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 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.
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Bollettino dell'Unione Matematica Italiana, 2018, 11 (1), pp.39-54. 〈10.1007/s40574-017-0119-z〉
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Julie Decaup, Adrien Dubouloz. Affine lines in the complement of a smooth plane conic. Bollettino dell'Unione Matematica Italiana, 2018, 11 (1), pp.39-54. 〈10.1007/s40574-017-0119-z〉. 〈hal-01394595〉

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