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

TL;DR

This paper classifies certain morphisms in characteristic two where the fibers are non-smooth genus two curves, addressing failures of the Bertini-Sard theorem in positive characteristic.

Contribution

It determines, up to birational equivalence, the structure of separable proper morphisms with non-smooth genus two fibers in characteristic two.

Findings

Classification of such morphisms up to birational equivalence

Identification of conditions for non-smooth fibers in characteristic two

Insight into failures of Bertini-Sard theorem in positive characteristic

Abstract

Looking in positive characteristic for failures of the Bertini-Sard theorem, we determine, up to birational equivalence, the separable proper morphisms of smooth algebraic varieties in characteristic two, whose fibres are non-smooth curves of arithmetic genus two.