non-isomorphic birationally equivalent calabi-yau varieties

Question

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.

Answer 1

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 $\mathbb{F}_2$.