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.