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

Jake Kettinger

TL;DR

The paper classifies all quasi-elliptic surfaces over an algebraically closed field in characteristics $2$ and $3$, and determines all blow-down sequences from such a surface $X$ to $\mathbb{P}^2_k$. It develops a Dynkin-diagram framework for the $(-2)$-curves on $X$, computes explicit change-of-basis maps, and identifies basepoint configurations corresponding to these fibrations. In characteristic $2$ it provides a complete classification of 8 configurations that yield unexpected plane cubics, with explicit pencils of cubics; in characteristic $3$ it shows, via a case-by-case analysis of Dynkin types, that no unexpected cubics can arise. Collectively, the work extends characteristic-zero insights to positive characteristics, giving a detailed bridge between quasi-elliptic fibrations and the existence (or nonexistence) of unexpected plane cubics in low characteristics.

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 $\mathbb{P}^2_k$. 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.

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

TL;DR

The paper classifies all quasi-elliptic surfaces over an algebraically closed field in characteristics and , and determines all blow-down sequences from such a surface to . It develops a Dynkin-diagram framework for the -curves on , computes explicit change-of-basis maps, and identifies basepoint configurations corresponding to these fibrations. In characteristic it provides a complete classification of 8 configurations that yield unexpected plane cubics, with explicit pencils of cubics; in characteristic it shows, via a case-by-case analysis of Dynkin types, that no unexpected cubics can arise. Collectively, the work extends characteristic-zero insights to positive characteristics, giving a detailed bridge between quasi-elliptic fibrations and the existence (or nonexistence) of unexpected plane cubics in low characteristics.

Abstract

In this paper, we categorize all isomorphism classes of quasi-elliptic surfaces over a field of characteristic 2 or 3. For every quasi-elliptic surface , we classify all possible sequences of blow-downs from to the projective plane . 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.

Paper Structure

This paper contains 4 sections, 9 theorems, 25 equations.

Table of Contents

  1. Introduction to quasi-elliptic fibrations
  2. Quasi-Elliptic Fibrations in Characteristic 2
  3. Unexpected Cubics in Characteristic 2
  4. Quasi-Elliptic Fibrations in Characteristic 3

Key Result

Lemma 1

Let $A,B,C,D\in F\subset \mathbb{P}^2$, a general plane cubic curve. Let $\langle P_1,\dots,P_n\rangle_{Q}$ denote the subgroup of $F$ generated by the points $P_1,\dots,P_n\in F$ with $Q\in F$ chosen as the zero point. Let $P_1+_Q P_2$, $-_Q P$, and $n\cdot_Q P$ denote addition, negation, and $\mat

Theorems & Definitions (19)

  • Definition 1
  • Lemma 1
  • proof
  • Corollary 1
  • Corollary 2
  • proof
  • Lemma 2
  • proof
  • Definition 2
  • Lemma 3
  • ...and 9 more