Proof. (Sketch.) This is a consequence of Kollár and Xu's Theorem, although we provide a brief sketch of the proof for convenience of the reader. By [Reference Abramovich and KaruAK00, Theorem 0.3], we can complete $(X \times T^{\times }, \mathcal {D}^{\times })/T^{\times }$ to a locally stable family $(\mathcal {X}, \mathcal {D})/T'$. Let $r = {q}/{p}$. Choosing an appropriate ample divisor $L$ on $\mathcal {X}$, one can run a relative minimal model program with scaling of $L + \mathcal {D}$ to obtain a minimal family over a smooth base, which is locally stable by [Reference Kollár, Nicaise and XuKNX18, Corollary 10]. Furthermore, the total space is $\mathbb {Q}$-factorial and one can show that $K_{\mathcal {X}^{\mathrm{min}}} + r\mathcal {D}^{\mathrm{min}}$ is relatively trivial and $\mathcal {D}^{\mathrm {min}}$ is relatively big, so by [Reference Hacon and XuHX13, Theorem 1.1] one can find a relative canonical model $(\mathcal {X}^{\rm c}, (r+\epsilon )\mathcal {D}^{\rm c})$. This is a stable family and one can show that $K_{\mathcal {X}^{\rm c}} + r\mathcal {D}^{\rm c}$ is relatively semiample and trivial on the generic fiber, so trivial on the central fiber. Because both $K_{\mathcal {X}^{\rm c}} + r\mathcal {D}^{\rm c}$ and $K_{\mathcal {X}^{\rm c}} + (r + \epsilon ) \mathcal {D}^{\rm c}$ are $\mathbb {Q}$-Cartier, we have $K_{\mathcal {X}^{\rm c}}$ and $\mathcal {D}^{\rm c}$ are both $\mathbb {Q}$-Cartier. Therefore, $(\mathcal {X}^{\rm c}, \mathcal {D}^{\rm c})$ is the desired family of H-stable pairs. Uniqueness follows from uniqueness of canonical models.

Proof. By a theorem of Matsusaka [Reference MatsusakaMat86, Theorem 2.4], there is a fixed integer $m$ such that all smooth Fano varieties $X$ with fixed anticanonical Hilbert polynomial $H_X(l) = \chi (X,-lK_X) = \chi (Z,-lK_Z)$ have $-mK_X$ very ample with no higher cohomology, hence a fixed projective space ${{\mathbb {P}}}^{n}$ containing all such $X$. In particular, $Z$ is one such $X$, as is any smooth variety $W$ deformation equivalent to $Z$. Let $r = {q}/{p}$. Similarly, we can choose $m$ such that $-mK_W|_D$ is very ample with no higher cohomology for any smooth $D \sim _{\mathbb {Q}} -r^{-1}K_W$. We can find a product $\mathrm {Hilb}$ of Hilbert schemes parameterizing embeddings $D \subset W \subset {{\mathbb {P}}}^{n}$. Let $B_0 \subset \mathrm {Hilb}$ be the subscheme parameterizing such pairs. Taking its closure $B_0 \subset B \subset \mathrm {Hilb}$, there is a universal family $(\mathcal {X}, \mathcal {D}) \to B$ whose general fiber is deformation equivalent to the smooth Fano variety $Z$ with a smooth divisor $D$. By [Reference Abramovich and KaruAK00, Theorem 0.3], we can apply weak semistable reduction to obtain a locally stable family $(\mathcal {Y}, \mathcal {D}_{\mathcal {Y}}) \to B'$, where $B' \to B$ is a modification of $B$ and $\mathcal {Y} \to \mathcal {X}' = \mathcal {X} \times _B B'$ is a modification of $\mathcal {X}'$. We can apply the techniques of proof of Theorems 2.6 and 2.7 to find the $K_{\mathcal {Y}} + (r + \epsilon ) \mathcal {D}_{\mathcal {Y}}$ canonical model over $B'$ for any $\epsilon$ sufficiently small; call it $(\mathcal {Y}^{\rm c}, \mathcal {D}^{\rm c})$. It is straightforward to show that every fiber satisfies the conditions of being an H-stable pair; now it suffices to show that every H-stable pair appears as a fiber.

Because the weak semistable reduction potentially modifies even smooth fibers, we first need to show the generic smooth fibers are isomorphic to the generic fibers of the canonical model. Let $(\mathcal {X}^{0}, \mathcal {D}^{0})$ be the universal family over $B_0$ and let $({\mathcal {X}^{0}}', {\mathcal {D}^{0}}')$ be the family over the preimage $B_0' \subset B'$. Let $(\mathcal {Y}^{0,c}, \mathcal {D}^{0,c}) \to B_0'$ be the fibers of the canonical model over $B_0'$. Because the fibers of $({\mathcal {X}^{0}}', {\mathcal {D}^{0}}') \to B_0'$ are smooth Fano varieties with smooth divisors $D$ over a smooth base, it follows that $({\mathcal {X}^{0}}', {\mathcal {D}^{0}}')$ is its own relative canonical model, and birational to $(\mathcal {Y}^{0,c}, \mathcal {D}^{0,c})$ by construction. Therefore, by uniqueness of canonical models, we must have $({\mathcal {X}^{0}}', {\mathcal {D}^{0}}') \cong (\mathcal {Y}^{0,c}, \mathcal {D}^{0,c}).$

Now, let $(Y,D_Y)$ be any H-stable pair. By the smoothability assumption, $(Y,D_Y)$ admits a smoothing over the germ of a curve to $(W,D_W)$ for some $D_W$, so $(Y, D_Y)$ is the central fiber of some family $(\mathcal {X}, \mathcal {D}) \to T$, where $T$ is a discrete valuation ring. Up to a finite base change, the generic fiber $(\mathcal {X}^{\times }, \mathcal {D}^{\times }) \to T^{\times }$ gives an embedding $T^{\times } \to B'$ which can be completed to a curve $T' \subset B'$. The family over $T'$ has the same generic fiber as $(\mathcal {X}, \mathcal {D}) \to T$, but up to a finite base change, both families are canonical models of a common semistable resolution. Therefore, their central fibers are isomorphic.

Proof. This follows directly from [Reference HackingHac04, Lemma 3.3].

Proof. Using the stronger inversion of adjunction result in [Reference PatakfalviPat16, Lemma 2.10], the proof of Lemma 3.4 in [Reference HackingHac04] applies directly.

Proof. Using the following lemma in place of Lemma 3.14 in the proof of Theorem 3.12 in [Reference HackingHac04], the same proof holds.

Proof. Recall that $D$ is an ample $\mathbb {Q}$-Cartier $\mathbb {Z}$-divisor. Although this is a standard consequence of Kawamata–Viehweg vanishing when $X$ is log-terminal, for the general case we apply [Reference FujinoFuj14, Theorem 1.7] to $(X,0)$, noting that $D$ and $-K_X$ are ample, so $D-K_X$ is ample. Note that the hypotheses in [Reference FujinoFuj14, Theorem 1.7] are satisfied because $(X,({q}/{p})D)$ is slc and, by definition, $D$ does not contain any component of the conductor of $X$ (see [Reference KollárKol13, Definition-Lemma 5.10]).

Proof. By [Reference HackingHac04, Remark after 3.9], the obstructions to extending $4$-$\mathbb {Q}$-Gorenstein deformations of $X$ are contained in sheaves $T^{2}_{QG,X}$ (see [Reference HackingHac04, Notation 3.6]). Furthermore, there is a spectral sequence $H^{p}({{\mathcal {T}}}^{q}_{QG,X}) \Rightarrow T^{p+q}_{QG,X}$. If, locally, $\pi : V \to X$ is the $4$-canonical cover where $X$ is the quotient by a group $G$, we can compute ${{\mathcal {T}}}^{q}_{QG,X} = (\pi _*{{\mathcal {T}}}^{q}_V)^{G}$. Furthermore, ${{\mathcal {T}}}^{0}_{QG,X} = {{\mathcal {T}}}^{0}_X, {{\mathcal {T}}}^{1}_V$ is supported on the singular locus of $V$, and ${{\mathcal {T}}}^{2}_V$ is supported where $V$ is not a local complete intersection. By Example 3.9, if $X$ is the cone over the anticanonical embedding of ${{\mathbb {P}}}^{1} \times {{\mathbb {P}}}^{1}$, then $V$ is the cone over the quadric surface. Because the singular locus of $V$ is a single point and $V$ is a hypersurface, we have $H^{1}({{\mathcal {T}}}^{1}_V) = 0$ and $H^{0}({{\mathcal {T}}}^{2}_V) = 0$. A computation shows that $H^{2}({{\mathcal {T}}}^{0}_X) = 0$, hence $T^{2}_{QG,X} = 0$.

Proof. By assumption, $R^{2} \phi _* \mathcal {F} = 0$ for any coherent sheaf $\mathcal {F}$ on $X$. By Grauert–Riemenschneider vanishing [Reference KollárKol13, Corollary 10.38], $R^{1} \phi _* \omega _X = 0$, and by [Reference Kawamata, Matsuda and MatsukiKMM87, Theorem 1-2-5], because $-K_X$ is $\phi$-ample, $R^{1} \phi _* \mathcal {O}_X = 0$. Then, consider any sheaf of ideals $J$ such that $\mathcal {O}_X / J$ is supported on a fiber $F$ of $\phi$:

\[ 0 \to J \to \mathcal{O}_X \to \mathcal{O}_X / J \to 0. \]

Pushing forward to $Y$, we see that $R^{1}\phi _* (\mathcal {O}_X / J) = H^{1}(F, \mathcal {O}_X / J |_F) = 0$. Similarly, we see that $H^{1}(F, \omega _X / J\omega _X |_F) = 0$ so $H^{1}(F, (\omega _X / J\omega _X)|_F /T) = 0$, where $T \subset (\omega _X / J\omega _X)|_F$ denotes torsion. Taking $J$ to be the ideal of $F$, we see that $F$ is a chain of $\mathbb {P}^{1}$s whose configuration is a tree. Then, consider an irreducible component $C \subset F$ and the sheaf $(\omega _X \otimes \mathcal {O}_C) / T$, where $T$ is the torsion in $\omega _X \otimes \mathcal {O}_C$. This is a torsion-free sheaf on $\mathbb {P}^{1}$, so must be a vector bundle of the form $(\omega _X \otimes \mathcal {O}_C) / T \cong \oplus \mathcal {O}_{\mathbb {P}^{1}}(a_i)$. The vanishing of $H^{1}$ given previously implies that $a_i \geqslant -1$ for each $i$. If $m$ is an integer such that $\omega _X^{[m]}$ is Cartier, we must have that $\omega _X^{[m]}\otimes \mathcal {O}_C = \mathcal {O}_{\mathbb {P}^{1}}(b)$ is a negative degree line bundle. However, there is a non-zero morphism from taking the double dual of $\omega _X$:

\[ (\omega_X \otimes \mathcal{O}_C / T)^{\otimes m} \to \omega_X^{[m]} \otimes \mathcal{O}_C \]

and $a_i \geqslant -1$ implies that $b \geqslant -1$. Therefore, $-1 \leqslant K_X \cdot C < 0$.

Proof. Let $\phi : X \to Y$ be the given contraction. Because $\phi$ is $K_X + D$ negative and divisorial, the negativity lemma implies that

\[ \phi^{*} (K_Y+D_Y) = K_X + D - aE, \]

where $D_Y = \phi _*D, E$ is the (irreducible) exceptional divisor, and $a > 0$. Because $D$ is Cartier in codimension 2 and normal by [Reference KollárKol13, Theorem 4.16], $K_X + D |_D = K_D$. By [Reference Kollár and MoriKM98, Corollary 3.44] and [Reference KollárKol13, Theorem 4.16], $D_Y$ is also normal, so restricting to $D$ (where, by abuse of notation, we still denote by $\phi$ the induced map $D \to D_Y$):

\[ \phi^{*}(K_{D_Y} + \mathrm{Diff}_{D_Y}(0)) = K_D - aE|_D, \]

where $\mathrm {Diff}_{D_Y}(0)$ is the correction term to the adjunction formula. This correction term is effective by [Reference KollárKol13, Proposition 4.5], and $aE|_D$ is effective, so $\phi |_D: D \to D_Y$ is an isomorphism if and only if $E|_D = 0$. This also shows that $\mathrm {Diff}_{D_Y}(0) = 0$.

Proof. Let $\pi : X \to Y$ be the contraction. If a divisor is contracted, then the morphism is either a divisorial contraction or Fano fiber contraction onto a variety with strictly lower dimension. If no curves in $D$ are contracted, the induced map $D \to \pi (D)$ is finite, but $(Y, \pi _*D)$ is dlt by [Reference Kollár and MoriKM98, Corollary 3.44] and $D$ is a prime divisor, hence $\pi _*D$ is normal by [Reference KollárKol13, Theorem 4.16]. Furthermore, because no curves in $D$ are contracted, the fibers have dimension at most one. Indeed, let $F$ be a fiber of $\pi$. Because $\pi$ contracts no curves in $D, F \cap D$ must be a collection of points. However, $\pi$ is a $D$-positive contraction, so $D|_F$ is ample, hence must be a divisor on $F$. Therefore, $F$ has dimension at most one.

However, if $\pi$ is divisorial, Lemma 4.8 implies $K_X \cdot C \geqslant -1$ for any irreducible curve $C$ contracted by $\pi$. Suppose that $C$ is an irreducible component of a general fiber of dimension one. By assumption, $D\cdot C > 0$, and because $D$ is Cartier in codimension two, $D$ is Cartier when restricted to $C$, so $D \cdot C \in \mathbb {Z}$. Therefore, $(K_X + D) \cdot C \geqslant 0$, a contradiction. Thus, the contraction cannot be birational and must be a fibration with general fiber $\mathbb {P}^{1}$. In this case, for general fiber $C, K_X \cdot C = -2$, so we must have $D \cdot C = 1$, so $\pi |_D: D \to \pi (D)$ is generically of degree one. Therefore, by Zariski's main theorem, and because $D$ is prime, $\pi _*D$ must be isomorphic to $D$ and $\pi : X \to Y$ is a Fano fiber contraction and $Y \cong D$.

Proof. If $X$ is terminal, the singular set has codimension at least three in $X$ (see [Reference KollárKol13, Corollary 2.30]), hence $D$ is Cartier in codimension two. If $-D|_D$ is nef, then any $D$-positive contraction contracts no curves in $D$, so by Corollary 4.11, the contraction of such a ray gives a Fano fibration $X \to D$.

Proof. Assume such a small contraction exists. Because this is a $K_X$-negative contraction, we consider the flip of $\phi$ as in the following diagram, where $Z$ is a log resolution of the rational map $X \dashrightarrow X^{+}$ (so both $Z$ and the strict transform of $D$ are smooth). The flip exists by [Reference Birkar, Cascini, Hacon and McKernanBCHM10, Corollary 1.4.1].

Note that the fiber of the contraction $\phi : X \to Y$ is not contained in $D$, by assumption. Because every $\pi$ exceptional divisor $E$ has non-negative discrepancy $a(E,X,D)$, if $D_Z = \pi ^{-1}_* D$, we have

\[ K_Z + D_Z = \pi^{*}(K_X + D) + \sum a_i E_i, \]

where $a_i > -1$ for each $i$, and $a_i \geqslant 0$ for any exceptional divisor $E_i$ such that $\text {center}_X(E_i) \not \subset D$. Restricting to $D$, because $D$ is Cartier in codimension two, we obtain

\[ K_{D_Z} = \pi|_D^{*}(K_D) + \sum a_i E_i|_{D_Z} . \]

However, by [Reference Kollár and MoriKM98, Lemma 3.38], flips can only improve singularities, so

\[ {\pi^{+}}^{*}(K_{X^{+}}+D^{+}) = \pi^{*}(K_X + D) - \sum c_iE_i, \]

where $c_i \geqslant 0$. Because the flip was $K_X$-negative, by the same lemma, $X^{+}$ is also terminal, so $D^{+}$ is Cartier in codimension two. Then, restricting to $D$ and $D^{+}$ we see that

\[ {\pi^{+}}|_{D^{+}}^{*}(K_{D^{+}}) = \pi|_D^{*}(K_D) - \sum c_iE_i|_{D_Z} . \]

Substituting, we see that

(4.1)\begin{equation} {\pi^{+}}|_{D^{+}}^{*}(K_{D^{+}}) = K_{D_Z} - \sum a_i E_i|_{D_Z} - \sum c_iE_i|_{D_Z} . \end{equation}

Because no curves in $D$ are contracted, the map $D \to \phi (D)$ is finite, and because $\phi$ is a small contraction, the map has degree one, so either $D \cong \phi (D)$ or $D$ is its normalization. In either case, because $X^{+}$ is dlt, $D^{+}$ is normal by [Reference KollárKol13, Theorem 4.16], so there is a morphism $f:D^{+} \to D$ making the diagram

commute. Observe that $f$ cannot be an isomorphism: if $D^{+} \cong D$, then $f^{*}(K_D) = K_{D^{+}}$, and hence $\pi |_D^{*}(K_D) = \pi ^{+}|_{D^{+}}^{*}(K_{D^{+}})$. However, this implies that $c_i = 0$ for all $i$. By [Reference Kollár and MoriKM98, Lemma 3.38], this implies that, for any exceptional divisor $E$ over $X, \phi \circ \pi = \phi ^{+} \circ \pi ^{+}$ is an isomorphism over the generic point of $\mathrm {center}_Y E$. This implies that neither $\pi$ nor $\pi ^{+}$ extract any divisors, so by normality of $X, Z,$ and $X^{+}$, and $\mathbb {Q}$-factoriality of $X$ and $X^{+}, X \cong X^{+}$, a contradiction.

Therefore, there is some exceptional divisor $E_0$ such $E_0|_{D_Z}$ is not contracted by $\pi ^{+}|_{D_Z}$ but is contracted by $\pi |_{D_Z}$. In this case, we must have $c_0 > 0$. If we can show that $\text {center}_X(E_0) \not \subset D$, this is a contradiction because it would imply the coefficient of $E_0|_{D_Z}$ in (1) is non-zero, but $E_0|_{D_Z}$ is not an exceptional divisor of $\pi ^{+}|_{D_Z}$.

To conclude the proof, assume for contradiction that $\text {center}_X(E_0) \subset D$ for any such divisor $E_0$. Then, for any exceptional divisor $E_0$ contracted to a curve in $D^{+}$, it must be contracted to a point in $D$, as the small contraction did not contract any curves in $D$. By definition of $Z$, there must exist a divisor $E_1$ whose image is a one-dimensional component of $C$, i.e. $\text {center}_X{E_1} \not \subset D$. By assumption, for any such divisor $E_1, \pi ^{+}(E_1) \cap D^{+}$ must be finite. However, let us compute discrepancies. Because $\pi (E_1)$ is a curve not contained in $D$ and $X$ has only isolated singularities, $K_Z = \pi ^{*}(K_X) + b_1E_1 + \text{other terms}$, where $b_1 \in \mathbb {Z}, b_1 > 0$. In addition, $\pi ^{*}(D) = D_Z + 0 \cdot E_1 + \text { other terms}$. Thus, the discrepancy $a_1$ of $E_1$ in the map $\pi$ is $a_1 = b_1 \in \mathbb {Z} > 0$. However, on $D^{+}$, we then obtain that $K_{D_Z} = \pi ^{+}|_{D^{+}}^{*}(K_{D^{+}}) + (1+c_1)E_1 + \text { other terms}$, where the other terms by assumption correspond to divisors with positive discrepancy in the same fashion as $E_1$. All discrepancies in the extraction $D_Z$ to $D^{+}$ are thus at least one. By assumption, $D_Z$ was smooth, so $D^{+}$ is therefore smooth. As $D_Z \to D^{+}$ can be factored as blow ups of smooth points [Reference HartshorneHar77, Chapter 5.5], the discrepancy of some $E_i$ (without loss of generality, assume it is $E_1$) must be $1$, and $c_1 = 0$. However, this contradicts the fact that $\phi : X \to Y$ is not an isomorphism above the generic point of $\text {center}_Y(E_1)$, as $E_1$ is contracted to a curve via $\pi$ (see [Reference Kollár and MoriKM98, Lemma 3.38]). Therefore, we have reached a contradiction and, thus, there exists an $E_0$ such that $\text {center}_X(E_0) \not \subset D$.

Proof. Because varieties with terminal singularities are singular only in codimension $\geqslant 3$, $D$ is Cartier in codimension two. By hypothesis, any curve $C \subset D$ has $K_X \cdot C \geqslant 0$, hence the contraction $\phi : X \to Y$ of a $K_X + D$-negative $D$-positive extremal ray cannot contract any curves in $D$. By Lemma 4.13, the contraction of $R$ cannot be a small contraction, so must contract a divisor. Applying Corollary 4.11, we obtain that $X\to Y$ is a Fano fibration and $Y \cong D$.

Proof. By a result of Hacon [Reference Kollár and KovácsKK10, Theorem 3.1], there is a minimal $\mathbb {Q}$-factorial dlt model of $Y \to X$. More precisely, there is a $\mathbb {Q}$-factorial variety $Y$ and a morphism $\pi : Y \to X$ extracting all divisors $E_i$ with discrepancy $a(E_i, X) = -1$ such that $K_Y$ is relatively nef. Let $E = \sum E_i$ and observe that $K_Y + E = \pi ^{*}K_X$. Because $-K_X$ is ample and $K_Y + E$ is numerically trivial on $E$ and negative on all curves not contained in $E$, there must exist a $K_Y + E$ negative, $E$ positive extremal ray $R$ in $\overline {NE}(Y)$. In fact, because $E \cdot R > 0$, there must be some component $E_0 \subset E$ such that $E_0 \cdot R >0$ and $(K_Y + E_0) \cdot R < 0$. Indeed, if $(K_Y + E_0) \cdot R \geqslant 0$, for any curve $[C] \in R$, we would have $C \subset E$, so $(K_Y+E) \cdot C = 0$, a contradiction.

Let $\phi : Y \to S$ be the contraction of $R$. By assumption, the pair $(Y,E)$ is canonical along the components of $E$ (because $a(F,Y,E) = a(F,X)$ for any exceptional divisor $F$ over $X$). By Corollary 4.11 applied to $(Y, E_0), \phi : Y \to S$ is a fiber contraction of relative dimension one and $S \cong E_0$. Note that, for a general fiber $l$ of $\phi, K_Y \cdot l = -2$ and $[l] \in R$, so $(K_Y + E) \cdot l < 0$. Choosing an appropriate fiber $l$ that misses the singular points of $Y$, one sees that $E_i \cdot l \in \mathbb {Z}$ for each $i$ because $l$ is contained in the smooth locus of $Y$, and $E_i$ cannot be contracted by $\phi$ as curves in $E_i$ are $K_Y+E$-trivial. Therefore, because $K_Y\cdot l = -2$ and $(K_Y + E) \cdot l < 0$, there is only one exceptional divisor $E_0 = E$. Because $(Y,E)$ is dlt, $E$ is normal and $\phi$ contracts no curves in $E$, hence $S \cong E$, giving $\phi : Y \to S$ the structure of a $\mathbb {P}^{1}$ bundle. However, as $E$ is contractible by $\pi : Y \to X$, we see that $X$ is a cone over $E$ (where ‘cone’ is only defined as the contraction of a section of a $\mathbb {P}^{1}$-bundle over $E$ to a point). We can further characterize $E$ by observing that $(K_Y + E)|_E = K_E$, hence $K_E$ is numerically trivial.

Proof. We proceed in a similar fashion as in the previous proof. By [Reference KollárKol13, Theorem 1.33, Corollary 1.37], there is a terminal model of $X$; a $\mathbb {Q}$-factorial variety $Y$ and a morphism $\pi : Y\to X$ extracting divisors $\Delta _i$ with discrepancy $a(\Delta _i,X) \leqslant 0$ such that $Y$ is terminal and $K_Y$ is relatively nef. Let $E = \sum \Delta _j$ be the sum over divisors $\Delta _j$ with discrepancy $-1$ and $F = \sum -a(\Delta _k,X)\Delta _k$ be the sum over divisors with discrepancy larger than $-1$. By construction of $Y$ (which is terminal, hence has finitely many singular points), these effective divisors are Cartier in codimension two, $\pi ^{*}(K_X) = K_Y + E + F$, and for any curves $C\subset \operatorname {Supp} (E+F)$ contracted by $\pi$, $K_Y\cdot C \geqslant 0$. By assumption on $X$, the general curve through $E$ has negative $K_X$-degree.

We would like to find an $E$ positive and $K_Y + E$ negative extremal ray in the cone of curves. If so, we proceed exactly as in the proof of Theorem 4.6 to conclude that the contraction of such a ray gives a Fano fibration $\phi : Y \to E$ (and $E$ consists of only one component, necessarily not rational nor ruled by assumption). Then, we conclude as in Theorem 4.6 that the log-canonical locus in $X$ consists of a single point $x\in X$, $E$ is a single component, and $X$ is birational to a ${{\mathbb {P}}}^{1}$ bundle over $E$.

In more generality, run a $K_Y$-minimal model program (MMP) (contracting $K_Y$ negative extremal rays) on $Y$. If at any point in the MMP we reach an intermediate variety $Y'$ with a $K_{Y'} + E'$-negative $E'$-positive extremal ray $R$ in $\overline {NE}(Y')$, the arguments in the previous paragraph imply that the MMP terminates in a fibration of dimension one over $E'$. As we show in the following arguments, the only components that could have been contracted in running the MMP up to this point are rational or ruled, so we obtain the desired result.

Finally, assume that we do not find such a ray in the course of the MMP. Because $X$ was a Fano threefold, the MMP must terminate with a Fano fibration $f: Y' \to S$ such that $\dim S < 3$. Because $Y$ has terminal singularities and at each stage of the MMP as we are contracting $K_Y$-negative extremal rays, by [Reference Kollár and MoriKM98, Corollary 3.43(2)], $Y'$ also has terminal singularities. We claim that the only components of $E$ that could be contracted by an MMP are rational or ruled.

In running the $K_Y$-MMP, first consider the case that in one of the steps one obtains a divisorial contraction $\phi : Y' \to Y''$ of a component $\Delta$ of $E'$. Because $(Y,E)$ is at worst log-canonical (and, hence, $(Y', E')$ is at worst log-canonical), $\Delta$ is a log-canonical surface, contracted by $\phi$ to a point or curve. If the divisorial contraction $\phi : Y' \to Y''$ contracted $\Delta$ to a curve, the general fiber is ${{\mathbb {P}}}^{1}$, so it is birationally ruled. Thus, in this case, the result follows. If instead $\Delta$ is contracted to a point, $\Delta$ is Fano, and a normal log-canonical surface, so only singular at points. By [Reference KollárKol92, Theorem 17.4], the locus of strictly log-canonical non-klt singularities is connected and hence a single point. Therefore, [Reference Hacon and MckernanHM07, Theorem 1.2] implies that $\Delta$ is rationally chain connected. Finally, by [Reference KollárKol96, Proposition 3.3.4], this implies that $\Delta$ must be uniruled, and any uniruled surface is birationally ruled by [Reference KollárKol96, Exercise 1.1.6.2], so the result follows.

Now, suppose no component of $E'$ is contracted until the termination of the MMP $f': Y' \to S$. If $Y'$ is a terminal Fano variety of Picard rank one, because $K_{Y} + E = \pi ^{*}(K_X)$ was negative on the generic curve in $Y$, we must have $K_{Y'} + E'$ negative. Hence, for any component $\Delta$ of $E', K_{Y'} + \Delta$ is negative and $\Delta$ is a log-canonical Fano surface, hence rationally connected, hence rational or birationally ruled by the previous argument.

If $Y'$ has Picard rank two and $S$ is a curve, if $\Delta$ is a fiber of $f'$, because $Y'$ is terminal, $\Delta$ must be log-terminal and Fano, and hence rational. If instead $f'|_{\Delta }: \Delta \to S$ is surjective, $\Delta$ is birationally ruled. Finally, if $\dim S = 2$, $\dim f'(\Delta ) = 0$ implies $\Delta$ is Fano and therefore rational or birationally ruled by the previous arguments. If $\dim f'(\Delta ) = 1$, $\Delta$ is again birationally ruled.

Therefore, if there exists a non-rational or birationally ruled component $\Delta$ of $E'$, it cannot be contracted in any way described previously, and thus we must be in the final remaining case: $f': Y' \to S$ is a Fano fibration and $\dim f'(\Delta ) = 2$. Because $f'$ is the contraction of a $K_{Y'}$-negative ray (and necessarily $K_{Y'} + E'$-negative, by consideration of the general fiber $l$, whose transform in $Y$ must be $K_Y+E$-negative), we have $K_{Y'}\cdot l = -2$ and, hence, must have $E' \cdot l = \Delta \cdot l = 1$, so $\Delta$ is birational to $S$. In particular, we are in the case of the Fano fibration over a surface birational to $\Delta$. By the computation $E' \cdot l = \Delta \cdot l = 1$, we see that $\Delta$ is the only such component that can map birationally to $S$, which implies the result.

Proof. By Theorem 4.6, Proposition 4.15, or Theorem 4.16, there is a birational model $Y \to X$ such that $\pi ^{*}K_X = K_Y + E + F$ and the minimal model program on $Y$ terminates in a Fano fibration $Y' \to S$, where $S$ is a surface birational to $E_0$, a component of $E$.

If $X$ is as in Theorem 4.6 or Proposition 4.15, from their proofs, we have $Y' = Y$. Consider a general fiber $l \subset Y$: we have $K_{Y} \cdot l = -2$, and $E \cdot l = 1$, so $\pi ^{*}(K_X)\cdot l = (K_Y+E)\cdot l = -1$. Hence, if $\bar {l}$ is the image of $l$ on $X$, $K_X \cdot \bar {l} = -1$. However, $D$ does not contain the strictly log-canonical locus of $X$, so choosing $l$ sufficiently generally implies that $D \cdot \bar {l} = n \in \mathbb {Z}$, hence the relationship $dK_X + 4D \sim 0$ implies $d(-1) + 4n = 0$, so $d$ is even.

In the next case, suppose $X$ is as in Theorem 4.16. Following the notation and proof, we obtain that the MMP results in a Fano fibration $f': Y' \to S$, where $S$ is birational to the unique non-rational or ruled component of $E'$ (and this component is generically a section of $f'$). Now, consider $F'$, the image of the divisors $F$ in $Y$ with discrepancy $> -1$. By [Reference Hacon and MckernanHM07, Theorem 1.2], these divisors are all rational or ruled. Hence, they cannot be birational to $S$, so for each component $\Delta _F \subset \operatorname {Supp} F'$, we have $\dim f'(\Delta _F) < 2$. Therefore, choosing a sufficiently general fiber $l$ of $f'$, $l$ does not intersect any component of $\operatorname {Supp} F'$ and $(K_{Y'} + E') \cdot l = -1$. Therefore, if $l'$ is the pre-image of $l$ in $Y$, $\pi ^{*} K_X \cdot l' = (K_Y+E+F)\cdot l' = -1$, and if $\bar {l}$ is the image of $l'$ on $X$, $K_X \cdot \bar {l} = -1$. However, $D$ does not contain the strictly log-canonical locus of $X$, so choosing $l$ sufficiently generally implies that $D \cdot \bar {l} = n \in \mathbb {Z}$, hence the relationship $dK_X + 4D \sim 0$ implies $d(-1) + 4n = 0$, so $d$ is even.

Proof. By [Reference KollárKol13, Theorem 1.33], there is a $\mathbb {Q}$-factorial variety $Y$ and a morphism $\pi : Y \to X$ extracting divisors $\Delta _j$ with discrepancy $a(\Delta _i,X) \leqslant 0$ such that $Y$ is terminal and $K_Y$ is relatively nef. Let $E = \sum \Delta _j$ be the sum over divisors $\Delta _j$ with discrepancy $-1$ and $F = \sum -a(\Delta _k,X)\Delta _k$ be the sum over divisors with discrepancy larger than $-1$. By construction of $Y$ (which is terminal, hence has finitely many singular points), these effective divisors are Cartier in codimension two, $\pi ^{*}(K_X) = K_Y + E + F$, and for any curves $C\subset \operatorname {Supp} (E+F)$ contracted by $\pi$, $K_Y\cdot C \geqslant 0$.

We run a $K_Y$-MMP to obtain the result. The proof is somewhat technical, so we first provide a sketch, pointing out the necessary technicalities. If there is a component of $E_0$ that is not contracted by the MMP, the MMP terminates in a fibration $Y' \to S$ and one can show that $S$ is birational to $E_0$. Now, to conclude that $d$ is even, we wish to perform a similar intersection theoretic computation to that above, finding a sufficiently generic fiber $l$ of $Y' \to S$ such that its preimage $l'$ in $Y$ satisfies $(K_Y+E_0)\cdot l' = -1$. However, to conclude that $\pi ^{*}(K_X)\cdot l' = -1$, we must show that either there do not exist other components of $\operatorname {Supp} E$ or $\operatorname {Supp} F$ that map finitely to $S$ or we can use the other components to conclude that $d$ is even.

If instead all components of $E$ are contracted by the MMP, we prove the necessary existence of certain components of $\operatorname {Supp} F$ intersecting $E$ so that the curves in $E$ (which, initially may be $K_Y$-positive) become $K_Y$-negative and thus contractible. Then, we make the following observation: because $dK_X + 4D \sim 0$ and the strictly log-canonical (non-klt) locus of $X$ is not contained in $D$, for any component $\Delta$ of $\operatorname {Supp} F$ that maps into the non-klt locus of $X$, the discrepancy of $\Delta$ must be of the form ${a}/{d}$ for some $a \in \mathbb {Z}$. Furthermore, on curves contained in components of $\Delta$ of $E$ contracted by $\pi$, we have $(K_Y + E + F)|_{\Delta } = 0$, so if $C_E = (E-\Delta )|_\Delta$ and $C_F = F|_\Delta$, we know $K_{\Delta } + C_E + C_F = 0$. If $\Delta$ is sufficiently nice, we use this relationship to show that either $d$ must be even or the discrepancies of the components of $F$ must be in a finite collection of exceptional possibilities, and we rule out these possibilities by hand.

We begin in the simpler case, when there exists a component $E_0 \subset E$ that is not contracted by the MMP.

Proof. In this case, the MMP must terminate with a morphism $Y \dashrightarrow Y' \to S$, and $\dim S = 2$. Exactly as in the proof of Theorem 4.6, we obtain that $S$ is birational to $E_0'$, where $E_0'$ is the image of $E_0$ in $Y'$, and by negativity of $K_Y + E = \pi ^{*}K_X$, the general fiber $l$ of $Y' \to S$ cannot intersect other components of $E$. We are done by the same argument as in Corollary 4.17 if $l$ intersects no components of $\operatorname {Supp} F$. However, a priori, $l$ could intersect components of $\operatorname {Supp} F$: suppose $\Delta _0' \to S$ is generically finite, where $\Delta _0'$ is the image of some $\Delta _0 \subset \operatorname {Supp} F$. Because $\Delta _0$ is contracted by $\pi$, let $C \subset S$ be the image of a general fiber of $\pi |_{\Delta _0}$, noting that $C$ must be rational as $\Delta _0$ contracts to a log-terminal singularity [Reference Hacon and MckernanHM07, Theorem 1.2]. Because $Y'$ is terminal, it has isolated singularities, so choosing $C$ sufficiently generally, we may assume that the pre-image of $C$ is a smooth ruled surface $P \to C$ with multi-section $\Delta _0'|_P$. As $E_0'$ is a section of $P$, and all fibers of $P$ are numerically equivalent with respect to $E_0'$, we see that there cannot be any singular fibers (using that $P$ is smooth). However, because $\Delta _0'|_P$ is a contractible multi-section in the smooth ruled surface $P$, it must, in fact, be a section. As there is only one contractible section in $P$, we see that $E_0'|_P$ cannot be contractible. Similar analysis implies, for any other component $\Delta _1$ of $\operatorname {Supp} F$ mapping finitely to $S, \Delta _1'|_P$ cannot be contractible.

However, for any such $\Delta _1'$, reversing roles of $\Delta _1'$ and $\Delta _0'$, we find a surface $P'$ with contractible section $\Delta _1'|_{P'}$ and non-contractible sections $\Delta _0'|_{P'}$ and $E_0'|_{P'}$. Computation shows that this would imply that $E_0$ is, in fact, not contractible in $Y$, a contradiction. Therefore, there can be at most one component $\Delta _0$ of $\operatorname {Supp}(F)$ that maps birationally onto $S$.

Thus, in the smooth ruled surface $P$, we may assume that $E_0'|_P$ is a positive section, $(E_0'|_P)^{2} = n$. Choosing another section $\gamma$ such that $\gamma \cap \Delta _0'|_P = \emptyset$, $\gamma ^{2} = E_0' \cdot \gamma = n$, and $K_P \cdot \gamma = K_{Y'} \cdot \gamma = -2 - n$, so that $K_X \cdot \gamma ' = (K_{Y'} + E_0')\cdot \gamma = -2$, where $\gamma '$ is the image of $\gamma$ on $X$. Provided that $C$ and $\gamma$ are chosen generically, $D \cdot \gamma ' = n \in \mathbb {Z}$, hence the relationship $dK_X + 4D \sim 0$ implies that $d(-2) + 4n = 0$ and $d$ is even.

Proof. Because $F$ is contracted to a curve and the discrepancy of the divisor $F$ is greater than $-1$, at the generic point of $C$, we reduce to studying log-terminal surface singularities. Because this curve is contained in $E_0$, these are cyclic quotient singularities [Reference Kollár and MoriKM98, Theorem 4.15]. Let $n$ be the index (and note that, by the previous discussion, $n \mid d$, so $n$ is odd). Then, the Cartier index of the divisor $K_{X^{\mathrm{dlt}}} + E^{\mathrm {dlt}}$ is a divisor of $n$, so all discrepancies take the form ${a_i}/{n}$.

Consider the map $g: E \dashrightarrow E^{\mathrm {dlt}}$. At the generic point of $C$, this is an isomorphism, so a local computation of the discrepancy of $F_i$ can be done using adjunction. Near the generic point of $C$, we have $K_Y + E + \sum a_i F_i = g^{*}(K_{X^{\mathrm {dlt}}} + E^{\mathrm {dlt}})$, and restricting to $E$, we obtain $K_E + \sum a_i F_i|_E = K_E + \mathrm {Diff}$, where $\mathrm {Diff}$ is supported on the codimension one part of singular locus of $X^{\mathrm {dlt}}$ contained in $E$, which is $C$. By [Reference KollárKol13, 4.4], the coefficients of the different are precisely $1 - {1}/{n}$. Thus, for $F_1$ such that $F_1 \cap E \ne \emptyset$, the discrepancy is as claimed.

Finally, the description of the divisors as a chain is [Reference Kollár and MoriKM98, Theorem 4.15(3)].