# 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.