Embeddings of affine spaces into quadrics

Blanc, Jérémy; van Santen, Immanuel

doi:10.1090/tran/7555

Embeddings of affine spaces into quadrics
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by Jérémy Blanc and Immanuel van Santen PDF

Trans. Amer. Math. Soc. 371 (2019), 8429-8465 Request permission

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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