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 \(\mathrm{SL}_2 \to \mathbb{A}^3\) 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.