Counterexamples over F̄_p to a Generic Equivalence Problem of Kraft and Russell

In Oberwolfach Report 01/2007, Kraft and Russell stated that two morphisms of varieties over an algebraically closed field of infinite transcendence degree, whose fibres over all closed points are isomorphic, become isomorphic after a dominant étale base change, and asked whether this holds over every algebraically closed field, or for counterexamples over F̄_p or Q̄. In 2014 they proved the statement for affine morphisms, with a dominant base change of finite degree. The étale form already fails over every algebraically closed field of positive characteristic, because of Russell's classical purely inseparable forms of the affine line; we record this, and note that it immediately answers a question on positive characteristic raised in a remark of Kaliman. We show that over F̄_p even the finite-degree form fails. For every prime p we give a pair of smooth affine families of threefolds over an open subset Y of the affine line, defined over F_p, whose fibres over each closed point of Y are isomorphic as F̄_p-varieties, but which do not become isomorphic after any base change U → Y whose image contains the generic point, whether it is étale, of finite degree, or neither. The fibres are affine modifications of G_m² × A¹ at two points, and their isomorphism classes are governed by GL₂(Z)-orbits. Over F̄_p every point of G_m² has finite order, and the Frobenius twist stays in the orbit at every closed point but not at the generic point. For p ≡ 1 (mod 4) we give smooth projective families with the same properties: blow-ups of E × E at two points, where E is the curve y² = x³ − x with complex multiplication by Z[i], parametrised by E × E minus the origin or by a curve in it. The case of Q̄ remains open. This is an unrefereed note.