# A non-prorepresentable deformation functor

I've been teaching a topics course on deformation theory this semester; so far we've covered quite a few interesting things -- the Tian-Todorov theorem and Green-Lazarsfeld's generic vanishing theorem, for example.  I just wanted to use this post to record an easy and fun example that I wasn't able to find in the literature.

Question: What is a good example of a smooth projective variety $X$ over a field $k$ such that the deformation function $\text{Def}_X$ is not pro-representable?  Here $$\text{Def}_X: \text{Art}/k\to \text{Sets}$$ is the functor sending a local Artin $k$-algebra $A$, with residue field $k$, to the set of isomorphism classes of flat $A$-schemes $Y$ equipped with an isomorphism $\phi: Y_k\overset{\sim}{\to} X$.

Answer: Let $X=\text{Bl}_Z(\mathbb{P}^2)$, where $Z$ is a set of $10$ points lying on a line.

It's not totally obvious, of course, that this works.  The idea of the proof follows.  Given a variety $X$, we let $$\text{DefAut}_X: \text{Art}/k\to \text{Sets}$$ be the deformation functor sending a local Artin $k$-algebra $A$, with residue field $k$, to the set of isomorphism classes of triples $Y, \psi, \phi$, where $Y$ is a flat $A$-scheme, $\phi: Y_k\overset{\sim}{\to}X$ is an isomorphism of the special fiber of $Y$ with $X$, and $\psi: Y\overset{\sim}{\to} Y$ is an automorphism of $Y$ which is the identity on $Y_k$.

Proposition.  Let $X$ be a proper variety.  The functor $\text{Def}_X$ is pro-representable if and only if the natural map $\text{DefAut}_X\to \text{Def}_X$ is formally smooth.  (Here the natural map simply forgets the automorphism $\psi$.)

This is somewhat standard; it's an easy application of Schlessinger's criteria, for example.  One way of saying this is that given a small extension $$0\to I\to B\to A\to 0,$$ and a deformation $Y$ of $X$ to $B$, there's a long exact sequence of sets $$0\to H^0(X, T_X)\otimes I\to \text{Aut}^0(Y)\to \text{Aut}^0(Y_A)\to H^1(X, T_X)\otimes I\to \text{Def}_X(B)\to \text{Def}_X(A)\to H^2(X, T_X)\otimes I.$$

Here $\text{Aut}^0(Y)$ is the set of automorphisms of $Y$ fixing $Y_k$.  The fact that this is a long exact sequence means that the first four non-zero terms are an exact sequence of groups; that $H^1(X, T_X)\otimes I$ acts transitively on the fibers of the map $\text{Def}_X(B)\to \text{Def}_X(A)$, with kernel exactly the image of $\text{Aut}^0(Y_A)$; and that an element of $\text{Def}_X(A)$ lifts to an element of $\text{Def}_X(B)$ if and only if it maps to zero in $H^2(X, T_X)\otimes I$.  Now formal smoothness of the map  $\text{DefAut}_X\to \text{Def}_X$ is the same as surjectivity of the map $\text{Aut}^0(Y)\to \text{Aut}^0(Y_A)$ for all small extensions and all choices of $Y$.  This is the same as faithfulness of the action of $H^1(X, T_X)\otimes I$ on the fibers of the map $\text{Def}_X(B)\to \text{Def}_X(A)$, which is a restatement of Schlessinger's pro-representability criterion in our setting.

So why does our example $X$ -- namely, $\mathbb{P}^2$ blown up at a bunch of points all lying on a line $\ell$ -- have non-prorepresentable deformation functor?  The point is that infinitesimal automorphisms of $X$ are precisely infinitesimal automorphisms of $\mathbb{P}^2$ fixing the line $\ell$.  But if one moves the points to general position, the infinitesimal automorphism group becomes trivial.  So the map $\text{DefAut}_X\to \text{Def}_X$ is far from smooth.  Pretty easy and cool.

Is there a reference with examples like these?  I'm sure they're well-known, but I wasn't able to find any smooth proper examples in the literature.