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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 $Def_R(C)$ 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$. Thanks a lot. 3 corrected typo I am reading the "Notes on stable maps and quantum cohomology" by Fulton and Pandharipande and I got stuck in the proof of the Theorem 2, p.27. The authors consider the space$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 $$0 \to H^0(C,T_C) \to Def_R(CDef_R(\mu) \to Def(CDef(\mu) \to 0,$$ where$Def_R(C)$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. 2 link to the article replaced 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 $$0 \to H^0(C,T_C) \to Def_R(C) \to Def(C) \to 0,$$ where$Def_R(C)$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\$.