Bollettino dell'Unione Matematica Italiana

, Volume 11, Issue 1, pp 39–54 | Cite as

Affine lines in the complement of a smooth plane conic



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.


Affine lines Smooth quadric surface Abhyankar-Moh problem Automorphisms 

Mathematics Subject Classification

14E07 14E25 14R25 14J50 



Some of the questions addressed in this article emerged during the workshop “Birational geometry of surfaces”, held at the Department of Mathematics of the University of Roma Tor Vergata in January 2016. The authors would like to thank the organizers of the workshop for the motivated but relaxed atmosphere of this workshop, as well as the other members of the “Afternoon Cremona Club”, Ciro Ciliberto, Alberto Calabri and Anne Lonjou, for stimulating discussions.


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Copyright information

© Unione Matematica Italiana 2017

Authors and Affiliations

  1. 1.Institut de Mathématiques de ToulouseUniversité Paul SabatierToulouse Cedex 9France
  2. 2.IMB UMR5584, CNRSUniv. Bourgogne Franche-ComtéDijonFrance

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