DOI: 10.1515/advgeom-2026-0016 ISSN: 1615-7168

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

Jake Kettinger
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Abstract

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

P k 2 . $\mathbb{P}_k^2 .$
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 this paper (based partly on the author’s thesis), the complete classification of unexpected plane cubic curves in characteristic 2 was open, as well as the question of the existence of unexpected plane cubic curves in characteristic 3.

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