# Infinitesimal deformation of strict transform

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.

• That usually is not true. For an example, consider the blowing up of the affine plane $\mathbb{A}^2_k$ at the origin, and consider $A$ equal to the first order neighborhood (i.e., $\mathcal{O}/\mathfrak{m}^2$) for the family of affine lines in the afine plane at a point parameterizing an affine line $L$ containing the origin. A first-order deformation of $L$ lifts to the blowing up if and only if the first-order deformation "vanishes" at the origin in $L$. – Jason Starr Mar 26 at 22:39

## 1 Answer

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?.

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