The classification of quasi-elliptic fibrations and unexpected plane cubics in characteristics 2 and 3
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912634395164672 |
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| author | Kettinger, Jake |
| author_facet | Kettinger, Jake |
| contents | 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 $\mathbb{P}^2_k$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_06365 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The classification of quasi-elliptic fibrations and unexpected plane cubics in characteristics 2 and 3 Kettinger, Jake Algebraic Geometry 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 $\mathbb{P}^2_k$. 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. |
| title | The classification of quasi-elliptic fibrations and unexpected plane cubics in characteristics 2 and 3 |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2510.06365 |