# Deformation Theory (Thursday February 27th) Big picture idea: We have moduli functors, such as \[ F_{S'}: (\Sch_{/k})\op &\to \Set \\ \hilb: S &\to \text{flat subschemes of } X_S \\ \pic: S &\to \pic(X_S)/\pic(S) \\ \mathrm{Def}: S &\to \text{flat families } / S,~ \text{smooth, finite, of genus } g .\] :::{.definition title="Deformation Theory"} Choose a point $f$ the scheme representing $F_{S'}$ with $\xi_0 \in F_{gl}(\spec K)$. Define \[ F_{\text{loc}}: (\text{Artinian local schemes} / K)\op \to \set .\] \begin{tikzcd} \spec(K) \arrow[rr, "i", hook] & & \spec(A) \arrow[rr] & & F(i)\inv(\xi_0) \arrow[rr] & & F_{gr}(\spec K) \arrow[dd, "F(i)"] \\ & & & & & & \\ & & & & & & F_{gl}(\spec K) \end{tikzcd} ::: :::{.definition title="Deformation Functors"} Let $F: (\Art_{/k}) \to \Set$ where $F(k)$ is a point. Denote $\hat{\Art}_{/k}$ the set of complete local $k\dash$algebras. Since $\Art_{/k} \subset \hat{\Art} / k$, we can make extensions $\hat F$ by just taking limits: \begin{tikzcd} & \Art_{/k} \arrow[rrr, "F"] & & & \Set \\ & & & & \\ \lim_{\from} R/\mfm_R^n = R \in & \hat{\Art}_{/k} \arrow[uu] \arrow[rrruu, "\hat F"] & & & \end{tikzcd} where we define \[ \hat{F}(R) \da \inverselim F(R/\mfm_R^n) .\] ::: :::{.question} When is $F$ pro-representable, which happens iff $\hat F$ is representable? In particular, we want $h_R \mapsvia{\cong} \hat F$ for $R\in \hat{\Art}_{/k}$, so \[ h_R = \hom_{\hat{\Art}_{/k}}(R, \wait) = \hom_{?}(\wait, \spec k) .\] ::: :::{.example title="?"} Let $F_{\text{gl}} = \hilb_{X_{_{/k}}}^p$, which is represented by $H_{/k}$. Then . \[ \xi_0 = F_{\text{gl}}(k) = H(k) = \theset{Z\subset X \suchthat P_z = f} .\] Then $F_{\text{loc} }$ is representable by $\hat \OO_{H/\xi_0}$. ::: :::{.definition title="Thickening"} Given an Artinian $k\dash$algebra $A \in \Art_{/k}$, a *thickening* is an $A' \in \Art_{/k}$ such that $0 \to J \to A' \to A \to 0$, so $\spec A \injects \spec A'$. ::: :::{.definition title="Small Thickening"} A **small thickening** is a thickening such that $0 = \mfm_{A'} J$, so $J$ becomes a module for the residue field, and $\dim_k J = 1$. ::: :::{.lemma title="?"} Any thickening of $A$, say $B\to A$, fits into a diagram: \begin{tikzcd} & & & & 0 & & & & \\ & & & & & & & & \\ & & J \arrow[rr] & & A' \arrow[uu] \arrow[rr] & & A \arrow[dd, Rightarrow] \arrow[rr] & & 0 \\ & & & & & & & & \\ 0 \arrow[rr] & & I \arrow[rr] \arrow[uu] & & B \arrow[uu] \arrow[rr] & & A \arrow[rr] & & 0 \\ & & & & & & & & \\ & & I' \arrow[rr, Rightarrow] \arrow[uu] & & I' \arrow[uu] & & & & \\ & & & & & & & & \\ & & 0 \arrow[uu] & & 0 \arrow[uu] & & & & \end{tikzcd} ::: :::{.proof title="of lemma"} We just need $I' \subset I$ with $\mfm_S I \subset J' \subset I \iff J \mfm_B = 0$. Choose $J'$ to be a preimage of a codimension 1 vector space in $I/\mfm_B I$. Thus $J = I/I'$ is 1-dimensional. ::: Thus any thickening $A$ can be obtained by a sequence of small thickenings. By the lemma, in principle $F$ and thus $\hat{F}$ are determined by their behavior under small extensions. ### Example Consider $\pic$, fix $X_{_{/k}}$, start with a line bundle $L_0 \in \pic(x) /\pic(k) = \pic(x)$ and the deformation functor $F(A)$ being the set of line bundles $L$ on $X_A$with $\restrictionof{L}{x} \cong L_0$, modulo isomorphism. Note that this yields a diagram \begin{tikzcd} x \arrow[rr] \arrow[dd, hook] & & k \arrow[dd, "\text{unique closed point}"] \\ & & \\ X_A \arrow[rr] & & \spec A \end{tikzcd} This is equal to $(I_x)\inv (L_0)$, where $\pic(X_a) \mapsvia{I_x} \pic(x)$. If \[ 0 \to J \to A' \to A \to 0 .\] is a small thickening, we can identify \begin{tikzcd} 0 \arrow[rr] & & J \tensor_x \OO_{x} \cong \OO_x \arrow[rr] & & \OO_{X_{A'}} \arrow[rr] & & \OO_{X_{A}} \arrow[rr] & & 0 & & \\ & & & & & & & & & &\in\text{AbSheaves} \\ 0 \arrow[rr] & & \OO_x \arrow[rr, "f\mapsto 1+f"] & & \OO_{X_{A'}}^* \arrow[rr] \arrow[uu, hook] & & \OO_{X_{A}}^* \arrow[rr] & & 0 & & \end{tikzcd} This yields a LES \begin{tikzcd}[column sep=tiny] 0 \arrow[rr] & & {H^0(X, \OO_x) = k} \arrow[rr] & & {H^0(X_{A'}, \OO_{x_{A'}}^*) = {A'}^*} \arrow[rr] & & {H^0(X_{A'}, \OO_{x_{A}}^*) = A^*} \arrow[lllldd] \arrow[rr] & & \therefore 0 \\ & & & & & & & & \\ \therefore 0 \arrow[rr] & & {H^1(X, \OO_{x})} \arrow[rr] & & {H^1(X_{A'}, \OO_{x_{A'}}^*) = \pic(X_{A'})} \arrow[rr, "\scriptsize\text{restriction to } X_A", outer sep=1em] & & {H^1(X_{A}, \OO_{x_{A}}^*) = \pic(X_A)} \arrow[lllldd, "\text{obs}"] & & \\ & & & & & & & & \\ & & {H^2(X, \OO_x)} \ar[rr] & &\cdots & & & & \end{tikzcd} :::{.remark} To understand $F$ on small extensions, we're interested in 1. Given $L \in F_{\text{loc}}(A)$, i.e. $L$ on $X_A$ restricting to $L_0$, when does it extend to $L' \in F_{\text{loc}}(A')$? I.e., does there exist an $L'$ on $X_{A'}$ restricting to $L$? 2. Provided such an extension $L'$ exists, how many are there, and what is the structure of the space of extensions? ::: :::{.question} We have an $L\in \pic(X_A)$, when does it extend? ::: By exactness, $L'$ exists iff $\text{obs}(L) = 0\in H^2(X, \OO_x)$, which answers 1. To answer 2, $(I_x)\inv(L)$ is the set of extensions of $L$, which is a torsor under $H^1(x, \OO_x)$. Note that these are fixed $k\dash$vector spaces. :::{.remark} $H^1(X, \OO_x)$ is interpreted as the **tangent space** of the functor $F$, i.e. $F_{\text{loc}}(K[\eps])$. Note that if $X$ is projective, line bundles can be unobstructed without the group itself being zero. ::: For (3), just play with $A = k[\eps]$, which yields $0 \to k \mapsvia{\eps} k[\eps] \to k \to 0$, then \begin{tikzcd} 0 \arrow[rr] & & {H^1(X, \OO_x)} \arrow[rr] & & {H^1(X_{k[\eps]}, \OO_{k[\eps]}^*)} \arrow[rr, "I_x"] & & {H^1(X, \OO_x^*)} \arrow[ll, bend right=49] \\ & & & & {(I_x)\inv(L_0) \in \pic(X_{k[\eps]})} & & L_0 \in \pic(x) \end{tikzcd} i.e., there is a canonical trivial extension $L_0[\eps]$. :::{.example title="?"} Let $X \supset Z_0 \in \hilb_{X_{_{/k}}}(k)$, we computed \[ T_{Z_0} \hilb_{X_{_{/k}}} = \hom_{\OO_x}(I_{Z_0}, \OO_z) .\] We took $Z_0 \subset X$ and extended to $Z' \subset X_{k[\eps]}$ by base change. In this case, $F_{\text{loc}}(A)$ was the set of $Z'\subset X_A$ which are flat over $A$, such that base-changing $Z' \cross_{\spec A} \spec k \cong Z$. This was the same as looking at the preimage restricted to the closed point, \[ \hilb_{X_{_{/k}}}(A) \mapsvia{i^*} \hilb_{X_{_{/k}}}(k) \\ (i^*)\inv(z_0) \mapsfrom z_0 .\] Recall how we did the thickening: we had $0 \to J \to A' \to A \to 0$ with $J^2 = 0$, along with $F$ on $X_A$ which is flat over $A$ with $X_{_{/k}}$ projective, and finally an $F'$ on $X_{A'}$ restricting to $F$. The criterion we had was $F'$ was flat over $A'$ iff $0 \to J\tensor_{A'} F' \to F'$, i.e. this is injective. Suppose $z\in F_{\text{loc}}(A)$ and an extension $z' \in F_{\text{loc}}(A')$. By tensoring the two exact sequences here, we get an exact grid: \begin{tikzcd} 0 \arrow[rr] \arrow[dd] & & I_{Z'} \arrow[rr] & & \OO_{X_{A'}} \arrow[rr] & & \OO_{Z'} \arrow[rr] & & 0 \\ & & 0 \arrow[d] & & 0 \arrow[d] & & 0 \arrow[d] & & \\ J \arrow[dd] & 0 \arrow[r] & I_{Z_0} \arrow[dd] \arrow[rr] & & \OO_X \arrow[dd] \arrow[rr] & & \OO_{Z_0} \arrow[dd] \arrow[r] & 0 & \\ & & & & & & & & \\ A' \arrow[dd] & 0 \arrow[r] & I_{Z'} \arrow[rr] \arrow[dd] & & \OO_{X_{A'}} \arrow[rr] \arrow[dd] & & \OO_{Z'} \arrow[dd] \arrow[r] & 0 & \\ & & & & & & & & \\ A \arrow[dd] & 0 \arrow[r] & I_Z \arrow[d] \arrow[rr] & & \OO_{X_A} \arrow[rr] \arrow[d] & & \OO_Z \arrow[d] \arrow[r] & 0 & \\ & & 0 & & 0 & & 0 & & \\ 0 & & & & & & & & \end{tikzcd} The space of extension should be a torsor under $\hom_{\OO_X}(I_{Z_0}, \OO_{Z_0})$, which we want to think of as $\hom_{\OO_X}(I_{Z_0}, \OO_{Z_0})$. Picking a $\phi$ in this hom space, we want to take an extension $I_{Z'} \mapsvia{\phi} I_{Z''}$. ::: > We'll cover how to make this extension next time.