Is there an example of a rational smooth projective variety over a perfect field of characteristic $p$, that is not liftable to characteristic zero?



Liftable rational varieties
Two such examples were given by Achinger and Zdanowicz [AZ17], both of which satisfy a whole bunch of other good properties (e.g. their classes in the Grothendieck ring of varieties are polynomials in the Lefschetz motive $\mathbb L = [\mathbb A^1]$). The easiest one to state is probably the following:


**Example.** (Achinger–Zdanowicz [AZ17, §4]) Let $k = \bar{\mathbb F}\_p$, let $P = \mathbb P^3\_k$, and construct $X$ from $P$ by first blowing up all $\mathbb F\_p$-points of $P$, and then blowing up all strict transforms of lines defined over $\mathbb F\_p$. Then $X$ cannot be lifted to any ring $A$ in which $pA \neq 0$.


The proof is relatively elementary, and relies on matroid theory and elementary projective geometry, as well as a little deformation theory.




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**References.** 


[AZ17] Achinger, Piotr; Zdanowicz, Maciej, [*Some elementary examples of non-liftable varieties*](http://dx.doi.org/10.1090/proc/13622). Proc. Amer. Math. Soc. **145** (2017). [ZBL06769127](https://zbmath.org/?q=an:06769127).


