Suppose that $C\cong P^1$ and $Def(f)$ denote the first order deformation of pointed stable map $(C,{p_i},f:C\longrightarrow X)$. I read that we have short exact sequence:

$0\longrightarrow H^0(C,T_C)\longrightarrow Def_R(f)\longrightarrow Def(f) \longrightarrow 0$

Where $Def_R(f)$ is the first order deformation of $(C,{p_i},f:C\longrightarrow X)$ with $C$ held rigid.

1)what does it mean(($C$ held rigid)? Does it mean we consider $C$ fixed?

2)Why we have this short exact sequence?

(If f is closed immersion and ignore marking and my comment on 1 is correct we have $0\longrightarrow T_C \longrightarrow f^*T_X \longrightarrow N_f\longrightarrow 0$ so we will get the short exact sequence because $P^1$ is rigid.But in general i cant reach to this short exact sequence )

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