Embeddings of Affine Spaces into Quadrics

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 ${\mathrm{S}\mathrm{L}}_2\to {\mathbb{A}}^3$ and the fibration ${\mathrm{S}\mathrm{L}}_2\to {\mathbb{A}}^3\to {\mathbb{A}}^1$ obtained by projections, as well as degenerations of variables of polynomial rings, and families of ${\mathbb{A}}^1$-fibrations.