# Obstruction and Deformation (Tuesday February 25th) Let $k$ be a field, $X_{_{/k}}$ projective, then the $k\dash$points $\hilb_{X_{_{/k}}}^P(k)$ corresponds to closed subschemes $Z\subset X$ with hilbert polynomial $P_z = P$. Given a $P$, we want to understand the local structure of $\hilb_{X_{_{/k}}}^p$, i.e. diagrams of the form \begin{tikzcd} & & & & \hilb_{X_{_{/k}}}^P \arrow[dd] \\ & & & & \\ \spec(k) \arrow[rrrruu, "p"] \arrow[rr] & & \spec(A) \arrow[rruu, "?", dashed] \arrow[rr] & & \spec(k) \\ & & & & \\ & & A_{/k} \text{ Artinian local} \arrow[uu] & & \end{tikzcd} :::{.example title="?"} For $A = k[\eps]$, the set of extensions is the Zariski tangent space. ::: :::{.definition title="Category of Artinian Algebras"} Let $(\Art_{/k})$ be the category of local Artinian $k\dash$algebras with local residue field $k$. ::: Note that these will be the types of algebras appearing in the above diagrams. :::{.remark} This category has fiber coproducts, i.e. there are pushouts: \begin{tikzcd} C \arrow[dd] \arrow[rr] & & A \arrow[dd, dashed] \\ & & \\ B \arrow[rr, dashed] & & A \tensor_C B \end{tikzcd} There are also fibered products, \begin{tikzcd} A \cross_C B \arrow[rr, dashed] \arrow[dd, dashed] & & B \arrow[dd] \\ & & \\ A \arrow[rr] & & C \end{tikzcd} Here, $A \cross_C B \definedas \theset{(a, b) \suchthat f(a) = g(b)} \subset A\cross B$. ::: :::{.example title="?"} If $A = B = k[\eps]/(\eps^2)$ and $C = k$, then $A\cross_C B = k[\eps_1, \eps_2]/(\eps_1, \eps_2)^2$ Note that on the $\spec$ side, these should be viewed as $$ \spec(A) \disjoint_{\spec(C)} \spec(B) = \spec(A\cross_C B) .$$ ::: :::{.definition title="Deformation Functor (Preliminary Definition)"} A *deformation functor* is a functor $F: (\Art_{/k}) \to \Set$ such that $F(k) = \pt$ is a singleton. ::: :::{.example title="?"} Let $X_{_{/k}}$ be any scheme and let $x\in X(k)$ be a $k\dash$point. We can consider the deformation functor $F$ such that $F(A)$ is the set of extensions $f$ of the following form: \begin{tikzcd} & & & & X \arrow[dd] \\ & & & & \\ \spec(k) \arrow[rrrruu, "x"] \arrow[rr, hook] & & \spec(A) \arrow[rruu, "f", dashed] \arrow[rr] & & \spec(k) \end{tikzcd} If $A' \to A$ is a morphism, then we define $F(A') \to F(A)$ is defined because we can precompose to fill in the following diagram \begin{tikzcd} & & & & & & & & X \arrow[ddd] \\ & & & & & & & & \\ & & & & & & & & \\ \spec(k) \arrow[rrd] \arrow[rrrrrrrruuu] & & & & & & & & \spec(k) \\ & & \spec(A) \arrow[rr] \arrow[rrrrrruuuu, "\exists \tilde f"] & & \spec(A') \arrow[rrrru] \arrow[rrrruuuu, "f", dashed] & & & & \end{tikzcd} So this is indeed a deformation functor. ::: :::{.example title="a motivating example"} The Zariski tangent space on the nodal cubic doesn't "see" the two branches, so we allow "second order" tangent vectors. ::: We can consider parametrizing the functors above as $F_{X, x}(A)$, which is isomorphic to $F_{\spec (\OO_x)_{X, x}}$ and further isomorphic to $F_{\spec \hat{\OO_x}_{x, X} }$. This is because for Artinian algebras, we have maps $$ \spec (\OO_{x, X})/\mfm^N \to \spec \OO_{X, x} \to X .$$ :::{.remark} $\hat{ \OO }_{X, x}$ will be determined by $F_{X, x}$. ::: :::{.example title="?"} Consider $y^2 = x^2(x+1)$, and think about solving this over $k[t]/t^n$ with solutions equivalent to $(0, 0) \mod t$. ![Image](figures/2020-02-25-13:20.png)\ Note that the 'second order' tangent vector comes from $\spec k[t]/t^3$. We can write $F_{X, x}(A) = \pi\inv(x)$ where $$ \hom_{\Sch_{/k}}(\spec k, X) \mapsvia{\pi} \hom_{\Sch_{/k}}(\spec k, x) \ni x .$$ Thus $$ F_{X, x}(A) = \hom_{\Sch_{/k}}(\spec A, \spec \OO_{x, X}) = \hom_{k\dash\alg}(\hat \OO_{X, x}, A) .$$ ::: :::{.example title="?"} Given any local $k\dash$algebra $R$, we can consider \[ h_R: (\Art_{/k}) &\to \Set \\ A &\mapsto \hom(R, A) .\] and \[ h_{\spec R}: (\Art\Sch_{/k})\op \to \Set \\ \spec(A) &\mapsto \hom(\spec A, \spec R) .\] ::: :::{.definition title="Representable Deformation"} A deformation $F$ is **representable** if it is of the form $h_R$ as above for some $R \in \Art_{/k}$. ::: :::{.remark} There is a Yoneda Lemma for $A\in \Art_{/k}$, \[ \hom_{\mathrm{Fun}}(h_A, F) = F(A) .\] We are thus looking for things that are representable in a larger category, which restrict. ::: :::{.definition title="Pro-Representability"} A deformation functor is *pro-representable* if it is of the form $h_R$ for $R$ a complete local $k\dash$algebra (i.e. a limit of Artinian local $k\dash$algebras). ::: :::{.remark} We will see that there are simple criteria for a deformation functor to be pro-representable. This will eventually give us the complete local ring, which will give us the scheme representing the functor we want. ::: :::{.remark} It is difficult to understand even $F_{X, x}(A)$ directly, but it's easier to understand small extensions. ::: :::{.definition title="Small Extensions"} A *small extension* is a SES of Artinian $k\dash$algebras of the form \[ 0 \to J \to A' \to A \to 0 .\] such that $J$ is annihilated by the maximal ideal fo $A'$. ::: :::{.lemma title="?"} Given any quotient $B\to A \to 0$ of Artinian $k\dash$algebras, there is a sequence of small extensions (quotients): \begin{tikzcd} 0 & & & & & & \\ & & & & & & \\ B_0 \arrow[uu] & & B_1 \arrow[lluu] & & \cdots & & B_n = A \arrow[lllllluu] \\ & & & & & & \\ B \arrow[uu] \arrow[rruu] \arrow[rrrrrruu] & & & & & & \end{tikzcd} This yields \begin{tikzcd} \spec A \arrow[rrrr, hook] \arrow[rrrrdddddd, Rightarrow] & & & & \spec B \\ & & & & \\ & & & & \spec B_0 \arrow[uu, hook] \\ & & & & \\ & & & & \vdots \arrow[uu, hook] \\ & & & & \\ & & & & \spec B_n \arrow[uu, hook] \end{tikzcd} where the $\spec B_i$ are all small. ::: :::{.remark} In most cases, extending deformations over small extensions is easy. ::: ## First Example of Deformation and Obstruction Spaces Suppose $k=\bar k$ and let $X_{_{/k}}$ be connected. We have a picard functor \[ \pic_{X_{_{/k}}}: (\Sch_{/k})\op &\to \Set \\ S &\mapsto \pic(X_S) / \pic(S) .\] If we take a point $x\in \pic_{X_{_{/k}}}(k)$, which is equivalent to line bundles on $X$ up to equivalence, we obtain a deformation functor \[ F \definedas F_{\pic_{ X_{_{/k}}, x }} &\to \Set\\ A \mapsto \pi\inv(x) \] where \[ \pi: \pic_{X_{_{/k}}}(\spec A) &\to \pic_{X_{_{/k}}} (\spec k) \\ \pi\inv(x) &\mapsto x .\] This is given by taking a line bundle on the thickening and restricting to a closed point. Thus the functor is given by sending $A$ to the set of line bundles on $X_A$ which restrict to $X_x$. That is, $F(A) \subset \pic_{X_{_{/k}}}(\spec A)$ which restrict to $x$. So just pick the subspace $\pic(X_A)$ (base changing to $A$) which restrict. There is a natural identification of $\pic(X_A) = H^1(X_A, \OO_{X_A}^*)$. If \[ 0\to J \to A' \to A \to 0 .\] is a thickening of Artinian $k\dash$algebras, there is a restriction map of invertible functions \[ \OO_{X_A}^* \to \OO_{X_A'}^* \to 0 .\] which is surjective since the map on structure sheaves is surjective and its a nilpotent extension. The kernel is then just $\OO_{X_{A'}} \tensor J$. If this is a small extension, we get a SES \[ 0 \to \OO_X \tensor J \to \OO_{X_{A'}}^* \to \OO_{x_A}^* \to 0 .\] Taking the LES in cohomology, we obtain \[ H^1 \OO_X \tensor J \to H^1 \OO_{X_{A'}}^* \to H^1\OO_{x_A}^* \to H^0 \OO_X \tensor J .\] Thus there is an obstruction class in $H^2$, and the ambiguity is detected by $H^1$. Thus $H^1$ is referred to as the **deformation space**, since it counts the extensions, and $H^2$ is the **obstruction space**.