Let $X$ be an affine, complex surface with isolated singularities and $\pi:\widetilde{X} \to X$ be a resolution of singularities (not necessarily minimal) i.e., $\widetilde{X}$ is non-singular and $\pi^{-1}(X\_{\mathrm{sm}}) \cong X\_{\mathrm{sm}}$, where $X\_{\mathrm{sm}}$ is the regular locus of $X$ and the isomorphism is simply the restriction of $\pi$. Denote by $E$ the exception divisor associated to the morphism $\pi$ (set-theoretically isomorphic to $\widetilde{X} \backslash \pi^{-1}(X\_{\mathrm{sm}})$).


Let $A$ be a local artinian ring, $f\_A: C\_A \to \mathrm{Spec}(A)$ a smooth family of irreducible, affine curves and an $A$-morphism $g\_A: C\_A \to X \times\_{\mathbb{C}} \mathrm{Spec}(A)$. Denote by $C\_o$ the special fiber of $C\_A$ (under the morphism $f\_A)$ and $g\_o:C\_o \to X$ the restriction of $g\_A$ to the special fiber. Suppose that $C\_o$ does not contract to a point on $X$. Since $C\_o$ is a curve, the universal property of blow-up implies that the morphism $g\_o$ lifts to $\widetilde{X}$ i.e., there exists a morphsim $h\_o:C\_o \to \widetilde{X}$ such that $g\_o=\pi \circ h\_o$. 


Is it then true that the morphism $g\_A$ also lifts to $\widetilde{X} \times \mbox{Spec}(A)$ i.e., there exists a morphism $h\_A: C\_A \to \widetilde{X} \times \mbox{Spec}(A)$ such that $g\_A=(\pi \times \mathrm{id}) \circ h\_A$? If not true in general, is there any known condition under which this holds true? Any hint/reference will be most useful.



Infinitesimal deformation of strict transform
This is an answer that uses the language of Artin rings as in Michael Schlessinger's papers, Michael Artin's papers, etc. The statements are relative to a fixed square-zero extension of Artinian local rings, $$0 \to M \hookrightarrow A'\twoheadrightarrow A \to 0.$$ Denote the residue field $A/\mathfrak{m} = A'/\mathfrak{m}'$ by $A\_0$. For every scheme $T'$ over $\text{Spec}\ A'$, denote $\text{Spec}\ A \times\_{\text{Spec}\ A'} T'$ by $T$, and denote $\text{Spec}\ A\_0\times\_{\text{Spec}\ A'} T'$ by $T\_0$.


Let $X'$ be an $A'$-scheme. Let $\mathcal{I}$ be a quasi-coherent ideal sheaf on $X'$. Denote the blowing up of $X'$ along $\mathcal{I}$ by $$\nu:\widetilde{X}'\to X'.$$ Denote the inverse image ideal sheaf of $\mathcal{I}$ by $\mathcal{O}\_{\widetilde{X}'}(-\underline{E}')$. This is an invertible sheaf on $\widetilde{X}'$. Recall the universal property of the blowing up: the pair $$(\nu:\widetilde{X}'\to X', \nu^\*\mathcal{I}\twoheadrightarrow \mathcal{O}\_{\widetilde{X}'}(-\underline{E}')),$$ of an $X'$-scheme and an invertible quotient of the pullback of $\mathcal{I}$ is universal among such pairs for which each induced homomorphism, $$ \nu^\*\text{Sym}^d\_{\mathcal{O}\_{X'}}(\mathcal{I}) \twoheadrightarrow \mathcal{O}\_{\widetilde{X}'}(-d\underline{E}), $$ factors through the quotient, $$\nu^\*\text{Sym}^d\_{\mathcal{O}\_{X'}}(\mathcal{I}) \to \nu^\*(\mathcal{I}^d).$$ For more on this, see the answer to the following MathOverflow question: [Which functor does the blowing up represent?](https://mathoverflow.net/questions/91357/which-functor-does-the-blowing-up-represent). 


**Definition.** A morphism to $\widetilde{X}'$ is $E'$-**flat** if the pullback of the following injective sheaf homomorphism is still injective, $$\mathcal{O}\_{\widetilde{X}'}(-\underline{E}') \hookrightarrow \mathcal{O}\_{\widetilde{X}'}.$$


Let $Y'\to \text{Spec}\ A'$ be a flat, finitely presented morphism. Let $$f':Y'\to X',$$ be a morphism of $A'$-schemes. Define $Z'\hookrightarrow Y'$ to be the closed subscheme defined by the inverse image ideal sheaf of $\mathcal{I}$. Denote by $\mathcal{T}$ the kernel of the induced morphism, $$M\otimes\_{A\_0} \mathcal{O}\_{Z\_0} \twoheadrightarrow M\cdot \mathcal{O}\_{Z'}.$$


**Proposition.** For every $E$-flat $X$-morphism, $$e:Y\to \widetilde{X},$$ there exists an $X'$-morphism $e':Y'\to \widetilde{X}'$ extending $e$ if and only if the closed subscheme $Z'$ of $Y'$ defined by the inverse image ideal sheaf of $\mathcal{I}$ is $A'$-flat, and in this case $e'$ is also $E'$-flat. Moreover, this holds if and only if $\mathcal{T}\to M\otimes\_{A\_0}\mathcal{O}\_{Z\_0}$ is the zero homomorphism.


**Proof.** By the local flatness criterion, every extension is $E'$-flat, and thus also $Z'$ is $A'$-flat. Conversely, if $Z'$ is $A'$-flat, then the ideal sheaf of $Z'$ is $A'$-flat. Since the restriction of this ideal sheaf to $Y$ is an invertible $\mathcal{O}\_Y$-module, the $A'$-flat ideal sheaf of $Z'$ is an invertible $\mathcal{O}\_{Y'}$-module. This invertible quotient of the pullback of $\mathcal{I}$ satisfies the universal property of the blowing up, and thus gives an extension. **QED** 


Among all $A\_0$-module quotients
$$M \twoheadrightarrow N,$$ such that the following composition is the zero homomorphism, $$\mathcal{T} \hookrightarrow M\otimes\_{A\_0} \mathcal{O}\_{Z\_0} \twoheadrightarrow N\otimes\_{A\_0} \mathcal{O}\_{Z\_0},$$ there exists an initial such quotient. Denote this initial $A\_0$-module quotient by $$q:M\twoheadrightarrow M\_e.$$ 


**Corollary.** The induced pushout of $A'$, $$A'\_e := (A'\oplus M\_e)/\Delta(M) = A'/\text{Ker}(q),$$ is the initial quotient of $A'$ such that $e$ extends to an $X'$-morphism on $Y\_e:=\text{Spec}\ A'\_e \times\_{\text{Spec}\ A}. Y'.$


For more on the "obstruction" to the extension of $e$ given by this element in $\text{Hom}(\mathcal{T},\mathcal{O}\_{Z\_0})\otimes\_{A\_0} M$, please confer Section 2 of the following.


MR2007396 (2004i:14002)   

Olsson, Martin; Starr, Jason   

Quot functors for Deligne-Mumford stacks.   

Special issue in honor of Steven L. Kleiman.   

Comm. Algebra 31 (2003), no. 8, 4069–4096.   

<https://math.berkeley.edu/~molsson/quot2a.pdf>


