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.



When does a stable map to a special fiber (locally) deform to a family of stable maps?
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, ...


