Affordable Access

Access to the full text

Affine lines in the complement of a smooth plane conic

Authors
  • Decaup, Julie1
  • Dubouloz, Adrien2
  • 1 Université Paul Sabatier, Institut de Mathématiques de Toulouse, 118 route de Narbonne, Toulouse Cedex 9, 31062, France , Toulouse Cedex 9 (France)
  • 2 Univ. Bourgogne Franche-Comté, IMB UMR5584, CNRS, Dijon, 21000, France , Dijon (France)
Type
Published Article
Journal
Bollettino dell'Unione Matematica Italiana
Publisher
Springer International Publishing
Publication Date
Mar 18, 2017
Volume
11
Issue
1
Pages
39–54
Identifiers
DOI: 10.1007/s40574-017-0119-z
Source
Springer Nature
Keywords
License
Yellow

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 automorphisms 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.

Report this publication

Statistics

Seen <100 times