A lemma for bend and break of Manifold with dimension higher than three

In this section, I would like to generate a lemma in [HOR2016] which has improtant meaning in constructing "bend and break" for Kahler three-fold. The statement below is for the surface:

Theorem (Lemma 5.5): 
Let $S'S be a smooth projective surface that is uniruled, and let $C' \subset S'$ be an irreducible curve.:
a) Suppose that $K_{S'}C < 0$. Then there exists an effective 1-cycle $\sum_{i} \alpha_{i} C_i$ with coefficients in $\mathbb{Q^+}$ such that $[\sum_{i} \alpha_{i} C_i] = [C'] $ and $C_1$ is a rational curve such that $K_{S'}.C_1 < 0$.
b) Suppose that $K_{S'}.C \leq -4$. Then there exists an effective 1-cycle $\sum_{i} \alpha_{i}C_{i}$ with $i \geq 2$ such that $[\sum_{i} \alpha_{i} C_i]=[C']$ such that $K_{S'}C_1 < 0$ and $K_{S'}.C_2 < 0$.




Now the statement for higher-dimensional will be:


Let $T$ be a smooth projective n-fold that is uniruled, and let $C \subset T$ be an irreducible curve.:
a) Suppose that $K_{S'}C < 0$. Then there exists an effective 1-cycle $\sum_{i} \alpha_{i} C_i$ with coefficients in $\mathbb{Q^+}$ such that $[\sum_{i} \alpha_{i} C_i] = [C'] $ and $C_1$ is a rational curve such that $K_{S'}.C_1 < 0$.
b) Suppose that $K_{T}.C \leq -n - 2$. Then there exists an effective 1-cycle $\sum_{i} \alpha_{i}C_{i}$ with $i \geq 2$ such that $[\sum_{i} \alpha_{i} C_i]=[C]$ such that $K_{T}C_1 < 0$ and $K_{T}.C_2 < 0$.

Now I will show the proof for $n=3$, with an assumption: exists $S$ uniruled s.t $S.C < 0$ .
proof: 
a)  By adjunction formula: $K_{S}=(K_{T}+T)|_{S}$, we see that $S$ contain $C$, and $(S.C) < 0$, then $(T.C)|_{S} = (S.C)< 0$ by a theorem in [Mum-Oda], hence $K_{S}.C < 0$. Now we will a proof of this theorem by induction on degree $d = H.C$ with $H$ is a fix ample invertible sheaf in $S$. With $d=0$ it is trivial. suppose it is true for $d > 0$. Consider $\widehat{S}$ is a composition of normalisation and desingularisation. Hence $\widehat{S}$ is smooth uniruled surface with morphism $\pi : \widehat{S} \rightarrow S$. By the original theorem, $\widehat{S}.\widehat{C} < 0$ with $\widehat{C}$ a strict transform of $C'$ implies $[\widehat{C}]= [\sum_{i} \alpha_{i} \widehat{C_i}]$ with $\widehat{C_1}$ rational and $(\widehat{S}.\widehat{C_1}) < 0$. Hence $[C] = \sum_{i} \alpha_{i} \pi_{*}[\widehat{C_i}]$ = $\sum_{i} \alpha_{i} [C_i]$ with $C'_1$ is rational or $i \geq 2$. For the first case, for the second case, $H.C_i < H.C$ for some i and then applies the induction.

b) For the same $\widehat{S}$ and $\pi$, we have $[\widehat{C'}]= \sum_{i} \alpha_{i} \widehat{C_i}]$ with $\widehat{C_1}$, $\widehat{C_2}$ rational and $(\widehat{S''}.\widehat{C_1}) < 0$, $(\widehat{S}.\widehat{C_2}) < 0$. Hence  $[C'] = \sum_{i} \alpha_{i} \pi_{*}[\widehat{C_i}]=[\sum_{i} \alpha_{i} C'_i]$ with $C'_1, C'_2$ are rational. By an arguement of the proof of the orginal lemma, $(K_{\widehat{S}}.\widehat{C'_1}) \leq (K_{\widehat{S}}.\widehat{C'_2})$ , hence $(K_{S}.C'_1) \leq (K_{S}.C'_2)$, if  $(K_{S}.C'_2) < 0$, we done, if not, we have $(H.C'_1) < (H.C')$, then applies induction.  

 Conference:

[HP16] Andreas Höring and Thomas Peternell. Minimal models for Kähler threefolds. Invent.
Math., 203(1):217–264, 2016.

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