John Lesieutre has just sent me an exciting new preprint in which he constructs a smooth projective variety \(X\) such that
- \(\text{Aut}(X)\) is discrete, and
- \(\text{Aut}(X)\) is not finitely generated.
Whether or not such varieties existed has been an open question (asked by many people) for a long time. John's construction is characteristically awesome. He observes that the set of automorphisms of \(\mathbb{P}^1_{k}\) which have a unique fixed point \(\infty\), namely the group of upper triangular unipotent matrices in \(PGL_2(k)\) contains many non-finitely generated subgroups as long as \(k\) is infinite --- in particular, this group is isomorphic to the additive group of \(k\). For convenience, we assume \(k\) has characteristic zero (which seems to be necessary for his construction as written, though I'm sure it can be adapted to positive characteristic). John observes that we can construct infinitely generated subgroups as follows:
If \(z_1, z_2, z_3, z_4\) are points in \(\mathbb{P}^1_k\), there is a unique involution \(\iota: \mathbb{P}^1_k\to \mathbb{P}^1_k\) such that $$\iota(z_1)=z_2, \iota(z_3)=z_4.$$
A picture of the involution described above, from John's paper
If \(p_1, p_2, p_3, p_4, p_5\in \mathbb{P}^1_k(k)\), we let \(\iota_{ij, kl}\) be the involution such that
$$\iota_{ij, kl}(p_i)= p_j, \iota_{ij, kl}(p_k)=p_l$$
Now if we let
\(p_1=0, p_2=1, p_3=2, p_4=3, p_5=6\)
the subgroup \(\Gamma_P\) of \(PGL_2=\text{Aut}(\mathbb{P}^1_k)\) generated by the \(\iota_{ij, kl}\) contains the matrices
$$\begin{pmatrix} 3 & 0\\ 0 & 1\end{pmatrix}, \begin{pmatrix} 1 & 1\\ 0 & 1\end{pmatrix}$$
and thus the matrices
$$\begin{pmatrix} 1 & \frac{1}{3^n} \\ 0 & 1\end{pmatrix}$$
which generate a non-finitely generated abelian group.
Now, John constructs a surface \(S\) containing a smooth rational curve \(C\) such that the group of automorphisms of \(S\) which preserve \(C\)
- is finite index in \(\text{Aut}(S)\), and
- has image containing \(\Gamma_P\) in \(\text{Aut}(C)\).
By various tricky product and blowup constructions, John then constructs a \(6\)-fold whose automorphism group is isomorphic to the group of automorphisms of \(S\) that
- preserve \(C\), and
- have the unique fixed point \(\infty\) in \(C\).
In particular, this automorphism group has an abelian quotient (the automorphisms of \(C\) fixing \(\infty\)), which contains the non-finitely generated group generated by the matrices
$$\begin{pmatrix} 1 & \frac{1}{3^n} \\ 0 & 1\end{pmatrix}$$
from before. Hence it is not finitely generated.
If I understand correctly, it's not too hard to find blowups \(Y\) of \(X\) such that the group of automorphisms of \(Y\) which descend to \(X\) has some given structure -- the hard part is to arrange that \(Y\) does not obtain any new automorphisms. John does this by introducing the notion of a \(\mathbb{P}^r\)-averse variety, i.e. a variety \(X\) so that any map \(\mathbb{P}^r\to X\) is constant. The point is that if \(X\) is \(\mathbb{P}^r\)-averse, any blowup \(Y\) of \(X\) along a smooth center \(Z\) of codimension greater than \(r\) satisfies
$$\text{Aut}(Y)=\text{Aut}(X; Z)$$
since any automorphism must preserve the fibers of the blowup (which are isomorphic to \(\mathbb{P}^s\) for some \(s\geq r\).)
John modifies his surface \(S\) (by replacing it with the \(6\)-fold \(S\times S\times T\), where \(T\) is some surface of general type). The variety thus obtained is \(\mathbb{P}^2\)-averse. This lets him retain control of the automorphism group of his variety even after blowing up the variety in codimension at least \(3\). He cleverly blows up some more or less randomly chosen things in codimension at least \(3\) to make sure any automorphisms of the blowup descends to an automorphism of the form
$$\text{id} \times \phi\times \text{id} \in \text{Aut}(S×S×T).$$
Then a clever final blowup reduces the automorphism group to the desired one.
In a previous paper of ours, John and I studied the issue of how blowups affect automorphism groups -- we use some \(p\)-adic methods to gain some control on this question. In particular, one thing we show is that if \(X\) is a variety such that $$\pi_0(\text{Aut}(X))$$ is finite, then blowups in high codimension, or along simple loci, do not alter the finiteness of this (component group of the) automorphism group. One thing John's construction is telling us, I think, is that if the automorphism group of \(X\) is more complicated, blowups in high codimension can have a rather more drastic effect.