal{Z}_{s}$ and $\delta$ is a sufficiently small positive number ([8, 27]).

2.5. Extremely non-canonical singularities

As we are interested in non-canonical singularities, we introduce the concept of extremely non-canonical singularities, which are the closest to terminal singularities among all non-canonical singularities.

Definition 2.1**.**

Let $X$ be a normal quasi-projective variety. We say that $X$ is extremely non-canonical if $X$ has ${\mathbb{Q}}$-factorial klt singularities and

1. (1)

there exists exactly one prime divisor $E_{0}$ over $X$ such that $a(E_{0};X)<1$; 2. (2)

there is no divisor $E$ over $X$ with $a(E;X)=1$.

Remark 2.2*.*

Suppose that $X$ is extremely non-canonical, then it is easy to see that $a(E_{0};X)={\text{\rm mld}}(X)$, and $X$ has terminal singularities outside the center of $E_{0}$ on $X$.

2.6. Cyclic quotient singularities and hyperquotient singularities

We recall the concept of hyperquotient singularities and the toric method which are useful in the classification of $3$-dimensional terminal singularities. Most of the contents come from [41, Section 4] except for Theorem 2.4.

Let $r$ be a positive integer. Let ${\bm{\mu}}_{r}$ denote the cyclic group of $r$-th roots of unity in ${\mathbb{C}}$. A cyclic quotient singularity is of the form $\mathbb{A}^{n+1}/{\bm{\mu}}_{r}$, where the action of ${\bm{\mu}}_{r}$ is given by

[TABLE]

for certain $a_{0},\ldots,a_{n}\in\mathbb{Z}/r$. Note that we may always assume that the action of ${\bm{\mu}}_{r}$ on $\mathbb{A}^{n+1}$ is small, that is, it contains no reflection ([23, Definition 7.4.6, Theorem 7.4.8]). We say that $\mathbb{A}^{n+1}/{\bm{\mu}}_{r}$ is of type $\frac{1}{r}(a_{0},\ldots,a_{n})$. Recall that this singularity is isolated if and only if $\gcd(a_{i},r)=1$ for every $0\leq i\leq n$ by [15, Remark 1].

The toric geometry interpretation of cyclic quotient singularities is as following ([41, (4.3)]): let $\overline{M}\simeq{\mathbb{Z}}^{n+1}$ be the lattice of monomials on ${\mathbb{A}}^{n+1}$, and $\overline{N}$ its dual. Define $N$ by $N=\overline{N}+{\mathbb{Z}}\cdot\frac{1}{r}(a_{0},\ldots,a_{n})$ and $M\subset\overline{M}$ the dual sublattice. Let $\sigma={\mathbb{R}}_{\geq 0}^{n+1}\subset N_{\mathbb{R}}$ be the positive quadrant and $\sigma^{\vee}\subset M_{\mathbb{R}}$ the dual quadrant. Then in toric geometry,

[TABLE]

and its quotient

[TABLE]

where $\Delta$ is the fan corresponding to $\sigma$.

Now we are interested in the hypersurface singularity $(Q\in Y):(f=0)\subset{\mathbb{A}}^{n+1}$ with an action of ${\bm{\mu}}_{r}$ which is free outside $Q$ and its quotient $(P\in X)=Y/{\bm{\mu}}_{r}$. It is known that the action of ${\bm{\mu}}_{r}$ extends to a small ${\bm{\mu}}_{r}$-action of ${\mathbb{A}}^{n+1}$ ([23, Lemma 8.3.8]). We still assume that the action of ${\bm{\mu}}_{r}$ on ${\mathbb{A}}^{n+1}$ is given by

[TABLE]

As $Y=(f=0)$ is fixed by the action of ${\bm{\mu}}_{r}$, we may write

[TABLE]

for certain $e\in\mathbb{Z}/r$. Such $(P\in X)$ is called a hyperquotient singularity of type $\frac{1}{r}(a_{0},\ldots,a_{n};e)$. Note that the action of ${\bm{\mu}}_{r}$ on the generator

[TABLE]

is given by

[TABLE]

Let $\alpha=(b_{0},\ldots,b_{n})\in N\cap\sigma$ be a vector, that is, $\alpha$ is a weighting such that $\alpha(x_{i})=b_{i}\in{\mathbb{Q}}$ on monomials such that

1. (1)

$\alpha\in N$, that is, $\alpha\equiv\frac{1}{r}(ja_{0},\ldots,ja_{n})\bmod{\mathbb{Z}}^{n+1}$ for some $j=0,1,\ldots,r-1$; 2. (2)

$\alpha\in\sigma$, that is, $b_{i}\geq 0$ for all $i$.

This weighting can be extended to ${\mathbb{C}}[x_{0},\ldots,x_{n}]$ in the following way: for ${\bf{x}}^{\bf m}=x_{0}^{m_{0}}\cdots x_{n}^{m_{n}}$, $\alpha({\bf{x}}^{\bf m})=\sum_{i=0}^{n}m_{i}\alpha(x_{i})=\sum_{i=0}^{n}m_{i}b_{i}$; and for a polynomial $f\in{\mathbb{C}}[x_{0},\ldots,x_{n}]$,

[TABLE]

Here ${\bf{x}}^{\bf m}\in f$ means that the monomial ${\bf{x}}^{\bf m}$ appears in $f$ with non-zero coefficient.

Proposition 2.3**.**

Consider $(Q\in Y):(f=0)\subset{\mathbb{A}}^{n+1}$ with an action of ${\bm{\mu}}_{r}$ which is free outside $Q$ and its quotient $(P\in X)=Y/{\bm{\mu}}_{r}$. Keep the above notation. Let $\alpha\in N\cap\sigma$ be a primitive vector and $\Delta(\alpha)$ be the star-shaped subdivision of $\Delta$ by $\alpha$, then the toric morphism $\phi_{\alpha}:T_{N}(\Delta(\alpha))\to T_{N}(\Delta)=\mathbb{A}^{n+1}/{\bm{\mu}}_{r}$ extracts an exceptional divisor $E_{\alpha}$. Denote $Z_{\alpha}=T_{N}(\Delta(\alpha))$ and $Z=T_{N}(\Delta)$, and let $X_{\alpha}\subset Z_{\alpha}$ be the strict transform of $X$ on $Z_{\alpha}$. Then

1. (1)

$K_{Z_{\alpha}}=\phi_{\alpha}^{*}K_{Z}+(\alpha(x_{0}\cdots x_{n})-1)E_{\alpha}$; 2. (2)

$X_{\alpha}=\phi_{\alpha}^{*}X-\alpha(f)E_{\alpha}$.

Proof.

This is standard, see [41, (4.8)] or [23, Proposition 8.3.11]. ∎

In the classification of $3$-dimensional terminal singularities, Proposition 2.3 is used to provide a necessary condition for a hyperquotient singularity being terminal (see [41, (4.6) Theorem]). As we are considering non-canonical singularities, in the following theorem we provide a necessary condition for an isolated hyperquotient singularity being extremely non-canonical by the toric method, which plays an essential role in the proof in Section 4.

Theorem 2.4**.**

Fix $0<\delta\leq\frac{1}{2}$. Consider $(Q\in Y):(f=0)\subset{\mathbb{A}}^{n+1}$ with an action of ${\bm{\mu}}_{r}$ which is free outside $Q$ and its quotient $(P\in X)=Y/{\bm{\mu}}_{r}$. Assume further that $(P\in X)$ is an isolated extremely non-canonical singularity with ${\text{\rm mld}}(X)>1-\delta$. Keep the above notation. Then

1. (1)

there exists at most one primitive vector $\beta\in N\cap\sigma$ such that

[TABLE] 2. (2)

for any primitive vector $\alpha\in N\cap\sigma$ such that $\alpha\neq\beta$,

[TABLE]

Furthermore, for any vector $\alpha^{\prime}\in N\cap\sigma$ such that $\alpha^{\prime}\neq\beta$,

[TABLE]

Proof.

Assume that there exists a primitive vector $\beta\in N\cap\sigma$ such that $\beta(x_{0}\cdots x_{n})-\beta(f)\leq 1$. To see the first two statements, it suffices to show that such $\beta$ is unique and

[TABLE]

Keep the notation in Proposition 2.3, we have

[TABLE]

which can be rewritten as

[TABLE]

where $t=1+\beta(f)-\beta(x_{0}\cdots x_{n})\geq 0$. Since $X$ has an isolated klt singularity, the pair $(Z,X)$ is plt by inversion of adjunction, which implies that $({Z_{\beta}},X_{\beta}+tE_{\beta})$ is also plt. By the subadjunction formula ([29, 16.6 Proposition, 16.7 Corollary]), there is a boundary $B_{\beta}$ on $X_{\beta}$ such that

[TABLE]

and the coefficients of $B_{\beta}$ are of the form $1-\frac{1}{l}+\frac{kt}{l}$ for some positive integers $l,k$, here $k>0$ since $E_{\beta}$ intersects $X_{\beta}$. By the assumption that $X$ is extremely non-canonical, coefficients of $B_{\beta}$ are positive since there is no exceptional divisor over $X$ with log discrepancy $1$, and in fact $B_{\beta}$ has exactly one component $F_{\beta}$ with coefficient $1-\frac{1}{l}+\frac{kt}{l}>0$. Since ${\text{\rm mld}}(X)>1-\delta$, $1-\frac{1}{l}+\frac{kt}{l}<\delta\leq\frac{1}{2}$. In particular, $l=1$ and $0<t<\delta$. This shows that

[TABLE]

To see the uniqueness of $\beta$, we look at the divisorial valuation $v_{F_{\beta}}$ on ${\mathbb{C}}(X)$, and the following proof is suggested by Jungkai Chen. Since $l=1$, from the subadjunction formula, we get $E_{\beta}|_{X_{\beta}}=B_{\beta}=kF_{\beta}$. Hence $v_{F_{\beta}}({\bf{x}}^{\bf m})=k\beta({\bf{x}}^{\bf m})$ for any monomial ${\bf{x}}^{\bf m}$ (${\bf m}\in M$). By the assumption that $X$ is extremely non-canonical, $v_{F_{\beta}}$ is unique. Hence such $\beta$ is unique by the primitivity.

For the last statement, for any non-primitive vector $\alpha^{\prime}\in N\cap\sigma$, we may write $\alpha^{\prime}=m\alpha$ where $m\geq 2$ is an integer and $\alpha\in N\cap\sigma$ is primitive. Then

[TABLE]

∎

2.7. ACC for minimal log discrepancies of cyclic quotient singularities

Recall that the ACC for minimal log discrepancies is proved for toric varieties [11, 4], in this paper we only need the following special case for cyclic quotient singularities:

Theorem 2.5** ([11]).**

Conjecture 1.1 holds for cyclic quotient singularities. In particular, fix a positive integer $d$, then the set of minimal log discrepancies of $d$-dimensional cyclic quotient singularities $(0\in W)$ at [math] satisfies the ascending chain condition.

As corollaries, $2$ and $1$ are not accumulation points of these sets from below, and we will only use this fact in dimensions $3$ and $5$.

Corollary 2.6**.**

There exists a positive constant $\delta_{3}>0$ such that for any isolated cyclic quotient singularity $(0\in W)$ in dimension $3$, if ${\text{\rm mld}}_{0}(W)<1$, then ${\text{\rm mld}}_{0}(W)\leq 1-\delta_{3}$.

Corollary 2.7**.**

There exists a positive constant $\delta_{5}>0$ such that for any cyclic quotient singularity $(0\in W)$ in dimension $5$, if ${\text{\rm mld}}_{0}(W)<2$, then ${\text{\rm mld}}_{0}(W)\leq 2-\delta_{5}$.

Note that in Corollary 2.6 we are only interested in isolated singularities, but in Corollary 2.7 the singularities are not necessarily isolated.

The following example is suggested by Alexeev:

Example 2.8**.**

Consider $(0\in W)$ to be a $3$-dimensional isolated cyclic quotient singularity of type $\frac{1}{13}(3,4,5)$, then ${\text{\rm mld}}(W)={\text{\rm mld}}_{0}(W)=\frac{12}{13}$. In particular, $\delta_{3}\leq\frac{1}{13}$.

Here for the computation of minimal log discrepancies of toric varieties, we refer to [4] (see also the proof of Lemma 2.12).

Remark 2.9*.*

In fact, it is not difficult to show that Example 2.8 is optimal, that is, we can take $\delta_{3}=\frac{1}{13}$ in Corollary 2.6. This can be done after some tedious but elementary calculation by hand. We will not give the proof nor use this fact in this paper. The value of $\delta_{5}$, on the other hand, seems to be more subtle as the dimension is higher and the singularities are not necessarily isolated.

2.8. The terminal lemma and the non-canonical lemma

In this subsection, we recall the terminal lemma by Morrison and Stevens [37] which plays an important role in the classification of $3$-dimensional terminal singularities. Here we only recall a special version for our application, for the full version we refer to [41, (5.4) Theorem]. Recall that $\overline{n}$ denotes the smallest non-negative residue modulo $r$.

Theorem 2.10** ([41, (5.4) Theorem, (5.6) Corollary]).**

Let $\frac{1}{r}(a_{1},\cdots,a_{4};e,1)$ be a $6$-tuple of rational numbers with denominator $r$ such that $q=\gcd(e,r)=\gcd(a_{4},r)$, and $a_{1},a_{2},a_{3}$ are coprime to $r$. Assume that for $k=1,\ldots,r-1$,

[TABLE]

If $q>1$, then $a_{4}\equiv e\bmod r$, and the remaining 4 elements can be paired together as $a_{1}\equiv 1,a_{2}+a_{3}\equiv 0\bmod r$ (or permutations); if $q=1$, then $\{a_{1},a_{2},a_{3},a_{4},-e,-1\}$ can be split up into $3$ disjoint pairs which add to $0\bmod r$ (for example, $a_{1}+a_{2}\equiv a_{3}+a_{4}\equiv-e-1\equiv 0\bmod r$).

Remark 2.11*.*

Note that in the statement of [41, (5.6) Corollary], the $q=1$ case is missing, but it can be easily derived from [41, (5.4) Theorem].

In order to study extremely non-canonical singularities by the toric method, we change the condition of the above terminal lemma and introduce the following “non-canonical” lemma.

Lemma 2.12**.**

*There exists a positive real number $\delta_{0}\leq\delta_{3}<1$ with the following property. Let $\frac{1}{r}(a_{1},\cdots,a_{4};e)$ be a $5$-tuple of rational numbers with denominator $r$ such that $q=\gcd(e,r)=\gcd(a_{4},r)$, and $a_{1},a_{2},a_{3}$ are coprime to $r$. Assume one of the following holds: *

1. (✩1)

$a_{1}+a_{2}\equiv e\bmod r$; 2. (✩2)

$2a_{4}\equiv e\bmod r$; 3. (✩3)

$2a_{1}\equiv e\bmod r$* and $q\leq 2$.*

Moreover, assume that

1. (1)

there exists a positive integer $k_{0}$ such that $1\leq k_{0}\leq r-1$ and

[TABLE] 2. (2)

for every integer $k$ such that $1\leq k\leq r-1$ and $k\neq k_{0}$,

[TABLE]

Then $\frac{k_{0}}{r}\leq 1-\delta_{0}$. Here $\delta_{3}$ is the constant from Corollary 2.6.

Proof.

We will show that we can take $\delta_{0}=\min\{\delta_{3},\delta_{5}\}>0$. Here $\delta_{3}$ and $\delta_{5}$ are constants from Corollaries 2.6 and 2.7.

Since $a_{1},a_{2},a_{3}$ are coprime to $r$, we know that $\overline{a_{1}k_{0}},\overline{a_{2}k_{0}},\overline{a_{3}k_{0}}$ are not [math]. Since $\gcd(e,r)=\gcd(a_{4},r)$, $\overline{a_{4}k_{0}}=0$ if and only if $\overline{ek_{0}}=0$.

First assume that $\overline{a_{4}k_{0}}\neq 0$ and $\overline{ek_{0}}\neq 0$. Consider $Z={\mathbb{A}}^{5}/{\bm{\mu}}_{r}$ to be a cyclic quotient singularity of type $\frac{1}{r}(a_{1},\cdots,a_{4},-e)$. It suffices to show that ${\text{\rm mld}}_{0}(Z)=1+\frac{k_{0}}{r}<2$. Keep the notation in Subsection 2.6. By the existence of log resolutions in toric category, we can compute the minimal log discrepancy by torus invariant divisors over $Z$. Recall that for the exceptional divisor $E_{\alpha}$ corresponding to a primitive vector $\alpha\in N\cap\sigma$, its log discrepancy is computed by $a(E_{\alpha};Z)=\alpha(x_{0}\cdots x_{4})$ (Proposition 2.3). This means that ([4])

[TABLE]

where $\text{relin}(\sigma)$ is the relative interior of $\sigma$. By the assumption, we can consider

[TABLE]

Assumption (1) gives

[TABLE]

On the other hand, take any $\alpha\in N\cap\text{relin}(\sigma)$ such that $\alpha\neq\beta$, recall that we can write $\alpha=(b_{0},\ldots,b_{4})$ such that $\alpha\equiv\frac{1}{r}(a_{1}j,\ldots,a_{4}j,-ej)\bmod{\mathbb{Z}}^{5}$ for some $j=0,1,\ldots,r-1$ and $b_{i}>0$ for all $i$. If $j=0$, then $\alpha(x_{0}\cdots x_{4})\geq 5$. If $j=k_{0}$, then $\alpha(x_{0}\cdots x_{4})\geq\beta(x_{0}\cdots x_{4})+1\geq 2$. If $1\leq j\leq r-1$ and $j\neq k_{0}$, then since $b_{4}>0$, we know that $b_{4}\geq\frac{1}{r}(r-\overline{je})$, and by assumption (2),

[TABLE]

Hence ${\text{\rm mld}}_{0}(Z)=1+\frac{k_{0}}{r}<2$. By Corollary 2.7, $\frac{k_{0}}{r}\leq 1-\delta_{5}$.

Then assume that $\overline{a_{4}k_{0}}=\overline{ek_{0}}=0$. Denote $q=\gcd(e,r)=\gcd(a_{4},r)$ and $r=pq$. Then $p$ divides $k_{0}$ and we can write $k_{0}=pk^{\prime}_{0}$. Now let $(\overline{n})_{q}$ be the smallest non-negative residue of $n$ modulo $q$, then $p(\overline{n})_{q}=\overline{pn}$. Hence we get new relations for $\frac{1}{q}(a_{1},a_{2},a_{3})$ and $k^{\prime}_{0}$ as the following: for every integer $k^{\prime}$ such that $1\leq k^{\prime}\leq q-1$ and $k^{\prime}\neq k^{\prime}_{0}$,

[TABLE]

on the other hand,

[TABLE]

Now we can consider $Z^{\prime}={\mathbb{A}}^{3}/{\bm{\mu}}_{q}$ to be a cyclic quotient singularity of type $\frac{1}{q}(a_{1},a_{2},a_{3})$. It is isolated since $a_{1},a_{2},a_{3}$ are coprime to $r$. By the same calculation as above, ${\text{\rm mld}}_{0}(Z^{\prime})=\frac{k^{\prime}_{0}}{q}=\frac{k_{0}}{r}<1$. To be more precise, we can consider

[TABLE]

here $N$ is the lattice corresponding to $Z^{\prime}$ by abusing the notation. Then

[TABLE]

On the other hand, take any $\alpha\in N\cap\text{relin}(\sigma)$ such that $\alpha\neq\beta^{\prime}$, recall that we can write $\alpha=(b_{0},b_{1},b_{2})$ such that $\alpha\equiv\frac{1}{r}(a_{1}j,a_{2}j,a_{3}j)\bmod{\mathbb{Z}}^{3}$ for some $j=0,1,\ldots,q-1$ and $b_{i}>0$ for all $i$. We may assume that $b_{i}<1$ for all $i$, otherwise $\alpha(x_{0}x_{1}x_{2})\geq 1$. Hence $\alpha=\frac{1}{q}((\overline{a_{1}j})_{q},(\overline{a_{2}j})_{q},(\overline{a_{3}j})_{q})$ with $1\leq j\leq q-1$ and $j\neq k^{\prime}_{0}$. In this case,

[TABLE]

Hence ${\text{\rm mld}}_{0}(Z^{\prime})=\frac{k^{\prime}_{0}}{q}<1$. By Corollary 2.6, $\frac{k_{0}}{r}\leq 1-\delta_{3}$. ∎

Remark 2.13*.*

In the proof of Lemma 2.12, assumptions (✩1–3) are not used. But we keep these assumptions for two reasons. For one thing, we always get one of (✩1–3) in our applications (see Propositions 4.3 and 4.4). For the other, these assumptions will be helpful when one tries to find an optimal or effective value for $\delta_{0}$ (and $\delta$ in Theorem 1.3). In fact, in a recent preprint by Liu and Xiao [33], they show that $\delta_{0}=\frac{1}{19}$ in Lemma 2.12 by some clever arguments with a help of computer program.

3. Reduction to extremely non-canonical singularities

In this section, we reduce the 1-gap conjecture to the case of extremely non-canonical singularities. During the preparation of this paper, we are informed by Jingjun Han and Jihao Liu that they also got similar result as Theorem 3.1 independently.

Theorem 3.1**.**

Let $X$ be a normal quasi-projective variety with klt singularities such that ${\text{\rm mld}}(X)<1$. Then there exists a projective birational morphism $Y\to X$ such that $Y$ is extremely non-canonical and ${\text{\rm mld}}(X)\leq{\text{\rm mld}}(Y)<1$.

Proof.

Let $X$ be a normal quasi-projective variety with klt singularities such that ${\text{\rm mld}}(X)<1$. Take $\mathfrak{E}$ to be the set of all exceptional prime divisors $E$ over $X$ with $a(E;X)\leq 1$, which is a finite set by [30, Proposition 2.36]. By [8, Corollary 1.4.3], there exists a projective birational morphism $\pi:W\to X$ with $W$ ${\mathbb{Q}}$-factorial such that $\mathfrak{E}$ is the set of exceptional divisors of $\pi$. We may write

[TABLE]

where $\Delta$ is a non-zero effective ${\mathbb{Q}}$-divisor as ${\text{\rm mld}}(X)<1$. Note that $(W,\Delta)$ is canonical by the construction. For a sufficiently small $\epsilon>0$, by [8], we can run a $(W,(1+\epsilon)\Delta)$-MMP over $X$, which terminates and reaches a minimal model over $X$ contracting $\text{Supp}(\Delta)$. Denote $W^{\prime}\to X$ to be the model obtained by the first divisorial contraction in this MMP. We will show that $W^{\prime}$ satisfies the requirement of the theorem.

Denote $E_{1}$ to be the prime divisor on $W$ contracted on $W^{\prime}$, and denote $\Delta^{\prime}$ to be the strict transform of $\Delta$ on $W^{\prime}$. Take a common resolution $p:Z\to W$, $p^{\prime}:Z\to W^{\prime}$. As this MMP is also a $\Delta$-MMP over $X$, we can write

[TABLE]

where $F$ is an effective ${\mathbb{Q}}$-divisor and ${\text{\rm mult}}_{E_{1}}(p_{*}F)>0$. On the other hand, we have

[TABLE]

as $K_{W}+\Delta\equiv_{X}0$. Hence

[TABLE]

Here $E_{1}^{Z}$ is the strict transform of $E_{1}$ on $Z$. Take any exceptional prime divisor $E\neq E_{1}$ over $W^{\prime}$, then $E$ is also exceptional over $W$, and hence

[TABLE]

Note that $W^{\prime}$ is ${\mathbb{Q}}$-factorial, so we conclude that $W^{\prime}$ is extremely non-canonical. The fact that ${\text{\rm mld}}(X)\leq{\text{\rm mld}}(W^{\prime})$ follows easily from ${\text{\rm mld}}(W^{\prime})=\text{ld}(W^{\prime})\geq\text{ld}(W^{\prime},\Delta^{\prime})={\text{\rm mld}}(X).$ ∎

4. The $1$-gap theorem for $3$-dimensional extremely non-canonical singularities: the hyperquotient case

In this section, we treat a special case of Theorem 1.3, where $X$ is an isolated extremely non-canonical singularity whose index $1$ cover is an isolated cDV singularity. This is the most technical part of this paper. In the proof, we mimic the classification of $3$-dimensional terminal singularities following the explanation given by Reid [41, Sections 6 and 7] case by case. Of course our situation is more complicated than the case of terminal singularities, but the strategy of [41, Sections 6 and 7] still works after some modifications. The essential differences in our proof are that we replace the criterion for a hyperquotient singularity to be terminal ([41, (4.6) Theorem]) by our new criterion for a hyperquotient singularity to be extremely non-canonical (Theorem 2.4), which leads to more non-trivial discussions in each case; and in order to apply the terminal lemma as in [41, Section 7], we need to first apply our new “non-canonical” lemma (Lemma 2.12) to exclude certain cases to guarantee the condition of the terminal lemma. We try to write down all the details to make the proof convincible and friendly to readers not familiar with [41].

The following is the main theorem of this section.

Theorem 4.1**.**

There exists a positive real number $\delta>0$ such that there is no $3$-dimensional hyperquotient singularity $(P\in X)=(Q\in Y)/{\bm{\mu}}_{r}$ satisfying the followings:

1. (1)

$(Q\in Y)$* is the canonical index $1$ cover of $(P\in X)$;* 2. (2)

$(Q\in Y)\subset{\mathbb{A}}^{4}$* is an isolated cDV singularity;* 3. (3)

$(P\in X)$* is an isolated extremely non-canonical singularity with ${\text{\rm mld}}(X)>1-\delta$.*

In fact, we can take $\delta=\delta_{0}$, where $\delta_{0}$ is the constant from Lemma 2.12.

Outline of the proof.

To the contrary, assume that such a $3$-dimensional hyperquotient singularity $(P\in X)=(Q\in Y)/{\bm{\mu}}_{r}$ exists. Suppose that $Y=(f=0)\subset{\mathbb{A}}^{4}$ and the hyperquotient is of type $\frac{1}{r}(a,b,c,d;e)$.

In Subsection 4.1, we introduce basic settings and restrictions on $f$ and $\frac{1}{r}(a,b,c,d;e)$, and roughly splits the possible $f$ into $5$ cases: $cA$, odd, $cD_{4}$, $cD_{n}$, $cE$.

In Subsection 4.2, using Lemma 2.12, we check that $\frac{1}{r}(a,b,c,d;e,1)$ satisfies the assumption of the terminal lemma (Theorem 2.10).

By applying the terminal lemma, we can get all possible values for $\frac{1}{r}(a,b,c,d;e)$ in each case. In Subsection 4.3, we exclude the $cA$ case. In Subsection 4.4, we exclude the odd case. In Subsection 4.5, we exclude the $cD_{4}$, $cD_{n}$, $cE$ cases. Then the nonexistence is proved. ∎

4.1. Settings and rules

In this subsection we introduce the settings and rules.

Throughout the remaining part of this section, we take $\delta=\delta_{0}$, where $\delta_{0}$ is the constant from Lemma 2.12. Recall that $\delta\leq\delta_{3}$, where $\delta_{3}$ is the constant from Corollary 2.6.

We assume that such a $3$-dimensional hyperquotient singularity $(P\in X)=(Q\in Y)/{\bm{\mu}}_{r}$ as in Theorem 4.1 exists, and we will exclude all the possibilities to get a contradiction.

As the index of $X$ is $r$ and ${\text{\rm mld}}(X)<1$, it is obvious that ${\text{\rm mld}}(X)\leq 1-\frac{1}{r}$. Since $\delta\leq\delta_{3}\leq\frac{1}{13}$ by Example 2.8, we always have $r>13$.

We will freely and frequently use the notation in Subsection 2.6. Set $(x,y,z,t)=(x_{1},x_{2},x_{3},x_{4})$ to be the local analytic coordinates on ${\mathbb{A}}^{4}$, $Y=(f=0)\subset{\mathbb{A}}^{4}$, and the action of ${\bm{\mu}}_{r}$ is given by

[TABLE]

We also identify $(a,b,c,d)=(a_{1},a_{2},a_{3},a_{4})$, and recall that these weights are viewed as elements in ${\mathbb{Z}}/r$. Note that all monomials in $f$ shall have the same weight $e\bmod r$ as $f(\xi x,\xi y,\xi z,\xi t)=\xi^{e}f(x,y,z,t)$ by the setting.

We will always assume the following rules in the proof, which are similar to that of [41, (6.6)] except that Rule I is changed according to our assumption:

Rule I: (i) There exists at most one primitive vector $\beta\in N\cap\sigma$ such that

[TABLE]

(ii) for any vector $\alpha\in N\cap\sigma$ such that $\alpha\neq\beta$,

[TABLE]

Rule II: (i) If $\gcd(a_{i},r)\neq 1$, then $a_{i}$ divides $e$, that is, $\gcd(a_{i},r)$ divides $\gcd(e,r)$;

(ii) $\gcd(a_{i},a_{j},r)=1$ for all $i\neq j$;

(iii) $a+b+c+d-e=1.$

Rule III: (i) After a ${\bm{\mu}}_{r}$-equivariant analytic change of coordinates, we may assume that $f=q(x_{1},\ldots,x_{k})+f^{\prime}(x_{k+1},\ldots,x_{4})$ with $q$ a nondegenerate quadratic form in $x_{1},\ldots,x_{k}$;

(ii) if the $3$-jet of $f$ is $x^{2}+y^{2}z$ then

[TABLE]

or if the $3$-jet is $x^{2}+y^{3}$ then

[TABLE]

Here Rule I is the conclusion of Theorem 2.4. Rules II and III are exactly the same with that in [41, Page 394]. As explained in [41], Rule II(i)(ii) are consequences of the fact that ${\bm{\mu}}_{r}$ acts freely on $Y$ outside $Q$; Rule II(iii) comes from the following: ${\bm{\mu}}_{r}$ acts on the generator $s\in\omega_{Y}$ by ${\bm{\mu}}_{r}\ni\xi:s\mapsto\xi^{a+b+c+d-e}s$, and the index of $K_{X}$ is $r$, which means that $a+b+c+d-e$ is coprime to $r$, so we may assume that $a+b+c+d-e=1$ by changing the choice of primitive root; Rule III is standard in singularity theory by taking analytic change of coordinates (see [41, Page 394–395]).

In fact, by Rule III, we can divide the possible $f$ into 5 cases by [41, (6.7)] as the following:

Proposition 4.2** ([41, (6.7) Proposition]).**

By making a ${\bm{\mu}}_{r}$-equivariant analytic change of coordinates and possibly permuting the coordinates, $f$ can be only in the following $5$ cases:

$cA$* case: $f=xy+g(z,t)$ with $g\in{\mathfrak{m}}^{2}$;*

odd case: $f=x^{2}+y^{2}+g(z,t)$ with $g\in{\mathfrak{m}}^{3}$ and $a\not\equiv b\bmod r$;

$cD_{4}$* case: $f=x^{2}+g(y,z,t)$ with $g\in{\mathfrak{m}}^{3}$ and $g_{3}$ is a reduced cubic;*

$cD_{n}$* case: $f=x^{2}+y^{2}z+g(z,t)$ with $g\in{\mathfrak{m}}^{4}$;*

$cE$* case: $f=x^{2}+y^{3}+yg(z,t)+h(z,t)$ with $g\in{\mathfrak{m}}^{3}$ and $h\in{\mathfrak{m}}^{4}$.*

Here ${\mathfrak{m}}$ is the maximal ideal of ${\mathbb{C}}[{x,y,z,t}]$ and $g_{3}$ is the cubic part of $g$.

For the proof we refer to that in [41] and we remark that the proof only uses Rule III.

4.2. Reduction to the terminal lemma

In this subsection, we check that $\frac{1}{r}(a,b,c,d;e,1)$ satisfies the assumption of the terminal lemma (Theorem 2.10), similar to [41, (7.2)]. But in our setting the existence of $\beta\in N\cap\sigma$ makes the situation more complicated. Usually $\beta$ and $\beta^{\prime}=(1,\ldots,1)-\beta$ should be considered separately from other vectors. Note that the coprimeness is not treated in [41, (7.2)] but later case by case, while in our situation, we should check the coprimeness in the middle of the proof before dealing with $\beta$. This is because we need to apply Lemma 2.12 to exclude certain cases of $\beta$, where the coprimeness is already needed.

We denote by $\square$ the unit cube of $N_{\mathbb{R}}$ removing 16 vertices, i.e., $\square=[0,1]^{4}\setminus\{0,1\}^{4}\subset{\mathbb{R}}^{4}$. For any $\alpha\in N\cap\square$, we will always use $\alpha^{\prime}$ to denote the vector $\alpha^{\prime}=(1,\ldots,1)-\alpha$. For $k=1,\ldots,r-1$, denote $\alpha_{k}=\frac{1}{r}(\overline{ak},\overline{bk},\overline{ck},\overline{dk})\in N\cap\square$, where $\overline{n}$ is the smallest non-negative residue modulo $r$. Note that $\alpha^{\prime}_{k}=\alpha_{r-k}$ if and only if none of $\overline{ak},\overline{bk},\overline{ck},\overline{dk}$ is [math]. Also note that for any $\alpha\in N\cap\square$, there exists some $k=1,\ldots,r-1$, such that $\alpha\equiv\alpha_{k}\bmod{\mathbb{Z}}^{4}$, and $\alpha=\alpha_{k}$ holds if and only if none of $\alpha(x),\alpha(y),\alpha(z),\alpha(t)$ is $1$.

Proposition 4.3**.**

Suppose that $xy\in f$. Then the followings hold.

1. (1)

For any $\alpha\in N\cap\square$ such that $\alpha\neq\beta,\beta^{\prime}$, one of the followings holds:

1. (i)

$\alpha(f)=\alpha(xy)\leq 1$* and $\alpha(zt)>1$, moreover, if $\alpha(xy)=1$, then one of $\alpha(z),\alpha(t)$ is $1$;* 2. (ii)

$\alpha(f)=\alpha(xy)-1$* and $\alpha(zt)<1$, moreover, if $\alpha(xy)=1$, then one of $\alpha(z),\alpha(t)$ is [math].*

The alternative cases are interchanged by the symmetry $\alpha\mapsto\alpha^{\prime}=(1,\ldots,1)-\alpha$. In particular, for $k=1,\ldots,r-1$, if $\alpha_{k}\neq\beta,\beta^{\prime}$, then these two cases imply

[TABLE]

or

[TABLE]

respectively. 2. (2)

Denote $q=\gcd(e,r)$. Then $q=\gcd(d,r)$, and $a,b,c$ are coprime to $r$ (after possibly interchanging $z$ and $t$). 3. (3)

If $\beta\in N\cap\square$, then there exists an integer $1\leq k_{0}\leq r-1$ such that $\beta=\alpha_{k_{0}}$. Moreover, $\beta(xy)\geq 1$ and $1-\delta<\beta(zt)<1$. In particular, $1-\delta<\frac{k_{0}}{r}<1$ and

[TABLE] 4. (4)

For any $k=1,\ldots,r-1$,

[TABLE]

Proof.

(1) As $xy\in f$, $a+b\equiv e$, and $c+d\equiv 1\bmod r$ by Rule II(iii). Since $a+b\equiv e\bmod r$, it is easy to see that $a$ and $b$ are coprime to $r$ by Rule II(i)(ii).

By $a+b\equiv e\bmod r$, $\alpha(f)\equiv\alpha(xy)\bmod{\mathbb{Z}}$ for all $\alpha\in N\cap\sigma$. Fix any $\alpha\in N\cap\square$. Since $0\leq\alpha(f)\leq\alpha(xy)<2$, either $\alpha(f)=\alpha(xy)$ or $\alpha(f)=\alpha(xy)-1$. Note that $\alpha(xy)<2$ because otherwise $\alpha(x)=\alpha(y)=1$, which contradicts the fact that $\gcd(a,b,r)=1$ by Rule II(ii). By Rule I, if $\alpha(f)=\alpha(xy)$ and $\alpha\neq\beta$, then $\alpha(zt)>1$.

Suppose that $\alpha\neq\beta,\beta^{\prime}$, certainly $\alpha^{\prime}\neq\beta,\beta^{\prime}$. There are two cases: (i) $\alpha(f)=\alpha(xy)$; (ii) $\alpha(f)=\alpha(xy)-1$.

Case (ii): Assume that $\alpha(f)=\alpha(xy)-1$, then $\alpha(xy)\geq 1$.

If $\alpha(xy)=1$, then $\alpha(f)=0$ and there is a monomial in $f$ with weight [math]. None of $\alpha(x),\alpha(y)$ is [math] since $a$ and $b$ are coprime to $r$, so one of $\alpha(z),\alpha(t)$ is [math], and in this case $\alpha(zt)<1$ holds.

If $\alpha(xy)>1$, then $\alpha^{\prime}(xy)<1$, and hence $\alpha^{\prime}(f)=\alpha^{\prime}(xy)$. This implies that $\alpha^{\prime}(zt)>1$ by Rule I and hence $\alpha(zt)<1$. This proves (ii).

Case (i): Assume that $\alpha(f)=\alpha(xy)$, then $\alpha(zt)>1$ by Rule I. Suppose that $\alpha(xy)>1$, then $\alpha^{\prime}(xy)<1$ which implies that $\alpha^{\prime}(f)=\alpha^{\prime}(xy)$ and $\alpha^{\prime}(zt)>1$, which contradicts $\alpha(zt)>1$. Hence $\alpha(xy)\leq 1$. On the other hand, if $\alpha(xy)=1$, then the same argument implies that $\alpha^{\prime}(xy)=1$ and $\alpha^{\prime}(f)=0$. By case (ii), one of $\alpha^{\prime}(z),\alpha^{\prime}(t)$ is [math], which implies that one of $\alpha(z),\alpha(t)$ is $1$. This proves (i).

Therefore, the former part of statement (1) is proved. Note that $\alpha(zt)<1$ if and only if $\alpha^{\prime}(zt)>1$, so the alternative cases are interchanged by the symmetry.

For the latter part, note that $\alpha_{k}(xy)=\frac{1}{r}(\overline{ak}+\overline{bk})\equiv\frac{1}{r}\overline{ek}\bmod{\mathbb{Z}}$ and $\alpha_{k}(zt)=\frac{1}{r}(\overline{ck}+\overline{dk})\equiv\frac{k}{r}\bmod{\mathbb{Z}}$. If $\alpha_{k}$ is in case (i), then $\alpha_{k}(xy)\neq 1$ since $\alpha_{k}(z),\alpha_{k}(t)<1$, therefore $\alpha_{k}(xy)<1$ and $1<\alpha_{k}(zt)<2$, which gives the first equation. If $\alpha_{k}$ is in case (ii), then $1\leq\alpha_{k}(xy)<2$ and $\alpha_{k}(zt)<1$, which gives the second equation.

Before proving (2), we note that if $\beta\in N\cap\square$, then $\beta(f)=\beta(xy)$, because otherwise $\beta(f)=\beta(xy)-1$, and $\beta(zt)<0$ by Rule I, which is absurd. It follows that $1-\delta<\beta(zt)<1$ by Rule I.

(2) Since $a+b\equiv e\bmod r$, it is easy to see that $a$ and $b$ are coprime to $r$ by Rule II(i)(ii). By Rule II(i), $\gcd(c,r)$ and $\gcd(d,r)$ divide $q=\gcd(e,r)$. So by Rule II(ii), it suffices to show that $q$ divides either $c$ or $d$. We may assume that $q>1$, and set $k_{1}=r/q\leq r/2$. If either $\overline{ck_{1}}=0$ or $\overline{dk_{1}}=0$, then $q$ divides either $c$ or $d$, and we are done. So we may assume that $\overline{ck_{1}}\neq 0,\overline{dk_{1}}\neq 0$ and try to get a contradiction. In particular this means that $\alpha_{r-k_{1}}=\alpha^{\prime}_{k_{1}}$. We need to consider 3 cases: $\alpha_{k_{1}}\neq\beta,\beta^{\prime}$; $\alpha_{k_{1}}=\beta$; $\alpha_{k_{1}}=\beta^{\prime}$.

If $\alpha_{k_{1}}\neq\beta,\beta^{\prime}$, then also $\alpha_{r-k_{1}}\neq\beta,\beta^{\prime}$. Hence by $\overline{ek_{1}}=\overline{e(r-k_{1})}=0$, we are in the second case of (1), that is,

[TABLE]

but this is absurd, since the sum of the left hand sides of the equations above should be $2r$ as $\overline{ck_{1}}\neq 0,\overline{dk_{1}}\neq 0$.

If $\alpha_{k_{1}}=\beta$, then $1-\delta<\beta(zt)<1$ implies that

[TABLE]

But this contradicts $k_{1}\leq r/2$.

If $\alpha_{k_{1}}=\beta^{\prime}$, then $\alpha_{r-k_{1}}=\beta$, and $1-\delta<\beta(zt)<1$ implies that

[TABLE]

This means that $k_{1}/r<\delta\leq\frac{1}{6}$ and hence $q>6$. For $j=2,3,5$, we may consider $jk_{1}=jr/q<r$ and consider the weighting $\alpha_{jk_{1}}$. Note that by the construction, $k_{1}<jk_{1}<r-k_{1}$, hence $\alpha_{jk_{1}}\neq\beta,\beta^{\prime}$ for $j=2,3,5$ (same holds for $\alpha_{r-jk_{1}}$). Hence by $\overline{ejk_{1}}=0$, we are in the second case of (1), that is,

[TABLE]

Since the sum of right hand sides of the equations above is $r$, either $\overline{cjk_{1}}=0$ or $\overline{djk_{1}}=0$ for each $j=2,3,5$. After possibly interchanging $z,t$, we may assume that $\overline{djk_{1}}=0$ holds for at least two $j\in\{2,3,5\}$, but this implies that $\overline{dk_{1}}=0$, a contradiction.

(3) Now suppose that $\beta\in N\cap\square$. Recall that $\beta(f)=\beta(xy)$ and $1-\delta<\beta(zt)<1$.

First we show that there exists an integer $1\leq k_{0}\leq r-1$ such that $\beta=\alpha_{k_{0}}$. Note that by definition there exists an integer $1\leq k_{0}\leq r-1$ such that $\beta\equiv\alpha_{k_{0}}\bmod{\mathbb{Z}}$. Since $a,b$ are coprime to $r$ by (2), $\beta(x),\beta(y)$ are not $1$, which means that $\beta=\alpha_{k_{0}}$. Note that in this case

[TABLE]

Then we will show that $\beta(xy)\geq 1$. Suppose that $\beta(xy)<1$, then we know that $\overline{ak_{0}}+\overline{bk_{0}}=\overline{ek_{0}}.$ Hence

[TABLE]

On the other hand, for any $1\leq k\leq r-1$ such that $k\neq k_{0}$, if $\alpha_{k}\neq\beta^{\prime}$, then by (1),

[TABLE]

if $\alpha_{k}=\beta^{\prime}$, then

[TABLE]

Hence $\frac{1}{r}(a,b,c,d;e)$ and $k_{0}$ satisfy the assumption of Lemma 2.12, which implies that $k_{0}/r\leq 1-\delta$, a contradiction. Here the coprimeness follows from (2) and (✩1) of Lemma 2.12 is satisfied. This concludes (3).

(4) For any $k=1,\ldots,r-1$, if $\alpha_{k}\neq\beta^{\prime}$, then the statement follows from (1) and (3). If $\alpha_{k}=\beta^{\prime}=\alpha^{\prime}_{k_{0}}$, then $k=r-k_{0}$; also we know that $\overline{ek_{0}}\neq 0$, because otherwise $\overline{dk_{0}}=0$ by (2), which contradicts $\alpha_{k}=\alpha^{\prime}_{k_{0}}$; therefore,

[TABLE]

This concludes (4). ∎

Proposition 4.4**.**

Suppose that $x^{2}\in f$. Then the followings hold.

1. (1)

For any $\alpha\in N\cap\square$ such that $\alpha\neq\beta,\beta^{\prime}$, one of the followings holds:

1. (i)

$\alpha(f)=2\alpha(x)\leq 1$* and $\alpha(yzt)>1+\alpha(x)$, moreover, if $2\alpha(x)=1$, then $(\alpha(y),\alpha(z),\alpha(t))$ is a permutation of $(\frac{1}{2},\frac{1}{2},1)$;* 2. (ii)

$\alpha(f)=2\alpha(x)-1$* and $\alpha(yzt)<1+\alpha(x)$, moreover, if $2\alpha(x)=1$, then $(\alpha(y),\alpha(z),\alpha(t))$ is a permutation of $(\frac{1}{2},\frac{1}{2},0)$.*

The alternative cases are interchanged by the symmetry $\alpha\mapsto\alpha^{\prime}=(1,\ldots,1)-\alpha$. In particular, for $k=1,\ldots,r-1$, if $\alpha_{k}\neq\beta,\beta^{\prime}$, then these two cases imply

[TABLE]

or

[TABLE]

respectively. 2. (2)

One of the followings holds (after possibly interchanging $y,z,t$):

1. (a)

$a\equiv e\equiv 0\bmod r$* and $b,c,d$ are coprime to $r$;* 2. (b)

$r$* is odd and $a,b,c,d,e$ are coprime to $r$;* 3. (c)

$q=2=\gcd(d,r)=\gcd(e,r)$* and $a,b,c$ are coprime to $r$.* 3. (3)

If $\beta\in N\cap\square$, then there exists an integer $1\leq k_{0}\leq r-1$ such that $\beta\equiv\alpha_{k_{0}}\bmod{\mathbb{Z}}^{4}$. Moreover, $1-\delta<\frac{k_{0}}{r}<1$ and

[TABLE] 4. (4)

For any $k=1,\ldots,r-1$,

[TABLE]

Proof.

(1) As $x^{2}\in f$, $2a\equiv e\bmod r$, and $b+c+d\equiv 1+a\bmod r$ by Rule II(iii). From the former one, $\alpha(f)\equiv 2\alpha(x)\bmod{\mathbb{Z}}$ for all $\alpha\in N\cap\sigma$. Fix any $\alpha\in N\cap\square$. Since $0\leq\alpha(f)\leq 2\alpha(x)\leq 2$, either $\alpha(f)=2\alpha(x)$ or $\alpha(f)=2\alpha(x)-1$. Note that here $\alpha(f)=2\alpha(x)-2$ is impossible since otherwise $\alpha(f)=0$ and $\alpha(x)=1$, but $\alpha(f)=0$ implies that at least one of $\alpha(y),\alpha(z),\alpha(t)$ is 0, and hence $x$ and one of $y,z,t$ have a common factor with $r$, which contradicts Rule II(ii). By Rule I, if $\alpha(f)=2\alpha(x)$ and $\alpha\neq\beta$, then $\alpha(yzt)>1+\alpha(x)$.

Suppose that $\alpha\neq\beta,\beta^{\prime}$, then $\alpha^{\prime}\neq\beta,\beta^{\prime}$. There are two cases: (i) $\alpha(f)=2\alpha(x)$; (ii) $\alpha(f)=2\alpha(x)-1$.

Case (ii): Assume that $\alpha(f)=2\alpha(x)-1$, then $2\alpha(x)\geq 1$.

If $2\alpha(x)=1$, then $\alpha(f)=0$ and hence one of $\alpha(y),\alpha(z),\alpha(t)$ is 0, say $\alpha(t)$. Recall that there exists an integer $1\leq k\leq r-1$ such that $\alpha\equiv\alpha_{k}\bmod{\mathbb{Z}}^{4}$. Then $2\alpha(x)=1$ and $\alpha(t)=0$ implies that $\overline{2ak}=\overline{dk}=0$. Since $\gcd(a,d,r)=1$, this implies that $d$ and $r$ are even and $k=\frac{r}{2}$. Since $\gcd(b,d,r)=\gcd(c,d,r)=1$, $\overline{bk}=\overline{ck}=\frac{1}{2}$. This means that $(\alpha(y),\alpha(z),\alpha(t))=(\frac{1}{2},\frac{1}{2},0)$, and in this case $\alpha(yzt)<1+\alpha(x)$ holds.

If $2\alpha(x)>1$, then $2\alpha^{\prime}(x)<1$ and hence $\alpha^{\prime}(f)=2\alpha^{\prime}(x)$. This implies that $\alpha^{\prime}(yzt)>1+\alpha^{\prime}(x)$ by Rule I and hence $\alpha(yzt)<1+\alpha(x)$. This proves (ii).

Case (i): Assume that $\alpha(f)=2\alpha(x)$, then $\alpha(yzt)>1+\alpha(x)$ by Rule I. Suppose that $2\alpha(x)>1$, then $2\alpha^{\prime}(x)<1$ which implies that $\alpha^{\prime}(f)=2\alpha^{\prime}(x)$, and $\alpha^{\prime}(yzt)>1+\alpha^{\prime}(x)$ by Rule I, but this contradicts $\alpha(yzt)>1+\alpha(x)$. Hence $2\alpha(x)\leq 1$. On the other hand, if $2\alpha(x)=1$, then the same argument implies that $\alpha^{\prime}(f)=0$ and $2\alpha^{\prime}(x)=1$. By case (ii), $(\alpha^{\prime}(y),\alpha^{\prime}(z),\alpha^{\prime}(t))=(\frac{1}{2},\frac{1}{2},0)$ after permutation, and this proves (i).

Hence the former part of statement (1) is proved. Note that $\alpha(yzt)>1+\alpha(x)$ if and only if $\alpha^{\prime}(yzt)<1+\alpha^{\prime}(x)$, so the alternative cases are interchanged by the symmetry.

The latter part follows easily by the fact that $2\alpha_{k}(x)=\frac{1}{r}(2\overline{ak})\equiv\frac{1}{r}\overline{ek}\bmod{\mathbb{Z}}$ and $\alpha_{k}(yzt)=\frac{1}{r}(\overline{bk}+\overline{ck}+\overline{dk})\equiv\frac{1}{r}(k+\overline{ak})\bmod{\mathbb{Z}}$. To be more precise, if $\alpha_{k}$ is in case (i), we get that $2\alpha_{k}(x)=\frac{\overline{ek}}{r}<1$, and $\alpha_{k}(yzt)=\frac{1}{r}(k+\overline{ak})+1$ or $\frac{1}{r}(k+\overline{ak})+2$. We need to show that $\alpha_{k}(yzt)=\frac{1}{r}(k+\overline{ak})+2$ can not happen. Suppose that it happens, then $\alpha^{\prime}_{k}(yzt)=1-\frac{1}{r}(k+\overline{ak})$, $\alpha^{\prime}_{k}(x)=1-\frac{\overline{ak}}{r}$, and $\alpha^{\prime}_{k}(f)=2\alpha^{\prime}_{k}(x)-1=1-\frac{2\overline{ak}}{r}$, but this contradicts Rule I. So this case we get the first equation. If $\alpha_{k}$ is in case (ii), we get that $2>2\alpha_{k}(x)\geq 1$, and $\alpha_{k}(yzt)=\frac{1}{r}(k+\overline{ak})$ or $\frac{1}{r}(k+\overline{ak})-1$. Hence $2\alpha_{k}(x)=\frac{\overline{ek}}{r}+1$. Here note that $\alpha_{k}(yzt)=\frac{1}{r}(k+\overline{ak})-1$ can not happen because it contradicts the fact that $\alpha_{k}(f)=\frac{2\overline{ak}}{r}-1$ and Rule I. So this case we get the second equation.

Note that if $\beta\in N\cap\square$, then we also have either $\beta(f)=2\beta(x)$ and $1-\delta<\beta(yzt)-\beta(x)<1$; or $\beta(f)=2\beta(x)-1$, and $-\delta<\beta(yzt)-\beta(x)<0$ by Rule I. In particular, if $\beta\equiv\alpha_{k}\bmod{\mathbb{Z}}^{4}$, then $\beta(yzt)-\beta(x)\equiv\frac{k}{r}\bmod{\mathbb{Z}}$, which implies that $1-\delta<\frac{k}{r}<1$ in both cases.

(2) First we show that if $\gcd(a,r)\neq 1$, then $\gcd(e,r)=\gcd(a,r)$ and $b,c,d$ are coprime to $r$. Suppose that $\gcd(a,r)=q_{1}>1$ and $\gcd(e,r)\neq\gcd(a,r)$, then since $e\equiv 2a\bmod r$, we know that $\gcd(e,r)=2q_{1}$. In particular, $r$ is even. Take $k_{1}=\frac{r}{2q_{1}}$, then $\overline{ek_{1}}=0$ and $\overline{ak_{1}}=\overline{a(r-k_{1})}=\frac{r}{2}$. Note that by Rule II(ii), $q_{1}$ does not divide $b,c,d$, hence $\overline{bk_{1}},\overline{ck_{1}},\overline{dk_{1}}$ can not be [math] or $\frac{r}{2}$. In particular, $\alpha^{\prime}_{k_{1}}=\alpha_{r-k_{1}}$. We need to consider $3$ cases: $\alpha_{k_{1}}\neq\beta,\beta^{\prime}$; $\alpha_{k_{1}}=\beta$; $\alpha_{k_{1}}=\beta^{\prime}$.

If $\alpha_{k_{1}}\neq\beta,\beta^{\prime}$, then $\alpha_{r-k_{1}}\neq\beta,\beta^{\prime}$, and we are in the second case of (1), which gives

[TABLE]

and

[TABLE]

But this is absurd since the sum of the left hand sides of the equations above is $3r$.

If $\alpha_{k_{1}}=\beta$, we get $1-\delta<\frac{k_{1}}{r}<1$. But this implies that $\delta>\frac{3}{4}$ since $k_{1}\leq\frac{r}{4}$, which is a contradiction.

If $\alpha_{k_{1}}=\beta^{\prime}$, then $\alpha_{r-k_{1}}=\beta$, and $1-\delta<\frac{r-k_{1}}{r}<1$. In particular, $\frac{k_{1}}{r}<\delta\leq\frac{1}{12}$ and hence $2q_{1}>12$. For $j=3,5,7,11$, we may consider $jk_{1}=\frac{jr}{2q_{1}}<r$ and consider the weighting $\alpha_{jk_{1}}$. Note that by the construction, $k_{1}<jk_{1}<r-k_{1}$, hence $\alpha_{jk_{1}}\neq\beta,\beta^{\prime}$ for $j=3,5,7,11$ (same holds for $\alpha_{r-jk_{1}}$). Hence by $\overline{ajk_{1}}=\frac{r}{2}$ and $\overline{ejk_{1}}=0$, we are in the second case of (1), that is,

[TABLE]

and

[TABLE]

Since the sum of the right hand sides of the equations above is $2r$, one of $\overline{bjk_{1}}$, $\overline{cjk_{1}}$, $\overline{djk_{1}}$ is 0 for each $j=3,5,7,11$. After possibly interchanging $y,z,t$, we may assume that $\overline{bjk_{1}}=0$ for at least two $j\in\{3,5,7,11\}$. But as these two $j$’s are coprime, this implies that $\overline{bk_{1}}=0$, a contradiction.

Therefore we showed that if $\gcd(a,r)\neq 1$, then $\gcd(e,r)=\gcd(a,r)$. In this case $b,c,d$ are coprime to $r$ by Rule II(i)(ii).

Now assume that $\gcd(a,r)=1$. Since $e\equiv 2a\bmod r$, it follows that $\gcd(e,r)=1$ if $r$ is odd, and $\gcd(e,r)=2$ if $r$ is even. Moreover, in the first case, $a,b,c,d$ are coprime to $r$ by Rule II(i).

Now consider $r$ is even and $\gcd(e,r)=2$. Note that at most one of $b,c,d$ is even by Rule II(ii). Suppose that none of them is even. Set $k_{2}=\frac{r}{2}$. Then $\overline{ek_{2}}=0$ and $\overline{ak_{2}}=\overline{bk_{2}}=\overline{ck_{2}}=\overline{dk_{2}}=\frac{r}{2}$. That is, $\alpha_{k_{2}}=(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2})$. Note that $\alpha_{k_{2}}\neq\beta,\beta^{\prime}$, otherwise $\beta=\beta^{\prime}$, $\beta(xyzt)=2$, and $2\beta(f)\in{\mathbb{Z}}$, which contradicts Rule I and $\delta<\frac{1}{2}$. Then we get a contradiction since $\alpha_{k_{2}}(yzt)=\alpha_{k_{2}}(x)+1$, which violates both situations in (1). Hence exactly one of $b,c,d$ is even, say $d$, and in this case $\gcd(d,r)=\gcd(e,r)=2$.

In summary, we showed that one of the followings holds:

1. (a’)

$\gcd(e,r)=\gcd(a,r)\neq 1$ and $b,c,d$ are coprime to $r$; 2. (b)

$r$ is odd and $a,b,c,d,e$ are coprime to $r$; 3. (c)

$q=2=\gcd(d,r)=\gcd(e,r)$ and $a,b,c$ are coprime to $r$.

To conclude statement (2), we only need to prove that if $\gcd(a,r)\neq 1$ then $a\equiv 0\bmod r$. But we will come back to this after proving (3) and (4).

(3) Now suppose that $\beta\in N\cap\square$. Recall that either $\beta(f)=2\beta(x)$ and $1-\delta<\beta(yzt)-\beta(x)<1$; or $\beta(f)=2\beta(x)-1$, and $-\delta<\beta(yzt)-\beta(x)<0$ by Rule I.

Note that there exists an integer $1\leq k_{0}\leq r-1$ such that $\beta\equiv\alpha_{k_{0}}\bmod{\mathbb{Z}}$.

If $\alpha_{k_{0}}\neq\beta,\beta^{\prime}$, then we get the desired equality by (1). If $\alpha_{k_{0}}=\beta^{\prime}$, then $\beta\equiv\beta^{\prime}\bmod{\mathbb{Z}}^{4}$, which implies that $\beta(x),\beta(y),\beta(z),\beta(t)\in\{0,\frac{1}{2},1\}$, and $2(\beta(yzt)-\beta(x))$ is an integer, but contradicts $\delta<\frac{1}{2}$.

Hence we may assume that $\alpha_{k_{0}}=\beta$. Recall that $\frac{k_{0}}{r}\equiv\beta(yzt)-\beta(x)\bmod{\mathbb{Z}}$ and $1-\delta<\frac{k_{0}}{r}<1$.

Suppose that $\beta(f)=2\beta(x)-1$, and $-\delta<\beta(yzt)-\beta(x)<0$. Then $2\beta(x)\geq 1$ and this implies that

[TABLE]

This gives

[TABLE]

On the other hand, for any $1\leq k\leq r-1$ such that $k\neq k_{0}$, if $\alpha_{k}\neq\beta^{\prime}$, then by (1),

[TABLE]

if $\alpha_{k}=\beta^{\prime}$, then

[TABLE]

Hence $\frac{1}{r}(a,b,c,d;e)$ and $k_{0}$ satisfy the assumption of Lemma 2.12 after possibly relabeling $a,b,c,d$ properly, which implies that $k_{0}/r\leq 1-\delta$, a contradiction. Here the coprimeness is guaranteed by one of (a’)(b)(c) in the part of (2) we just proved; (✩2) of Lemma 2.12 is satisfied in case (a’) by labeling $a=a_{4}$; (✩3) of Lemma 2.12 is satisfied in cases (b)(c).

Suppose that $\beta(f)=2\beta(x)$, and $1-\delta<\beta(yzt)-\beta(x)<1$. Note that in this case, $\beta(yzt)-\beta(x)=\frac{k_{0}}{r}$, and hence

[TABLE]

Then we will show that $2\beta(x)\geq 1$. Suppose that $2\beta(x)<1$, then we know that $2\overline{ak_{0}}=\overline{ek_{0}}$ and

[TABLE]

On the other hand, for any $1\leq k\leq r-1$ such that $k\neq k_{0}$, if $\alpha_{k}\neq\beta^{\prime}$, then by (1),

[TABLE]

if $\alpha_{k}=\beta^{\prime}$, then

[TABLE]

Hence $\frac{1}{r}(a,b,c,d;e)$ and $k_{0}$ satisfy the assumption of Lemma 2.12 after possibly relabeling $a,b,c,d$ properly, which implies that $k_{0}/r\leq 1-\delta$, a contradiction. Here the coprimeness is guaranteed by one of (a’)(b)(c) in the part of (2) we just proved; (✩2) of Lemma 2.12 is satisfied in case (a’) by labeling $a=a_{4}$; (✩3) of Lemma 2.12 is satisfied in cases (b)(c).

Hence $2\beta(x)\geq 1$ and we have $2\overline{ak_{0}}=\overline{ek_{0}}+r$. This concludes (3).

(4) For any $k=1,\ldots,r-1$, if $\alpha_{k}\neq\beta^{\prime}$, then the statement follows from (1) and (3). If $\alpha_{k}=\beta^{\prime}=\alpha^{\prime}_{k_{0}}$, then $k=r-k_{0}$; also we know that $\overline{ek_{0}}\neq 0$, because otherwise $\overline{ak_{0}}=0$ or $\overline{dk_{0}}=0$ by case (a’) or (c) of (2), which contradicts $\alpha_{k}=\alpha^{\prime}_{k_{0}}$; therefore,

[TABLE]

This concludes (4).

(2) (continued) Now we are ready to show that if $\gcd(a,r)\neq 1$ then $a\equiv 0\bmod r$. If $\gcd(a,r)\neq 1$, then we are in case (a’) of (2), and by (4), the assumption of Theorem 2.10 is satisfied. Hence we get $a\equiv e\bmod r$. On the other hand, $2a\equiv e\bmod r$. Hence $a\equiv e\equiv 0\bmod r$. ∎

Remark 4.5*.*

Before discussing case by case, we explain the strategy again. By Proposition 4.3(2)(4) or 4.4(2)(4), we checked that the $6$-tuple $\frac{1}{r}(a,b,c,d;e,1)$ satisfies the terminal lemma (Theorem 2.10). So we can list all possible values for $\frac{1}{r}(a,b,c,d;e)$ in each case. Then we can apply Proposition 4.3(1) or 4.4(1) to some special $\alpha_{k_{1}}$ to get more restrictions on monomials in $f$, which leads to the final conclusion. For the smart choice of $\alpha_{k_{1}}$ we just follow [41, Section 7], but again the existence of $\beta$ gets in the way. So we have to consider the case that $\alpha_{k_{1}}=\beta$ or $\beta^{\prime}$, in which we can not apply Proposition 4.3(1) or 4.4(1). In this case, we should consider to choose other special $\alpha_{k_{2}}$, $\alpha_{k_{3}}$, etc., and make more discussions.

4.3. The $cA$ case

In this subsection, we consider case $cA$ in Proposition 4.2: $f=xy+g(z,t)$ with $g\in{\mathfrak{m}}^{2}$.

By Proposition 4.3(2), $q=\gcd(d,r)=\gcd(e,r)$, and $c$ is coprime to $r$, this means that $q$ divides the degree of $z$ in each monomial in $g$, that is, we may write $g=g(z^{q},t)$ by abusing the notation.

By Proposition 4.3(2)(4), we can list all possible types by Theorem 2.10, and one of the followings holds (after possibly interchanging $x,y$ or $z,t$):

If $q>1$,

1. (A)

$a+b\equiv 0,c\equiv 1,d\equiv e\bmod r$; that is, $\frac{1}{r}(a,-a,1,0;0)$; 2. (B)

$a\equiv 1,b+c\equiv 0,d\equiv e\bmod r$; that is, $\frac{1}{r}(1,b,-b,b+1;b+1)$.

If $q=1$,

1. (C)

$\frac{1}{r}(a,1,-a,a+1;a+1)$; 2. (D)

$\frac{1}{r}(a,-a-1,-a,a+1;-1)$.

This list can be easily derived from Theorem 2.10 and for the proof we refer to [41, (7.7)]. In each case, we may always assume that $0<a<r$ or $0<b<r$ accordingly.

We will discuss case by case.

Case (A): This gives an isolated hyperquotient singularity of type $\frac{1}{r}(a,-a,1,0;0)$ and $f=xy+g(z^{r},t)$ (note that $q=r$ in this case), where $g\in{\mathfrak{m}}^{2}$ and $a,r$ are coprime. But such a singularity is terminal by [31, Theorem 6.5], so this case can be excluded.

Case (C): Since $a$, $a+1$ are coprime to $r$, we can take an integer $1<k_{1}<r$ such that $\overline{k_{1}(a+1)}=1$. Consider

[TABLE]

Then $\alpha_{k_{1}}(zt)=\frac{k_{1}}{r}<1$. There are 3 cases: (C.1) $\alpha_{k_{1}}\neq\beta,\beta^{\prime}$; (C.2) $\alpha_{k_{1}}=\beta$; (C.3) $\alpha_{k_{1}}=\beta^{\prime}$.

Case (C.1): If $\alpha_{k_{1}}\neq\beta,\beta^{\prime}$, then by Proposition 4.3(1), $\alpha_{k_{1}}(f)=\alpha_{k_{1}}(xy)-1=\frac{1}{r}$. So there is a monomial ${\bf{x}}^{\bf m}\in g\in(z,t)^{2}$ with $\alpha_{k_{1}}({\bf{x}}^{\bf m})=\frac{1}{r}$, but this is absurd.

Case (C.2): If $\alpha_{k_{1}}=\beta$, then by Proposition 4.3(3), $k_{1}=k_{0}$ and $\frac{3}{4}<1-\delta<\frac{k_{1}}{r}<1$. Then $2k_{1}>\frac{3}{2}r>r+3$. Note that $r-k_{1}<2k_{1}-r<k_{1}$. Consider

[TABLE]

Then $\alpha_{2k_{1}-r}(zt)=\frac{2k_{1}-r}{r}<1$ and $\alpha_{2k_{1}-r}\neq\beta,\beta^{\prime}$. By Proposition 4.3(1), $\alpha_{2k_{1}-r}(f)=\alpha_{2k_{1}-r}(xy)-1=\frac{2}{r}$. So there is a monomial ${\bf{x}}^{\bf m}\in g\in(z,t)^{2}$ with $\alpha_{2k_{1}-r}({\bf{x}}^{\bf m})=\frac{2}{r}$, but this is absurd since by definition $\alpha_{2k_{1}-r}(z)\geq\alpha_{2k_{1}-r}(t)=\frac{2}{r}$. Therefore, this case can be excluded.

Case (C.3): If $\alpha_{k_{1}}=\beta^{\prime}$, then $\alpha_{r-k_{1}}=\alpha^{\prime}_{k_{1}}=\beta$, which implies that $k_{1}=r-k_{0}$ and $1-\delta<\frac{r-k_{1}}{r}<1$ by Proposition 4.3(3). Hence $\frac{k_{1}}{r}<\delta\leq\frac{1}{3}$ and $k_{1}<2k_{1}<r-k_{1}$. Consider

[TABLE]

In particular, $\alpha_{2k_{1}}(zt)=\frac{2k_{1}}{r}<1$ and $\alpha_{2k_{1}}\neq\beta,\beta^{\prime}$. So by Proposition 4.3(1), $\alpha_{2k_{1}}(f)=\alpha_{2k_{1}}(xy)-1=\frac{2}{r}$. So there is a monomial ${\bf{x}}^{\bf m}\in g\in(z,t)^{2}$ with $\alpha_{2k_{1}}({\bf{x}}^{\bf m})=\frac{2}{r}$, but this is absurd since by definition $\alpha_{2k_{1}}(z)\geq\alpha_{2k_{1}}(t)=\frac{2}{r}$. Therefore, this case can be excluded.

Case (D): We can take the integer $k_{1}=r-1$. Then $\alpha_{k_{1}}{(zt)}=\frac{k_{1}}{r}<1$. There are 3 cases: (D.1) $\alpha_{k_{1}}\neq\beta,\beta^{\prime}$; (D.2) $\alpha_{k_{1}}=\beta^{\prime}$; (D.3) $\alpha_{k_{1}}=\beta$.

Case (D.1): If $\alpha_{k_{1}}\neq\beta,\beta^{\prime}$, then by Proposition 4.3(1), $\alpha_{k_{1}}(f)=\alpha_{k_{1}}(xy)-1=\frac{1}{r}$. So there is a monomial ${\bf{x}}^{\bf m}\in g\in(z,t)^{2}$ with $\alpha_{k_{1}}({\bf{x}}^{\bf m})=\frac{1}{r}$, but this is absurd.

Case (D.2): If $\alpha_{k_{1}}=\beta^{\prime}$, then $\alpha_{r-k_{1}}=\alpha^{\prime}_{k_{1}}=\beta$, which implies that $k_{0}=r-k_{1}=1$ and $1-\delta<\frac{1}{r}<1$ by Proposition 4.3(3), but this is absurd as $\delta<\frac{1}{2}$.

Case (D.3): If $\alpha_{k_{1}}=\beta$, then by Proposition 4.3(3), $k_{1}=k_{0}=r-1$. Now consider $\alpha_{r-2}$, then it is easy to see that $\alpha_{r-2}\neq\beta,\beta^{\prime}$ as $r>3$. Note that $\alpha_{r-2}{(zt)}\equiv\frac{r-2}{r}\bmod{\mathbb{Z}}$.

If $\alpha_{r-2}{(zt)}<1$, then $\alpha_{r-2}{(zt)}=\frac{r-2}{r}$ and by Proposition 4.3(1), $\alpha_{r-2}(f)=\alpha_{r-2}(xy)-1=\frac{2}{r}$. So there is a monomial ${\bf{x}}^{\bf m}\in g\in(z,t)^{2}$ with $\alpha_{r-2}({\bf{x}}^{\bf m})=\frac{2}{r}$. As $r>4$, either $z^{2}\in g$, $\alpha_{r-2}(z)=\frac{1}{r}$, $\alpha_{r-2}(t)=\frac{r-3}{r}$; or $t^{2}\in g$, $\alpha_{r-2}(z)=\frac{r-3}{r}$, $\alpha_{r-2}(t)=\frac{1}{r}$. We only deal with the former case, the latter one can be reduced to the former one by symmetry by interchanging $x$ with $y$, $z$ with $t$, and $a$ with $-a-1$. The former case implies that $\overline{2a}\equiv 1\bmod r$, which means that $2a=r+1$. The type becomes

[TABLE]

Now consider

[TABLE]

as $r>9$. Recall that $k_{0}=r-1$, hence $\alpha_{r-3}\neq\beta,\beta^{\prime}$. Since $\alpha_{r-3}{(zt)}=\frac{r-3}{r}$, by Proposition 4.3(1), $\alpha_{r-3}(f)=\alpha_{r-3}(xy)-1=\frac{3}{r}$. So there is a monomial ${\bf{x}}^{\bf m}\in g\in(z,t)^{2}$ with $\alpha_{r-3}({\bf{x}}^{\bf m})=\frac{3}{r}$. But this is absurd since $\alpha_{r-3}{(z)}>\alpha_{r-3}{(t)}=\frac{r-9}{2r}>\frac{3}{2r}$ as $r>12.$

If $\alpha_{r-2}{(zt)}>1$, then $\alpha_{r-2}{(zt)}=\frac{2r-2}{r}$ and $\overline{2a}+\overline{-2a-2}=2r-2$. Since $a,a+1$ are coprime to $r$, the only solution to this equation is $a\equiv\frac{r-1}{2}\bmod r$. The type becomes

[TABLE]

Now consider

[TABLE]

Recall that $k_{0}=r-1$, hence $\alpha_{r-3}\neq\beta,\beta^{\prime}$. Since $\alpha_{r-3}{(zt)}=\frac{r-3}{r}$, by Proposition 4.3(1), $\alpha_{r-3}(f)=\alpha_{r-3}(xy)-1=\frac{3}{r}$. So there is a monomial ${\bf{x}}^{\bf m}\in g\in(z,t)^{2}$ with $\alpha_{r-3}({\bf{x}}^{\bf m})=\frac{3}{r}$. But this is absurd since $\alpha_{r-3}{(z)}=\alpha_{r-3}{(t)}=\frac{r-3}{2r}>\frac{3}{2r}$ as $r>6.$

Case (B): Consider $\frac{1}{r}(1,b,-b,b+1;b+1)$ with $b$ coprime to $r$ and $q=\gcd(b+1,r)>0$. If $b+1\equiv 0\bmod r$, then we get type $\frac{1}{r}(1,-1,1,0;0)$ which is in case (A), and we already excluded this case. Hence from now on we assume $1\leq b<b+1<r$. Consider

[TABLE]

Note that $\alpha_{r-1}(zt)=\frac{r-1}{r}<1$. We consider 3 cases: (B.1) $\alpha_{r-1}=\beta^{\prime}$; (B.2) $\alpha_{r-1}\neq\beta,\beta^{\prime}$; (B.3) $\alpha_{r-1}=\beta$.

Case (B.1): If $\alpha_{r-1}=\beta^{\prime}$, then $\alpha_{1}=\beta$ and $k_{0}=1$, which contradicts $\frac{k_{0}}{r}>1-\delta$.

Case (B.2): If $\alpha_{r-1}\neq\beta,\beta^{\prime}$, then by Proposition 4.3(1), $\alpha_{r-1}(f)=\alpha_{r-1}(xy)-1=\frac{r-b-1}{r}$. Hence there is a monomial ${\bf{x}}^{\bf m}\in g(z^{q},t)\in(z,t)^{2}$ with $\alpha_{r-1}({\bf{x}}^{\bf m})=\frac{r-b-1}{r}$. Obviously no multiple of $t$ will work, so this monomial has to be $z^{n}$ with $nb=r-b-1$ for some $n\geq 2$. In particular, $r\geq 3b+1\geq 2b+2$. We should further consider

[TABLE]

with $\alpha_{r-2}(zt)=\frac{r-2}{r}<1$. Again, there are 3 cases: (B.2.1) $\alpha_{r-2}=\beta^{\prime}$; (B.2.2) $\alpha_{r-2}\neq\beta,\beta^{\prime}$; (B.2.3) $\alpha_{r-2}=\beta$.

Case (B.2.1): If $\alpha_{r-2}=\beta^{\prime}$, then $\alpha_{2}=\beta$ and $k_{0}=2$, which contradicts $\frac{k_{0}}{r}>1-\delta$.

Case (B.2.2): If $\alpha_{r-2}\neq\beta,\beta^{\prime}$, then by Proposition 4.3(1), $\alpha_{r-2}(f)=\alpha_{r-2}(xy)-1=\frac{r-2b-2}{r}$. Hence there is a monomial ${\bf{x}}^{\bf m}\in g(z^{q},t)\in(z,t)^{2}$ with $\alpha_{r-2}({\bf{x}}^{\bf m})=\frac{r-2b-2}{r}$. This time $z^{n}$ can not work (because $2nb>r-2b-2$), nor can any multiple of $zt$, so this monomial has to be $t^{n^{\prime}}$ with $n^{\prime}(r-2b-2)=r-2b-2$ for some $n^{\prime}\geq 2$. This implies that $r=2b+2$. Combining with $r\geq 3b+1$, this implies that $r=4$, a contradiction.

Case (B.2.3): If $\alpha_{r-2}=\beta$, then we should further consider $\alpha_{r-3}$. Recall that $r\geq 3b+1$. Note that $r\neq 3b+2$ as $\gcd(r,b+1)>1$. Hence there are two cases:

[TABLE]

if $r\geq 3b+3$, or

[TABLE]

if $r=3b+1$.

In the first case, we have $\alpha_{r-3}(zt)=\frac{r-3}{r}<1$ and $\alpha_{r-3}\neq\beta,\beta^{\prime}$ as $r>5$. Then by Proposition 4.3(1), $\alpha_{r-3}(f)=\alpha_{r-3}(xy)-1=\frac{r-3b-3}{r}$ and there is a monomial ${\bf{x}}^{\bf m}\in g(z^{q},t)\in(z,t)^{2}$ with $\alpha_{r-3}({\bf{x}}^{\bf m})=\frac{r-3b-3}{r}$. This time $z^{n}$ can not work, nor can any multiple of $zt$, so this monomial has to be $t^{n^{\prime}}$ with $n^{\prime}(r-3b-3)=r-3b-3$ for some $n^{\prime}\geq 2$. This implies that $r=3b+3$. Combining with $nb=r-b-1$ for some $n\geq 2$, this implies that $n\geq 3$ and $r\leq 9$, a contradiction.

In the second case, we consider further

[TABLE]

(this holds since $b\geq 3$). We have $\alpha_{r-5}(zt)=\frac{r-5}{r}<1$ and $\alpha_{r-5}\neq\beta,\beta^{\prime}$ as $r>7$. Then by Proposition 4.3(1), $\alpha_{r-5}(f)=\alpha_{r-5}(xy)-1=\frac{2r-5b-5}{r}$ and there is a monomial ${\bf{x}}^{\bf m}\in g(z^{q},t)\in(z,t)^{2}$ with $\alpha_{r-5}({\bf{x}}^{\bf m})=\frac{2r-5b-5}{r}$. This time $z^{n}$ can not work, nor can any multiple of $zt$, so this monomial has to be $t^{n^{\prime}}$ with $n^{\prime}(2r-5b-5)=2r-5b-5$ for some $n^{\prime}\geq 2$. This implies that $2r=5b+5$. Combining with $r=3b+1$, this implies that $r=10$, a contradiction. Therefore case (B.2) is excluded.

Case (B.3): If $\alpha_{r-1}=\beta$, then $k_{0}=r-1$. We consider further $\alpha_{r-2}\neq\beta,\beta^{\prime}$ as $r>3$. Note that $r\neq 2b,2b+1$ as $\gcd(r,b)=1$ and $\gcd(r,b+1)>1$. Hence there are 2 cases:

[TABLE]

if $r\geq 2b+2$; or

[TABLE]

if $r\leq 2b-1$.

Case (B.3.1): In this case, $\alpha_{r-2}(zt)=\frac{r-2}{r}<1$ and by Proposition 4.3(1), $\alpha_{r-2}(f)=\alpha_{r-2}(xy)-1=\frac{r-2b-2}{r}$. Hence there is a monomial ${\bf{x}}^{\bf m}\in g(z^{q},t)\in(z,t)^{2}$ with $\alpha_{r-2}({\bf{x}}^{\bf m})=\frac{r-2b-2}{r}$. Arguing as before, either $r=2b+2$, or this monomial is $z^{n}$ with $2nb=r-2b-2$ and $n\geq 2$.

In the first case, we further consider

[TABLE]

(this holds since $b\geq 2$). Then $\alpha_{r-3}(zt)=\frac{r-3}{r}<1$ and $\alpha_{r-3}\neq\beta,\beta^{\prime}$ as $r>4$. Then by Proposition 4.3(1), $\alpha_{r-3}(f)=\alpha_{r-3}(xy)-1=\frac{2r-3b-3}{r}$ and there is a monomial ${\bf{x}}^{\bf m}\in g(z^{q},t)\in(z,t)^{2}$ with $\alpha_{r-3}({\bf{x}}^{\bf m})=\frac{2r-3b-3}{r}$. As $2r-3b-3>0$, this monomial has to be $z^{n^{\prime}}$ with $n^{\prime}(3b-r)=2r-3b-3$ for some $n^{\prime}\geq 2$. Combing with $r=2b+2$, this implies that $b\leq 5$ and $r\leq 12$, a contradiction.

In the second case, $r\geq 6b+2$. We further consider

[TABLE]

Then $\alpha_{r-3}(zt)=\frac{r-3}{r}<1$ and $\alpha_{r-3}\neq\beta,\beta^{\prime}$ as $r>4$. Then by Proposition 4.3(1), $\alpha_{r-3}(f)=\alpha_{r-3}(xy)-1=\frac{r-3b-3}{r}$ and there is a monomial ${\bf{x}}^{\bf m}\in g(z^{q},t)\in(z,t)^{2}$ with $\alpha_{r-3}({\bf{x}}^{\bf m})=\frac{r-3b-3}{r}$. But $z^{n}$ will not work, nor any multiple of $t$ as $r\geq 6b+2$, a contradiction.

Case (B.3.2): In this case, $\alpha_{r-2}(zt)=\frac{r-2}{r}<1$ and by Proposition 4.3(1), $\alpha_{r-2}(f)=\alpha_{r-2}(xy)-1=\frac{2r-2b-2}{r}$. Hence there is a monomial ${\bf{x}}^{\bf m}\in g(z^{q},t)\in(z,t)^{2}$ with $\alpha_{r-2}({\bf{x}}^{\bf m})=\frac{2r-2b-2}{r}$. Since $r>b+1$, this monomial is $z^{n}$ with $n(2b-r)=2r-2b-2$ and $n\geq 2$. This implies that $2r\geq 3b+1$. Note that $2r\neq 3b+2$ as $\gcd(r,b+1)>1$. Hence there are two cases for $\alpha_{r-3}$:

[TABLE]

if $2r\geq 3b+3$; or

[TABLE]

if $2r=3b+1$.

In the first case, $\alpha_{r-3}(zt)=\frac{r-3}{r}<1$ and $\alpha_{r-3}\neq\beta,\beta^{\prime}$ as $r>4$. Then by Proposition 4.3(1), $\alpha_{r-3}(f)=\alpha_{r-3}(xy)-1=\frac{2r-3b-3}{r}$ and there is a monomial ${\bf{x}}^{\bf m}\in g(z^{q},t)\in(z,t)^{2}$ with $\alpha_{r-3}({\bf{x}}^{\bf m})=\frac{2r-3b-3}{r}$. But $z^{n}$ or any multiple of $zt$ can not work, hence this monomial is $t^{n^{\prime}}$ for some $n^{\prime}\geq 2$ which implies that $2r=3b+3$. Combining with $n(2b-r)=2r-2b-2$ for $n\geq 2$, this implies that $n\geq 3$. If $n\geq 4$, then it is easy to show that $r\leq 12$ by this two equations, a contradiction. If $n=3$, then $(r,b+1)=(18,12)$. But recall that $q=\gcd(b+1,r)$ divides $n$ by construction as $z^{n}\in g(z^{q},t)$, this is also absurd.

In the second case, further consider

[TABLE]

(since $b\geq 5$). Since $\alpha_{r-4}(zt)=\frac{r-4}{r}<1$ and $\alpha_{r-4}\neq\beta,\beta^{\prime}$ as $r>5$, then by Proposition 4.3(1), $\alpha_{r-4}(f)=\alpha_{r-4}(xy)-1=\frac{3r-4b-4}{r}$ and there is a monomial ${\bf{x}}^{\bf m}\in g(z^{q},t)\in(z,t)^{2}$ with $\alpha_{r-4}({\bf{x}}^{\bf m})=\frac{3r-4b-4}{r}$. If $3r-4b-4\neq 0$, then any multiple of $t^{2}$ or $zt$ can not work, hence this monomial is $z^{n^{\prime}}$ for some $n^{\prime}\geq 2$ which implies that $3r-4b-4\geq 2(4b-2r)$. In conclusion, $3r-4b-4=0$ or $3r-4b-4\geq 2(4b-2r)$ holds. Combining with $2r=3b+1$, it is easy to see that $r\leq 8$, a contradiction.

Therefore, the $cA$ case is excluded.

4.4. The odd case

In this subsection, we consider the odd case in Proposition 4.2: $f=x^{2}+y^{2}+g(z,t)$ with $g\in{\mathfrak{m}}^{3}$ and $a\neq b$.

By Proposition 4.4(2)(4), we can list all possible types by Theorem 2.10, and one of the followings holds: $\frac{1}{2}(0,1,1,1;0)$; or $\frac{1}{r}(1,\frac{r+2}{2},\frac{r-2}{2},2;2)$ with $4|r.$ For the proof we refer to [41, (7.10)].

Here we only need to exclude the second case. Recall that $r>12$. Consider $\alpha_{r-2}=\frac{1}{r}(r-2,r-2,2,r-4)$. We need to consider $2$ cases: (1) $\alpha_{r-2}\neq\beta,\beta^{\prime}$; (2) $\alpha_{r-2}=\beta$ or $\beta^{\prime}$.

Case (1): If $\alpha_{r-2}\neq\beta,\beta^{\prime}$, then since $\alpha_{r-2}(yzt)=\frac{2r-4}{r}<\alpha_{r-2}(x)+1=\frac{2r-2}{r}$, by Proposition 4.4(1), $\alpha_{r-2}(f)=2\alpha_{r-2}(x)-1=\frac{r-4}{r}$ and there is a monomial ${\bf{x}}^{\bf m}\in g\in(z,t)^{2}$ with $\alpha_{r-2}({\bf{x}}^{\bf m})=\frac{r-4}{r}$. Note that the only possible monomial with weight $\frac{r-4}{r}$ is $z^{\frac{r-4}{2}}$, but it is not in the same eigenspace as $f$ since $\frac{r-4}{2}c=\frac{r-4}{2}\cdot\frac{r-2}{2}\equiv 2-\frac{r}{2}\not\equiv 2\equiv e\bmod r$.

Case (2): If $\alpha_{r-2}=\beta$ or $\beta^{\prime}$, then we further consider $\alpha_{r-4}=\frac{1}{r}(r-4,r-4,4,r-8)$. Note that $\alpha_{r-4}\neq\beta,\beta^{\prime}$ as $r>6$, and $\alpha_{r-4}(yzt)=\frac{2r-8}{r}<\alpha_{r-4}(x)+1=\frac{2r-4}{r}$. Then by Proposition 4.4(1), $\alpha_{r-4}(f)=2\alpha_{r-4}(x)-1=\frac{r-8}{r}$ and there is a monomial ${\bf{x}}^{\bf m}\in g\in(z,t)^{2}$ with $\alpha_{r-4}({\bf{x}}^{\bf m})=\frac{r-8}{r}$. Note that the only possible monomial with weight $\frac{r-8}{r}$ is $z^{\frac{r-8}{4}}$, but it is not in the same eigenspace as $f$ since $\frac{r-8}{4}c\not\equiv 2\equiv e\bmod r$. This can be seen by $\frac{r-8}{4}c=\frac{r-8}{4}\cdot\frac{r-2}{2}=2-r+\frac{r^{2}-2r}{8}$ where $\frac{r-2}{8}$ is not an integer.

Therefore, the odd case is excluded.

4.5. The $cD$-$E$ case

In this subsection, we consider the remaining cases in Proposition 4.2: $f=x^{2}+g(y,z,t)$ with $g\in{\mathfrak{m}}^{3}$.

By Proposition 4.4(2)(4), we can list all possible types by Theorem 2.10, and one of the followings holds (after possibly interchanging $y,z,t$):

If $a\equiv e\equiv 0\bmod r$ and $b,c,d$ are coprime to $r$, then

1. (a)

$\frac{1}{r}(0,b,-b,1;0)$ with $b$ coprime to $r$.

If $q=2=\gcd(d,r)=\gcd(e,r)$ and $a,b,c$ are coprime to $r$, then

1. (b)

$\frac{1}{r}(a,-a,1,2a;2a)$ with $r$ even and $a$ coprime to $r$; 2. (c)

$\frac{1}{r}(1,b,-b,2;2)$ with $r$ even and $b$ coprime to $r$.

If $r$ is odd and $a,b,c,d,e$ are coprime to $r$, then

1. (d)

$\frac{1}{r}(\frac{r-1}{2},\frac{r+1}{2},c,-c;-1)$ with $r$ odd and $c$ coprime to $r$; 2. (e)

$\frac{1}{r}(a,-a,2a,1;2a)$ with $r$ odd and $a$ coprime to $r$; 3. (f)

$\frac{1}{r}(1,b,-b,2;2)$ with $r$ odd and $b$ coprime to $r$.

In each case, we may always assume that $0<a,b,c<r$ accordingly.

We will discuss case by case.

Case (b): Take $0<k_{1}<r$ such that $\overline{k_{1}a}=\frac{r+2}{2}<r$, then $\alpha_{k_{1}}=\frac{1}{r}(\frac{r+2}{2},\frac{r-2}{2},k_{1},2)$. We need to consider $2$ cases: (b.1) $\alpha_{k_{1}}\neq\beta,\beta^{\prime}$; (b.2) $\alpha_{k_{1}}=\beta$ or $\beta^{\prime}$.

Case (b.1): If $\alpha_{k_{1}}\neq\beta,\beta^{\prime}$, since $\alpha_{k_{1}}(yzt)<\alpha_{k_{1}}(x)+1$, by Proposition 4.4(1), $\alpha_{k_{1}}(f)=2\alpha_{k_{1}}(x)-1=\frac{2}{r}$, but no monomial in ${\mathfrak{m}}^{3}$ has weight $\leq\frac{2}{r}$, a contradiction.

Case (b.2): If $\alpha_{k_{1}}=\beta$ or $\beta^{\prime}$, then we consider further $0<k_{2}<r$ such that $\overline{k_{2}a}=\frac{r+4}{2}<r$, then $\alpha_{k_{2}}=\frac{1}{r}(\frac{r+4}{2},\frac{r-4}{2},k_{2},4)$. Note that $\alpha_{k_{2}}\neq\beta,\beta^{\prime}$, otherwise $k_{1}+k_{2}\equiv 0\bmod r$, which implies that $\frac{r+2}{2}+\frac{r+4}{2}\equiv 0\bmod r$, a contradiction. Since $\alpha_{k_{2}}(yzt)<\alpha_{k_{2}}(x)+1$, by Proposition 4.4(1), $\alpha_{k_{2}}(f)=2\alpha_{k_{2}}(x)-1=\frac{4}{r}$, but the only possible monomial in ${\mathfrak{m}}^{3}$ has weight $\leq\frac{4}{r}$ is $z^{4}$ with $k_{2}=1$ (recall that $r>12$). In this case, $a\equiv\frac{r+4}{2}\bmod r$ and hence $k_{1}\frac{r+4}{2}\equiv\frac{r+2}{2}\bmod r$. This implies that $r|4k_{1}-2$. $r>12$ implies that $k_{1}\geq 4$. Note that $r>k_{1}$ and $r$ is even, so either $r=4k_{1}-2$ or $3r=4k_{1}-2$, in particular, in both case, $3r\geq 4k_{1}-2\geq\frac{7}{2}k_{1}$. Recall that $\alpha_{k_{1}}=\beta$ or $\beta^{\prime}$ implies that $k_{1}=k_{0}$ or $r-k_{0}$. If $k_{1}=r-k_{0}$, then $\frac{k_{0}}{r}>1-\delta$ implies that $r>4k_{1}$, a contradiction. If $k_{1}=k_{0}$, then $\frac{k_{0}}{r}\leq\frac{6}{7}$ by the above calculation, again a contradiction.

Case (c): This case is similar to case (b). Take $k_{1}=\frac{r+2}{2}$ and consider $\alpha_{k_{1}}=\frac{1}{r}(\frac{r+2}{2},\overline{k_{1}b},r-\overline{k_{1}b},2)$. We need to consider $2$ cases: (c.1) $\alpha_{k_{1}}\neq\beta,\beta^{\prime}$; (c.2) $\alpha_{k_{1}}=\beta$ or $\beta^{\prime}$.

Case (c.1): In this case the argument is the same as case (b.1). If $\alpha_{k_{1}}\neq\beta,\beta^{\prime}$, since $\alpha_{k_{1}}(yzt)<\alpha_{k_{1}}(x)+1$, by Proposition 4.4(1), $\alpha_{k_{1}}(f)=2\alpha_{k_{1}}(x)-1=\frac{2}{r}$, but no monomial in ${\mathfrak{m}}^{3}$ has weight $\leq\frac{2}{r}$, a contradiction.

Case (c.2): If $\alpha_{k_{1}}=\beta$ or $\beta^{\prime}$, then $k_{1}=k_{0}$ or $r-k_{0}$. Recall that $k_{1}=\frac{r+2}{2}$, so $\frac{k_{0}}{r}\leq\frac{r+2}{2r}\leq\frac{5}{6}$ as $r>2$, and get a contradiction by $\frac{k_{0}}{r}>1-\delta$.

Case (d): Take $k_{1}=r-1$, then $\alpha_{k_{1}}=\frac{1}{r}(\frac{r+1}{2},\frac{r-1}{2},\overline{k_{1}c},r-\overline{k_{1}c})$. We need to consider $2$ cases: (d.1) $\alpha_{k_{1}}\neq\beta,\beta^{\prime}$; (d.2) $\alpha_{k_{1}}=\beta$ or $\beta^{\prime}$.

Case (d.1): If $\alpha_{k_{1}}\neq\beta,\beta^{\prime}$, since $\alpha_{k_{1}}(yzt)<\alpha_{k_{1}}(x)+1$, by Proposition 4.4(1), $\alpha_{k_{1}}(f)=2\alpha_{k_{1}}(x)-1=\frac{1}{r}$, but no monomial in ${\mathfrak{m}}^{3}$ has weight $\leq\frac{1}{r}$, a contradiction.

Case (d.2): If $\alpha_{k_{1}}=\beta$ or $\beta^{\prime}$, then we consider $0<k_{2},k_{3}<r$ such that $\overline{k_{2}a}=\frac{r+3}{2}<r$ and $\overline{k_{3}a}=\frac{r+5}{2}<r$. Then $\alpha_{k_{2}}=\frac{1}{r}(\frac{r+3}{2},\frac{r-3}{2},\overline{k_{2}c},r-\overline{k_{2}c})$ and $\alpha_{k_{3}}=\frac{1}{r}(\frac{r+5}{2},\frac{r-5}{2},\overline{k_{3}c},r-\overline{k_{3}c})$. Note that $\alpha_{k_{2}}\neq\beta$ or $\beta^{\prime}$, otherwise $k_{1}+k_{2}\equiv 0$ and $\overline{ak_{1}+ak_{2}}=2\equiv 0\bmod r$, which is absurd. Since $\alpha_{k_{2}}(yzt)<\alpha_{k_{2}}(x)+1$, by Proposition 4.4(1), $\alpha_{k_{2}}(f)=2\alpha_{k_{2}}(x)-1=\frac{3}{r}$, but the only possible monomial in ${\mathfrak{m}}^{3}$ has weight $\frac{3}{r}$ is $z^{3}$ with $\overline{k_{2}c}=1$ (after possibly interchanging $z,t$). Similarly, $\alpha_{k_{3}}\neq\beta$ or $\beta^{\prime}$, otherwise $k_{1}+k_{3}\equiv 0$ and $\overline{ak_{1}+ak_{3}}=3\equiv 0\bmod r$, which is absurd. Since $\alpha_{k_{3}}(yzt)<\alpha_{k_{3}}(x)+1$, by Proposition 4.4(1), $\alpha_{k_{3}}(f)=2\alpha_{k_{3}}(x)-1=\frac{5}{r}$. In order to have a monomial in ${\mathfrak{m}}^{3}$ with weight $\frac{5}{r}$, one of $\overline{k_{3}c}$ and $r-\overline{k_{3}c}$ is 1. Therefore $k_{2}\pm k_{3}\equiv 0\bmod r$ and this implies that $\frac{r+3}{2}\pm\frac{r+5}{2}\equiv 0\bmod r$, a contradiction.

Case (e): Take $0<k_{1}<r$ such that $\overline{k_{1}a}=\frac{r+1}{2}$, then $\alpha_{k_{1}}=\frac{1}{r}(\frac{r+1}{2},\frac{r-1}{2},1,k_{1})$. We need to consider $2$ cases: (e.1) $\alpha_{k_{1}}\neq\beta,\beta^{\prime}$; (e.2) $\alpha_{k_{1}}=\beta$ or $\beta^{\prime}$.

Case (e.1): In this case the argument is the same as case (d.1). If $\alpha_{k_{1}}\neq\beta,\beta^{\prime}$, since $\alpha_{k_{1}}(yzt)<\alpha_{k_{1}}(x)+1$, by Proposition 4.4(1), $\alpha_{k_{1}}(f)=2\alpha_{k_{1}}(x)-1=\frac{1}{r}$, but no monomial in ${\mathfrak{m}}^{3}$ has weight $\leq\frac{1}{r}$, a contradiction.

Case (e.2): If $\alpha_{k_{1}}=\beta$ or $\beta^{\prime}$, then we consider $0<k_{2},k_{3}<r$ such that $\overline{k_{2}a}=\frac{r+3}{2}<r$ and $\overline{k_{3}a}=\frac{r+5}{2}<r$. Then $\alpha_{k_{2}}=\frac{1}{r}(\frac{r+3}{2},\frac{r-3}{2},3,k_{2})$ and $\alpha_{k_{3}}=\frac{1}{r}(\frac{r+5}{2},\frac{r-5}{2},5,k_{3})$. We can get a contradiction similarly as case (d.2). Note that $\alpha_{k_{2}}\neq\beta$ or $\beta^{\prime}$, otherwise $k_{1}+k_{2}\equiv 0$ and $\overline{ak_{1}+ak_{2}}=2\equiv 0\bmod r$, which is absurd. Since $\alpha_{k_{2}}(yzt)<\alpha_{k_{2}}(x)+1$, by Proposition 4.4(1), $\alpha_{k_{2}}(f)=2\alpha_{k_{2}}(x)-1=\frac{3}{r}$, but the only possible monomial in ${\mathfrak{m}}^{3}$ has weight $\frac{3}{r}$ is $t^{3}$ with $k_{2}=1$. Similarly, $\alpha_{k_{3}}\neq\beta$ or $\beta^{\prime}$, otherwise $k_{1}+k_{3}\equiv 0$ and $\overline{ak_{1}+ak_{3}}=3\equiv 0\bmod r$, which is absurd. Since $\alpha_{k_{3}}(yzt)<\alpha_{k_{3}}(x)+1$, by Proposition 4.4(1), $\alpha_{k_{3}}(f)=2\alpha_{k_{3}}(x)-1=\frac{5}{r}$, but the only possible monomial in ${\mathfrak{m}}^{3}$ has weight $\frac{5}{r}$ is $t^{5}$ with $k_{3}=1$. This is absurd as $k_{2}\neq k_{3}$.

Case (f): Take $k_{1}=\frac{r+1}{2}$, then $\alpha_{k_{1}}=\frac{1}{r}(\frac{r+1}{2},\overline{k_{1}b},r-\overline{k_{1}b},1)$. We need to consider $2$ cases: (f.1) $\alpha_{k_{1}}\neq\beta,\beta^{\prime}$; (f.2) $\alpha_{k_{1}}=\beta$ or $\beta^{\prime}$.

Case (f.1): In this case the argument is the same as case (d.1). If $\alpha_{k_{1}}\neq\beta,\beta^{\prime}$, since $\alpha_{k_{1}}(yzt)<\alpha_{k_{1}}(x)+1$, by Proposition 4.4(1), $\alpha_{k_{1}}(f)=2\alpha_{k_{1}}(x)-1=\frac{1}{r}$, but no monomial in ${\mathfrak{m}}^{3}$ has weight $\leq\frac{1}{r}$, a contradiction.

Case (f.2): If $\alpha_{k_{1}}=\beta$ or $\beta^{\prime}$, then we consider $0<k_{2},k_{3}<r$ such that $\overline{k_{2}a}=\frac{r+3}{2}<r$ and $\overline{k_{3}a}=\frac{r+5}{2}<r$. Then $\alpha_{k_{2}}=\frac{1}{r}(\frac{r+3}{2},\overline{k_{2}b},r-\overline{k_{2}b},3)$ and $\alpha_{k_{3}}=\frac{1}{r}(\frac{r+5}{2},\overline{k_{3}b},r-\overline{k_{3}b},5)$. We can get a contradiction similarly as case (d.2). Note that $\alpha_{k_{2}}\neq\beta$ or $\beta^{\prime}$, otherwise $k_{1}+k_{2}\equiv 0$ and $\overline{ak_{1}+ak_{2}}=2\equiv 0\bmod r$, which is absurd. Since $\alpha_{k_{2}}(yzt)<\alpha_{k_{2}}(x)+1$, by Proposition 4.4(1), $\alpha_{k_{2}}(f)=2\alpha_{k_{2}}(x)-1=\frac{3}{r}$, but the only possible monomial in ${\mathfrak{m}}^{3}$ has weight $\frac{3}{r}$ is $z^{3}$ with $r-\overline{k_{2}b}=1$. Similarly, $\alpha_{k_{3}}\neq\beta$ or $\beta^{\prime}$, otherwise $k_{1}+k_{3}\equiv 0$ and $\overline{ak_{1}+ak_{3}}=3\equiv 0\bmod r$, which is absurd. Since $\alpha_{k_{3}}(yzt)<\alpha_{k_{3}}(x)+1$, by Proposition 4.4(1), $\alpha_{k_{3}}(f)=2\alpha_{k_{3}}(x)-1=\frac{5}{r}$, but the only possible monomial in ${\mathfrak{m}}^{3}$ has weight $\frac{5}{r}$ is $z^{5}$ with $r-\overline{k_{3}b}=1$. This is absurd as $k_{2}\neq k_{3}$.

Case (a): Finally we consider case (a), $\frac{1}{r}(0,b,-b,1;0)$ with $b$ coprime to $r$. Note that $\alpha_{k}=\frac{1}{r}(0,\overline{bk},r-\overline{bk},k)$ for $1\leq k\leq r-1$ and $\alpha_{k}(g)\equiv 2\alpha_{k}(x)=0\bmod{\mathbb{Z}}$. So $\alpha_{k}(g)\in{\mathbb{Z}}_{>0}$. In this case we can get the following condition for $g(y,z,t)$.

Claim 4.6**.**

In case (a), for $1\leq k\leq r-1$ such that $\alpha_{k}\not\equiv\beta\bmod{\mathbb{Z}}^{4}$, $\alpha_{k}(g)=1$.

Proof.

Take $i=\frac{\alpha_{k}(g)}{2}$ or $\frac{\alpha_{k}(g)+1}{2}$ respectively if $\alpha_{k}(g)$ is even or odd. Consider $\gamma=\alpha_{k}+(i,0,0,0)$. Then $\gamma(f)=\min\{\alpha_{k}(g),2i\}=\alpha_{k}(g)$ and $\gamma(xyzt)=\alpha_{k}(xyzt)+i=\frac{r+k}{r}+i$. By the assumption, $\gamma\neq\beta$, hence by Rule I, $\gamma(xyzt)>\gamma(f)+1$, which implies that $\alpha_{k}(g)<\frac{k}{r}+i<1+i$. Writing out the definition of $i$, it is easy to see that $\alpha_{k}(g)=1$ is the only solution. ∎

Now come back to case (a). Take $0<k_{1}<r$ such that $\overline{k_{1}b}=\frac{r-1}{2}$. We may assume that $k_{1}\geq\frac{r}{2}$ by possibly interchanging $z,t$. Consider $\alpha_{k_{1}}=\frac{1}{r}(0,\frac{r-1}{2},\frac{r+1}{2},k_{1})$.

If $\alpha_{k_{1}}\not\equiv\beta\bmod{\mathbb{Z}}^{4}$, then by Claim 4.6, $\alpha_{k_{1}}(g)=1$, which means that there is a monomial in $(y,z,t)^{3}$ with weight $1$. But all monomials in $(y,z,t)^{3}$ have weights $\geq\frac{3(r-1)}{2r}>1$, a contradiction.

If $\alpha_{k_{1}}\equiv\beta\bmod{\mathbb{Z}}^{4}$, then take $0<k_{2}<r$ such that $\overline{k_{2}b}=\frac{r-3}{2}$. Consider $\alpha_{k_{2}}=\frac{1}{r}(0,\frac{r-3}{2},\frac{r+3}{2},k_{2})$ and $\alpha_{r-k_{2}}=\frac{1}{r}(0,\frac{r+3}{2},\frac{r-3}{2},r-k_{2})$. It is easy to see that $\alpha_{k_{2}},\alpha_{r-k_{2}}\not\equiv\beta\bmod{\mathbb{Z}}^{4}$ as $\frac{r-3}{2}\pm\frac{r-1}{2}\not\equiv 0\bmod r$. Hence by Claim 4.6, $\alpha_{k_{2}}(g)=\alpha_{r-k_{2}}(g)=1$. If $k_{2}\geq\frac{r}{2}$, then all monomials in $(y,z,t)^{3}$ have $\alpha_{k_{2}}$-weights $\geq\frac{3(r-3)}{2r}>1$; if $k_{2}<\frac{r}{2}$, then all monomials in $(y,z,t)^{3}$ have $\alpha_{r-k_{2}}$-weights $\geq\frac{3(r-3)}{2r}>1$ as $r>9$. So this case is excluded.

Therefore, the $cD$-$E$ case is excluded.

5. The $1$-gap theorem for $3$-dimensional non-canonical singularities: the general case

Firstly we prove the $1$-gap theorem for surfaces, which may be well-known to experts.

Lemma 5.1**.**

Let $S$ be a normal quasi-projective ${\mathbb{Q}}$-Gorenstein surface. Assume that ${\text{\rm mld}}(S)<1$, then ${\text{\rm mld}}(S)\leq\frac{2}{3}$.

Proof.

We may assume that $S$ has klt singularities. Take $\pi:S^{\prime}\to S$ to be the minimal resolution of $S$ and write $K_{S^{\prime}}+\sum_{i}a_{i}C_{i}=\pi^{*}K_{S}$ where $C_{i}$ are distinct exceptional curves and $1>a_{i}\geq 0$. Since ${\text{\rm mld}}(S)<1$, it has worse than du Val singularities, and hence there exists an exceptional curve $C$ with $C^{2}\leq-3$. We may assume that $C_{1}=C$. Then by the genus formula,

[TABLE]

This implies that $a(C;S)=1-a_{1}\leq\frac{2}{3}$. ∎

Remark 5.2*.*

The number $\frac{2}{3}$ is optimal in Lemma 5.1. In fact, the minimal log discrepancy of a cyclic quotient singularity of type $\frac{1}{3}(1,1)$ is $\frac{2}{3}$.

Now we are ready to prove Theorem 1.3, the $1$-gap theorem for $3$-dimensional non-canonical singularities.

Proof of Theorem 1.3.

Take $\delta=\delta_{0}$ as in Theorem 4.1, where $\delta_{0}$ is the constant from Lemma 2.12. Recall that $\delta\leq\delta_{3}$, where $\delta_{3}$ is the constant from Corollary 2.6.

Assume that there is a normal quasi-projective ${\mathbb{Q}}$-Gorenstein $3$-fold $X$ with $1-\delta<{\text{\rm mld}}(X)<1$, in particular, $X$ is klt. By Theorem 3.1, after replacing $X$, we may assume that $X$ is extremely non-canonical. Let $E_{0}$ be the unique exceptional divisor over $X$ such that $a(E_{0};X)<1$. Let $c_{E_{0}}(X)$ denote the center of $E_{0}$ on $X$. By definition, $X$ is terminal outside $c_{E_{0}}(X)$. As $3$-dimensional terminal singularities are isolated, by shrinking $X$, we may assume that $X$ is smooth outside $c_{E_{0}}(X)$.

If the center $c_{E_{0}}(X)=C$ is a curve, we can take a general hyperplane section $H\subset X$ intersecting $C$. Here $H$ is a normal quasi-projective ${\mathbb{Q}}$-Gorenstein surface and ${\text{\rm mld}}(H)\geq{\text{\rm mld}}(X)$ by the Bertini theorem ([30, Lemma 5.17]). On the other hand, by the inversion of adjucntion ([8, Corollary 1.4.5]), ${\text{\rm mld}}(H)\leq a(E_{0};X,H)=a(E_{0};X)={\text{\rm mld}}(X)$. Hence $1-\delta<{\text{\rm mld}}(H)<1$. But this contradicts Lemma 5.1.

So we may assume that $c_{E_{0}}(X)=P$ and $(P\in X)$ is an isolated extremely non-canonical klt singularity with ${\text{\rm mld}}(X)>1-\delta$. Denote $r$ to be the minimal positive integer such that $rK_{X}$ is Cartier and take $(Q\in Y)$ to be the canonical index $1$ cover of $(P\in X)$. Then $(Q\in Y)$ is an isolated index one canonical singularity. By the classification of $3$-dimensional index one canonical singularities (see [40] or [30, 5.3]), there are 3 cases: $(Q\in Y)$ is smooth; $(Q\in Y)$ is an isolated cDV singularity; $(Q\in Y)$ is an isolated non-cDV singularity.

If $(Q\in Y)$ is an isolated non-cDV singularity, then there exists an exceptional divisor $E^{\prime}$ over $Y$ centered at $Q$ such that $a(E^{\prime};Y)=1$. Hence by the ramification formula (see, for example, the calculation in [29, (20.3) Proposition] or [30, Proposition 5.20]), there exists an exceptional divisor $E$ over $X$ such that $n\cdot a(E;X)=a(E^{\prime};Y)=1$ for some positive integer $n$. Since $X$ is extremely non-canonical, $n>1$. Therefore ${\text{\rm mld}}(X)\leq a(E;X)\leq\frac{1}{2}$, a contradiction.

If $(Q\in Y)$ is smooth, then $(P\in X)$ is an isolated cyclic quotient singularity and this contradicts Corollary 2.6.

If $(Q\in Y)$ is an isolated cDV singularity, then we get a contradiction by Theorem 4.1.

In summary, such a normal quasi-projective ${\mathbb{Q}}$-Gorenstein $3$-fold $X$ with $1-\delta<{\text{\rm mld}}(X)<1$ does not exist, and the theorem is proved. ∎

6. Boundedness of global indices of klt Calabi–Yau $3$-folds

In this section, we give applications for Theorem 1.3. We show that the set of all non-canonical klt Calabi–Yau $3$-folds are bounded modulo flops, and the global indices of all klt Calabi–Yau $3$-folds are bounded from above. To be more precise, we show the followings:

Theorem 6.1**.**

The set of non-canonical klt Calabi–Yau $3$-folds forms a bounded family modulo flops.

Corollary 6.2**.**

There exists a positive integer $m$ such that for any klt Calabi–Yau $3$-fold $X$, $mK_{X}\sim 0$.

Recall that a variety is uniruled if it is covered by rational curves. The following lemma may be well-known to experts.

Lemma 6.3**.**

Let $X$ be a klt Calabi–Yau variety. Then $X$ is non-canonical if and only if $X$ is uniruled.

Proof.

Suppose that $X$ is uniruled. Then by taking a resolution $\phi:Y\to X$, $Y$ is again uniruled, which implies that $K_{Y}$ is not pseudo-effective. Assume, to the contrary, that $X$ is canonical, then $K_{Y}\geq\phi^{*}K_{X}$, and therefore $K_{X}$ is not psuedo-effective, which contradicts $K_{X}\equiv 0$.

Suppose that $X$ is non-canonical, then there exists an exceptional divisor $E$ over $X$ with log discrepancy $<1$. By [8, Corollary 1.4.3], there is a projective birational morphism $\phi:Y\to X$ extracting $E$. We can write $K_{Y}+aE=\phi^{*}K_{X}\equiv 0$ with $a>0$, which means that $K_{Y}$ is not psuedo-effective. But this implies that $Y$ is uniruled by [12], and so is $X$. ∎

The key is to show the following proposition (comparing with [13, Corollary 4.2]).

Proposition 6.4**.**

Fix positive real numbers $\epsilon$, $\delta$. Then, the set of log pairs $(X,B)$ satisfying

1. (1)

$(X,B)$* is an $\epsilon$-lc log Calabi–Yau pair of dimension $3$, * 2. (2)

there is a component of $\text{\rm Supp}\,B$ which is uniruled, and 3. (3)

the non-zero coefficients of $B$ are at least $\delta$,

forms a log bounded family modulo flops.

Proof.

We may replace $X$ by its small ${\mathbb{Q}}$-factorialization (by [8, Corollary 1.4.3]) and assume that $X$ is ${\mathbb{Q}}$-factorial. We may write $B=B^{\prime}+dD$, where $D$ is a uniruled component of $B$ and $d>0$. We can run a $(K_{X}+B^{\prime})$-MMP with scaling of an ample divisor which ends with a Mori fiber space $f:Y\to Z$. Denote by $B_{Y}$, $D_{Y}$ the strict transforms of $B$, $D$ on $Y$. Since $K_{X}+B^{\prime}+dD\equiv 0$, and we are running a $(K_{X}+B^{\prime})$-MMP, it follows that $D_{Y}$ is uniruled and dominates $Z$. Also note that $(Y,B_{Y})$ is again an $\epsilon$-lc log Calabi–Yau pair, and coefficients of $B_{Y}$ are at least $\delta$.

We claim that $Z$ is in a bounded family. If $Z$ is a point, then there is nothing to prove. If $\dim Z=1$, then according to Ambro’s canonical bundle formula (see [16, Theorem 3.1]), $-K_{Z}$ is pseudo-effective, which means that $Z$ is either ${\mathbb{P}}^{1}$ or an elliptic curve, which is in a bounded family. If $\dim Z=2$, then by [5, Corollary 1.7], there exists an effective ${\mathbb{R}}$-divisor $\Delta$ such that $(Z,\Delta)$ is $\epsilon^{\prime}$-klt and $K_{Z}+\Delta\sim_{\mathbb{R}}0$, where $\epsilon^{\prime}$ is a positive number depending only on $\epsilon$. Since $D_{Y}$ dominates $Z$, $Z$ is also uniruled. In particular, by Lemma 6.3, we are not in the case that $K_{Z}\equiv 0$, $\Delta=0$, and $Z$ has canonical singularities. Therefore, by [2, Theorem 6.9], such $Z$ is in a bounded family.

As $Z$ is in a bounded family, we may find a very ample divisor $A$ on $Z$, and a positive integer $r$ independent of $X$ such that $A^{\dim Z}\leq r$. Here if $Z$ is a point, we just formally define $A^{\dim Z}=1$. Therefore, $(Y,B_{Y})\to Z$ is a $(3,r,\epsilon)$-Fano type log Calabi–Yau fibration in the sense of [7, Definition 1.1]. Hence by [7, Theorem 1.2], such $Y$ is in a bounded family. Here instead of using [7, Theorem 1.2], we can also use [6, Theorem 1.4] and [13, Theorem 4.6] to conclude the boundedness of $Y$. As the coefficients of $B_{Y}$ are at least $\delta$, it is easy to see that the pair $(Y,B_{Y})$ is in a log bounded family. In fact, as $Y$ is bounded, we can find a very ample divisor $H$ on $Y$ such that $H^{3}\leq r^{\prime}$ and $H^{2}\cdot(-K_{Y})\leq r^{\prime}$ for some positive integer $r^{\prime}$ independent of $Y$, then $H^{2}\cdot{\text{\rm Supp}}(B_{Y})\leq\frac{1}{\delta}H^{2}\cdot B_{Y}=\frac{1}{\delta}H^{2}\cdot(-K_{Y})\leq\frac{r^{\prime}}{\delta}$, and we can use [6, Lemma 2.20] to conclude the log boundedness. For any prime divisor $E$ on $X$ which is exceptional over $Y,$ we have

[TABLE]

Hence, $(X,B)$ is in a log bounded family modulo flops by [13, Proposition 4.8] by extracting all such $E$ simultaneously in the log bounded family of $(Y,B_{Y})$. ∎

Now we are ready to present the proof of Theorem 6.1. It is almost the same as that of [13, Theorem 5.1], the essential modifications are that we remove the condition on minimal log discrepancies by Theorem 1.3, and remove the rational connectedness condition by Proposition 6.4.

Proof of Theorem 6.1.

Consider a non-canonical klt Calabi–Yau $3$-fold $X$. By Theorem 1.3, there exists a constant $0<\delta<1$ independent of $X$ such that ${\text{\rm mld}}(X)\leq 1-\delta$. By [8, Corollary 1.4.3], we may take a projective birational morphism $\pi:Y\to X$ extracting only one exceptional divisor $E$ with log discrepancy $a=a(E;X)\leq 1-\delta$. We can write

[TABLE]

Also by Global ACC [19, Theorem 1.5] (see [13, Lemma 3.12]), there exists a constant $\epsilon\in(0,\frac{1}{2})$ such that $X$ is $(2\epsilon)$-lc, and therefore $(Y,(1-a)E)$ is a $(2\epsilon)$-lc log Calabi–Yau pair with $1-a\geq\delta>0$. Here $E$ is uniruled by [18].

Now we can apply Proposition 6.4 to see that the pairs $(Y,(1-a)E)$ are log bounded modulo flops. That is, there are finitely many quasi-projective normal varieties $\mathcal{W}_{i}$, a reduced divisor $\mathcal{E}_{i}$ on $\mathcal{W}_{i}$, and a projective morphism $\mathcal{W}_{i}\to S_{i}$, where $S_{i}$ is a normal variety of finite type and $\mathcal{E}_{i}$ does not contain any fiber, such that for every $(Y,(1-a)E)$, there is an index $i$, a closed point $s\in S_{i}$, and a small birational map $f:{\mathcal{W}_{i,s}}\dashrightarrow Y$ such that $\mathcal{E}_{i,s}=f_{*}^{-1}E$. We may assume that the set of points $s$ corresponding to such $Y$ is dense in each $S_{i}$. We may just consider a fixed index $i$ and ignore the index in the following argument.

Now we are going to prove that $X$ is bounded modulo flops by contracting $E$ simultaneously in the bounded family $(\mathcal{W},\mathcal{E})$. The argument is exactly the same as the latter half of [13, Theorem 5.1].

For the point $s$ corresponding to $(Y,(1-a)E)$,

[TABLE]

and therefore $({\mathcal{W}_{s}},(1-a)f_{*}^{-1}E)$ is a $(2\epsilon)$-lc log Calabi–Yau pair. Now consider a log resolution $g:\mathcal{W}^{\prime}\to\mathcal{W}$ of $(\mathcal{W},\mathcal{E})$ and denote by $\mathcal{E}^{\prime}$ the strict transform of $\mathcal{E}$ and the sum of all $g$-exceptional reduced divisors on $\mathcal{W}^{\prime}$. Consider the log pair $(\mathcal{W}^{\prime},(1-\epsilon)\mathcal{E}^{\prime})$. There exists an open dense set $U\subset S$ such that for the point $s\in U$ corresponding to $(Y,(1-a)E)$, $g_{s}:\mathcal{W}^{\prime}_{s}\to\mathcal{W}_{s}$ is a log resolution and we can write

[TABLE]

where the coefficients of $B_{s}$ are $\leq 1-2\epsilon$ and its support is contained in $\mathcal{E}^{\prime}_{s}=\mathcal{E}^{\prime}|_{{\mathcal{W}^{\prime}_{s}}}$. We have

[TABLE]

Note that the support of $(1-\epsilon)\mathcal{E}^{\prime}_{s}-B_{s}$ coincides with that of $\mathcal{E}^{\prime}_{s}$ which are precisely the divisors on $\mathcal{W}^{\prime}_{s}$ exceptional over $X$. Hence $(K_{\mathcal{W}^{\prime}}+(1-\epsilon)\mathcal{E}^{\prime})$ is of Kodaira dimension zero on the fiber $\mathcal{W}^{\prime}_{s}$ and we can run a $(K_{\mathcal{W}^{\prime}}+(1-\epsilon)\mathcal{E}^{\prime})$-MMP with scaling of an ample divisor over $S$ to get a relative minimal model $\tilde{\mathcal{W}}$ over $S$. Such MMP terminates by [21, Corollary 2.9, Theorem 2.12]. Note that for the point $s\in U$ corresponding to $(Y,(1-a)E)$, $\mathcal{E}^{\prime}_{s}$ is contracted by this MMP and hence $\tilde{\mathcal{W}}_{s}$ is isomorphic to $X$ in codimension one. This gives a bounded family modulo flops, over $U$. Applying Noetherian induction on $S$, the family of all such $X$ is bounded modulo flops. ∎

Before proving Corollary 6.2, we show the boundedness of global indices in a bounded family.

Lemma 6.5**.**

Let $\mathcal{D}$ be a bounded family of projective varieties. Then there exists a positive integer $m$ such that if $Y\in\mathcal{D}$ is a klt Calabi–Yau variety, then $mK_{Y}\sim 0$.

Proof.

Without loss of generality, we may assume that all varieties in $\mathcal{D}$ is of dimension $d$ for some positive integer $d$. Note that by Global ACC [19, Theorem 1.5] (see [13, Lemma 3.12]), there exists a constant $\epsilon\in(0,1)$ such that $Y$ is $\epsilon$-lc for any klt Calabi–Yau variety $Y$ in $\mathcal{D}$.

By definition, there is a quasi-projective scheme $\mathcal{Z}$ and a projective morphism $h:\mathcal{Z}\to T$, where $T$ is of finite type, such that for every $X\in\mathcal{D}$, there is a closed point $t\in T$ and an isomorphism $f:\mathcal{Z}_{t}\to X$. Replacing $T$ by disjoint union of locally closed subsets while taking log resolutions of $\mathcal{Z}$, we may assume that there are finitely many smooth varieties $T_{i}$ and projective morphisms $(\mathcal{W}_{i},\mathcal{E}_{i})\to\mathcal{Z}_{i}\to T_{i}$ such that $(\mathcal{W}_{i},\mathcal{E}_{i})$ is log smooth over $T_{i}$ and for every $t\in T_{i}$, the fiber $(\mathcal{W}_{i,t},\mathcal{E}_{i,t})$ is a log resolution of $\mathcal{Z}_{i,t}$ with $\mathcal{E}_{i,t}$ the reduced exceptional divisor, and every $X\in\mathcal{D}$ is isomorphic to a fiber of $\mathcal{Z}_{i}\to T_{i}$ for some $i$.

Note that for any $t\in T_{i}$ such that the fiber $\mathcal{Z}_{i,t}$ is an $\epsilon$-lc Calabi–Yau variety, and for any positive integer $m$, we have

[TABLE]

By [20, Theorem 4.2], the left hand side is independent of $t$ for fixed $i$ and $m$. On the other hand, $h^{0}(\mathcal{Z}_{i,t},mK_{\mathcal{Z}_{i,t}})=1$ if and only if $mK_{\mathcal{Z}_{i,t}}\sim 0$. Hence for each $i$, the global index of $\mathcal{Z}_{i,t}$, where $\mathcal{Z}_{i,t}$ is an $\epsilon$-lc Calabi–Yau variety, is independent of $t\in T_{i}$. As there are only finitely many such families, there exists a uniform positive integer $m$ such that if $Y\in\mathcal{D}$ is a klt Calabi–Yau variety, then $mK_{Y}\sim 0$. ∎

Proof of Corollary 6.2.

Consider a klt Calabi–Yau $3$-fold $X$.

If $X$ has canonical singularities, then we can take a terminalization $\pi:X^{\prime}\to X$ such that $X^{\prime}$ has terminal singularities and $K_{X^{\prime}}=\pi^{*}K_{X}$. By [26, 36], there exists a positive integer $m_{1}$ independent of $X^{\prime}$ such that $m_{1}K_{X^{\prime}}\sim 0$, which implies that $m_{1}K_{X}\sim 0$.

If $X$ has worse than canonical singularities, then by Theorem 6.1, $X$ is bounded modulo flops, that is, there exists a bounded family of varieties $\mathcal{D}$ such that there is a normal projective variety $Y\in\mathcal{D}$ isomorphic to $X$ in codimension one. Moreover, $Y$ is also a Calabi–Yau $3$-fold. Hence by Lemma 6.5, there exists a uniform positive integer $m_{2}$ such that $m_{2}K_{Y}\sim 0$, which implies that $m_{2}K_{X}\sim 0$ as $X$ and $Y$ are isomorphic in codimension one. ∎

Acknowledgment

The author is grateful to Valery Alexeev for fruitful discussions. This paper (especially Section 4) could never be finished without his warm encouragement and support. The author would like to thank Jungkai Alfred Chen, Jingjun Han, Yujiro Kawamata, James McKernan, Jihao Liu, Miles Reid, Vyacheslav Shokurov, and Chenyang Xu for discussions and comments during the preparation of this paper. Part of this paper was written during the author’s visit to Johns Hopkins University in April 2019, and the author appreciates the support and hospitality. This work was supported by the National Science Foundation under Grant No. DMS-1440140 while the author were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2019 semester. The author also thanks Jihao Liu for discussions on [33]. The author thanks the referees for carefully checking the details and many useful suggestions.