# Embeddings of Affine Spaces into Quadrics

## TL;DR

This paper constructs infinitely many distinct algebraic embeddings of affine spaces into smooth quadrics over any field, using birational morphisms, fibrations, and degenerations, expanding understanding of affine space embeddings.

## Contribution

It introduces new methods to produce infinitely many non-equivalent embeddings of affine spaces into quadrics, utilizing birational and fibration techniques.

## Key findings

- Infinitely many non-equivalent embeddings of $\\mathbb{A}^1$ into 2D quadrics.
- Infinitely many non-equivalent embeddings of $\mathbb{A}^2$ into 3D quadrics.
- Use of degenerations and fibrations to distinguish embeddings.

## 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.