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.