Proof. See [Reference Fulton19, Theorem 1.4].

Covering Theorem 2.3 [Reference Iitaka22, Theorem 10.5]

Let $f\colon Y \to X$ be a proper surjective morphism between normal complete varieties. If D is a Cartier divisor on X and E an effective f-exceptional divisor on Y, then

$$ \begin{align*}\kappa(Y,f^{*}D + E) = \kappa(X,D).\end{align*} $$

Proof. To prove the connectedness of M, we consider the pencil $\Phi \colon X\dashrightarrow \mathbb {P}^{1}$ induced by M and $M_0$ . Let $\Gamma \subset X\times \mathbb {P}^{1}$ be the closure of the graph of $\Phi $ with projection $\widetilde \Phi \colon \Gamma \to \mathbb {P}^1$ . Denote by $\widetilde {M_0}\subset \Gamma $ the strict transform of $M_0$ . Then $\widetilde \Phi $ has a fiber $\widetilde {M_0}+E$ with E exceptional over X. Since the fiber $\widetilde {M_0} + E$ is connected and non-multiple, by Stein factorization, each fiber of $\widetilde \Phi $ is connected. Therefore, M is connected.

Proof. Since D is nef and big, there exists an effective $\mathbb {Q}$ -Cartier divisor N such that $D-N$ is ample. Therefore, the $\mathbb {Q}$ -divisor $A_k:=D - \frac {1}{k}N=\frac {k-1}{k}D+\frac {1}{k}(D-N)$ is ample for $k\geq 1$ . As $D=A_k+ \frac {1}{k}N$ , it is easy to find the desired divisor $D_\epsilon $ .

Proof. Applying the flattening trick of [Reference Raynaud and Gruson30] (cf. [Reference Witaszek36, Theorem 2.3]), there exists a commutative diagram

where $h_1$ is a projective birational morphism, $f_1$ is flat, $X_1$ is the closure of the generic fiber $X_{K(S)}$ of f in $X\times _S S_1$ , $h_2$ is the normalization morphism and $X_2$ is the normalization of $X_1\times _{S_1}S_2$ . Let $h = h_1\circ h_2$ and $h' = h^{\prime }_1\circ h^{\prime }_2$ . Now $f_2$ is equidimensional, therefore by [Reference Witaszek36, Lemma 2.18], there exists a $\mathbb {Q}$ -divisor $D_2$ on $S_2$ such that $h^{\prime }* L \sim _{\mathbb {Q}} f_2^{*} D_2$ . Since $f_2^{*}D_2$ is $\mathbb {Q}$ -Cartier and $f_2$ is equidimensional, $D_2$ is $\mathbb {Q}$ -Cartier too (see [Reference Druel13, Lemma 2.6]). Then, since $f_2^{*}D_2$ is nef, by [Reference Kleiman25, § I.4, Proposition 1], $D_2$ is nef, and it follows that $D := h_*D_2$ is pseudo-effective. Take $S^{\circ } \subseteq S^{\text {reg}}$ to be a big open subset over which $h_1$ is an isomorphism, then $L|_{f^{-1}(S^{\circ })}\sim _{\mathbb {Q}} f^{*}(D|_{S^{\circ }})$ .

Proof. Applying the adjunction formula of Lemma 2.7, we may write that

$$ \begin{align*}\bigl( (K_X+ \Delta) + (1-a)T \bigr)|_{T^\nu}\sim_{\mathbb{Q}} (K_X + T + \Delta')|_{T^\nu} \sim_{\mathbb{Q}} K_{T^\nu} + \Delta_{T^\nu} + \Delta'|_{T^\nu},\end{align*} $$

where $\Delta _{T^\nu }\ge 0$ . We then deduce (1) by setting $B_{T^\nu } = \Delta _{T^\nu } + \Delta '|_{T^\nu }$ .

Proof. The statement (1) follows from [Reference Zhang37, Theorem 4.1], where X is assumed to be smooth, but the proof also applies to normal varieties.

To show (2), let $\kappa (X,K_X) = 0$ . We may chose $K_Y$ so that $\rho _*K_Y = K_X$ , then $\kappa (X,K_X)\ge \kappa (Y,K_Y) \ge 0$ . Thus $\kappa (Y,K_Y) = 0$ . By [Reference Hacon, Patakfalvi and Zhang20, Theorem 0.2], the Albanese morphism $a_Y\colon Y\to A_Y$ is birational. Note that, the composition morphism $Y\to X\to A$ factors into $Y\buildrel a_Y\over \to A_Y \buildrel \pi \over \to A$ . We see that $\pi \colon A_Y \to A$ is surjective. Moreover, since $a_X\colon X\to A$ is generically finite, $\pi $ is generically finite. Then, as $\pi $ is a morphism of abelian varieties, it is finite. It follows that $A_Y$ is the normalization of A in $K(Y)=K(X)$ . Thus, there is a birational morphism $X\to A_Y$ .

The statement (3) is [Reference Ejiri and Patakfalvi15, Proposition 3.2].

Proof. Since the statement involves only codimension one points, we may restrict to a big open subset of S and assume that S is regular.

By assumption we have $p_a(X_{K(S)})=0$ , thus $\deg _{K(S)} (M_0+\Delta ) = 2$ . In the following, we take $M\in \mathfrak M$ to be a general member and T one of its irreducible horizontal components. Write $M= T + G$ . We get $G|_{T^\nu } \succeq _{\mathbb {Q}} 0$ by the following lemma.

Write $\Delta =\alpha T + \Delta '$ with $T\wedge \Delta ' = 0$ , and then $0\le \alpha \le 1$ as $\deg _{K(S)} (M_0+\Delta ) = 2$ . We may write

$$\begin{align*}G+\Delta = G+\alpha T+\Delta' = (1-\alpha)G+\alpha M +\Delta', \end{align*}$$

thus $(G+\Delta )|_{T^\nu } \succeq _{\mathbb {Q}} 0$ .

Case (1): $\deg _{K(S)} T =1$ . Then $T \to S$ is a birational section. By adjunction formula, there exists an effective divisor $E_{T^\nu }$ on $T^\nu $ such that

$$ \begin{align*}(K_X+ M+\Delta)|_{T^\nu} = (K_X + T + G + \Delta)|_{T^\nu} \sim_{\mathbb{Q}} K_{T^\nu} + E_{T^\nu}.\end{align*} $$

Denote by $\sigma \colon T^\nu \to S$ the natural birational morphism. Now we have $K_{T^\nu } + E_{T^\nu } \sim _{\mathbb {Q}} \sigma ^{*}D$ , and then by Lemma 2.2 we obtain

$$ \begin{align*}D \sim_{\mathbb{Q}} \sigma_*(K_{T^\nu} + E_{T^\nu}) = K_S + E_S,\end{align*} $$

where $E_S \geq 0$ .

Case (2): $\deg _{K(S)} T =2$ .

Case (2.1): $T \to S$ is a separable morphism. As in case (1), there exists $E_{T^\nu } \geq 0$ such that

(13) $$ \begin{align} \sigma^{*}D\sim_{\mathbb{Q}} (K_X+ M + \Delta)|_{T^\nu}\sim_{\mathbb{Q}} K_{T^\nu} + E_{T^\nu}. \end{align} $$

Since $\sigma \colon T^\nu \to S$ is separable, $K_{T^\nu } \succeq \sigma _*K_S+R_{T^\nu }+N_{T^\nu }$ , where $R_{T^\nu }\ge 0$ and $N_{T^\nu }$ is exceptional. Now, pushing forward (13) via $\sigma $ gives

$$ \begin{align*}D \sim_{\mathbb{Q}} \frac12\sigma_*(K_{T^\nu} + E_{T^\nu}) \succeq_{\mathbb{Q}} K_S.\end{align*} $$

Case (2.2): $T \to S$ is inseparable, which only happens when $p=2$ . Let $S_1$ be the normalization of S in $K(T)$ . In other words, the natural morphisms $T^\nu \to S_1 \to S$ form the Stein factorization of $\sigma $ . Hence, $T^\nu \to S_1$ is birational and $S_1\to S$ is finite. Since $K(S_1)/K(S)$ is a purely inseparable extension of degree two, $X_{K(S_1)}$ is integral by Proposition 4.2. By shrinking S again, we may assume that $K_{S_1}$ is $\mathbb {Q}$ -Cartier.

Case (2.2.1): $X_{K(S_1)}$ is normal. Consider the following commutative diagram

The morphism $\pi \colon X_1 \to X$ is finite and purely inseparable, and the normalization morphism ${\nu \colon X_1 \to X_{S_1}}$ induces an isomorphism of the generic fibers over $S_1$ . Since $X_{S_1} \to S_1$ has a birational section which is mapped to T, the prime divisor $T_1$ supported on $\pi ^{-1}T$ is a birational section, and $\pi ^{*}T = 2T_1$ . Applying Proposition 3.5 and Remark 3.6, we have

(14) $$ \begin{align} \pi^{*}(K_X - f^{*}K_S) \sim_{\mathbb{Q}} K_{X_1} - g^{*}K_{S_1} + E_1 + N, \end{align} $$

where $E_1$ is effective and g-vertical, and N is exceptional over S. Write $\pi ^{*}M = 2T_1 + E_1'$ . Then

$$ \begin{align*}\pi^{*}(K_X+ M+\Delta) \sim_{\mathbb{Q}} K_{X_1} + 2T_1 + g^{*}\tau^{*}K_S - g^{*}K_{S_1} + E_1 + \pi^{*}\Delta + E_1' + N.\end{align*} $$

Denote by $\sigma _1\colon T^\nu _1 \buildrel \nu \over \to T_1 \to S_1$ the composition morphism. Consider the restriction on $T^\nu _1$ . Applying the adjunction formula $(K_{X_1} + T_1)|_{T^\nu _1} \sim _{\mathbb {Q}} K_{T^\nu _1} + \Delta _{T^\nu _1}$ , we have

(15) $$ \begin{align} \begin{aligned} \tau^{*}D|_{T^\nu_1} &\sim_{\mathbb{Q}}(K_{X_1} + T_1 + T_1 + g^{*}\tau^{*}K_S - g^{*}K_{S_1} +E_1 + \pi^{*}\Delta + E_1' + N)|_{T^\nu_1}\\ & \sim_{\mathbb{Q}} (K_{T^\nu_1} - \sigma_1^{*}K_{S_1}) + \Delta_{T^\nu_1} +(T_1+E_1' + E_1+\pi^{*}\Delta)|_{T^\nu_1}+ \sigma_1^{*}\tau^{*}K_S + N|_{T^\nu_1}. \end{aligned} \end{align} $$

Note that

• ○ $(T_1+E_1')|_{T^\nu _1}\succeq _{\mathbb {Q}} 0$ by Lemma 5.2 since $\pi ^{*}M = 2T_1 + E_1'$ ;

• ○ both $K_{T^\nu _1} - \sigma _1^{*}K_{S_1}$ and $N|_{T^\nu _1}$ are exceptional over S;

• ○ both $E_1$ and $\Delta $ are vertical over S, and consequently, the restriction $(E_1+\pi ^{*}\Delta )|_{T_1^\nu }$ is effective.

Applying $\tau _*\sigma _{1*}$ to (15), by Lemma 2.2 we obtain that $D \sim _{\mathbb {Q}} K_S + E_S$ for some divisor $E_S \geq 0$ .

Case (2.2.2): $X_{K(S_1)}$ is not normal. This means that $X_{K(S)}$ is a non-smooth conic as described in Proposition 4.2 (3). In this case, f is inseparable and we assume that one of the conditions (b, c) holds.

Applying Proposition 4.2, we see that $X_{K(S_1)}$ is not normal along the preimage of the generic point of T, and $K(S_1)$ is not algebraically closed in $K(X)\otimes _{K(S)}K(S_1)$ . Let $X_1:=(X_{S_1})^\nu $ and denote by $S_1'$ the normalization of $S_1$ in $X_1$ . Let $T_1$ be the irreducible divisor corresponding to $\pi ^{-1}T$ . Denote by $\rho \colon T^\nu _1 \to T^\nu $ the induced morphism by the normalizations. These varieties fit into the following commutative diagram

The conductor divisor of $X_1 \to X_{S_1}$ is like $aT_1 + C'$ where $a\geq 1$ , namely, $\nu ^{*}K_{X_{S_1}}= K_{X_1} + aT_1 + C'$ . Similarly to (14), we have

$$ \begin{align*}\pi^{*}K_X \sim_{\mathbb{Q}} K_{X_1} + aT_1 + E_1 + N_1 - \nu^{*}f_1^{*}K_{S_1/S} ,\end{align*} $$

where $E_1$ is effective and $N_1$ is g-vertical. We may write that $\pi ^{*}T = bT_1$ and $\pi ^{*}M = bT_1 + E_1'$ . Then

$$ \begin{align*}\pi^{*}(K_X+ M+\Delta) \sim_{\mathbb{Q}} K_{X_1} + (a+b)T_1 + E_2 + N_1 - \nu^{*}f_1^{*}K_{S_1/S},\end{align*} $$

where $E_2 = E_1 + \pi ^{*}\Delta + E^{\prime }_1$ is effective. Since $(K_X+ M+\Delta )_{K(S)} \sim 0$ , we conclude that $a=b=1$ and that $T_1 \to S^{\prime }_1$ is birational. Note that $T\not \subseteq \operatorname {\mathrm {Supp}}\Delta $ , thus Lemma 2.8 gives

$$ \begin{align*}T_1|_{T^\nu_1} = \rho^{*}(T|_{T^\nu}) \sim_{\mathbb{Q}} \rho^{*}(K_{T^\nu} + B_{T^\nu} + M|_{T^\nu}) - \pi^{*}(K_X+M+\Delta)|_{T^\nu_1},\end{align*} $$

where $B_{T^\nu }\ge 0$ . Applying the adjunction formula on $T_1$ we have

$$ \begin{align*} \begin{aligned} \pi^{*}(K_X+ M+\Delta)|_{T^\nu_1} & \sim_{\mathbb{Q}} (K_{X_1} + T_1)|_{T^\nu_1} + T_1|_{T^\nu_1} + (E_2 + N_1 - \nu^{*}f_1^{*}K_{S_1/S})|_{T^\nu_1} \\ &\sim_{\mathbb{Q}} K_{T^\nu_1} + \Delta_{T^\nu_1} + \rho^{*}(K_{T^\nu} + B_{T^\nu}+M|_{T^\nu}) - \pi^{*}(K_X+M+\Delta)|_{T^\nu_1} \\ & \qquad + (E_2 + N_1 - \nu^{*}f_1^{*}K_{S_1/S})|_{T^\nu_1} \end{aligned} \end{align*} $$

It follows that

(16) $$ \begin{align} \begin{aligned} 2\pi^{*}(K_X+ M+\Delta)|_{T^\nu_1} \sim_{\mathbb{Q}} {}& (\pi^{*}f^{*}K_S)|_{T_1^\nu} + K_{T^\nu_1} \\ &+ \Delta_{T^\nu_1} + \rho^{*}B_{T^\nu} + \pi^{*}M|_{T^\nu_1}+ E_2|_{T^\nu_1} + N_2 \end{aligned} \end{align} $$

where $N_2= N_1|_{T^\nu _1} + \rho ^{*}(K_{T^\nu }-\delta ^{*}K_{S_1})$ is exceptional over S.

If condition (b) holds, then there is a generically finite morphism $T_1^\nu \to A$ , where A is an abelian variety. Using Nagata’s compactification over A, we can embed $T_1^\nu $ into a proper and normal birational model $\overline {T}_1$ over A. Since $T_1^\nu $ is quasi-projective, by Chow’s lemma ([Reference Conrad11, Corollary 2.6]), we may assume that $\overline {T}_1$ is projective. Since $\overline {T}_1$ admits a generically finite morphism to an abelian variety, $\overline {T}_1$ is of m.A.d. Applying Proposition 2.9, we have that $K_{\overline {T}_1} \succeq _{\mathbb {Q}} 0$ , and thus $K_{T_1^\nu } \succeq _{\mathbb {Q}} 0$ . Note that $2\pi ^{*}(K_X+ M+\Delta )|_{T^\nu _1} = 2\tau ^{*}D|_{T^\nu _1}$ . Pushing down the equation (16) via the morphism $T^\nu _1 \to S$ yields $D \sim _{\mathbb {Q}} \frac {1}{2}K_S + E_S$ for some divisor $E_S \geq 0$ .

If the condition (c) holds, then $S_1'\to S$ is of height one and of degree $\geq p^2=4$ , this implies . It follows that where $N_3$ is a divisor on $T^\nu _1$ exceptional over S. Pushing down the equation (16) via the morphism $T^\nu _1 \to S$ yields $D \sim _{\mathbb {Q}} \frac {3}{4}K_S + E_S$ for some divisor $E_S \geq 0$ .

Proof. Let $M \in \mathfrak M$ be a general divisor and write $M = T + M' + V$ where T is a horizontal irreducible component, V is the vertical part, and $M'$ is the remaining part. Since $\deg _{K(S)} M_0 \leq 2$ , we have $M'=0$ , $M'=T$ , or $M'$ is another horizontal component. Since T is dominant over S, it is of m.A.d., and we have $K_{T^\nu } \succeq _{\mathbb {Q}} 0$ . We have $(M'+ V)|_{T^\nu } \succeq _{\mathbb {Q}} 0$ by Lemma 5.2.

For convenience, we first establish a lemma.

Proof of the lemma

We first remark that

1. (*) Suppose $V \neq 0$ , as M varies in $\mathfrak M$ we get a family of numerically equivalent horizontal divisors $\mathfrak T$ and a family of numerically equivalent vertical divisors $\mathfrak V$ , which contains T and V respectively. Both of the families of divisors cover X, therefore, $\operatorname {\mathrm {Supp}} T \cap \operatorname {\mathrm {Supp}} V \neq \emptyset $ .

By the adjunction formula, we have

(17) $$ \begin{align} (K_{X} + M +\Delta)|_{T^\nu} = (K_{X} + T+ (M'+V)+ \Delta)|_{T^\nu} \sim_{\mathbb{Q}} K_{T^\nu} + \Delta_{T^\nu} + (M'+V)|_{T^\nu} + \Delta|_{T^\nu}, \end{align} $$

which is anti-nef. Since $K_{T^\nu }, \Delta _{T^\nu }, (M'+ V)|_{T^\nu }$ and $\Delta |_{T^\nu }$ (note that $T\wedge \Delta =0$ for a general M) are $\mathbb {Q}$ -effective, we see that

$$ \begin{align*}(K_X + M+\Delta)|_{T^\nu} \equiv 0,\quad K_{T^\nu} \sim_{\mathbb{Q}} (M' + V)|_{T^\nu}\sim_{\mathbb{Q}} 0 \text{ \ and \ } \Delta_{T^\nu} = \Delta|_{T^\nu} = 0.\end{align*} $$

We show that $V|_{T^\nu } \sim _{\mathbb {Q}} 0$ by distinguishing the following two cases:

1. 1. If $M'\neq T$ , then $M'|_{T^\nu } \succeq _{\mathbb {Q}} 0$ and thus $V|_{T^\nu } \sim _{\mathbb {Q}} 0$ .

2. 2. If $M'=T$ , we can take another element $M_1\in |M_0|$ so that $T \not \subseteq M_1$ . Then by

   $$\begin{align*}0 \sim_{\mathbb{Q}} (2T+2V)|_{T^\nu} \sim (M_1 + V)|_{T^\nu} \sim_{\mathbb{Q}} M_1|_{T^\nu} + V|_{T^\nu} \succeq_{\mathbb{Q}} 0 \end{align*}$$
   it follows that $V|_{T^\nu } \sim _{\mathbb {Q}} 0$ .

We conclude that

• ○ $\operatorname {\mathrm {Supp}} T\cap \operatorname {\mathrm {Supp}} V =\varnothing $ , hence $(*)$ implies that a general $M \in \mathfrak M$ has no vertical part;

• ○ by Lemma 2.7 and Proposition 2.9, $T = T^\nu $ is isomorphic to an abelian variety;

• ○ $(K_{X} + M +\Delta )|_{M} \equiv 0$ .

This completes the proof of the lemma.

We proceed with the proof of the proposition. We first prove the assertions (i) and (ii) by dividing the argument into two cases based on whether $\deg _{K(S)} (K_X + M_0 + \Delta )$ equals zero or not.

In the following we take a general divisor $M \in \mathfrak M$ . Remark that to prove that $M_0$ is semi-ample, it suffices to verify that on any irreducible component T of M the restriction $ M|_{T} \equiv 0$ , which implies that $|M_0|$ is base-point-free with $\nu (M_0) =1$ .

Case (1): $\deg _{K(S)} (K_X + M_0 + \Delta ) <0$ . In this case $\deg _{K(S)} M_0 =1$ , a general $M\in |M_0|$ is reduced and irreducible, and $(X_{K(S)}, \Delta _{K(S)})$ is klt. Let $ N = -(K_{X} + M +\Delta )$ . Then $ N|_{M^\nu } \equiv 0$ by Lemma 5.4.

Take an ample divisor H on S. For any rational number $\epsilon>0$ , the divisor $N+ \epsilon f^{*}H$ is nef and big, thus $\mathbb {Q}$ -effective. Therefore

$$ \begin{align*}K_X+ M +\Delta+ \text{(effective)} \sim_{\mathbb{Q}} \epsilon f^{*}H.\end{align*} $$

Thus, by Theorem 5.1, we have $\epsilon H\succeq _{\mathbb {Q}} \frac 12 K_{S}$ . But $K_S$ is $\mathbb {Q}$ -effective by Proposition 2.9 (1), thus letting $\epsilon \to 0$ , we obtain $K_S\sim _{\mathbb {Q}} 0$ . Therefore, S is an abelian variety by Proposition 2.9 (3).

Next, we show that $M_0$ is semi-ample by verifying that the restriction $ M|_{M} \equiv 0$ . Take a surface W which is the intersection of $\dim X -2$ very general hyperplane sections on X intersecting the base locus of $|M|$ properly. It suffices to show that $\nu (M|_W) =1$ . Otherwise, we have $(M |_W)^2>0$ , since both $M|_W$ and $N|_W$ are nef but not numerically trivial and $( M |_W)\cdot ( N |_W)=0$ by Lemma 5.4, Hodge Index Theorem implies that $( N|_W)^2 <0$ , which contradicts the fact that $ N|_W$ is nef.

Case (2): $\deg _{K(S)} (K_X + M_0 + \Delta ) = 0$ .

Since $(K_X + M_0 + \Delta )|_{X_{K(S)}}\sim _{\mathbb {Q}}0$ , by Lemma 2.6 there exists a big open subset $S^{\circ }\subseteq S$ and a pseudo-effective $\mathbb {Q}$ -Cartier $\mathbb {Q}$ -divisor D on S such that $-(K_X + M_0 + \Delta )|_{X_{S^{\circ }}}\sim _{\mathbb {Q}} f^{*}D^{\circ }$ , where $D^{\circ }:=D|_{S^{\circ }}$ . By Theorem 5.1, we have $-D^{\circ } \succeq _{\mathbb {Q}} \frac 12 K_{S^{\circ }}$ . On the other hand, since S is of m.A.d., $K_{S^{\circ }} \succeq _{\mathbb {Q}} 0$ . We conclude that $K_S \sim _{\mathbb {Q}} 0$ which implies that S is an abelian variety.

Next, we prove that $M_0$ is semi-ample. With the notation at the beginning $M= T+M'$ , by Lemma 5.4, $M'|_{T} \equiv 0$ , thus we only need to show that $T|_{T} \equiv 0$ . We divide the argument into the following cases.

Case (2.I): $M' \neq 0$ . If $M'= T$ then we are done since $M'|_{T^\nu }\equiv 0$ . Assume $M' \neq T$ , then $M' \cap T = \emptyset $ . We apply a similar strategy as in Case (1). Take a surface W which is the intersection of $\dim X -2$ very general hyperplane sections on X intersecting the base locus of M properly. Since both $M'|_W$ and $T|_W$ are nef but not numerically trivial and $(M'|_W)\cdot (T|_W)=0$ , we can apply Hodge Index Theorem to show that $(T|_W)^2 = (M'|_W)^2=0$ as in in Case (1). This indicates $T|_T \equiv 0$ .

In the remaining cases, we assume that M is reduced and irreducible.

Case (2.II): $\deg _{K(S)} M_0=1$ . In this case, $\deg _{K(S)} \Delta> 0$ , thus $\operatorname {\mathrm {Supp}}\Delta $ must contain a horizontal component $T_1$ . By Lemma 5.4, we know that $T_1$ does not intersect a general M, but the family of divisors in $|M_0|$ covers X, thus $T_1$ is a component of certain $M_1 \in |M_0|$ and $\deg _{K(S)} T_1 =1$ . We may write that

$$ \begin{align*}M_1 = T_1+ V_1\ \ \text{and}\ \ \Delta = aT_1+ \Delta',\quad 0<a\le 1\end{align*} $$

where $V_1$ is a vertical divisor and a is the coefficient of $T_1$ in $\Delta $ . Then by the adjunction formula, we have

(18) $$ \begin{align} \begin{aligned} (K_X + M_1+ \Delta)|_{T^\nu_1} &\sim_{\mathbb{Q}} (K_X + T_1 + aM_1 + (1-a)V_1 + \Delta' )|_{T^\nu_1} \\ &\sim_{\mathbb{Q}} K_{T^\nu_1} + \Delta_{T^\nu_1}+ aM_1|_{T^\nu_1} + (1-a)V_1|_{T^\nu_1} + \Delta'|_{T^\nu_1}. \end{aligned} \end{align} $$

Since $K_{T^\nu _1}, M_1|_{T_1^\nu }, V_1|_{T^\nu _1},\Delta '|_{T_1^\nu } \succeq _{\mathbb {Q}} 0$ and $K_X + M_1+ \Delta $ is anti-nef, we conclude that

$$ \begin{align*}(K_X + M_1+ \Delta)|_{T^\nu_1}\equiv0,\quad K_{T^\nu_1} \sim_{\mathbb{Q}} \Delta_{T^\nu_1} \sim_{\mathbb{Q}} M_1|_{T^\nu_1} \sim_{\mathbb{Q}} (1-a)V_1|_{T^\nu_1} \sim_{\mathbb{Q}} \Delta'|_{T^\nu_1} \sim_{\mathbb{Q}} 0.\end{align*} $$

Since $M_1=T_1+V_1$ is connected by Lemma 2.4, thus if $a<1$ we obtain $V_1=0$ by $V_1|_{T^\nu _1} \sim _{\mathbb {Q}} 0$ ; and if $a=1$ , then by $M|_{T^\nu _1} \sim _{\mathbb {Q}} (T_1+V_1)|_{T^\nu _1} \sim _{\mathbb {Q}} 0$ and by the assumption that $T_1|_{T^\nu _1}$ is pseudo-effective, we also have $V_1|_{T^\nu _1} \sim _{\mathbb {Q}} 0$ and thus $V_1=0$ . In turn we conclude that $M_1 = T_1$ and $M|_{T_1^\nu } \sim _{\mathbb {Q}} 0$ . Therefore $|M|$ is semi-ample with $\nu (M)=1$ .

Case (2.III): $\deg _{K(S)} M_0 =2$ . We fall into one of the following two cases:

1. (III-1) The generic fiber $X_{K(S)}$ is not geometrically normal (this can happen only when $p=2$ ), that is, $X_{K(S)}$ is a non-smooth conic over $K(S)$ .

2. (III-2) $X_{K(S)}$ is smooth over $K(S)$ .

Fix a general divisor $M_1\in |M_0|$ , which is a prime divisor. By Lemma 5.4, $M_1$ is an abelian variety. Let us rename it by $S':= M_1$ and consider the base change via the degree two morphism $S'\to S$ . Note that in each of the above two cases, $X_{K(S')}$ is integral by Proposition 4.2 or [Reference Schröer31, Lemma 1.3]. Let $\nu \colon X_1:=(X_{S'})^{\nu }\to X_{S'}$ be the normalization morphism. Let $f_1\colon X_1 \to S_1$ be the fibration arising from the Stein factorization of $X_1\to S'$ . We have the following commutative diagram

In case (III-1), by Proposition 4.2 (3), $X_{S'}$ is not normal along $\pi ^{\prime }{-1}M_1$ , and $S_1 \to S'$ is a finite purely inseparable morphism of degree two. Moreover, if we denote by $T_1$ the prime divisor such that $T_1 = \operatorname {\mathrm {Supp}} \pi ^{*}M_1$ , then $T_1$ is a birational section over $S_1$ and $\pi ^{*}M_1 = T_1$ . We may write that

$$ \begin{align*}\pi^{*}K_{X} \sim K_{X_1} + T_1 + V_1,\end{align*} $$

where $V_1 \geq 0$ (by Proposition 3.4) is a vertical divisor over $S_1$ . Hence

$$ \begin{align*}\pi^{*}(K_{X} + M + \Delta) \sim_{\mathbb{Q}} K_{X_1} + T_1 + \pi^{*}M + V_1 + \pi^{*}\Delta .\end{align*} $$

Applying the adjunction formula on $T_1$ , we have

$$ \begin{align*}(K_{X} + M + \Delta) |_{T^\nu_1} \sim_{\mathbb{Q}} (K_{X_1} + T_1 + \pi^{*}M + V_1 + \pi^{*}\Delta)|_{T^\nu_1} \sim_{\mathbb{Q}} K_{T^\nu_1} + \Delta_{T^\nu_1} + (\pi^{*}M + V_1 + \pi^{*}\Delta)|_{T^\nu_1}.\end{align*} $$

Since $K_{X} + M + \Delta $ is anti-nef and $K_{T^\nu _1}, \Delta _{T^\nu _1}, \pi ^{*}M|_{T^\nu _1}, (V_1 + \pi ^{*}\Delta )|_{T^\nu _1}\ge 0$ , we conclude that $\pi ^{*}M|_{T^\nu _1}=0$ . Since $\pi ^{*}M\sim T_1$ , we have $T_1|_{T_1^{\nu }} \succeq _{\mathbb {Q}} 0$ , which implies that $M|_M \equiv 0$ .

In case (III-2), we have $S_1=S'$ and

$$ \begin{align*}\pi^{*}K_{X} \sim K_{X_1} + V_1\end{align*} $$

for some effective divisor $V_1$ vertical over $S_1$ .

If $M_1 \to S$ is purely inseparable, then $\pi ^{*}M_1 = 2T_1$ where $T_1$ is a birational section over $S_1$ . Applying the adjunction formula, we have

$$ \begin{align*}(K_{X} + M + \Delta) |_{T^\nu_1} \sim_{\mathbb{Q}} (K_{X_1} + T_1 + T_1 + V_1 + \pi^{*}\Delta)|_{T^\nu_1} \sim_{\mathbb{Q}} K_{T^\nu_1} + \Delta_{T^\nu_1} + (T_1 + V_1 + \pi^{*}\Delta)|_{T^\nu_1}.\end{align*} $$

Since $T_1|_{T^\nu _1} \sim _{\mathbb {Q}} \frac {1}{2}\pi ^{*}M_1|_{T^\nu _1} \succeq _{\mathbb {Q}} 0$ , we conclude as before that $\pi ^{*}M|_{T^\nu _1} \sim _{\mathbb {Q}} 0$ , which implies that $M|_{M} \sim _{\mathbb {Q}} 0$ .

Now consider the case that for general $M \in |M_0|$ , $M \to S$ is a separable morphism of degree two. Since M is isomorphic to an abelian variety, the natural morphism $M \to S$ is an étale morphism of abelian varieties. Note that

• ○ up to isomorphism, there are only finitely many abelian varieties étale over S of degree two;

• ○ if $M\in |M_0|$ is birationally equivalent to $M_1$ then $K(M_1)\otimes _{K(S)} K(M) \cong K(M)\times K(M)$ , hence $\pi ^{*}M$ splits into two distinct components.

From this we conclude that general M belongs to the same birational equivalent class, and $\pi ^{*}M$ splits into the sum of two divisors $T_1 + T_2$ which varies as M varies. We may consider the fibration $X_{S'} \to S'$ and apply the argument of Case (2.I) to conclude that $M|_{M}\equiv 0$ .

We now prove the statement (iii). By (ii), the linear system $|M_0|$ induces a fibration $g\colon X\to C$ onto a smooth curve C. Since the generic fiber $X_\eta $ of $f\colon X\to S$ has arithmetic genus zero and is dominant over C, the curve C is isomorphic to $\mathbb {P}^{1}$ .

Finally, let us prove the statement (iv). Denote by $G_t$ the fiber of $g\colon X\to \mathbb {P}^{1}$ over $t\in \mathbb {P}^{1}$ . Let T be an f-horizontal component of $G_t$ . Write that

$$\begin{align*}G_t = aT + G_t' \text{ \ and \ } \Delta = bT + \Delta' \end{align*}$$

with $(G_t' + \Delta ')\wedge T = 0$ , where $0\le b\le 1$ .

If $b<1$ , setting $c = (1-b)/a$ , we have

$$\begin{align*}(K_X + cG_t + \Delta)|_{T^\nu} = (K_X + T + cG_t' + \Delta')|_{T^\nu} \sim_{\mathbb{Q}} K_{T^\nu} + \Delta_{T^\nu} + cG_t'|_{T^\nu} + \Delta'|_{T^\nu}, \end{align*}$$

which is anti-nef since $G_t|_{T^\nu } \sim 0$ . Since $\Delta _{T^\nu },cG_t'|_{T^\nu },\Delta '|_{T^\nu },K_{T^\nu } \succeq _{\mathbb {Q}}0$ , it follows that $\Delta _{T^\nu }=G_t'|_{T^\nu }=\Delta '|_{T^\nu }=0$ and $K_{T^\nu } \equiv 0$ . By Lemma 2.7 and Proposition 2.9, T is normal and isomorphic to an abelian variety. Moreover, as $c>0$ , we have $G_t' |_{ T } \sim _{\mathbb {Q}} 0$ , which implies $G_t' = 0$ since $G_t$ is connected. Therefore, $G_t$ is isomorphic to either an abelian variety or a multiple of an abelian variety. By the above argument, we have $\Delta |_{T} \equiv 0$ ; in turn we conclude that $\text {Supp}\, \Delta $ is contained in finitely many closed fibers of g. Moreover, since g is a fibration onto a curve, the generic fiber of g is geometrically reduced (see [Reference Schröer31, Corollary 2.5]). Therefore, a general fiber $G_t$ of g is reduced and thus is isomorphic to an abelian variety.

If $b=1$ , then by assumption, $T|_{T^\nu }$ is pseudo-effective, and thus so is $\Delta |_{T^\nu }$ . Applying the adjunction formula, we have

$$\begin{align*}(K_X + G_t + \Delta)|_{T^\nu} = (K_X + T + G_t' + \Delta)|_{T^\nu} \sim_{\mathbb{Q}} K_{T^\nu} + \Delta_{T^\nu} + G_t'|_{T^\nu} + \Delta|_{T^\nu}. \end{align*}$$

By a similar argument as in the case $b<1$ , we can finish the proof.

Proof. Since the statements involve only codimension one points, we may assume that S and $S_1$ are both regular. Here, once $\bar T$ is constructed over an open subset of S, we can replace it with the normalization of S in $K(\bar T)$ . This ensures that $\tau $ is finite. The same applies to $\bar T'$ and $\tau ^{\prime }_1$ .

Let $\nu \colon X'\to X_{S_1}$ be the normalization morphism. Since $X_{K(S_1)}$ is not normal, we can find an f-horizontal irreducible component $T'$ on $X'$ of the conductor divisor. Denote by $\pi \colon X'\to X$ the induced morphism, which is purely inseparable of height one. Let $f'\colon X' \to S'$ be the fibration arising from the Stein factorization of $X' \to S_1$ . These varieties fit into the following commutative diagram

By Proposition 3.5, we have

(19) $$ \begin{align} \pi^{*}(K_X + \Delta) \sim_{\mathbb{Q}} K_{X'}+ (p-1)(\gamma_1T' + E') + \pi^{*}\Delta + f_1^{*}(\tau_1^{*}K_S - K_{S_1}) + N, \end{align} $$

where $E'$ is effective, $T'\wedge E'=0$ and N is exceptional over S. Since $p_a(X^{\prime }_{K(S')})=0$ , we have $\deg _{K(S')}(p-1)(\gamma _1 T' + E' + \pi ^{*} \Delta ) = 2$ , thus $\deg _{K(S')}T'=\gamma _1=1$ if $p=3$ and $\gamma _1\cdot \deg _{K(S')} T' \le 2$ if $p=2$ . We may assume $N=0$ since it does not affect our result.

Let $\rho \colon T^{\prime }\nu \to T^\nu $ be the morphism arising from the normalization of $T'\to T$ . We may write $\pi ^{*}T = \gamma _2T'$ where $\gamma _2 = 1$ or p. Write $\Delta = \alpha T+\Delta '$ with $T\wedge \Delta '=0$ . By the adjunction formula there exists an effective $\mathbb {Q}$ -divisor $\Delta _{T^{\prime }\nu }$ on $X'$ such that $(K_{X'} + T')|_{T^{\prime }\nu } \sim _{\mathbb {Q}} K_{T^{\prime }\nu } + \Delta _{T^{\prime }\nu }$ . By the relation (19) we have

(20) $$ \begin{align} \begin{aligned} \pi^{*}(K_X+\Delta)|_{T^{\prime}\nu} &{}\sim_{\mathbb{Q}} (K_{X'} + T')|_{T^{\prime}\nu} + \bigl((p-1)\gamma_1+\alpha \gamma_2-1\bigr)T'|_{T^{\prime}\nu} \\ & \quad \quad+\bigl((p-1)E'+\pi^{*}\Delta'\bigr)|_{T^{\prime}\nu} + f_1^{*}(\tau_1^{*}K_S - K_{S_1})|_{T^{\prime}\nu}\\ &{}\sim_{\mathbb{Q}} K_{T^{\prime}\nu} + \Delta_{T^{\prime}\nu} + \bigl((p-1)\gamma_1+\alpha \gamma_2-1\bigr)T'|_{T^{\prime}\nu} \\ &\quad \quad+\bigl((p-1)E'+\pi^{*}\Delta'\bigr)|_{T^{\prime}\nu} + f_1^{*}(\tau_1^{*}K_S - K_{S_1})|_{T^{\prime}\nu}. \end{aligned} \end{align} $$

By Lemma 2.8, there exists an effective divisor $B_{T^\nu }$ on $T^\nu $ such that

(21) $$ \begin{align} (1-\alpha )\gamma_2T'|_{T^{\prime}\nu} \sim (1-\alpha )\rho^{*}(T|_{T^\nu}) \sim_{\mathbb{Q}} \rho^{*}(K_{T^\nu} + B_{T^\nu})|_{T^{\prime}\nu} - \pi^{*}(K_X+\Delta)|_{T^{\prime}\nu}. \end{align} $$

Multiplying the equations (20) and (21) by $(1-\alpha )\gamma _2$ and $\bigl ((p-1)\gamma _1+\alpha \gamma _2-1\bigr )$ respectively and then summing up, we obtain

(22) $$ \begin{align} \begin{aligned} &\bigl((1-\alpha )\gamma_2+ (p-1)\gamma_1 + \alpha \gamma_2-1\bigr)\pi^{*}(K_X+\Delta)|_{T^{\prime}\nu} \\ &\sim_{\mathbb{Q}} (1-\alpha )\gamma_2K_{T^{\prime}\nu} + \bigl((p-1)\gamma_1+ \alpha \gamma_2-1\bigr)\rho^{*}K_{T^\nu} \\ & + (1-\alpha )\gamma_2f_1^{*}(\tau_1^{*}K_S - K_{S_1})|_{T^{\prime}\nu} +E_{T^{\prime}\nu}, \end{aligned} \end{align} $$

where $E_{T^{\prime }\nu }:= (1-\alpha )\gamma _2\Delta _{T^{\prime }\nu } + (1-\alpha )\gamma _2\bigl ((p-1)E'+\pi ^{*}\Delta '\bigr )|_{T^{\prime }\nu } + \bigl ((p-1)\gamma _1+\alpha \gamma _2-1\bigr )\rho ^{*}B_{T^\nu }\ge 0$ .

Finally we denote by $\bar {T}, \bar {T}'$ the normalization of S in $T^\nu , T^{\prime }\nu $ respectively. By the construction the varieties $\bar {T}, \bar {T}', S_1, S$ fit into the commutative diagram

as claimed in the theorem. We push down the relation (22) via $\sigma $ to $\bar {T}'$ and obtain the relation

$$ \begin{align*} &(\tau_1'\circ\tau_1)^{*}\bigl( \gamma_2 + (p-1)\gamma_1 -1 \bigr)D \\ &\qquad\sim_{\mathbb{Q}} (1-\alpha)\gamma_2K_{\bar{T}'} +\bigl((p-1)\gamma_1+\alpha \gamma_2-1\bigr)\tau^{\prime}*K_{\bar{T}} + (1-\alpha)\gamma_2\tau_1^{\prime}*(\tau_1^{*}K_S - K_{S_1}) + \sigma_*E_{T^{\prime}\nu}. \end{align*} $$

The rational numbers $a,b,c$ and the divisor $E_{\bar {T}'}$ are determined by the above equation. In particular, if $\theta $ denotes the maximum of the coefficients of $\Delta _{K(S)}$ , then we have

$$ \begin{align*}c=\frac{(1-\alpha)\gamma_2}{ \gamma_2+ (p-1)\gamma_1 -1 }\ge \frac{1-\theta}{p(p-1)}.\end{align*} $$

Moreover, if f is a separable fibration, applying Proposition 4.5, we know that both $\tau $ and $\tau _1'$ are purely inseparable by construction.

Proof. If the generic fiber of f is smooth then we may apply [Reference Witaszek36, Theorem 3.4], which tells that there exists a finite purely inseparable morphism $\tau \colon T \to S$ such that

$$ \begin{align*}\tau^{*}D \sim_{\mathbb{Q}} t\tau^{*}K_S + (1-t)(K_T+\Delta_T)\end{align*} $$

for some rational number $t\in [0,1]$ and some effective $\mathbb {Q}$ -divisor $\Delta _T$ on T. Here we remark that when $(X_{K(S)}, \Delta _{K(S)})$ is klt, the argument of [Reference Witaszek36, Theorem 3.4] actually shows that $t\ge c_0> 0$ , where $c_0$ relies on the maximal coefficient of prime divisors in $\Delta _{K(S)}$ . We may set $\bar {T}' = \bar {T} = T$ to get our assertion as a special case.

Now assume that the generic fiber of f is not smooth. Since f is separable, if we set , then $X_{K(S_1)}$ is integral but not smooth. We may apply Theorem 6.1 and obtain the assertion once noticing that $\tau _1^{*}K_S - K_{S_1} \sim _{\mathbb {Q}} \tau _1^{*}\bigl (\frac {p-1}{p}K_S\bigr )$ .