Affine lines in the complement of a smooth plane conic

TL;DR

This paper classifies affine lines in the complement of a smooth conic in the plane over fields of characteristic zero, showing there are exactly two such lines up to certain automorphisms, and explores differences in positive characteristic.

Contribution

It provides a complete classification of affine lines in the complement of a smooth conic over characteristic zero fields and constructs explicit automorphisms, also highlighting differences in positive characteristic.

Findings

Exactly two affine lines up to automorphisms over characteristic zero

Existence of exotic embeddings in positive characteristic

Explicit generators for the automorphism group of the complement

Abstract

We classify closed curves isomorphic to the affine line in the complement of a smooth rational projective plane conic Q. Over a field of characteristic zero, we show that up to the action of the subgroup of the Cremona group of the plane consisting of birational endomorphisms restricting to biregular auto-morphisms outside Q, there are exactly two such lines: the restriction of a smooth conic osculating Q at a rational point and the restriction of the tangent line to Q at a rational point. In contrast, we give examples illustrating the fact that over fields of positive characteristic, there exist exotic closed embeddings of the affine line in the complement of Q. We also determine an explicit set of birational endomorphisms of the plane whose restrictions generates the automorphism group of the complement of Q over a field of arbitrary characteristic.