Affine lines in the complement of a smooth plane conic

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.
Type de document :
Pré-publication, Document de travail
2016
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https://hal.archives-ouvertes.fr/hal-01394595
Contributeur : Adrien Dubouloz <>
Soumis le : mercredi 9 novembre 2016 - 14:36:44
Dernière modification le : lundi 14 novembre 2016 - 11:52:36
Document(s) archivé(s) le : mercredi 15 mars 2017 - 04:30:54

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  • HAL Id : hal-01394595, version 1
  • ARXIV : 1611.03248

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Julie Decaup, Adrien Dubouloz. Affine lines in the complement of a smooth plane conic . 2016. <hal-01394595>

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