I am reading the "Notes on stable maps and quantum cohomology" by Fulton and Pandharipande and I got stuck in the proof of the Theorem 2, p.27. The authors consider the space $Def(\mu)$ of first order deformations of a pointed stable map $\mu = (C, p_i, \mu: C \to X)$. For $C = \mathbb P^1$ they write down an exact sequence $$ 0 \to H^0(C,T_C) \to Def_R(\mu) \to Def(\mu) \to 0, $$ where $Def_R(\mu)$ is the space of first order deformations of $\mu$ with $C$ held rigid.

What does they mean by

held rigid?

Later $C$ is a tree of $\mathbb P^1$'s. Let $G$ be the dual graph of $C$. Now they consider the space $Def_G(\mu) \subset Def(\mu)$ of first order deformations of the pointed stable map $\mu$ preserving the dual graph.

What does they mean by

preserving the dual graph?

If I understand it correctly, $Def(\mu)$ consists of all $(\mathcal C \to Spec\ \mathbb C[\varepsilon], \bar p_i, \bar \mu: \mathcal C \to X)$ such that $\mathcal C \otimes_{\mathbb C[\varepsilon]} \mathbb C \cong C, \bar p_i \otimes \mathbb C = p_i$ and $C \to \mathcal C \xrightarrow{\bar \mu} X$ coincides with $\mu$. So, a geometric fiber of $\mathcal C$ should have the same dual graph as $C$.

Thanks a lot.