On minimal 4-folds of general type with $p_g \geq 2$

1.   Introduction

Studying the behavior of pluricanonical maps of projective varieties has been one of the fundamental tasks in birational geometry. For varieties of general type, an interesting and critical problem is to find a positive integer $m$ so that the pluricanonical map $\varphi_m$ is birational onto the image. A momentous theorem given by Hacon-McKernan [13], Takayama [19] and Tsuji [20] says that for any integer $n>0$, there is some constant $r_n$ (we assume $r_n$ to be the smallest one) such that the pluricanonical map $\varphi_m$ is birational onto its image for all $m \geq r_n$ and for all minimal projective $n$-folds of general type. By using the birationality principle (see, for example, Theorem 2.2), an explicit upper bound of $r_{n+1}$ is determined by that of $r_n$. Therefore, finding the explicit constant $r_n$ for smaller $n$ is the next problem. However, $r_n$ is known only for $n\leq 3$, namely, $r_1 = 3$, $r_2 = 5$ by Bombieri [2] and $r_3\leq 57$ by Chen-Chen [4,5,6] and Chen [11].

The first partial result concerning the explicit bound of $r_4$ was due to [6,Theorem 1.11] by Chen and Chen that $\varphi_{35}$ is birational for all nonsingular projective 4-folds of general type with $p_g\geq 2$. It is mysterious whether the numerical bound "35" is optimal under the same assumption.

In this paper, we go on studying this question and prove the following theorem:

Theorem 1.1. Let $V$ be a nonsingular projective 4-fold of general type with $p_g(V) \geq 2$. Then

(1) $\varphi_m$ is birational for all $m\geq 33$;

(2) ${\rm Vol}(V)\geq \frac{1}{480}$.

Remark 1.2. As pointed out by Brown and Kasprzyk [3], the requirement on $p_g$ in Theorem 1.1(2) is indispensable from the following list of canonical fourfolds, which are hypersurfaces in weighted projective spaces with at worst canonical singularities:

1. $X_{78}\subset {\mathbb P}(39,14,9, 8,6,1)$, ${\rm Vol}(X_{78}) = 1/3024$;

2. $X_{78}\subset {\mathbb P}(39,13,10,8,6,1)$, ${\rm Vol}(X_{78}) = 1/3120$;

3. $X_{72}\subset {\mathbb P}(36,11,9,8,6,1)$, ${\rm Vol}(X_{72}) = 1/2376$;

4. $X_{70}\subset {\mathbb P}(35,14,10,6,3,1)$, ${\rm Vol}(X_{70}) = 1/1260$;

5. $X_{70}\subset {\mathbb P}(35,14,10,5,4,1)$, ${\rm Vol}(X_{70}) = 1/1400$;

6. $X_{68}\subset {\mathbb P}(34, 12,8,7,5,1)$, ${\rm Vol}(X_{68}) = 1/1680$.

Moreover, the following two hypersurfaces has $p_g \geq 2$ and $\varphi_{17}$ is non-birational, so one may expect that $18$ is the optimal lower bound of $m$ such that $\varphi_m$ is birational for a nonsingular projective 4-fold of general type with $p_g \geq 2$:

(1) $X_{36}\subset {\mathbb P}(18,6,5,4,1,1)$;

(2) $X_{36}\subset {\mathbb P}(18,7,5,3,1,1)$.

Throughout this paper, all varieties are defined over an algebraically closed field $k$ of characteristic $0$. We will frequently use the following symbols:

$\diamond$'$\sim$' denotes linear equivalence or $\mathbb{Q}$-linear equivalence;

$\diamond$'$\equiv$' denotes numerical equivalence;

$\diamond$'$|A| \succcurlyeq |B|$' or, equivalently,'$|B| \preccurlyeq |A|$' means $|A| \supseteq |B|+$ fixed effective divisors.

2.   Preliminaries

Let $V$ be a nonsingular projective 4-fold of general type with geometric genus $p_g(V): = \dim_kH^0(V, \mathcal{O}_V(K_V)) \geq 2$, where $K_V$ is a canonical divisor of $V$. By the minimal model program (see, for instance [1,16,17,18]), we can find a minimal model $Y$ of $V$ with at worst $\mathbb{Q}$-factorial terminal singularities. Since the properties, which we study on $V$, are birationally invariant in the category of normal varieties with canonical singularities, we shall focus our study on $Y$ instead.

2.1.   Convention

For an arbitrary linear system $|D|$ of positive dimension on a normal projective variety $Z$, we define a generic irreducible element of $|D|$ in the following way. We have $|D| = \rm{Mov}|D|+\rm{Fix}|D|$, where $\rm{Mov}|D|$ and $\rm{Fix}|D|$ denote the moving part and the fixed part of $|D|$ respectively. Consider the rational map $\varphi_{|D|} = \varphi_{\rm{Mov}|D|}$. We say that $|D|$ is composed of a pencil if $\dim \overline{\varphi_{|D|}(Z)} = 1$; otherwise, $|D|$ is not composed of a pencil. A generic irreducible element of $|D|$ is defined to be an irreducible component of a general member in $\rm{Mov}|D|$ if $|D|$ is composed of a pencil or, otherwise, a general member of $\rm{Mov}|D|$.

Keep the above settings. We say that $|D|$ can distinguish different generic irreducible elements $X_1$ and $X_2$ of a linear system $|M|$ on $Z$ if neither $X_1$ nor $X_2$ is contained in $\rm{Bs}|D|$, and if $\overline{\varphi_{|D|}(X_1)} \nsubseteq \overline{\varphi_{|D|}(X_2)}, \overline{\varphi_{|D|}(X_2)} \nsubseteq \overline{\varphi_{|D|}(X_1)}$.

A nonsingular projective surface $S$ of general type with $K_{S_0}^2 = u$ and $p_g(S_0) = v$ is referred to as a $(u,v)$-surface, where $S_0$ is the minimal model of $S$.

2.2.   Setup for the map $\varphi_{1,Y}$

Fix an effective divisor $K_1 \sim K_Y$. By Hironaka's theorem, we may take a series of blow-ups along nonsingular centers to obtain the model $\pi: Y' \rightarrow Y$ satisfying the following conditions:

(ⅰ) $Y'$ is nonsingular and projective;

(ⅱ) the moving part of $|K_{Y'}|$ is base point free so that

is a non-trivial morphism;

(ⅲ) the union of $\pi^*(K_1)$ and all those exceptional divisors of $\pi$ has simple normal crossing supports.

Take the Stein factorization of $g_1$. We get

and hence the following diagram commutes:

We may write

where $E_{\pi}$ is a sum of distinct exceptional divisors with positive rational coefficients. Denote by $|M_1|$ the moving part of $|K_{Y'}|$. Since $Y$ has at worst $\mathbb{Q}$-factorial terminal singularities, we may write

where $E_1$ is an effective $\mathbb{Q}$-divisor as well. One has $1\leq \dim(\Gamma) \leq 4$.

If $\dim(\Gamma) = 1$, we have $M_1 \sim \sum\limits_{i = 1}^{b} F_i\equiv bF$, where $F_i$ and $F$ are smooth fibers of $f_1$ and $b = \deg {f_1}_*\mathcal{O}_{Y'}(M_1) \geq p_g(Y)-1 \geq 1$. More specifically, when $g(\Gamma) = 0$, we say that $|M_1|$ is composed of a rational pencil and when $g(\Gamma)>0$, we say that $|M_1|$ is composed of an irrational pencil.

If $\dim(\Gamma)>1$, by Bertini's theorem, we know that general members $T_i \in |M_1|$ are nonsingular and irreducible.

Denote by $T'$ a generic irreducible element of $|M_1|$. Set

So we naturally get

2.3.   Notations

Pick a generic irreducible element $T'$ of $|M_1|$. Modulo further blow-ups on $Y'$, which is still denoted as $Y'$ for simplicity, we may have a birational morphism $\pi_T = \pi|_{T'}: T'\rightarrow T$ onto a minimal model $T$ of $T'$. Let $t_1$ be the smallest positive integer such that $P_{t_1}(T): = \dim_kH^0(T, \mathcal{O}_T(t_1K_T)) \geq 2$. Modulo a further blow-up of $Y'$, we may assume that $\rm{Mov}|t_1K_{T'}|$ is base point free.

Set $|N| = \rm{Mov}|t_1K_{T'}|$ and let $\varphi_{t_1, T}$ be the $t_1$-canonical map: $T \dashrightarrow \mathbb{P}^{P_{t_1}(T)-1}$. Similar to the 4-fold case in Section 2.2, take the Stein factorization of the composition:

Denote by $j$ the induced projective morphism with connected fibers from $\varphi_{t_1, T} \circ \pi_T$ by Stein factorization. Set

where $c = \deg{j_*\mathcal{O}_{T'}(N)}\geq P_{t_1}(T)-1$. Let $S$ be a generic irreducible element of $|N|$. Then we have

where $E_{N}$ is an effective $\mathbb{Q}$-divisor. Denote by $\sigma: S \rightarrow S_0$ the contraction morphism of $S$ onto its minimal model $S_0$.

Suppose that $|H|$ is a base point free linear system on $S$. Let $C$ be a generic irreducible element of $|H|$. As $\pi_T^*(K_T)|_S$ is nef and big, by Kodaira's lemma, there is a rational number $\tilde{\beta} >0$ such that $\pi_T^*(K_T)|_S \geq \tilde{\beta} C$.

Set

2.4.   Technical preparation

We will use the following theorem which is a special form of Kawamata's extension theorem (see [15,Theorem A]).

Theorem 2.1. (cf. [12,Theorem 2.2]) Let $Z$ be a nonsingular projective variety on which $D$ is a smooth divisor. Assume that $K_Z+D\sim A+B$ where $A$ is an ample $\mathbb{Q}$-divisor and $B$ is an effective $\mathbb{Q}$-divisor such that $D \not\subseteq {\rm{Supp}}{\rm{(}}B{\rm{)}}$. Then the natural homomorphism

is surjective for any integer $m>1$.

In particular, when $Z$ is of general type and $D$, as a generic irreducible element, moves in a base point free linear system, the conditions of Theorem 2.1 are automatically satisfied. Keep the settings as in 2.2 and 2.3. Take $Z = Y'$ and $D = T'$.

If $|M_1|$ is composed of an irrational pencil, by [9,Lemma 2.5], we have

If $|M_1|$ is not composed of an irrational pencil, then for a sufficiently large and divisible integer $n>0$, we have

and the homomorphism

is surjective. By [17,Theorem 3.3], Mov$|nK_{T'}|$ is base point free, so one has

It follows that

where the latter inequality holds by [7,Lemma 2.7]. Together with (1), we get the canonical restriction inequality:

Similarly, one has

We will tacitly use the following type of birationality principle.

Theorem 2.2. (cf. [5,2.7]) Let $Z$ be a nonsingular projective variety, $A$ and $B$ be two divisors on $Z$ with $|A|$ being a base point free linear system. Take the Stein factorization of $\varphi_{|A|}$: $Z \overset{h} \longrightarrow W \longrightarrow {\mathbb P}^{h^0(Z,A)-1}$ where $h$ is a fibration onto a normal variety $W$. Then the rational map $\varphi_{|B+A|}$ is birational onto its image if one of the following conditions is satisfied:

(i) $\dim\varphi_{|A|}(Z)\geq 2$, $|B|\neq \emptyset$ and $\varphi_{|B+A|}|_D$ is birational for a general member $D$ of $|A|$.

(ii) $\dim\varphi_{|A|}(Z) = 1$, $\varphi_{|B+A|}$ can distinguish different general fibers of $h$ and $\varphi_{|B+A|}|_F$ is birational for a general fiber $F$ of $h$.

2.5.   Some useful lemmas

The following results on surfaces will be used in our proof.

Lemma 2.3. (see [6,Lemma 2.5]) Let $\sigma: S \longrightarrow S_0$ be the birational contraction onto the minimal model $S_0$ from a nonsingular projective surface $S$ of general type. Assume that $S$ is not a $(1,2)$-surface and that $\tilde{C}$ is a curve on $S$ passing through a very general point. Then $(\sigma^*(K_{S_0})\cdot \tilde{C})\geq 2$.

Lemma 2.4. ([10,Lemma 2.5]) Let $S$ be a nonsingular projective surface. Let $L$ be a nef and big $\mathbb{Q}$-divisor on $S$ satisfying the following conditions:

(1) $L^2>8$;

(2) $(L \cdot C_{x})\geq 4$ for all irreducible curves $C_{x}$ passing through a very general point $x\in S$.

Then $|K_S+\ulcorner{{L}}\urcorner|$ gives a birational map.

3.   Proof of the main theorem

As an overall discussion, we keep the same settings as in 2.2 and 2.3.

3.1.   Separation properties of $\varphi_{m,Y}$

Lemma 3.1. Let $Y$ be a minimal 4-fold of general type with $p_g(Y)\geq 2$. Then $|mK_{Y'}|$ can distinguish different generic irreducible elements of $|M_1|$ for all $m \geq 3$.

Proof. Suppose $m \geq 3$. As we have $mK_{Y'} \geq M_1$, we may just consider the case when $|M_1|$ is composed of a pencil. In particular, when $|M_1|$ is composed of a rational pencil, which is the case when $\Gamma \cong \mathbb{P}^1$, the global sections of ${f_1}_*\mathcal{O}_{Y'}(M_1)$ can distinguish different points of $\Gamma$. So $|M_1|$, and consequently $|mK_{Y'}|$ can distinguish different general fibers of $f_1$. Hence we may just deal with the case when $|M_1|$ is composed of an irrational pencil. We have $M_1 \sim \sum\limits_{i = 1}^{b} T_i$, where $T_i$ are smooth fibers of $f_1$ and $b \geq 2$. Pick two different generic irreducible elements $T_1$, $T_2$ of $|M_1|$. Then by Kawamata-Viehweg vanishing theorem ([14,21]), one has

and the surjective map

Since $p_g(Y)\geq 2$, both $K_{T_i}$ and $\pi^*(K_Y)$ are effective. So for general $T_i$, $\pi^*(K_Y)|_{T_i}$ is effective. As $T_i$ is moving and $M_1|_{T_i}\sim 0$, both groups in (4) and (5) are non-zero. Therefore, $|mK_{Y'}|$ can distinguish different generic irreducible elements of $|M_1|$.

Lemma 3.2. Let $Y$ be a minimal 4-fold of general type with $p_g(Y)\geq 2$. Pick a generic irreducible element $T'$ of $|M_1|$. Then $|mK_{Y'}||_{T'}$ can distinguish different generic irreducible elements of $|N|$ for all

Proof. Suppose $m \geq 2t_1+4$. We have $K_{Y'} \geq \pi^*(K_Y) \geq T'$. Similar to the proof of Lemma 3.1, we consider the following two situations: (i) $|N|$ is not composed of a pencil or $|N|$ is composed of a rational pencil; (ii) $|N|$ is composed of an irrational pencil.

For (i), one has

By Theorem 2.1, one has

As $(t_1+1)K_{T'} \geq N,$ $|mK_{Y'}||_{T'}$ can distinguish different generic irreducible elements of $|N|$.

For (ii), it holds that

Using Theorem 2.1 again, one gets

where $S_1$ and $S_2$ are two different generic irreducible elements of $|N|$. The vanishing theorem implies the surjective map

where we note that $(N-S_{i})|_{S_i}$ is linearly trivial for $i = 1,2$. Since $p_g(T)>0$, both groups in (6) and (7) are non-zero. So $|mK_{Y'}||_{T'}$ can distinguish different generic irreducible elements of $|N|$ for any $m \geq 2t_1+4$.

Lemma 3.3. Let $Y$ be a minimal 4-fold of general type with $p_g(Y)\geq 2$. Pick a generic irreducible element $T'$ of $|M_1|$ and a generic irreducible element $S$ of $|N|$. Define

Then $|mK_{Y'}||_S$ can distinguish different generic irreducible elements of $|H|$ for all $m \geq 4(t_1+1).$

Proof. Similar to the proof of Lemma 3.2, we have

Since $t_1K_{T'}\geq S$, we have

As $p_g(S)>0$, $|H|$ is not composed of an irrational pencil, concluding the proof.

3.2.   Two useful propositions

Proposition 3.4. Let $Y$ be a minimal 4-fold of general type with $p_g(Y)\geq 2$. Pick a generic irreducible element $T'$ of $|M_1|$ and a generic irreducible element $S$ of $|N|$. If $S$ is not a $(1,2)$-surface, then $\varphi_{m,Y}$ is birational for all

Proof. Suppose $m>(2\sqrt{2}+1)(\frac{t_1}{a_{t_1,T}}+1)(1+\frac{1}{\theta_1})$. Since

is nef and big, and it has simple normal crossing support, Kawamata-Viehweg vanishing theorem implies

where $Q_{m,T'} = ((m-1)\pi^*(K_Y)-T'-\frac{1}{\theta_1}E_1)|_{T'} \equiv (m-1-\frac{1}{\theta_1})\pi^*(K_Y)|_{T'}$ is nef and big and has simple normal crossing support.

By the canonical restriction inequality (2), we may write

where $E_{1, T'}$ is certain effective ${\mathbb Q}$-divisor. As $t_1\pi_T^*(K_T)\equiv a_{t_1,T}S+E_{N}$, one may obtain that

is nef and big and has simple normal crossing support. So by Kawamata-Viehweg vanishing theorem, one has

where

By (3), we have

for some effective ${\mathbb Q}$-divisor $E_{t_1,S}$ on $S$, together with (9), one has

where $U_{m,S}$ is nef and big. Hence the statement clearly follows from Lemma 2.3, Lemma 2.4, Lemma 3.1, Lemma 3.2 and Theorem 2.2.

Proposition 3.5. Let $Y$ be a minimal 4-fold of general type with $p_g(Y)\geq 2$. Pick a generic irreducible element $T'$ of $|M_1|$ and a generic irreducible element $S$ of $|N|$. If $S$ is neither a $(1,2)$-surface nor a $(2,3)$-surface, then $\varphi_{m,Y}$ is birational for all

Proof. Suppose $m \geq 6(t_1+1)$. Following the same procedures as in the proof of Lemma 3.2 and Lemma 3.3, one has

and

Furthermore, one has

By virtue of Bombieri's result in [2] that $|3K_S|$ gives a birational map unless $S$ is a $(1,2)$-surface or a $(2,3)$-surface, together with Lemma 3.1, Lemma 3.2 and Theorem 2.2, the statement holds.

3.3.   The case of $\dim(\Gamma) \geq 2$

We follow Chen-Chen's approach in [6,Theorem 8.2] to deal with the case of $\dim(\Gamma) \geq 2$.

Theorem 3.6. ([6,Theorem 8.2]) Let $Y$ be a minimal 4-fold of general type with $p_g(Y)\geq 2$. Assume that $\dim(\Gamma) \geq 2$, then $\varphi_{m,Y}$ is birational for all $m\geq 15$.

Proof. By Theorem 2.2, we may just consider $\varphi_{m,Y'}|_{T'}$ for a general member $T' \in |M_1|$. As $\theta_1 = 1$, (2) gives

Pick a generic irreducible element $S$ of $|M_1|_{T'}|$. It follows that

Modulo $\mathbb{Q}$-linear equivalence, we have

Using Theorem 2.1, we get

Thus, combining (10) and (12), one gets

By (8), we already have

where $Q_{m,T'} = ((m-1)\pi^*(K_Y)-T'-\frac{1}{\theta_1}E_1)|_{T'} \equiv (m-2)\pi^*(K_Y)|_{T'}$. As $\pi^*(K_Y)|_{T'} \equiv S+E_S$ for some effective $\mathbb{Q}$-divisor $E_S$ on $T'$ and

is nef and big, Kawamata-Viehweg vanishing theorem implies

where

Since $R_{m,S}' \equiv \frac{m-3}{3}\sigma^*(K_{S_0})+E_{m,S}'$, where $E_{m,S}'$ is an effective $\mathbb{Q}$-divisor on $S$, by Lemma 2.4, $|K_S+\ulcorner{{R_{m,S}'}}\urcorner|$ gives a birational map whenever $m \geq 15$.

Since $\rm{Mov}|K_{T'}| \succcurlyeq |M_1|_{T'}|$, we may take $t_1 = 1$ and by the proof of Lemma 3.2 we know that $|mK_{Y'}||_{T'}$ distinguishes different generic irreducible elements of $|M_1|_{T'}|$ for $m\geq 6$. Therefore, $\varphi_{m,Y}$ is birational for all $m\geq 15$ in this case.

3.4.   The case of $\dim(\Gamma)=1$

Theorem 3.7. Let $Y$ be a minimal 4-fold of general type with $p_g(Y) \geq 2$. Assume that $\dim(\Gamma) = 1$, then $\varphi_{m,Y}$ is birational for all $m \geq 33$.

Proof. We have $\theta_1 \geq 1$ and $p_g(T')>0$. By Lemma 3.1, $|mK_{Y'}|$ distinguishes different generic irreducible elements of $|M_1|$ for all $m \geq 3$.

As an overall discussion, we study the linear system $|mK_{Y'}||_C$ for generic irreducible element $C$ of $|H|$. Recall that, by (8) and (9), we already have

where $U_{m,S}\equiv \big((m-1-\frac{1}{\theta_1})\frac{\theta_1}{\theta_1+1}-\frac{t_1}{a_{t_1,T}}\big)\pi_T^*(K_T)|_S$ is a nef and big ${\mathbb Q}$-divisor on $S$. As we have $\pi_T^*(K_T)|_S \sim \beta C+E_H$ for some effective ${\mathbb Q}$-divisor $E_H$ on $S$, applying Kawamata-Viehweg vanishing theorem, we may get

where $\mathcal{D}_m = \ulcorner{{U_{m,S}-C-\frac{1}{\beta}E_H}}\urcorner|_C$ with

Thus, whenever $m>\big(\frac{2}{\xi}+\frac{t_1}{a_{t_1,T}}+\frac{1}{\beta}+1\big)\cdot \frac{\theta_1+1}{\theta_1}$, $|mK_{Y'}||_C$ gives a birational map.

Therefore, by Lemma 3.2, Lemma 3.3 and Theorem 2.2, $\varphi_{m,Y}$ is birational provided that

Now we study this problem according to the value of $p_g(T)$.

Case 1. $p_g(T) \geq 2$

Clearly, we may take $t_1 = 1$ and so $a_{t_1,T} = 1$. From [8,Section 2,Section 3], we know that one of the following cases occurs:

$(1)$ $\beta \geq 1$, $\xi\geq 1$; (correspondingly, $d \geq 2$ in [8,3.1,3.2])

$(2)$ $\beta \geq \frac{1}{2}$, $\xi \geq 2$; (correspondingly, $d = 1$ and $b>0$ in [8,2.8,2.10,3.3])

$(3)$ $\beta \geq \frac{1}{4}$, $\xi \geq 1$; (correspondingly, $d = 1$, $b = 0$ and $(1,1)$-surface case in [8,2.13,3.8])

$(4)$ $\beta \geq \frac{1}{2}$, $\xi \geq \frac{2}{3}$; (correspondingly, $d = 1$, $b = 0$ and $(1,2)$-surface case in [8,2.15,3.7])

$(5)$ $\beta \geq \frac{1}{2}$, $\xi \geq 1$; (correspondingly, $d = 1$, $b = 0$ and $(2,3)$-surface case in [8,2.12,3.6])

$(6)$ $\beta \geq \frac{1}{4}$, $\xi \geq 2$. (correspondingly, $d = 1$, $b = 0$ and other surface case in [8,2.11,3.5])

So $\varphi_{m,Y}$ is birational for all $m\geq 17$.

Case 2. $p_g(T) = 1$

According to [6,Corollary 4.10], $T$ must be of one of the following types: (i) $P_4(T) = 1, P_5(T) \geq 3$; (ii) $P_4(T) \geq 2$.

For Type (i), we have $t_1 = 5$ and set $|N| = \rm{Mov}|5K_{T'}|$. When $|N|$ is composed of a pencil, we have $a_{t_1,T}\geq 2$ and $S$ is exactly the general fiber of the induced fibration from $\varphi_{t_1, T} \circ \pi_T$. If $S$ is not a $(1,2)$-surface, by Proposition 3.4, $\varphi_{m,Y}$ is birational for all $m\geq 27$. If $S$ is a $(1,2)$-surface, by [12,Proposition 4.1,Case 2], we have $\beta\geq \frac{2}{7}$ and $\xi\geq \frac{2}{7}$, and hence $\varphi_{m,Y}$ is birational for $m\geq 29$. When $|N|$ is not composed of a pencil, we have $a_{t_1,T}\geq 1$. Using the case by case argument of [12,Proposition 4.2,Proposition 4.3], to give an exact list, $(\beta,\xi)$ must be one of the following: $(1/5,3/7)$, $(1/5,2/3)$, $(1/3,1/3)$, $(1/5,5/13)$, $(1/5,1)$, $(1/2,1/3)$, $(1/5,1/2)$, $(2/5,1/3)$, $(1/4,1/3)$. Hence $\varphi_{m,Y}$ is birational for all $m\geq 33$.

For Type (ii), we have $t_1 = 4$ and set $|N| = \rm{Mov}|4K_{T'}|$. When $|N|$ is composed of a pencil and the generic irreducible element $S$ of $|N|$ is neither a $(1,2)$-surface nor a $(2,3)$-surface, by Proposition 3.5, $\varphi_{m,Y}$ is birational for all $m\geq 30$. When $P_4(T) = h^0(T,\mathcal{O}_T(4K_T)) = 2$ and $|N|$ is composed of a rational pencil of $(1,2)$-surfaces, the case by case argument of [12,Proposition 4.6,Proposition 4.7] tells that $(\beta,\xi)$ must be one of the following: $(2/7,2/7)$, $(1/5,2/5)$, $(2/5,2/7)$, $(1/5, 1/3)$, $(1/5, 2/3)$, $(1/5, 5/12)$, $(1/3, 2/7)$, $(1/4, 2/7)$. Hence $\varphi_{m,Y}$ is birational for all $m\geq 33$. Otherwise, the case by case argument of [12,Proposition 4.5] tells that $(\beta,\xi)$ must be one of the following: $(1/4,2/5)$, $(1/5,2/5)$, $(1/3,1/3)$. Hence $\varphi_{m,Y}$ is birational for all $m\geq 31$.

In conclusion, $\varphi_{m, Y}$ is birational for all $m \geq 33$.

3.5.   The canonical volume of 4-folds

Theorem 3.8. Let $Y$ be a minimal 4-fold of general type with $p_g(Y) \geq 2$. Then $\mathit{\rm{Vol}}(Y) \geq \frac{1}{480}$.

Proof. We have $\rm{Vol}(Y) = K_Y^4 = (\pi^*(K_Y))^4$.

Recall that $\pi^*(K_Y) \equiv \theta_1T'+E_1$. One has

As we also have (2) and $t_1\pi_T^*(K_T) \equiv a_{t_1,T}S+E_{N}$, it follows that

By (3) and $\pi_T^*(K_T)|_S \geq \beta C$, we may get

and

Now we estimate the canonical volume according to the same classification of $T$ and $S$ as in Subsection 3.3 and Subsection 3.4.

(Ⅰ) The case of $\dim(\Gamma) \geq 2$

Remember that in this case, $\theta_1 = 1, t_1 = 1, a_{t_1,T} = 2$ (by (11)) and $\pi_T^*(K_T)|_S \geq \frac{2}{3} \sigma^*(K_{S_0})$ (by (12)). So we have $\rm{Vol}(Y) \geq \frac{1}{9}.$

(Ⅱ) The case of $\dim(\Gamma) = 1$

Subcase (Ⅱ-1). $p_g(T) \geq 2$.

As in Theorem 3.7, Case 1, $t_1 = 1, a_{t_1,T} = 1$, so we correspondingly have the estimation as follows:

$(1)$ $\beta \geq 1$, $\xi\geq 1$, then $\rm{Vol}(Y) \geq \frac{1}{8}$;

$(2)$ $\beta \geq \frac{1}{2}$, $\xi \geq 2$, then $\rm{Vol}(Y) \geq \frac{1}{8}$;

$(3)$ $\beta \geq \frac{1}{4}$, $\xi\geq 1$, then $\rm{Vol}(Y) \geq \frac{1}{32}$;

$(4)$ $\beta \geq \frac{1}{2}$, $\xi\geq \frac{2}{3}$, then $\rm{Vol}(Y) \geq \frac{1}{24}$;

$(5)$ $\beta \geq \frac{1}{2}$, $\xi\geq 1$, then $\rm{Vol}(Y) \geq \frac{1}{16}$;

$(6)$ $\beta \geq \frac{1}{4}$, $\xi\geq 2$, then $\rm{Vol}(Y) \geq \frac{1}{16}$.

Subcase (Ⅱ-2). $p_g(T) = 1$.

We follow the same classification of $T$ as in Theorem 3.7, Case 2.

Recall that for Type (ⅰ), we have $t_1 = 5$. When $|N|$ is composed of a pencil and the general fiber $S$ of the induced fibration from $\varphi_{t_1, T} \circ \pi_T$ is not a $(1,2)$-surface, we have $a_{t_1,T}\geq 2$, $\beta \geq \frac{1}{7}$, $\xi \geq (\frac{2}{7}\sigma^*(K_{S_0}) \cdot C) \geq \frac{4}{7}$, and thus $\rm{Vol}(Y) \geq \frac{1}{245}$. When $|N|$ is composed of a pencil and the general fiber $S$ is a $(1,2)$-surface, we have $a_{t_1,T}\geq 2$, $\beta\geq \frac{2}{7}$, $\xi\geq \frac{2}{7}$, and hence $\rm{Vol}(Y) \geq \frac{1}{245}$. When $|N|$ is not composed of a pencil, we have $a_{t_1,T}\geq 1$. The two cases corresponding to $(\beta,\xi) = (\frac{1}{5}, \frac{5}{13})$ in Theorem 3.7 Case 2 both have $K_{S_0}^2 \geq 4$, where $S_0$ is the minimal model of a generic irreducible element $S$ of $|N|$. So $\rm{Vol}(Y) \geq \frac{1}{360}$. The corresponding lower bounds of $\rm{Vol}(Y)$ to those of $(\beta,\xi)$ (except $(\beta,\xi) = (\frac{1}{5}, \frac{5}{13})$) are as follows: $\frac{3}{1400}, \frac{1}{300}, \frac{1}{360}, \frac{1}{200}, \frac{1}{240}, \frac{1}{400}, \frac{1}{300}, \frac{1}{480}$.

For Type (ⅱ), we have $t_1 = 4$. When $|N|$ is not composed of a pencil, then $\beta \geq \frac{1}{4}, \xi \geq \frac{2}{5}$ and $\rm{Vol}(Y) \geq \frac{1}{320}$. When $|N|$ is composed of a pencil of $(2,3)$-surfaces, then $\beta \geq \frac{1}{5}, \xi \geq \frac{2}{5}$ and $\rm{Vol}(Y) \geq \frac{1}{400}$. When $|N|$ is composed of a pencil of surfaces with $K_{S_0}^2 \geq 2$, then $\rm{Vol}(Y) \geq \frac{1}{400}$. When $P_4(T) = h^0(T,\mathcal{O}_T(4K_T)) = 2$ and $|N|$ is composed of a rational pencil of $(1,2)$-surfaces, the corresponding lower bounds of $\rm{Vol}(Y)$ to those of $(\beta,\xi)$ are as follows: $\frac{1}{392}, \frac{1}{400}, \frac{1}{280}, \frac{1}{480}, \frac{1}{240}, \frac{1}{384}, \frac{1}{336},\frac{1}{448}.$ Otherwise, $\beta \geq \frac{1}{3}, \xi \geq \frac{1}{3}$ and $\rm{Vol}(Y) \geq \frac{1}{288}$.

So we have shown $\rm{Vol}(Y) \geq \frac{1}{480}$.

3.6.   Proof of Theorem 1.1

Proof. Theorem 3.6, Theorem 3.7 and Theorem 3.8 directly implies Theorem 1.1.

Acknowledgments

The author would like to express her gratitude to Meng Chen for his guidance over this paper and his encouragement to the author. The author would also like to thank Dr. Yong Hu for pointing out the nonexistence of a kind of fibration in Subcase(Ⅱ-2) in the previous version, which improves her result in the previous version. The author would also like to thank the referees for valuable suggestions which improve the result of the lower bound of the canonical volume and make the paper more well-organized.