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I'm sure the answer to my question is well-known -- I'm mostly looking for a reference.

Suppose I have a nonsingular variety $X$ which fibers over $\mathbb{A}^1$. Moreover, suppose I have a stable map $C_0 \stackrel{f_0}{\to}X_0$, where $X_0$ is the fiber of $X$ over $0 \in \mathbb{A}^1$. When does this map deform to a family of stable maps $$ \begin{array}{c} C & \stackrel{f}{\longrightarrow} & X \\ \downarrow & & \downarrow \\ \mathbb{A}^1 & \longrightarrow & \mathbb{A}^1 \end{array} $$ where $C_0$ is the fiber of $C$ over $0 \in \mathbb{A}^1$? I expect that such deformations correspond to global sections of $f_0^*\mathcal{T}_X$ on $C_0$ -- is this correct?

I'm especially interested in the case where $X = \{ \alpha w = \beta x, \alpha y = \beta t\} \in \mathbb{A}^4 \times \mathbb{P}^1$ and $X \stackrel{t}{\to} \mathbb{A}^1$. Here, $[\alpha:\beta]$ give coordinates on $\mathbb{P}^1$.

Thanks for your help.

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Asking for a deformation over $\mathbb{A}^1$ is quite restrictive. Even asking for formal deformations / deformations over an étale cover of $\mathbb{A}^1$ is nontrivial. The "standard" obstruction group for deforming a stable map is the hyper-Ext group $\mathbf{R}Hom^2_{\mathcal{O}_{C_0}}(L^\bullet_{f_0},\mathcal{O}_{C_0})$, where $L^\bullet_{f_0}$ is the dualizing complex of $f$, i.e., it is (globally) quasi-isomorphic to the two-term complex, $$ NL^{\bullet}_{f_0}: \ f_0^*\Omega_{X_0/k} \to \Omega_{C_0/k}, $$ concentrated in degrees $-1$ and $0$.

In some form, this is described in Behrend-Fantechi, particularly the last few sections. I also recommend the first chapter of Kollár's textbook, "Rational Curves on Algebraic Varieties". Sernesi's book on Hilbert schemes is also great. Debarre's book is wonderful, and very readable, ...

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