Fibrations by non-smooth projective curves of arithmetic genus two in characteristic two

Looking in positive characteristic for failures of the Bertini–Sard theorem, we determine, up to birational equivalence, the separable proper morphisms between smooth algebraic varieties in characteristic two, whose fibres are non-smooth curves of arithmetic genus two. We show that almost all fibres of such a morphism are geometrically elliptic if and only if the function field of the generic fibre is separable over its canonical quadratic rational subfield. We discover a smooth sixfold Z⊂P4×A5Z⊂P4×A5 such that almost all fibres of the projection morphism π:Z→A5π:Z→A5 are cuspidal geometrically elliptic curves of arithmetic genus two. The main theorem of the paper states that each proper separable morphism between smooth algebraic varieties, whose fibres are geometrically elliptic curves of arithmetic genus two, is birational equivalent to a base extension of the fibration π.