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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Embeddings of affine spaces into quadrics


Authors: Jérémy Blanc and Immanuel van Santen
Journal: Trans. Amer. Math. Soc. 371 (2019), 8429-8465
MSC (2010): Primary 14R10, 14R25, 14J70, 14J50, 14E05
DOI: https://doi.org/10.1090/tran/7555
Published electronically: March 28, 2019
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Abstract: This article provides, over any field, infinitely many algebraic embeddings of the affine spaces $ \mathbb{A}^1$ and $ \mathbb{A}^2$ into smooth quadrics of dimension two and three, respectively, which are pairwise non-equivalent under automorphisms of the smooth quadric. Our main tools are the study of the birational morphism $ \mathrm {SL}_2 \to \mathbb{A}^3$ and the fibration $ \mathrm {SL}_2 \to \mathbb{A}^3 \to \mathbb{A}^1$ obtained by projections, as well as degenerations of variables of polynomial rings, and families of $ \mathbb{A}^1$-fibrations.


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Additional Information

Jérémy Blanc
Affiliation: Departement Mathematik und Informatik, Universität Basel, Spiegelgasse 1, CH-4051 Basel, Switzerland
Email: jeremy.blanc@unibas.ch

Immanuel van Santen
Affiliation: Fachbereich Mathematik der Universität Hamburg, Bundesstraße 55, DE-20146 Hamburg, Germany
Email: immanuel.van.santen@math.ch

DOI: https://doi.org/10.1090/tran/7555
Received by editor(s): March 6, 2017
Received by editor(s) in revised form: October 10, 2017
Published electronically: March 28, 2019
Additional Notes: The second author was born Immanuel Stampfli
Article copyright: © Copyright 2019 American Mathematical Society