# non-isomorphic birationally equivalent calabi-yau varieties

What are some examples of pairs of birationally equivalent smooth projective Calabi-Yau $F_p$-varieties that are not isomorphic? By Calabi-Yau, I only mean the condition of trivial canonical bundle. I think examples among abelian varieties and K-3 surfaces cannot be found.

• There are famous examples of Namikawa, see the following MO post: mathoverflow.net/questions/140695/open-torelli-problems/140741 – Jason Starr Dec 29 '16 at 14:40
• Oops, I read your question too quickly. Namikawa's examples are more profound: they have equal Hodge structures, but they are not birational. Birationally equivalent but non-isomorphic Calabi-Yau's are much easier, e.g., blow up one of the 2875 lines in a "sufficiently general" quintic threefold (many of these do exist over finite fields), and then blow down the exceptional divisor $E\cong \mathbb{P}^1\times \mathbb{P}^1$ in the "other direction". – Jason Starr Dec 29 '16 at 14:42
• Your first answer is actually right: there are famous examples of Namikawa. But you want the paper "On the Birational Structure of Certain Calabi-Yau Threefolds" instead. The example there was (as I understand it) one impetus for the Kawamata-Morrison cone conjecture. (This gives an example of birational CY3's which are connected by flops, as indeed any such examples must be, but he shows that there are only finitely many isomorphism types within the birational class.) – user47305 Dec 30 '16 at 2:45
• (However, I am not so sure the Namikawa examples I mentioned can be carried out over $\mathbb F_p$: it requires some nondegeneracy of certain elliptic fibrations that might not be possible to arrange over a finite field.) – user47305 Dec 30 '16 at 2:47
• Why not just take a quintic threefold in projective four-space containing a projective plane, $L$. If the quintic is general, then there will be 16 ODPs along the plane. There are two small resolutions, one of which is obtained by blowing up the plane. This should work for sufficiently large $p$. It is easy to see these two models are not isomorphic. – Mark Gross Dec 30 '16 at 20:32

Mark is clearly more expert in this than I, and I am happy to delete this answer if Mark posts an answer. However, there is one construction that I really like, and it has to do with Hilbert schemes of stacks.

Let $k$ be any field with $\text{char}(k)\neq 2$. Let $\mathbb{P}^3_k$ denote $\text{Proj}\ k[r,t,u,v]$. Denote by $G(r,t,u,v)\in k[r,t,u,v]$ the following degree $4$, homogeneous polynomial, $$G(r,t,u,v) = 2r^2(t^2+u^2+v^2) - (t^4+u^4+v^4).$$ The associated surface $S=\text{Zero}(G)\subset \mathbb{P}^3_k$ has a unique singularity -- an ordinary double point $p$ at $[r,t,u,v]=[1,0,0,0]$. The smooth locus of $S$ is the open subset $S_o=S\setminus\{p\}$. There is a crepant desingularization $$\nu:\widetilde{S}\to S,$$ that is an isomorphism over $S_o$ obtained by blowing up the ideal sheaf of $p$. The surface $\widetilde{S}$ is a smooth, projective, K3 surface.

There is also a smooth, proper, $2$-dimensional Deligne-Mumford stack $\Sigma$ and a $1$-morphism, $$\mu:\Sigma\to S,$$ that is an isomorphism over $S_o$ and that identifies $S$ with the coarse moduli space of $\Sigma$. The preimage of $p$ is a stacky point whose stabilizer group is cyclic of order $2$ (note that $2$ is prime to the characteristic), and the action of the stabilizer on the tangent space is via the special linear group. In other words, $\Sigma$ is a tame, stacky K3 surface. Because $\Sigma$ is smooth, it is a global quotient stack (the quotient of an iterated frame bundle of the tangent bundle of $\Sigma$).

For every integer $n\geq 1$, as first discovered by Beauville, the Hilbert scheme $\widetilde{S}^{[n]}=\text{Hilb}^n_{\widetilde{S}/k}$ is a smooth, irreducible, projective $k$-scheme of dimension $2n$ that has an everywhere nondegenerate element $$\widetilde{w} \in H^0\left(\widetilde{S}^{[n]},\ \Omega^2_{\widetilde{S}^{[n]}/k}\right).$$

Following earlier work of Nakamura, Olsson and I also constructed Hilbert schemes of Deligne-Mumford stacks. In particular, since $\Sigma$ is a tame, global quotient stack with projective coarse moduli space, the Hilbert scheme $\Sigma^{[n]}$ is a smooth, irreducible, projective $k$-scheme of dimension $2n$ that has an everyhwere nondegenerate element $$\varpi \in H^0\left(\Sigma^{[n]},\ \Omega^2_{\Sigma^{[n]}/k}\right).$$ The Hilbert scheme $(S_o)^{[n]} = \text{Hilb}^n_{S_o/k}$ is a common dense open in both $\widetilde{S}^{[n]}$ and in $\Sigma^{[n]}$. However, for $n>1$, this birational equivalence is not a regular isomorphism.

There should be similar examples over $\mathbb{F}_2$.

• I just spent a week trying to extend this to the non-tame case, but I could not do it. Probably there are examples in characteristic $2$, but with $\mu_3$-quotient, Gorenstein singularities instead of quotient singularities that are cyclic of order $2$. – Jason Starr Jan 5 '17 at 20:33