Proof. To prove the first statement, we may replace $X$ by a birational model for which the canonical map for $K_{X} + L$ is a morphism and replace $L$ with its pullback to this birational model. Thus, it suffices to prove that the last statement holds.

Let $\pi : X \to W'$ denote the canonical map for $K_{X} + L$. By condition (iii), there is a rational map $g: W' \dashrightarrow W$ such that $\psi = g \circ \pi$ as rational maps. By condition (ii), a general fiber of $\psi$ is contracted by $\pi$. Together with condition (i) these show that $g$ is birational, yielding the desired claim.

Proof. [Reference BirkarBir21, Theorem 1.1] shows that as we vary over all such pairs the set of underlying varieties $X$ is bounded. We show that in fact the set of pairs $(X,\Delta )$ is log bounded. Using the boundedness of the underlying $X$, we can find a family of very ample divisors $A$ on these varieties $X$ such that the space of sections of $A$ is bounded in dimension. Since the coefficient set of $\Delta$ is finite, it suffices to prove that the degree of $\Delta$ against $A$ is bounded. This follows from

\[ \Delta \cdot A^{\dim X-1} < -K_{X} \cdot A^{\dim X-1} \]

and the boundedness of the varieties $X$.

Since the pairs $(X,\Delta )$ are log bounded, by [Reference BirkarBir19, Lemma 2.24] and the fact that the coefficient set of $\Delta$ is finite we deduce the desired statement on the Cartier index.

Proof. Theorem 3.3 yields an upper bound $C_{1}$ on $(-K_{X} - \Delta )^{\dim X}$ that only depends on $d$ and $\epsilon$. Choose a positive integer $C_{2}$ as in Theorem 3.5 so that $-C_{2}(K_{X} + \Delta )$ is a Cartier ample divisor. By [Reference KollárKol93, 1.1 Theorem and 1.2 Lemma] there is a positive integer $M = M(d)$ such that $-MC_{2}(K_{X} + \Delta )$ is very ample. Set $C_{3} = C_{1}(C_{2}M)^{d-1}$. Then one can find a suitable curve $C$ by taking intersections of general elements in $|-MC_{2}(K_{X} + \Delta )|$.

Proof. We first prove the result when the base field is algebraically closed. Let $I$ be the coefficient set of $\Delta$. Suppose that $\phi : X \dashrightarrow X'$ is a run of the $(K_{X} + \Delta )$-minimal model program with scaling resulting in a Mori fiber space $\pi : X' \to Z'$. Let $Y$ denote a general fiber of $\pi$. Then $\phi _{*}\Delta |_{Y}$ has coefficients in $I$ and $(Y,\phi _{*}\Delta |_{Y})$ is an $\epsilon$-lc Fano pair. Altogether we see that there is a universal bound $C_{3}$ as in Lemma 3.6 for all such pairs $(Y,\Delta |_{Y})$ obtained in this way.

By [Reference AraujoAra10, Theorem 1.1] we have

\[ \overline{\mathrm{Eff}}_{1}(X)_{K_{X} + \Delta \geqslant 0} + \mathrm{Nef}_{1}(X) = \overline{\mathrm{Eff}}_{1}(X)_{K_{X} + \Delta \geqslant 0} + \overline{\sum_{\alpha \in \Sigma} \mathbb{R}_{\geqslant 0}\alpha} \]

where each $\alpha$ is obtained by running a $(K_{X} + \Delta )$-minimal model program with scaling to obtain $\phi : X \dashrightarrow X'$, where $X'$ carries a Mori fiber space structure $\pi : X' \to Z'$, and setting $\alpha$ to be the numerical pullback (as in [Reference AraujoAra10, § 4]) of the class of a curve in a general fiber $Y$ of $\pi$. As discussed earlier, we know that for any such $Y$ there is a movable curve $C'$ in $Y$ avoiding the $\phi ^{-1}$-indeterminacy locus and satisfying $-(K_{X'} + \phi _{*}\Delta ) \cdot C' \leqslant C_{3}$. Since $C'$ avoids the $\phi ^{-1}$-indeterminacy locus, its strict transform $C$ in $X$ satisfies $-(K_{X} + \Delta ) \cdot C \leqslant C_{3}$. Note that the numerical pullback of the corresponding class $\alpha$ is represented by the movable curve $C$ because $\rho (X'/Z') =1$.

There are only finitely many classes of curves $C$ which satisfy both ${(K_{X} + \Delta + A) \cdot C \leqslant 0}$ and $-(K_{X} + \Delta ) \cdot C \leqslant C_{3}$. This proves that the set of such $\alpha$ as above satisfying ${(K_{X} + \Delta + A) \cdot \alpha < 0}$ is finite, finishing the proof when the ground field is algebraically closed.

For a general ground field $F$ of characteristic $0$, let $\{\overline {X}_{j}\}_{j=1}^{s}$ denote the components of $X_{\overline {F}/F}$. Note that

\[ N_{1}(X) = (\oplus_{j} N_{1}(\overline{X}_{j}))^{\mathrm{Gal}(\overline{F}/F)} \]

and that the action of $\mathrm {Gal}(\overline {F}/F)$ on $\oplus _{j} N_{1}(\overline {X}_{j})$ factors through a finite group $G$. We can apply the statement of the theorem to each component $\overline {X}_{j}$ and take direct sums to see that $\overline {\mathrm {Eff}}_{1}(X_{\overline {F}})_{K_{X} + \Delta \geqslant 0} + \mathrm {Nef}_{1}(X_{\overline {F}})$ admits only finitely many $(K_{X} + \Delta + A)$-negative extremal rays generated by numerical classes $\{ \overline {\alpha }_{i} \}_{i=1}^{r}$.

Define $\alpha _{i} = ({1}/{|G|}) \sum _{g \in G} g\overline {\alpha }_{i}$. Since the pseudo-effective and nef cones of curves are preserved by the $G$-action, it is clear that $\overline {\mathrm {Eff}}_{1}(X)_{K_{X} + \Delta \geqslant 0} + \mathrm {Nef}_{1}(X)$ has only finitely many extremal rays with negative intersection against $K_{X} + \Delta + A$, and these rays are generated by a subset of $\{\alpha _{i}\}_{i=1}^{r}$. Furthermore, we can construct an irreducible movable curve over the ground field representing some multiple of $\alpha _{i}$ by taking the Galois orbit of an irreducible curve representing $\overline {\alpha }_{i}$ over $\overline {F}$ and descending to $F$.

Proof. By the birational invariance of the $a$-invariant, it suffices to prove this when $\pi$ is a morphism and $Y$ is a smooth fiber of $\pi$. The equality $a(Y,L|_{Y}) = a(X,L)$ follows from the fact that the restriction of $K_{X} + a(X,L)L$ to a general fiber $Y$ has Iitaka dimension $0$, hence lies on the boundary of the pseudo-effective cone.

Proof. After applying an embedded resolution of singularities, we have a birational model $X'$ over $F'$ such that the strict transform $Y'$ of $Y$ is smooth. Let $\sigma _*X'$ be the pullback of $X' \to \mathrm {Spec} (F')$ by $\sigma ^{-1} : \mathrm {Spec} (F') \to \mathrm {Spec} (F')$. We have a natural isomorphism as schemes $\sigma : X' \to \sigma _*X'$ and we denote the image of $Y'$ by $\sigma (Y')$. Then $\sigma (Y')$ is a smooth model of $\sigma (Y)$. Let $K_{Y'}$ be the canonical divisor on $Y'$. Then we have $\sigma (K_{Y'}) = K_{\sigma (Y')}$. Moreover, for any integer $m$ such that $ma(Y,L|_{Y})L$ is a Cartier divisor we have

\[ m(K_{Y'} + a(Y,L_{F'}|_{Y})L_{F'}|_{Y'}) \sim D \geqslant 0 \iff m(K_{\sigma(Y')} + a(Y, L_{F'}|_{Y})L_{F'}|_{\sigma(Y')}) \sim \sigma(D) \geqslant 0. \]

This shows that $K_{Y} + a(Y,L_{F'}|_{Y})L_{F'}|_{Y}$ and $K_{\sigma (Y')} + a(Y, L_{F'}|_{Y})L_{F'}|_{\sigma (Y')}$ have the same Iitaka dimension, proving the claims.

Proof. It suffices to prove the statement when $F'/F$ is either an algebraic extension or a purely transcendental extension since any extension of fields is a composition of these two types.

Suppose first that $F'/F$ is algebraic. Let $\widetilde {F}/F'$ be an extension such that $\widetilde {F}/F$ is Galois. Then we can prove the desired statement for $F'/F$ by verifying it for both $\widetilde {F}/F$ and $\widetilde {F}/F'$. Thus, it suffices to prove the statement under the additional condition that $F'/F$ is Galois.

When the extension is Galois then $\mathrm {Gal}(F'/F)$ acts transitively on the components of $X_{F'}$. By Proposition 4.4 we see that every component of $X_{F'}$ has the same $a$-invariant and thus this $a$-invariant is equal to $a(X,L)$. Furthermore, since base change commutes with taking spaces of sections, we see that for every integer $m$ such that $m(K_{X} + a(X,L)L)$ is a Cartier divisor we have

\[ \dim H^{0}(X,m(K_{X} + a(X,L)L)) = \dim H^{0}(X_{F'},m(K_{X_{F'}} + a(X,L)L_{F'})). \]

Since $X_{F'}$ is a disjoint union of Galois translates of $X'$, we have

\[ \dim H^{0}(X_{F'},m(K_{X_{F'}} + a(X,L)L_{F'})) = r \cdot \dim H^{0}(X',m(K_{X'} + a(X,L)L_{F'}|_{X'})) \]

where $r$ denotes the number of components of $X_{F'}$. Since $r$ does not change as $m$ increases, the Iitaka dimensions coincide.

Now suppose that $F'/F$ is purely transcendental. This implies that $X_{F'}$ is smooth and irreducible. Since sections are preserved by base change, we deduce that for any positive rational number $a$ we have $\kappa (X,K_{X} + aL) = \kappa (X_{F'},K_{X_{F'}} + aL_{F'})$. This implies that the $a$-invariants over $F$ and $F'$ also coincide.

Proof. Over an algebraically closed field this is [Reference Lehmann and TanimotoLT17, Theorem 4.3]. Over a general $F$, we can take the base change to $\overline {F}$, replace $\mathcal {X}$ with any component of $\mathcal {X}_{\overline {F}}$, and replace the morphism to $W_{\overline {F}}$ with its Stein factorization. Since the space of sections associated to a divisor is stable under flat base change, the result over an arbitrary field of characteristic $0$ follows from the result over its algebraic closure combined with Corollary 4.5.

Proof. After birational modifications to $X$ and $Y$ we may assume that $X$ and $Y$ are smooth. By the ramification formula, we have

\[ a(X, L)f^*L + K_Y = a(X, L)f^*L + f^*K_X + R \]

where $R \geqslant 0$ is the ramification divisor. Thus, $K_{Y} + a(X, L)f^*L$ is pseudo-effective and our assertion follows.

Proof. This is stated in [Reference Lehmann, Tanimoto and TschinkelLTT18, Proposition 4.1] assuming the base field is algebraically closed. Its generalization to arbitrary fields of characteristic $0$ follows from Corollary 4.5.

Proof. By Lemma 4.3 we have $a(\Gamma,f^{*}L|_{\Gamma }) = a(Y,f^*L)$ and $\Gamma$ is adjoint rigid with respect to $f^{*}L|_{\Gamma }$. Since $f|_{\Gamma }$ is generically finite for a general $\Gamma$, Lemma 4.7 shows that $a(S,L|_{S}) \geqslant a(\Gamma,f^{*}L|_{\Gamma })$. However, as we vary $\Gamma$ the images $S$ form a dominant family of subvarieties of $X$, so by Lemma 4.8 we must have $a(S,L|_{S}) \leqslant a(X,L)$. By combining these inequalities with the fact that $a(Y,f^{*}L) = a(X,L)$ we deduce that we have equalities of $a$-values everywhere.

Let $g: \Gamma ' \to S'$ be a birational model of $f|_{\Gamma }: \Gamma \to S$ such that $\Gamma '$ and $S'$ are smooth and let $L'$ denote the pullback of $L$ to $S'$. By the ramification formula we have

\[ a(S',L')g^*L' + K_{\Gamma'} = a(S', L')g^{*}L' + g^*K_{S'} + R \]

for some effective divisor $R$. Since the $a$-values of $\Gamma '$ and $S'$ are the same, we deduce that

\[ \kappa(S',K_{S'}+a(S',L')L') \leqslant \kappa(\Gamma',K_{\Gamma'}+a(\Gamma',g^{*}L')g^{*}L') = 0. \]

Thus, $S$ is adjoint rigid with respect to $L|_{S}$.

Proof. For any geometric component of $X'$ the Néron–Severi space is spanned by the $\overline {f}$-exceptional divisors and the restriction of $\overline {f}^{*}N^{1}(\overline {X})$ to the component. Since $X'$ is integral we know that $\mathrm {Gal}(\overline {F}/F)$ acts transitively on the geometric components of $X'$. We conclude that $N^{1}(X')$ is spanned by $f$-exceptional divisors and $f^{*}N^{1}(X)$.

We next argue that $f_{*}$ actually maps $\mathcal {F}_{X',f^{*}L}$ to $\mathcal {F}_{X,L}$. Note that there is an effective $\phi$-exceptional divisor $E$ on $X'$ such that $K_{X'} + a(X',f^{*}L)f^{*}L = f^{*}(K_{X} + a(X,L)L) + E$. Thus, any nef curve class with vanishing intersection against $K_{X'} + a(X',f^{*}L)f^{*}L$ will also have vanishing intersection against $f^{*}(K_{X} + a(X,L)L)$ and vanishing intersection against $E$. The former property shows that $f_{*}$ maps $\mathcal {F}_{X',f^{*}L}$ to $\mathcal {F}_{X,L}$.

Next note that $E$ contains every $f$-exceptional divisor with a positive coefficient. Hence, any class in $\mathcal {F}_{X',f^{*}L}$ has zero intersection against every $f$-exceptional divisor. Therefore, every curve class in $\mathcal {F}_{X',f^{*}L}$ is pulled back from $\mathcal {F}_{X,L}$. Thus, the pullback for curve classes induces a map $({1}/{\deg (f)}) f^* : \mathcal {F}_{X,L} \to \mathcal {F}_{X',f^{*}L}$ inverse to $f_{*}$ and our assertion follows.

Proof. This follows from the fact that $b(F, X, L)$ is given by

\[ \dim \mathrm{Nef}_1(\overline{X})^{\mathrm{Gal}(\overline{F}/F)} \cap (K_{X} + a(X,L)L)^{\perp}. \]

Proof. By Lemma 3.1, $a(X,L)L$ is $\mathbb {Q}$-linearly equivalent to a sum $\Delta + A$ where $(X,\Delta +A)$ is terminal and $A$ is ample. Let $C$ be a compact polyhedral convex hull of a finite set of ample $\mathbb {Q}$-divisors such that the interior of $C$ contains $A$. Let $U$ denote the set of classes in $N^{1}(X)$ of the form

\[ U = \big\{ K_{X} + \Delta + \frac{1}{2}A + \lambda D \bigm| D \in C, \lambda \geqslant 0 \big\}. \]

In particular, $U$ is a closed rational polyhedral set containing $K_{X} + a(X,L)L$ in its interior.

(i) We apply Lemma 3.7 to the pair $(X,\Delta )$ and the ample $\mathbb {Q}$-divisor $\frac {1}{4}A$ to obtain a finite set of $(K_{X} + \Delta + \frac {1}{4}A)$-negative extremal rays. Note that these are the only extremal rays of $\mathrm {Nef}_{1}(X)$ which can have vanishing intersection against an element of $U$ that lies on the boundary of the pseudo-effective cone. Thus, $\overline {\mathrm {Eff}}^1(X) \cap U$ is a rational polyhedral set, since it is cut out in $U$ by finitely many rational linear inequalities. In particular, every face of $\overline {\mathrm {Eff}}^{1}(X)$ intersecting the interior of $U$ must be a supported face, proving (i).

(ii) By [Reference Birkar, Cascini, Hacon and McKernanBCHM10, Theorem D] (and the fact that $\mathbb {R}$-linear equivalence can be detected after change of base field), every pseudo-effective class contained in $U$ is represented by an effective divisor. By (i) we know that $K_{X} + a(X,L)L$ is in the relative interior of $\mathcal {F}^{X,L}$. Thus, if $E$ is an effective divisor representing a class in $\mathcal {F}^{X,L} \cap U$, there is some positive constant $c$ such that $K_{X} + a(X,L)L - cE$ also has class in $\mathcal {F}^{X,L} \cap U$ and is thus represented by an effective divisor. The statement follows.

Proof. We may resolve $\pi$ to be a morphism without affecting $\pi ^{-1}(W^{\circ })$, and since by Lemma 4.12 blowing up induces an isomorphism of the faces $\mathcal {F}_{X,L}$ the statement for the blow-up is equivalent to the statement for the original variety.

(i) We first show that there are only finitely many divisors $\{ E_{i} \}_{i=1}^r$ which dominate $W$ and satisfy $K_{X} + a(X,L)L - c_iE_{i} \in \overline {\mathrm {Eff}}^{1}(X)$ for some $c_i > 0$. The geometric generic fiber $\overline {X}_{\overline {\eta }}$ is adjoint rigid with respect to $L$. If we restrict $E_{i}$ to $\overline {X}_{\overline {\eta }}$, then the support must lie in the unique effective divisor numerically equivalent to $K_{\overline {X}_{\overline {\eta }}} + a(X,L)L|_{\overline {X}_{\overline {\eta }}}$. Thus, there are only finitely many divisors $E_{i}$ of this type.

By [Reference AndréAnd96, Théorème 5.2] and [Reference Maulik and PoonenMP12, Theorem 1.1] we know that for some geometric point $\overline {s} \in \overline {W}^{\circ }$ the specialization map $N^{1}(\overline {X}_{\overline {\eta }}) \to N^{1}(\overline {X}_{\overline {s}})$ is an isomorphism where $\overline {\eta }$ denotes the geometric generic point of $W$. Furthermore, by [Reference Maulik and PoonenMP12, Proposition 3.3] this isomorphism is compatible with restriction from $N^{1}(\overline {X})$ and is a map of $\pi ^{\unicode{x00E9} \textrm{t}}_1(\overline {W^{\circ }},\overline {s})$-modules. Note that the image of the map

\[ N^1(\overline{X}) \twoheadrightarrow N^{1}(\overline{X}_{\eta}) \to N^{1}(\overline{X}_{\overline{\eta}}) \]

is exactly the $\mathrm {Gal}(\overline {\overline {F}(\overline {W})}/\overline {F}(\overline {W}))$-invariant part, or equivalently, the $\pi ^{\unicode{x00E9} \textrm{t}}_1(\overline {W^{\circ }},\overline {s})$-invariant part. Altogether we conclude that under the restriction map $N^{1}(\overline {X})$ will surject onto $N^{1}(\overline {X}_{\overline {s}})^{\pi ^{\unicode{x00E9} \textrm{t}}_1(\overline {W^{\circ }},\overline {s})}$ for a single fiber $\overline {X}_{\overline {s}}$.

We claim that $N^{1}(\overline {X}) \to N^{1}(\overline {X}_{\overline {t}})^{\pi ^{\unicode{x00E9} \textrm{t}}_1(\overline {W^{\circ }},\overline {t})}$ is surjective for every $\overline {t} \in \overline {W}^{\circ }$. Indeed, consider the sequence of maps

\[ N^1(\overline{X}) \to N^{1}(\overline{X}_{\eta}) \to N^{1}(\overline{X}_{\overline{\eta}}) \to N^{1}(\overline{X}_{\overline{t}}) \]

where [Reference Maulik and PoonenMP12, Proposition 3.6] defines the last map and shows it is injective. The map in the middle is also injective. Since $\overline {X}_{\eta }$ is rationally connected, the kernel of the leftmost map is generated by the kernel of $\mathrm {Pic}(\overline {X}) \to \mathrm {Pic}(\overline {X}_{\eta })$. Altogether we see that for every $\overline {t}$ the kernel of $N^{1}(\overline {X}) \to N^{1}(\overline {X}_{\overline {t}})$ is the subspace of $N^{1}(\overline {X})$ spanned by all $\pi$-vertical divisors. To prove our claim it suffices to show that the rank of $N^{1}(\overline {X}_{\overline {t}})^{\pi ^{\unicode{x00E9} \textrm{t}}_1(\overline {W^{\circ }},\overline {t})}$ is constant. Since the generic fiber of $\overline {\pi }$ is rationally connected, the Picard rank of the geometric fibers of $\pi$ is constant. By taking monodromy invariants we obtain the claim.

In particular, we have a surjection $N^{1}(\overline {X}) \to N^{1}(\overline {X}_{\overline {w}})^{\pi ^{\unicode{x00E9} \textrm{t}}_1(\overline {W^{\circ }},\overline {w})}$. Note that the action of $\mathrm {Gal}(\overline {F}/F)$ on $N^{1}(\overline {X})$ and on $N^{1}(\overline {X}_{w})$ factors through a finite group. Thus, we see that the restriction map

(4.1)\begin{align} N^{1}(\overline{X})^{\mathrm{Gal}(\overline{F}/F)} \to N^{1}(X_w)\cap N^1(\overline{X}_{\overline{w}})^{\pi^{\unicode{x00E9} \textrm{t}}_1(\overline{W^{\circ}},\overline{w})} \end{align}

is surjective.

By Lemma 4.14, $b(F,X,L)$ is the dimension of the quotient of $N^{1}(X)$ by all effective irreducible divisors $E$ satisfying $K_{X} + a(X,L)L - c E \in \overline {\mathrm {Eff}}^{1}(X)$ for some $c>0$. Note that $E$ lies in the kernel of the restriction map to $N^{1}(X_{w})$ if and only if $\pi (E) \subsetneq W$ and $E$ satisfies $K_{X} + a(X,L)L - cE \in \overline {\mathrm {Eff}}^{1}(X)$ for some $c > 0$. Thus, if we further quotient by the finitely many $E_i$ which dominate $W$ and satisfy $K_{X} + a(X,L)L - c E \in \overline {\mathrm {Eff}}^{1}(X)$ for some $c>0$, the dimension of the resulting space is $b(F,X,L)$.

(ii) First we claim that for every fiber $X_{w}$ over a closed point of $W^{\circ }$ an effective divisor on $X$ will restrict to a divisor on $X_{w}$ linearly equivalent to an effective divisor. Indeed, since the invertible sheaf associated to any effective divisor $D$ on $X$ is flat over $W^\circ$, the function $w \mapsto h^0(X_{w}, \mathcal {O}_X(D)|_{X_w})$ satisfies upper semicontinuity. Since this space of sections has positive dimension for a general fiber, it must be positive for all points on $W^\circ$. Then, dually, nef curve classes on $X_w$ push forward to nef curve classes on $X$. Using adjunction and the assumption that $a(X,L) = a(X_w,L|_{X_w})$, we see that $f_*$ maps $\mathcal {F}_{X_w,L|_{X_{w}}}$ to $\mathcal {F}_{X,L}$.

(iii) Finally, we show that when $X_{w}$ is general, $i_{*}$ induces a surjection of faces. By Lemma 3.1, $L$ is $\mathbb {Q}$-linearly equivalent to a sum $\Delta + A$ where $A$ is ample and $(X,\Delta + A)$ is terminal. By Lemma 3.7 the face $\mathcal {F}_{X,L}$ only contains finitely many extremal rays of $\mathrm {Nef}_{1}(X)$ and each such ray is generated by the class of a movable curve $C_{j}$ on $X$. Recall that we have modified $X$ so that the canonical map $\pi$ is a morphism on $X$. Thus, there is an ample divisor $H$ on $W$ such that $K_{X} + a(X,L) L - \pi ^{*}H$ is $\mathbb {Q}$-linearly equivalent to an effective divisor. This implies that each $C_{j}$ must be $\pi$-vertical. In particular, for each $C_{j}$, a general fiber of $\pi$ will contain an irreducible deformation of $C_{j}$ which is a movable curve on that fiber. Thus, the class of each such curve is nef on $X_{w}$ and, in particular, is contained in $\mathcal {F}_{X_{w},L|_{X_{w}}}$. This proves that $i_* : \mathcal {F}_{X_w,L|_{X_{w}}} \to \mathcal {F}_{X,L}$ is surjective.

Proof. Let $\phi : X' \to X$ be a smooth birational model such that the canonical map for $K_{X'} + a(X',\phi ^{*}L)\phi ^{*}L$ is a morphism on $X'$. Let $\pi ': X' \to W$ be the composition $\pi \circ \phi$. Note that $\pi '$ agrees with the canonical map over an open subset of $W$. We first shrink $W^{\circ }$ to ensure that $\pi '$ coincides with the canonical map over $W^{\circ }$. After shrinking $W^{\circ }$ further we may ensure that $W^{\circ }$ is smooth and that $\pi '$ is smooth over $W^{\circ }$, and by Lemma 4.3 we may also ensure that every fiber $X'_{w}$ over a closed point $w \in W^{\circ }$ satisfies $a(X'_{w},\phi ^{*}L|_{X'_{w}}) = a(X',\phi ^{*}L)$. We may now apply Lemma 4.15(iii) to deduce that for a general $w \in W^{\circ }$ the map $i_{*}: \mathcal {F}_{X'_{w},\phi ^{*}L|_{X'_{w}}} \to \mathcal {F}_{X',\phi ^{*}L}$ is a surjection. By Lemma 4.12 the map $i_{*}: \mathcal {F}_{X_{w},L|_{X_{w}}} \to \mathcal {F}_{X,L}$ is also surjective. On the other hand, the monodromy assumption shows that $N^1(X) \rightarrow N^1(X_w)$ is surjective by (4.1). Dually $i_{*}: N_1(X_w) \to N_1(X)$ is an injection, proving the claim.

Proof. Suppose that we have a subvariety $\overline {Y} \subset \overline {X}$ which is adjoint rigid with respect to $L$, not contained in $\mathbf {B}_{+}(L_{\overline {X}})$, and has $a$-value at least as large as $a(X,L)$. Let $\psi : \widetilde {\overline {Y}} \to \overline {Y}$ be a resolution. By [Reference Lehmann and TanimotoLT17, Theorem 3.5], there is a birational contraction $\phi : \widetilde {\overline {Y}} \dashrightarrow \overline {Y}'$ where $\overline {Y}'$ is a $\mathbb {Q}$-factorial terminal weak Fano variety such that $a(\overline {Y},L|_{\overline {Y}})\phi _{*}\psi ^{*}L|_{\overline {Y}} \equiv -K_{\overline {Y}'}$. By [Reference BirkarBir21, Theorem 2.11] (whose validity is established in all dimensions by induction), we see that there is some constant $C$ depending only on the dimension of $X$ such that

\begin{align*} C \geqslant (-K_{\overline{Y}'})^{\dim \overline{Y}'} & = a(\overline{Y},L|_{\overline{Y}})^{\dim \overline{Y}} \big(\phi_{*}\psi^{*}L|_{\overline{Y}}\big)^{\dim \overline{Y}} \\ & \geqslant a(\overline{Y},L|_{\overline{Y}})^{\dim \overline{Y}} (\psi^{*}L|_{\overline{Y}})^{\dim \overline{Y}} \\ & = a(\overline{Y},L|_{\overline{Y}})^{\dim \overline{Y}} L|_{\overline{Y}}^{\dim \overline{Y}}. \end{align*}

Thus, we have

\[ L|_{\overline{Y}}^{\dim \overline{Y}} \leqslant C/a(\overline{Y},L|_{\overline{Y}})^{\dim \overline{Y}} \leqslant C/a(\overline{X},L)^{\dim \overline{Y}}. \]

By applying [Reference Lehmann, Tanimoto and TschinkelLTT18, Lemma 4.7] to each irreducible component of $\overline {X}$ we conclude that such subvarieties $\overline {Y}$ are parametrized by a bounded subset of $\mathrm {Hilb}(\overline {X})$. We let $\overline {M} \subset \mathrm {Hilb}(\overline {X})$ denote a finite union of components which contains this locus (equipped with the reduced structure).

Let $\overline {N} \subset \overline {M}$ denote the constructible subset over which the universal family has irreducible and reduced fibers. By repeatedly resolving singularities in the fibers and stratifying the base, we find a finite union of locally closed subsets $\overline {N}_{i} \subset \overline {N}$ and smooth morphisms $\overline {p}_{i}: \overline {\mathcal {U}}_{i} \to \overline {N}_{i}$ whose fibers are smooth projective varieties birational to the subvarieties of $\overline {X}$ parametrized by points of $\overline {N}$. We next replace each $\overline {N}_{i}$ by the open subset parametrizing subvarieties not contained in $\mathbf {B}_{+}(L)$; this ensures that $L$ is relatively big and nef on the universal family over $\overline {N}_{i}$. Applying Theorem 4.6 and further stratifying the $\overline {N}_{i}$, we may suppose that the $a$-invariant and Iitaka dimension of the adjoint pair are constant in each family. In particular, there is a constructible sublocus $\overline {T} \subset \overline {M}$ parametrizing the subvarieties which are not contained in $\mathbf {B}_{+}(L)$, have $a$-invariant at least $a(X,L)$, and are adjoint rigid with respect to $L$.

Now we note that $\overline {T}$ is defined over the ground field and thus descends to a subset $T$ of $\mathrm {Hilb}(X)$. Indeed, given any subvariety $Y$ parametrized by a point of $\overline {T}$ and any $\sigma \in \mathrm {Gal}(\overline {F}/F)$, $\sigma (Y)$ is also parametrized by $\overline {T}$ by Proposition 4.4. Furthermore, $\mathbf {B}_{+}(L_{\overline {X}})$ descends to $\mathbf {B}_{+}(L)$. By repeatedly resolving singularities of fibers of the universal family over $T$ and taking finer stratifications of the base, we construct smooth families $p_{i}: \mathcal {U}_{i} \to T_{i}$ whose fibers are birational to the subvarieties parametrized by $T$. We can further stratify $T$ so that after base change to $\overline {F}$ we obtain a substratification of the original stratification of $\overline {T}$; since the $a$-invariant and Iitaka dimension are not affected by base change (see Corollary 4.5) these families will now have the desired properties. Altogether we have established all claims in the statement except the last claim.

Assume that $s_i : \mathcal {U}_i \to X$ is dominant. We claim that a general member $Y$ of $p_i : \mathcal {U}_i \rightarrow T_i$ is contracted by $\pi$ where $\pi : X \dashrightarrow W$ is the canonical map associated to $a(X, L)L + K_X$. Indeed, take a general complete intersection $T \subset T_i$ of dimension $\dim (X) - \dim (Y)$. Then the base change $\mathcal {U}_{i, T}$ has an evaluation map $s_i' : \mathcal {U}_{i, T} \to X$ which is dominant and generically finite. The ramification formula implies that there exists an effective divisor $R \geqslant 0$ on $\mathcal {U}_{i, T}$ such that for a general member $Y$ of $\mathcal {U}_{i, T}$ we have

\begin{align*} a(X, L)s_i^*L|_Y + K_Y &= (a(X, L)(s_i')^*L + K_{\mathcal{U}_{i, T}})|_Y\\ &= (a(X, L)(s_i')^*L + (s_i')^*K_X + R)|_Y. \end{align*}

Since this divisor has Iitaka dimension $0$, we conclude that $(a(X, L)(s_i')^*L + (s_i')^*K_X)|_{Y}$ also has Iitaka dimension $0$, proving our claim.

Since general fibers of $\pi : X \dashrightarrow W$ are adjoint rigid, [Reference Lehmann and TanimotoLT17, Proposition 4.14] applied to a general fiber of $\pi$ shows that the dominant family of adjoint rigid subvarieties contained in that fiber will have a generically finite evaluation map. We deduce that the evaluation map $s_{i}$ will also be generically finite.

Proof. Statement (i) follows immediately from the finiteness of the families in Theorem 4.17. To see (ii), we first combine the finiteness of the families in Theorem 4.17 with Lemma 4.8 to see that $V^{a}$ is not Zariski dense in $X$. This argument also shows that each component of the Zariski closure of $V^{a}$ which is not contained in $\mathbf {B}_{+}(L)$ admits a dominant family of subvarieties $Y$ such that $a(Y,L|_{Y}) > a$. By again applying Lemma 4.8 we see that each component of the Zariski closure of $V^{a}$ which is not contained in $\mathbf {B}_{+}(L)$ has $a$-value larger than $a$. Furthermore, [Reference NakamayeNak00] shows that each component $B_{d}$ of $\mathbf {B}_{+}(L)$ satisfies $a(B_{d},L|_{B_{d}}) = \infty$. Together these imply that $V^{a}$ is closed and that $a(V^{a}_{d}, L|_{V^{a}_{d}}) > a$ for every component $V^{a}_{d}$.

Proof. First, we replace the loci $T_{i} \subset \mathrm {Hilb}(X)$ in Theorem 4.17 with their projective closures $W_{i}$ (equipped with the reduced structure). Let $p_{i}: \mathcal {U}_{i} \to W_{i}$ be the universal family equipped with evaluation maps $s_{i}: \mathcal {U}_{i} \to X$. To start with we set $V = \mathbf {B}_{+}(L)$. Then, for each $i$ such that $s_{i}: \mathcal {U}_{i} \to X$ is not dominant, we add the image of $s_{i}$ to $V$ and remove $\mathcal {U}_{i}$ from our set of families. By Theorem 4.18, any subvariety with $a$-invariant larger than $X$ will be contained in $V$. The desired properties of the $\mathcal {U}_{i}$ then follow directly from Theorem 4.17.

Proof. Let $V \subset X$ denote the closed subset defined by Theorem 4.19. Assume for a contradiction that $f(Y) \not \subset V$. In particular, this implies that $a(Y,f^{*}L) = a(f(Y),L|_{f(Y)}) = a(X,L)$. Let $\Gamma$ denote the closure of a general fiber of the canonical model for $(Y,a(Y,f^{*}L)f^{*}L)$ and let $S$ denote its image in $X$. By Lemma 4.9 we see that $S$ has the same $a$-invariant as $X$ and is adjoint rigid with respect to the restriction of $L$. Since the general such $S$ is not contained in $V$, as we vary $\Gamma$ the images $S$ are parametrized by a family $p_{i}: \mathcal {U}_{i} \to W_{i}$ as in Theorem 4.19 such that the evaluation map $s_{i}$ is dominant (even if $f$ is not dominant). To summarize this discussion, we will obtain the desired contradiction if we can prove that there is no dominant family of subvarieties $S$ such that $(S,L|_{S})$ is adjoint rigid, $a(S,L|_{S}) = a(X,L)$, and ${\dim (S) > d(X,L)}$.

To prove this, we may replace $X$ by any birational model. In particular, we may suppose that the canonical model map for $K_{X} + a(X,L)L$ is a morphism $\pi : X \to W$. Thus, there exist an ample $\mathbb {Q}$-divisor $H$ on $W$ and an effective $\mathbb {Q}$-divisor $E$ such that $K_{X} + a(X,L)L$ is $\mathbb {Q}$-linearly equivalent to $\pi ^{*}H + E$. Suppose we have a diagram of smooth varieties

such that $g$ is generically finite and dominant and the fibers of $q$ are smooth varieties $S_t$ satisfying $a(S_t,g^{*}L|_{S_t}) = a(X,L)$ and $\dim (S_t) > d(X,L)$. We can write

\[ K_{\mathcal{S}} + a(X,L)g^{*}L = g^{*}(K_{X} + a(X,L)L) + R \sim_{\mathbb{Q}} g^{*}\pi^{*}H + (g^{*}E + R) \]

for some effective divisor $R$. Note that the restriction of $g^{*}\pi ^{*}H$ to a general fiber of $q$ has Iitaka dimension at least $1$, so that the general fiber of $q$ cannot possibly be adjoint rigid with respect to the restriction of $L$. This proves the desired contradiction.

Assumption 4.24 Let $X$ be a geometrically uniruled geometrically integral smooth projective variety defined over a field $F$ of characteristic $0$ and let $L$ be a big and nef $\mathbb {Q}$-divisor on $X$. Let $f: Y \to X$ be a morphism of smooth projective varieties that is generically finite onto its image. Suppose that either

1. (i) $f$ is dominant and $a(Y,f^{*}L) = a(X,L)$, or

2. (ii) $a(Y,f^{*}L) = a(X,L)$, $d(Y,f^{*}L) = d(X,L)$, and there is a commuting diagram

   
   where $\pi _{Y}$ and $\pi _{X}$ are the canonical models for the adjoint pairs and the general fiber of $\pi _{Y}$ maps onto a fiber of $\pi _{X}$ which has the same $a$-value as $X$ and is contained in the locus where $\pi _{X}$ is smooth.

Proof. In case (i) $f$ is a dominant map, so by the Riemann–Hurwitz formula we have $K_{Y} \geqslant f^{*}K_{X}$. Since the $a$-invariants are the same we have $K_{Y} + a(Y,f^{*}L)f^{*}L \geqslant f^{*}(K_{X} + a(X,L)L)$. Thus, any nef curve class on $Y$ with vanishing intersection against $K_{Y} + a(Y,f^{*}L)L$ also has vanishing intersection against $f^{*}(K_{X} + a(X,L)L)$, showing that $f_{*}\mathcal {F}_{Y,f^{*}L} \subset \mathcal {F}_{X,L}$.

Suppose that we are in case (ii). Fix a general closed point $w$ in the $(\pi _{X}\circ f)$-image of $Y$. By assumption $X_{w}$ is smooth. Let $Y_{t}$ denote the fiber over a closed point $t \in T$ mapping to $w$. Since the fiber $Y_{t}$ is general, Lemma 4.15(iii) shows that pushforward induces a surjection $\mathcal {F}_{Y_{t},f^{*}L|_{Y_{t}}} \to \mathcal {F}_{Y,f^{*}L}$. By Lemma 4.15(ii) pushforward induces a map $\mathcal {F}_{X_w,L|_{X_{w}}} \to \mathcal {F}_{X,L}$. Thus, it suffices to prove that $f|_{Y_{t}*} \mathcal {F}_{Y_{t},f^{*}L|_{Y_{t}}} \subset \mathcal {F}_{X_{w},L|_{X_{w}}}$.

Since $f|_{Y_{t}}: Y_{t} \to X_{w}$ is a dominant generically finite map of smooth varieties, by the Riemann–Hurwitz formula we have that $K_{Y_{t}} - f|_{Y_{t}}^{*}K_{X_{w}}$ is effective. Since by Lemma 4.3 the $a$-invariants are the same, this implies that $f|_{Y_{t}*}\mathcal {F}_{Y_{t},f^{*}L|_{Y_{t}}} \subset \mathcal {F}_{X_{w},L|_{X_{w}}}$ by the same argument as before.