Embedding theorem for flexible varieties

TL;DR

This paper establishes criteria for embedding affine algebraic varieties into smooth flexible varieties, particularly when the target is a special linear group with sufficient dimension, expanding understanding of algebraic embeddings.

Contribution

It introduces new conditions under which an affine variety can be embedded into a smooth flexible variety, especially when the target is a special linear group.

Findings

Embedding criteria for affine varieties into flexible varieties.

Conditions involving dimension and tangent space for embeddings.

Application to special linear groups as embedding targets.

Abstract

Let $Z$ be an affine algebraic variety and $X$ be a smooth flexible variety. We develop some criteria under which $Z$ admits a closed embedding into $X$. In particular, we show that if $X$ is isomorphic (as an algebraic variety) to a special linear group and $\dim X \geq \max(2\dim Z+1, \dim TZ)$, then $Z$ admits a closed embedding into $X$.