{0 \ar[d] & 0 \ar[d] \\
{\mathcal T}_{C_a} (-p -q_a -r_a) \oplus {\mathcal T}_{C_b} (-p -q_b -r_b) \ar[d] & f^* {\mathcal T}_S \ar[d] \\
{\mathcal T}_{C_a} (-q_a -r_a) \oplus {\mathcal T}_{C_b} (-q_b -r_b) \ar[d] ^{\alpha } \ar[r] & 
\tilde f ^* {\mathcal T}_S \ar[d] ^{\varepsilon } \\
{\mathcal T}_{C_a, p} \oplus {\mathcal T}_{C_b, p} \ar[d] \ar[r] ^{f_*} & {\mathcal T}_{S, f(p)} \ar[d] \\
0 & 0}$$ where the unlabeled horizontal map is extension by zero. Since $C_a$ and $C_b$ are rational curves, their tangent bundles have degree 2 and $\alpha $ is surjective on global sections; $f_*$ is surjective by assumption. It follows that $\varepsilon $ is also surjective on global sections. [ $\Box $ ]{}

[*Remark*]{}. The second condition in the lemma is certainly satisfied if $f|_{C_a}$ and $f|_{C_b}$ are birational and the intersection of $f(C_a)$ and $f(C_b)$ is transverse at $f(p)$.

The Conormal Sheaf {#conormale}
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Let $f:C \rightarrow X$ be a morphism from a connected, projective, at worst nodal curve $C$ to a smooth projective variety $X$.

The morphism $f : C \rightarrow X$ is called a stable map if $C$ is a connected, projective, at worst nodal curve and every contracted component of geometric genus zero 