The classification of quasi-elliptic fibrations and unexpected plane cubics in characteristics 2 and 3

In this paper, we categorize all isomorphism classes of quasi-elliptic surfaces over a field $k$ of characteristic 2 or 3. For every quasi-elliptic surface $X$, we classify all possible sequences of blow-downs from $X$ to the projective plane ${\mathbb{P}}_k^2$. We then use these categorizations to identify all unexpected plane cubic curves in characteristic 2 and present a proof of the lack of unexpected cubics in characteristic 3. Before the work in this paper – based partly on the author’s thesis – the complete classification of unexpected plane cubic curves in characteristic 2 was unknown, as well as the question of the existence of unexpected plane cubic curves in characteristic 3.