Original proof of bend and break lemma
In this section, I will give a half of a proof of Theorem 4 in [Mor79]. We usually call this"bend and break" lemma which widely used in birational geometry.
Theorem: Let $f: \mathbb{P}^1 \rightarrow C$ be a morphism such that $f(0) \neq f(\infty)$. Suppose there exists a smooth connected curve $D$, a closed point $t_0 \in D$ and a morphism $\varphi: \mathbb{P}^1 \times D \rightarrow C$ such that:
$\varphi |_{\mathbb{P}^1 \times t_0}=f$
$\varphi(0 \times D) = f(0)$
$\varphi(\infty \times D) = F(\infty)$
$dim \varphi(\mathbb{P}^1 \times D) = 2$
Suppose exists a closed point $t_1$ in the smooth compactification $\overline{D}$ of $D$ with $t_1 \notin D$ and morphism $\pi: Y \rightarrow \overline{D}$ and $\xi: Y \rightarrow C$ such that:
$(\xi_{*} Y_t)_{cycle} = (f_{*} \mathbb{P}^1)_{cycle}$ for $ t = t_0$ or cycle with at least two rational curves or with multiple rational curves for $t = t_1$
In particularly, $(f_{*} \mathbb{P}^1)_{cycle}$ is numerical equivalence to cycle with at least two rational curves or with multiple rational curve as components.
proof of theorem: (cf. [Mum, pro. 9.1.10] and [Koll-Mor98, pro. 1.9]). First, consider a ruled surface $q: S \rightarrow \overline{D}$, we see that $q$ will be a $\mathbb{P}^1$-bundle morphism and $q^{-1} ( D) = \mathbb{P}^1 \times D$. Because there exists section $\alpha$ from $\overline{D}$ to $S$ and we define a projection morphism $(0,\alpha): \mathbb{P}^1 \times D \rightarrow D$ hence $q$ compatible with $p_2: \mathbb{P}^1 \times D \rightarrow D$ by $(0,\alpha)$. For the morphism $\varphi: \mathbb{P}^1 \times D \rightarrow C$, we get an extension map $\overline{\varphi}: S \dashrightarrow C$.
Now we prove the theorem by induction on the number of blow-up in $Y \rightarrow S$ s.t $Y \rightarrow C$ is a morphism.
We first prove that $\overline{\varphi}$ is not morphism. First, Let $C_0$ and $C_{\inf}$ are two divisors extending $0 \times D$ and $\inf \times D$. We see that $\overline{\varphi}$ maps $C_0$ and $C_{\infty}$ to two points $f(0)$ and $f(\infty)$. Let $H$ be an ample sheaf on $S$. We see that dim$\overline{\varphi} (S) $=dim $\varphi( \mathbb{P}^1 \times D) = 2$ because $ \mathbb{P}^1 \times D$ is dense in $S$ and then $(\overline{\varphi}*H.\overline{\varphi}*H)=(H.H) > 0$ $(\overline{\varphi}*H.C_0) = (H.\overline{\varphi}_{*}C_0) > 0$, $(\overline{\varphi}*H.C_{\infty}) = (H.\overline{\varphi}_{*}C_{\infty})=0$ by projection formula over surface. Hence, by Hodge index theorem, we get: $(C_0^{2}) < 0 $, $(C_{\infty}^{2}) < 0$ and $C_0.C_{\infty} = 0$ (because these two don't intersect anywhere). Hence, there are there linear independence divisor, but picard number over a surface is only two, which means when you consider $C^1(S)$ as a vector space with dimension 2, there can not be there divisor which can be linear independence, and thus we arrived a contradiction.
Let $r: Y \rightarrow S$ be a composition of blow-up sequence which eliminating indeteminacy of $\overline{\varphi}$ and gives a morphism $\xi: Y \rightarrow C$. Denote $\pi = q \circ r: Y \rightarrow \overline{D}$. Now we will prove our problem by induction on the number of points blow-up to eliminating indeterminacy.
Let $\sigma: S' \rightarrow S$ be a first blow up at $P$ . Denote $t_1 = q(P) \in \overline{D} - D$. Then $\pi ^{-1}(t_1)$ irreducible components must be a rational curve by taking inverse among exceptional divisor or strict transform of $\mathbb{P}^1$. Then $\xi(\pi^{-1}(t_1))$ is a union of rational curves by Liiroth’s theorem as $q$ induced a surjective from $\mathbb{P}^1$ to each curve. If $\xi(\pi^{-1}(t_1))$ is reducible or non-reduce, then $\xi(Y_j)_{red}$ will be rational curve with $Y_j$ are irreducible component of $\pi^{-1}(t_1)$. Now we prove that $\xi(\pi^{-1}(t_1))$ can not be reduced and irreducible. Assume it's so. We see that $\overline{\varphi}$ is define at $q^{-1}(t_1) - P$. If it's not at a $P'$, then $\xi(\pi^{-1}(t_1)) \supset \xi(r^{-1}(P)) \cup \xi(r^{-1}(P')) $ which contradict to irreducile assumption.
Now let $E$ and $F$ be the exceptional curve for $\sigma: S' \rightarrow C$ and strict transform of $q^{-1}(t_1)$ by $\sigma$ respectedly. Thus: $(q \circ \sigma)^{-1}(t_1)=F \cup E$ with $Q = E \cap F$. $\varphi: \mathbb{P} \times D \rightarrow D$ give rise to a rational map $\varphi ' : S \dashrightarrow C$. Denote $r = r' \circ \sigma$ with $r': Y \rightarrow S'$ being the composition of the point blow-up other than $\sigma$.
To finish the proof, we must claim that $\varphi '$ is defined at $Q$. By blowing up we see that $r'^{-1} (Q)$ must have all irreducible components of multiplicity at least 2 because those components belong to two curves $E$ and $F$, any of those irreducible components must appear two times, one as a cycle related to $E$, one as a cycle related to $F$. But for any reduce and irreducible scheme, the cycle associated to it must not have any component with multiplicity larger than 1, because its Ass set containing all minimal point. Hence $\xi$ must contracted $r^{-1}(Q)$. Hence $\varphi'$ must define at $Q$ and hence a morphism.
Now, for a case that $S'$ is the $Y$ we need, we can see that $\varphi'_{*}S'_{t_1}_{cycle}$ will s.t the demand of theorem because $\varphi'(\pi^{-1}(t_1))$ is the union of at least more than 2 rational curves. For $t_0 \in D$, $\pi^{-1}(t_0)$ must be $\mathbb{P}^1$ by property of ruled surface, hence $ \varphi'_{*}(\pi^{-1}(t_0))_{cycle} = $f_{*}(\mathbb{P}^1)_{cycle}$.
Now the statement true for 1, now we suppose it's true for $n-1$.
We have $F=\mathbb{P}^1$ with $(F^2)=-1$. Hence by Castelnuovo's criterion, $F$ can e contracted to a point y $\sigma ': S' \rightarrow S''$, which gives us another ruled surfce by an elementary transformation. The rational map $\varphi '': S'' \dashrightarrow C$ needs one less point blow-ups for elimination indeterminacy, since $\varphi$ define along $F$ of $\sigma' : S' \rightarrow S$. Thus we are done by induction.
References:
[Mor79] S.Mori, Projective Manifolds with Ample Tangent Bundles, Annals of mathematics, second series Vol. 110, No. 3 (Nov. 1979), pp. 593 - 606.
[Kol-Mor98] S.Mori, J.Kollar, Birational Geometry of Algebraic Varieties, Cambridge tracts in mathematics.
Theorem: Let $f: \mathbb{P}^1 \rightarrow C$ be a morphism such that $f(0) \neq f(\infty)$. Suppose there exists a smooth connected curve $D$, a closed point $t_0 \in D$ and a morphism $\varphi: \mathbb{P}^1 \times D \rightarrow C$ such that:
$\varphi |_{\mathbb{P}^1 \times t_0}=f$
$\varphi(0 \times D) = f(0)$
$\varphi(\infty \times D) = F(\infty)$
$dim \varphi(\mathbb{P}^1 \times D) = 2$
Suppose exists a closed point $t_1$ in the smooth compactification $\overline{D}$ of $D$ with $t_1 \notin D$ and morphism $\pi: Y \rightarrow \overline{D}$ and $\xi: Y \rightarrow C$ such that:
$(\xi_{*} Y_t)_{cycle} = (f_{*} \mathbb{P}^1)_{cycle}$ for $ t = t_0$ or cycle with at least two rational curves or with multiple rational curves for $t = t_1$
In particularly, $(f_{*} \mathbb{P}^1)_{cycle}$ is numerical equivalence to cycle with at least two rational curves or with multiple rational curve as components.
proof of theorem: (cf. [Mum, pro. 9.1.10] and [Koll-Mor98, pro. 1.9]). First, consider a ruled surface $q: S \rightarrow \overline{D}$, we see that $q$ will be a $\mathbb{P}^1$-bundle morphism and $q^{-1} ( D) = \mathbb{P}^1 \times D$. Because there exists section $\alpha$ from $\overline{D}$ to $S$ and we define a projection morphism $(0,\alpha): \mathbb{P}^1 \times D \rightarrow D$ hence $q$ compatible with $p_2: \mathbb{P}^1 \times D \rightarrow D$ by $(0,\alpha)$. For the morphism $\varphi: \mathbb{P}^1 \times D \rightarrow C$, we get an extension map $\overline{\varphi}: S \dashrightarrow C$.
Now we prove the theorem by induction on the number of blow-up in $Y \rightarrow S$ s.t $Y \rightarrow C$ is a morphism.
We first prove that $\overline{\varphi}$ is not morphism. First, Let $C_0$ and $C_{\inf}$ are two divisors extending $0 \times D$ and $\inf \times D$. We see that $\overline{\varphi}$ maps $C_0$ and $C_{\infty}$ to two points $f(0)$ and $f(\infty)$. Let $H$ be an ample sheaf on $S$. We see that dim$\overline{\varphi} (S) $=dim $\varphi( \mathbb{P}^1 \times D) = 2$ because $ \mathbb{P}^1 \times D$ is dense in $S$ and then $(\overline{\varphi}*H.\overline{\varphi}*H)=(H.H) > 0$ $(\overline{\varphi}*H.C_0) = (H.\overline{\varphi}_{*}C_0) > 0$, $(\overline{\varphi}*H.C_{\infty}) = (H.\overline{\varphi}_{*}C_{\infty})=0$ by projection formula over surface. Hence, by Hodge index theorem, we get: $(C_0^{2}) < 0 $, $(C_{\infty}^{2}) < 0$ and $C_0.C_{\infty} = 0$ (because these two don't intersect anywhere). Hence, there are there linear independence divisor, but picard number over a surface is only two, which means when you consider $C^1(S)$ as a vector space with dimension 2, there can not be there divisor which can be linear independence, and thus we arrived a contradiction.
Let $r: Y \rightarrow S$ be a composition of blow-up sequence which eliminating indeteminacy of $\overline{\varphi}$ and gives a morphism $\xi: Y \rightarrow C$. Denote $\pi = q \circ r: Y \rightarrow \overline{D}$. Now we will prove our problem by induction on the number of points blow-up to eliminating indeterminacy.
Let $\sigma: S' \rightarrow S$ be a first blow up at $P$ . Denote $t_1 = q(P) \in \overline{D} - D$. Then $\pi ^{-1}(t_1)$ irreducible components must be a rational curve by taking inverse among exceptional divisor or strict transform of $\mathbb{P}^1$. Then $\xi(\pi^{-1}(t_1))$ is a union of rational curves by Liiroth’s theorem as $q$ induced a surjective from $\mathbb{P}^1$ to each curve. If $\xi(\pi^{-1}(t_1))$ is reducible or non-reduce, then $\xi(Y_j)_{red}$ will be rational curve with $Y_j$ are irreducible component of $\pi^{-1}(t_1)$. Now we prove that $\xi(\pi^{-1}(t_1))$ can not be reduced and irreducible. Assume it's so. We see that $\overline{\varphi}$ is define at $q^{-1}(t_1) - P$. If it's not at a $P'$, then $\xi(\pi^{-1}(t_1)) \supset \xi(r^{-1}(P)) \cup \xi(r^{-1}(P')) $ which contradict to irreducile assumption.
Now let $E$ and $F$ be the exceptional curve for $\sigma: S' \rightarrow C$ and strict transform of $q^{-1}(t_1)$ by $\sigma$ respectedly. Thus: $(q \circ \sigma)^{-1}(t_1)=F \cup E$ with $Q = E \cap F$. $\varphi: \mathbb{P} \times D \rightarrow D$ give rise to a rational map $\varphi ' : S \dashrightarrow C$. Denote $r = r' \circ \sigma$ with $r': Y \rightarrow S'$ being the composition of the point blow-up other than $\sigma$.
To finish the proof, we must claim that $\varphi '$ is defined at $Q$. By blowing up we see that $r'^{-1} (Q)$ must have all irreducible components of multiplicity at least 2 because those components belong to two curves $E$ and $F$, any of those irreducible components must appear two times, one as a cycle related to $E$, one as a cycle related to $F$. But for any reduce and irreducible scheme, the cycle associated to it must not have any component with multiplicity larger than 1, because its Ass set containing all minimal point. Hence $\xi$ must contracted $r^{-1}(Q)$. Hence $\varphi'$ must define at $Q$ and hence a morphism.
Now, for a case that $S'$ is the $Y$ we need, we can see that $\varphi'_{*}S'_{t_1}_{cycle}$ will s.t the demand of theorem because $\varphi'(\pi^{-1}(t_1))$ is the union of at least more than 2 rational curves. For $t_0 \in D$, $\pi^{-1}(t_0)$ must be $\mathbb{P}^1$ by property of ruled surface, hence $ \varphi'_{*}(\pi^{-1}(t_0))_{cycle} = $f_{*}(\mathbb{P}^1)_{cycle}$.
Now the statement true for 1, now we suppose it's true for $n-1$.
We have $F=\mathbb{P}^1$ with $(F^2)=-1$. Hence by Castelnuovo's criterion, $F$ can e contracted to a point y $\sigma ': S' \rightarrow S''$, which gives us another ruled surfce by an elementary transformation. The rational map $\varphi '': S'' \dashrightarrow C$ needs one less point blow-ups for elimination indeterminacy, since $\varphi$ define along $F$ of $\sigma' : S' \rightarrow S$. Thus we are done by induction.
References:
[Mor79] S.Mori, Projective Manifolds with Ample Tangent Bundles, Annals of mathematics, second series Vol. 110, No. 3 (Nov. 1979), pp. 593 - 606.
[Kol-Mor98] S.Mori, J.Kollar, Birational Geometry of Algebraic Varieties, Cambridge tracts in mathematics.
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