Are there non-trivial (i.e. excluding concepts that can be defined only for $p>0$) statements in algebraic geometry that hold for all fields of characteristic $p$ for all prime $p$ but are known to be false in characteristic zero?



Characteristic zero and characteristic $p$ in algebraic geometry
Here are two examples.


The moduli space of dimension $g$ principally polarized abelian varieties $A\_g$ contains complete codimension $g$ subvarieties in any positive characteristic $p$ (for instance, the locus of abelian varieties with no nontrivial $p$-torsion points), but not in characteristic $0$ (by Keel and Sadun arXiv:math/0204229).


The other example is also a resul of Keel (arXiv:math/9901149). It states that a nef and big line bundle $L$ on a projective variety over a field of positive characteristic is semi-ample if and only if its restriction to the exceptional locus (i.e. the union of subvarieties $Z$ such that $L|\_Z$ is not big) is semi-ample. This criterion is not true in characteristic $0$.


