Proof. Consider the tensor product of the function fields $R:= K(X) \otimes _{K(S)} K(T)$ . Since f is a fibration, $K(S)$ is algebraically closed in $K(X)$ , so $\operatorname {\mathrm {Spec}} R$ is irreducible by [Reference Grothendieck25, Proposition 4.3.2]. Since f is separable, $\operatorname {\mathrm {Spec}} R$ is reduced by [Reference Grothendieck25, Proposition 4.3.5]. Therefore, $\operatorname {\mathrm {Spec}} R$ is integral, which implies that $Y := X \times _S T$ is generically integral (in the sense that it becomes integral restricting to certain open subsets of $X,S,T$ ). Since the base change morphism is flat, Y satisfies Serre’s condition $(S_2)$ by [Reference Matsumura41, Corollary of Theorem 23.3]. Consequently, Y satisfies $(S_1)+(R_0)$ , and thus it is reduced. Furthermore, since Y is $(S_2)$ and generically irreducible, it is irreducible overall according to Hartshorne’s connectedness lemma [Reference Tate58, Tag 0FIV]. Therefore, Y is integral. Since Y is ( $S_2$ ), it is normal if and only if it satisfies ( $R_1$ ) condition, which is equivalent to the condition that the conductor divisor is zero.

Proof. First, we consider the case when X is smooth. Since $\kappa (D) =1$ , by replacing D with its multiple, we can assume that the linear system $|D|$ contains a movable part. Write that $|D|= |M| + V,$ where $|M|$ denotes the movable part and V the fixed part. Since X is smooth, intersections of divisors make sense. By restricting on the intersection of $\dim X-2$ general hypersurfaces, since M is movable, from $D^2\equiv M(M+V) + V\cdot D\equiv 0,$ we deduce that $M^2 \equiv 0$ and $M\cdot V\equiv 0$ . From this, we conclude that:

• • the linear system $|M|$ has no base point, hence it induces a fibration $f\colon X \to B,$ where B is a smooth projective curve;

• • the fixed part V is contained in finitely many closed fibers of f.

Moreover, since D is nef, we see that V is nef, and V must be like $a_1F_1 + \cdots + a_rF_r$ , where $F_i$ are closed fibers of f and $a_i \in \mathbb {Q}^+$ . In conclusion, there exists an effective $\mathbb {Q}$ -divisor $D_B$ such that $D\sim _{\mathbb {Q}}f^*D_B$ . Thus, D is semi-ample.

In general, we can take a smooth alteration $\pi \colon Y\to X$ [Reference de Jong14, Theorem 4.1], so that Y is smooth, projective, and dominant over X. We see that $\pi ^*D$ is nef, and $\nu (\pi ^*(D)) = \kappa (Y, \pi ^*(D))= \nu (D) =1$ [Reference Cascini, Hacon, Mustaţă and Schwede10, Remark 4.6]. In the previous paragraph, we have proved that $\pi ^*D$ is semi-ample on Y, which implies that D is semi-ample.

Proof. By Proposition 2.8(1), we have $K_{D^\nu } \succeq _{\mathbb Q} 0$ . In turn, applying the adjunction formula (Proposition 2.7), we have

$$\begin{align*}(K_X + D + \Delta)|_{D^\nu} \sim_{\mathbb Q} K_{D^\nu} + \Delta_{D^\nu} + \Delta|_{D^\nu}\succeq_{\mathbb Q} 0, \end{align*}$$

which is the assertion (1).

Next, assuming $(K_X + D+\Delta )|_{D^\nu } \sim _{\mathbb Q} 0$ , from the above equation, we see that

$$ \begin{align*}K_{D^\nu} \sim_{\mathbb{Q}} \Delta_{D^\nu} \sim_{\mathbb{Q}} \Delta|_{D^\nu} \sim_{\mathbb{Q}} 0.\end{align*} $$

Then, the assertion (2) follows immediately from Proposition 2.8(2) and Proposition 2.7(2).

Finally, assuming $D|_{D^\nu } \succeq _{\mathbb Q} 0$ , applying the adjunction formula (Proposition 2.7) again, we have

$$\begin{align*}0\sim_{\mathbb Q} (K_X + D + aD + \Delta)|_{D^\nu} \sim_{\mathbb{Q}} K_{D^\nu} + \Delta_{D^\nu} + aD|_{D^\nu} + \Delta|_{D^\nu}\succeq_{\mathbb Q} 0. \end{align*}$$

It follows that $K_{D^\nu } \sim _{\mathbb {Q}} \Delta _{D^\nu }\sim _{\mathbb {Q}} D|_{D^\nu }\sim _{\mathbb {Q}} \Delta |_{D^\nu } \sim _{\mathbb {Q}} 0$ . Then, the assertion (3) follows by the same argument as above.

Proof. By doing a semi-stable reduction (see [Reference Liu40, Proposition 10.2.33]), we obtain the following commutative diagram:

where $C'$ is a smooth curve, $\tau $ is a finite morphism, $X_{C'} := X\times _C C'$ is the fiber product, $\nu $ is the normalization, $\mu $ is a minimal resolution, $\pi = \tau '\circ \nu \circ \mu $ is the composition, and g is a semi-stable elliptic fibration. We have $K_{Y/C'} = (\tau '\circ \nu )^* K_{X/C} - \mathfrak C,$ where $\mathfrak C$ is the conductor divisor of $\nu $ , and $K_{\widetilde Y} = \mu ^*K_Y - \widetilde E,$ where $\widetilde E$ is an effective $\mu $ -exceptional divisor. It follows that

$$\begin{align*}K_{\widetilde Y/C'} = \pi^* K_{X/C} - \mu^*\mathfrak C - \widetilde E. \end{align*}$$

We have $K_{\widetilde Y/C'} \succeq _{\mathbb Q} 0$ by Theorem 2.10, in turn, combining with the assumption $K_{X/C} \sim _{\mathbb Q} 0$ we can show that $K_{\widetilde Y/C'}\sim _{\mathbb Q}0$ and $\mathfrak C = \widetilde E = 0$ . It follows that Y has at most canonical singularities, and that $X_{C'}$ is normal by Lemma 2.4, that is, $Y=X_{C'}$ . Since $K_{\widetilde Y/C'}\sim _{\mathbb Q}0$ , we can apply [Reference Chen and Zhang12, Theorem 2.14] to obtain a finite morphism $C_1\to C'$ from a smooth curve $C_1$ such that $\widetilde Y \times _{C'} C_1\cong C_1 \times F$ . In particular, all fibers of g are irreducible, which implies that $\widetilde Y \to Y$ is an isomorphism. We complete the proof by taking $\sigma $ to be the composition $C_1\to C'\to C$ .

Proof. We only need to prove that for every ample $\mathbb {Q}$ -divisor A on X, $K_X + \Delta + L + A$ is nef. Fix an ample divisor A on X. By [Reference Patakfalvi50, Lemma 3.15], we can find an ample $\mathbb {Q}$ -divisor $A'> 0$ , such that:

1. (i) $A-A'$ is ample,

2. (ii) $K_X + \Delta '$ is a $\mathbb {Z}_{(p)}$ -Cartier divisor, where $\Delta '= \Delta + A'$ , and

3. (iii) $(X_{\eta }, \Delta ^{\prime }_{\eta })$ is strongly F-regular.

Here, we remark that in order to apply Theorem 3.1, we only require the pair $(X, \Delta )$ to be F-pure over an open subset of Y, which is guaranteed by condition (iii). Therefore, since $K_X + \Delta ' + L$ is f-nef, we can apply the same proof of [Reference Ejiri and Patakfalvi20, Theorem 7.1] to the pair $(X,\Delta ')$ and L, which yields that $K_X + \Delta ' + L\sim _{\mathbb Q} K_X + \Delta + L + A'$ is nef. From this, we conclude that $K_X + \Delta + L$ is nef.

Proof. First, we borrow the argument of [Reference Ejiri and Patakfalvi20, Theorem 7.3] to show that the divisor D is nef as follows. Take an ample $\mathbb {Q}$ -divisor H on Y. Let us show that $D+ f^*H$ is nef. By the construction of D, the divisor $D+ f^*H$ is big and f-ample. Take an effective divisor $ \Gamma \sim _{\mathbb Q} D+ f^*H$ and a small rational number $\epsilon>0$ such that $(X, \Delta '= \Delta + \epsilon \Gamma )$ is strongly F-regular on $X_{\eta }$ . Since $-(K_X + \Delta )$ is nef, Corollary 3.2 applies and shows that

$$ \begin{align*}\epsilon (D+ f^*H) \equiv K_X + \Delta' + (-(K_X + \Delta))\end{align*} $$

is a nef divisor. Therefore, we conclude that D is nef, and hence $\nu (D) = n-1$ by $D^n=0$ . This proves (i).

Next, we prove (ii): $g(Y) = 1$ . Otherwise, $H = K_Y$ is big. Since D is nef and f-ample and $D-(K_{X/Y}+\Delta ) -f^*K_Y = D-(K_{X}+\Delta ) $ is nef, we can apply [Reference Zhang60, Theorem 1.5] to show that D is big, which contradicts $\nu (D) = n-1$ .

Having proved $\nu (D) = n-1$ and $g(Y)=1$ , by the same argument of [Reference Ejiri and Patakfalvi20, Theorems 7.4 and 7.5], we can show that $f_*\mathcal {O}_X(mD)$ is numerically flat. This proves (iii).

Finally, we can conclude the assertion (iv) from the proof of [Reference Ejiri18, Theorem 6.1] under these conditions.

Proof. By assumption (a), we may fix an integer $g>0$ such that $(p^g - 1)\Delta $ is integral. For an integer $e>0$ divisible by g, setting $D_e=(1-p^e)(K_X + \Delta )+ p^eD$ , we have the trace map

$$ \begin{align*}\mathrm{Tr}^{e}_D: \mathcal{F}^{e}:=f_*(F_{X*}^{e}{\mathcal O}_X((1-p^e)(K_X + \Delta)) \otimes {\mathcal O}_X(D)) \cong f_*(F_{X*}^{e}{\mathcal O}_X(D_e)) \to f_*{\mathcal O}_X(D),\end{align*} $$

and we denote its image by $\mathcal {F}_0^e$ . Note that for positive integers $e'>e$ divisible by g, the trace map $\mathrm {Tr}^{e'}_D$ factors through the trace map [Reference Zhang61, Section 2.7]

$$ \begin{align*}\mathrm{Tr}^{e'-e}_{D_e}: \mathcal{F}^{e'}=f_*(F_{X*}^{e'-e}{\mathcal O}_X((D_{e'}))) \to \mathcal{F}^{e}=f_*{\mathcal O}_X(D_e),\end{align*} $$

which implies that $\mathcal {F}^{e'}_0\subseteq \mathcal {F}^{e}_0$ . Therefore, there exists a positive integer $e_0$ such that for all $e \geq e_0$ divisible by g, the sheaf $\mathcal {F}_0^e$ has constant rank r, which means that

1. (c^′) the trace map $\mathrm {Tr}^{e}_D\colon \mathcal {F}^{e} \to \mathcal {F} :=\mathcal {F}^{e_{0}}_0 $ is generically surjective.

We shall apply [Reference Zhang61, Theorem 3.7], and we need to verify the three conditions required there. We refer to [Reference Zhang61, Section 3] for the related notions of Fourier–Mukai transform.

With the assumption that $D-(K_X+\Delta )$ is nef and relatively ample over A, the proof of the vanishing condition (C2) in [Reference Zhang61, Section 4.2], if substituting the pair $(X,B)$ with $(X,\Delta )$ and the divisor $l(K_X+B)$ with D, still works and yields the following.

Claim. If H is an ample line bundle on $\widehat {A}$ , then for any $i>0$ and sufficiently divisible integer $e>0$ ,

$$ \begin{align*}H^i(A, \mathcal{F}^{e} \otimes \widehat{H}^*) = 0.\end{align*} $$

By induction, we can find integers $e_0< e_1 < e_2 < \cdots <e_{d}$ divisible by g, and ample line bundles $H_0, H_1, \ldots , H_{d-1}$ on $\widehat {A}$ such that, if setting

$$ \begin{align*}\mathcal{F}_0= \mathcal{F},\, \mathcal{F}_1=\mathcal{F}^{e_{1}},\, \mathcal{F}_2=\mathcal{F}^{e_{2}},\ldots,\, \mathcal{F}_d=\mathcal{F}^{e_{d}},\end{align*} $$

we have

1. (a^′) for $0\leq l \leq d-1$ and every i, the sheaf $R^i\Phi _{\mathcal {P}}D_A(\mathcal {F}_{l}) \otimes H_l$ is globally generated, and if $j>0,$ then $H^j(\widehat {A}, R^i\Phi _{\mathcal {P}}D_A(\mathcal {F}_{l}) \otimes H_l) = 0$ ;

2. (b^′) for $0\leq l < l' \leq d$ , if $j>0,$ then $H^j(A, \mathcal {F}_{l'}\otimes \widehat {H}_l^*) = 0$ .

We see that the conditions (a, b, c) of [Reference Zhang61, Theorem 3.7] are guaranteed by the above conditions (a $'$ , b $'$ , c $'$ ), respectively. Therefore, we can apply [Reference Zhang61, Theorem 3.7]:

• • by [Reference Zhang61, Theorem 3.7(i)] the homomorphism $\alpha _{\mathcal {F}}\colon \mathcal {F}^{*} \to (-1)_A^*R^{0}\Psi _{\mathcal {P}}R^{0}\Phi _{\mathcal {P}}D_A(\mathcal {F})$ is injective, and in turn, applying [Reference Zhang61, Proposition 3.4(2)] we can show that

  $$ \begin{align*}V^0(f_*\mathcal{O}_X(D)) =\mathrm{Supp}(-1)_{\widehat{A}}^*R^{0}\Phi_{\mathcal{P}}D_A(\mathcal{F}) \neq \emptyset;\end{align*} $$

• • by [Reference Zhang61, Theorem 3.7(ii)], the second assertion of the theorem follows.

Proof. Since $B|_{B^\nu } \equiv 0$ , B is a nef divisor with numerical dimension $\nu (B) =1$ . To show the semi-ampleness of B, it suffices to find an integer $l>0$ and a numerically trivial line bundle L such that $h^0(X, lB +L)>1$ . Indeed, granted this, by Lemma 2.5, the divisor $lB +L$ is semi-ample and induces a fibration $g\colon X\to C$ to a curve, then since $lB\equiv lB+L$ and B is irreducible by assumption, we see that a multiple of B coincides with a fiber of g, and hence B is semi-ample.

Since $(K_X+B)|_{B^\nu } \equiv 0$ , by Lemma 2.9, B is isomorphic to an abelian variety and thus $B\to A$ is a finite morphism. Note that since X has relative dimension one over A, the divisor B is relatively big over A. Let $\mathbb E(B)$ denote the relatively exceptional locus with respect to B, namely, the union of f-exceptional irreducible varieties on which the restriction of B is not f-big.

The f-relative semi-ample divisor B induces a birational contraction $\sigma \colon X\to Y$ , which is isomorphic both near B and on the generic fiber $X_{\eta }$ of f, and there exists a $\mathbb {Q}$ -Cartier divisor $B_Y$ on Y such that $B=\sigma ^*B_Y$ .

Since $B|_{X_{\eta }}$ is an ample divisor on $X_{\eta }$ , we may take a sufficiently divisible integer $l>0$ such that $lB$ is Cartier and $r:=\dim _{K(\eta )}S^0(X_{\eta }, lB)> 1$ . Set $\mathcal L = \mathcal {O}_X(lB)$ . By construction, we have $B_Y$ is relatively ample over A, and $K_Y \sim _{\mathbb Q} 0$ . If setting $D =lB_Y$ , then $D-K_Y$ is relatively ample over A. Therefore, Theorem 3.6 applies to Y and $D =lB_Y$ and yields that $V^0(f_*\mathcal {L}) \neq \emptyset $ .

If $\dim V^0(f_*\mathcal {L})>0$ , then we can apply the argument Step 1 of the proof of [Reference Zhang61, Theorem 4.2] to show that $\kappa (X, B+ f^*L)\geq 1$ for some $L \in \mathrm {Pic}^0(A)$ , which is sufficient to conclude the proof.

Now, assume $\dim V^0(f_*\mathcal {L}) =0$ . Then, by Theorem 3.6 (2), there exist a subsheaf $\mathcal {F} \subseteq f_*\mathcal {L}$ of rank r, an isogeny $\pi \colon A_1\to A$ of abelian varieties, some $P_1, \ldots , P_r \in \mathrm {Pic}^0(A_1),$ and a generically surjective homomorphism

$$ \begin{align*}\beta\colon \bigoplus_iP_i \to \pi^*\mathcal{F}.\end{align*} $$

Applying the covering theorem as in Step 2 of the proof of [Reference Zhang61, Theorem 4.2], we show that there exist an integer $m>0$ and some $L_1, \ldots , L_r \in \operatorname {\mathrm {Pic}}^0(A)$ such that:

• • $H^0(X, mB + f^*L_i) \neq 0$ ;

• • the sub-linear system of $|(mB)_{K(\eta )}|$ corresponding to the subspace $\sum _iH^0(X, mB + L_i)\otimes _k K(\eta ) \subseteq H^0(X_{\eta }, mB|_{X_{\eta }})$ defines a non-trivial map.

We may assume each $h^0(X, mB + f^*L_i)=1$ , thus there exists a unique effective divisor $D_i \sim mB + f^*L_i$ .

Write $D_i = aB + D'$ such that $B\not \subseteq \operatorname {\mathrm {Supp}} D'$ . We have $D_i|_{B}\equiv 0$ . By $\nu (B)=1$ , we conclude that $\operatorname {\mathrm {Supp}} B\cap \operatorname {\mathrm {Supp}} D' = \emptyset $ , thus $D'|_{B} \sim 0$ . Moreover, by Lemma 2.9, $B|_{B} \sim _{\mathbb Q} (K_X + B)|_{B} \succeq _{\mathbb Q} 0$ . Then, it follows that $D_i|_{B} \sim _{\mathbb Q} 0$ .

Take $D_1 \neq D_2$ . By $D_1 - D_2 \sim f^*(L_1-L_2)$ , we conclude that $f^*(L_1-L_2) |_{B} \sim _{\mathbb Q} 0$ . Since $B \to A$ is dominant, we have $L_1 \sim _{\mathbb Q} L_2$ , that is, there exists some $N>0$ such that $NL_1\sim NL_2$ . But then $ND_1 \sim ND_2$ . This shows $h^0(X, NmB+ NL_1) \geq 2$ , which concludes the proof.

Proof. The assertion (i) is well known. Let us prove the remaining ones.

Note that $X \times _A A^{\frac {1}{p}}$ is not normal by Proposition 2.16. Let $X_1:=(X \times _A A^{\frac {1}{p}})^{\nu }$ be the normalization, and let $\pi \colon X_1\to X$ , $f_1\colon X_1\to A^{\frac 1p}$ be the natural morphisms. We have the following commutative diagram:

By results of Section 2.5, we can write that

(5.2) $$ \begin{align} \pi^* K_X \sim K_{X_1} + (p-1)\mathcal C, \quad\text{with } \mathcal C\ge0, \end{align} $$

where $\mathcal C$ can be chosen such that $\mathcal C|_{X_{1,\eta }}$ coincides with the conductor of the normalization of the generic fiber of $f'$ . Hereafter, we fix such $\mathcal C$ . According to Proposition 2.15(1), $\deg _{K(A_1)}(p-1)\mathcal C =2$ . Write $(p-1)\mathcal C = H + V$ , where H is the $f_1$ -horizontal part and V the vertical part of $\mathcal C$ . Note that H is irreducible by Proposition 2.16(1), thus $H=nC$ , where C is reduced and $n=1$ or $2$ . Let D be the reduced divisor supported on $\pi (C)$ . We have $\pi ^*D = C$ or $pC$ by Proposition 2.16.

Step 1. We prove that both C and D are semi-ample with numerical dimension one.

Since $X_1\to X$ is a finite purely inseparable morphism, the semi-ampleness of one of the divisors C or D implies the semi-ampleness of the other. We shall show that D is semi-ample with numerical dimension one. By Theorem 3.7, it suffices to verify that $D|_{D^{\nu }}\sim _{\mathbb Q} 0$ , which is equivalent to that $C|_{C^{\nu }}\sim _{\mathbb Q} 0$ .

Claim. Let T be an $f_1$ -horizontal prime divisor of $X_1$ . Then, $T|_{T^\nu } \succeq _{\mathbb Q} 0$ .

To prove this claim, set $\bar {T} := \pi (T)$ . By Lemma 2.9(1), $\bar {T}|_{\bar {T}^{\nu }} \sim _{\mathbb Q} (K_X+ \bar {T})|_{\bar {T}^{\nu }} \succeq _{\mathbb Q} 0$ . Then, since $X_1 \to X$ is finite and purely inseparable, we have $T|_{T^{\nu }} \succeq _{\mathbb Q} 0$ .

Consequently, since C is $f_1$ -horizontal, we have $C|_{C^{\nu }}\succeq _{\mathbb Q} 0$ . Thus, applying Lemma 2.9 to

(5.3) $$ \begin{align} 0\sim_{\mathbb Q} (K_{X_1} + \mathcal C)|_{C^\nu} \sim_{\mathbb Q} (K_{X_1} + nC+V)|_{C^\nu} \end{align} $$

we see that C is an abelian variety, $V|_{C^\nu }\sim _{\mathbb Q} 0$ , and moreover, in case $n=2$ , we have $C|_{C^\nu } =C|_{C}\sim _{\mathbb Q} 0$ . Thus, we only need to show $C|_{C} \sim _{\mathbb Q} 0$ in case $n=1$ . In this case, we have $p=2$ , and $C\to A_1$ is purely inseparable of degree $2$ since $C|_{X_{K(A_1)}}$ is a point which is inseparable of degree $2$ over $\operatorname {\mathrm {Spec}} K(A_1)$ by Proposition 2.16. By doing the base change $C \to A_1$ and setting $Z=(X_1\times _{A_1} C)^{\nu }$ , we have the following commutative diagram:

where $\phi , \pi ',f_2$ denote the natural morphisms fitting into the above diagram. Since $C\times _{A_1} C$ is non-reduced and $f_1\colon X_1\to A_1$ is smooth over the generic point of $A_1$ , we see that $\phi ^*C=2E$ for some $\mathbb Z$ -divisor E on Z. Then, we have

(5.4) $$ \begin{align} \pi^{\prime*}K_X = K_Z + 2E + V', \end{align} $$

where $V'\ge 0$ is vertical over C. Consequently, $(K_Z + 2E + V')|_{E^\nu } \sim _{\mathbb Q} 0$ . Since $C|_{C}\succeq _{\mathbb Q}0$ , we have $E|_{E^\nu }\succeq _{\mathbb Q} 0$ . It follows from Lemma 2.9 that $E|_{E^\nu } \sim _{\mathbb Q} 0$ , which is equivalent to that $C|_{C} \sim _{\mathbb Q} 0$ .

In summary, the semi-ample divisor D (resp., C) induces a fibration $g\colon X\to B$ (resp., $g_1\colon X_1\to B_1$ ) to a curve. Since general fibers of f and $f_1$ are rational curves, we have $g(B) = g(B_1) =0$ . In summary, there is a commutative diagram:

Step 2. We prove the following statements:

1. (a) D is isomorphic to an abelian variety (we have shown this for C in the last step).

2. (b) X (resp., $X_1$ ) is regular at codimension one points of D (resp., C).

3. (c) $V = 0$ .

4. (d) A general fiber of g (resp., $g_1$ ) is an abelian variety, and a special fiber of g (resp., $g_1$ ) is a multiple of an abelian variety.

First, since C and D are irreducible and $C|_{C^\nu } \sim _{\mathbb Q} D|_{D^\nu } \sim _{\mathbb Q} 0$ , we have $(K_X + D)|_{D^\nu } \sim _{\mathbb Q} (K_{X_1} + C)|_{C^\nu } \sim _{\mathbb Q} 0$ , thus the statements (a, b) follow from Lemma 2.9.

To show the last two statements, we denote by $G_t$ (resp., $G^1_t$ ) the fiber of g (resp., $g_1$ ) over $t\in \mathbb P^1$ . We first consider the fibration $g_1\colon X_1\to \mathbb P^1$ . Write $G^1_t=mT+V'$ , where T is an $f_1$ -horizontal component and $V'$ is the remaining part with $T\not \subset \mathrm { Supp}\ V'$ . By the claim in Step 1, we have $T|_{T^\nu }\succeq _{\mathbb Q} 0$ . Since C is irreducible, we see that $C\sim _{\mathbb Q} rG^1_t$ for some positive rational number r, and it follows that $C|_{T^\nu }\sim _{\mathbb Q} rG^1_t|_{T^\nu }\sim _{\mathbb Q} 0$ . Thus,

$$\begin{align*}0 \sim_{\mathbb Q} (K_{X_1} + nC + V+G^1_t)|_{T^\nu}\sim_{\mathbb Q} (K_{X_1} + mT+V'+ V)|_{T^\nu}.\end{align*}$$

By Lemma 2.9(2), we see that T is an abelian variety, $V|_{T^\nu }=V'|_{T^\nu }=0$ , which implies that $\mathrm { Supp}\ V' \cap \mathrm {Supp}\ T= \mathrm {Supp}\ V \cap \mathrm {Supp}\ T = \emptyset $ . Since the fiber $G^1_t$ is connected, we have $V'=0$ , and consequently $G^1_t=mT$ is (a multiple of) an abelian variety. It follows that $\mathrm {Supp}\ G^1_t \cap \mathrm {Supp}\ V =\emptyset $ , and thus $V=0$ , which is the statement (c). Moreover, since $g_1$ is a fibration to a curve, by [Reference Bădescu1, Corollary 7.3], a general fiber of g is integral, so we obtain the statement (d) for $g_1$ .

Finally, by writing $G_t = mT + V'$ and using $(K_X + G_t)|_{T^\nu } \sim _{\mathbb Q} (K_X + mT + V')|_{T^\nu } \sim _{\mathbb Q} 0$ , Lemma 2.9 implies that $G_t$ is also (a multiple of) an abelian variety. Thus, we obtain the statement (d) for g.

Step 3. We show that there exists an isogeny $\tau \colon B\to A_1$ of abelian varieties such that $X_1 \times _{A_1} B \cong B \times \mathbb P^1$ . In turn, $f_1\colon X_1 \to A_1$ is a smooth morphism.

Take a general fiber $G^1_t$ of $g_1\colon X_1\to \mathbb P^1$ over $t\in \mathbb P^1$ , which is an abelian variety as established in the previous step. We denote by n the degree $\deg (G^1_t\to A_1)$ , which is independent of $t\in \mathbb P^1$ . Then, the morphism $\times n\colon B:=A_1 \to A_1$ factors through the isogeny $G^1_t\to A_1$ for a general t (see [Reference Mumford46, p. 169, Remark]).

By Lemma 2.4, the fiber product $X_{1,B} := X_1\times _{A_1}B$ is integral. Let W be the normalization of $X_{1,B}$ . We have the following commutative diagram:

where q is the fibration resulting from the Stein factorization of $W\to X_1 \buildrel g_1\over \to \mathbb P^1$ . By our choice of the base change $B\to A_1$ , a general fiber $Q_{t}$ of q is a birational section of p. Since each fiber of $g_1$ contains no vertical components over $A_1$ and $\nu $ is finite, each fiber of q contains no vertical components over B. Thus, for each fiber $Q_t$ of q, the morphism $Q_t\to B$ is birational. Denote by $\mathcal N$ the conductor divisor of $\nu $ . Remark that, since $X_1\to A_1$ is generically smooth, $\mathcal N$ is vertical over B. It follows that $K_W + (p-1)\pi ^*\mathcal C + \mathcal N \sim _{\mathbb Q} \pi ^*(K_{X_1} + (p-1)\mathcal C) \sim _{\mathbb Q} 0$ . Thus,

$$\begin{align*}( K_W + (p-1)\pi^*\mathcal C + \mathcal N + Q_{t}) |_{Q^\nu_t} \sim_{\mathbb Q} 0. \end{align*}$$

Applying Lemma 2.9, we see that $Q_t = Q^\nu _t$ is an abelian variety and $\mathcal N|_{Q_t} \sim _{\mathbb Q} 0$ . In turn, we conclude that:

• • $\mathcal N = 0$ , thus $X_{1,B}$ is normal by Lemma 2.4;

• • for each $t\in \mathbb P^1$ , the projection $Q_t\to B$ is an isomorphism, hence the morphism $X_{1,B}\cong W\to B\times \mathbb P^1$ is an isomorphism by Zariski’s main theorem.

Step 4. We consider the case $(p-1)\mathcal C=2C$ and prove the statement (iii-1).

We first show that $(p+1) C|_{C} \sim 0$ . Denote by $\tilde \pi \colon C\to D$ the induced finite morphism of abelian varieties and note that $\pi ^*D = pC$ by Proposition 2.16. By Step 2 (c), $K_X+D$ is Cartier at codimension-one points of D. Therefore, the adjunction formula gives

(5.5) $$ \begin{align} (K_X + D)|^w_{D} \sim K_D \sim 0, \end{align} $$

where $|^w$ is the restriction on D by first considering a big open subset $D^\circ \subset D$ over which $K_X + D$ is Cartier and then extending it (see [Reference Kollár37, pp. 173–174]). Thus, $\pi ^* (K_X + D) |_{C} = \tilde \pi ^* ( (K_X+D)|^w_{D} )\sim 0$ (here the pullback of $\pi $ makes sense since $\pi $ is finite). On the other hand, we have

$$ \begin{align*} \pi^* (K_X + D) |_{C} \sim (K_{X_1}+ 2C +\pi^*D)|_{C} \sim (p+1)C|_{C}. \end{align*} $$

It follows that $(p+1) C|_{C} \sim 0$ .

Next, we equip the smooth morphism $f_1\colon X_1\to A_1$ with a projective bundle structure over $A_1$ . Note that since $C\to A_1$ is birational and C is isomorphic to an abelian variety, $C\to A_1$ is an isomorphism; this gives a section $s\colon A_1\to C$ of $f_1$ which is $f_1$ -ample. Set $\mathcal {E}=f_{1*}\mathcal {O}_{X_1}(C)$ . Since each fiber of $f_1$ is isomorphic to $\mathbb {P}^1$ , by exactly the same proof of [Reference Hartshorne29, Proposition V.2.2], we can show that $f_1^*\mathcal E\to {\mathcal O}_{X_1}(C)$ is surjective and induces an isomorphism $X_1 \buildrel \sim \over \to \mathbb P(\mathcal {E})$ . Under this isomorphism, the section $s\colon A_1\to C$ corresponds to the exact sequence