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 $X$.

Lines in affine toric varieties

Embedding theorem for flexible varieties

/pdf/the-transversality-of-a-general-translate-xyg77wggba.pdf

The transversality of a general translate

© Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1970, tous droits réservés. L’accès aux archives de la revue « Annales scientifiques de l’É.N.S. » (http://www. elsevier.com/locate/ansens) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

/pdf/sous-groupes-algebriques-de-rang-maximum-du-groupe-de-t8tkk7u3pj.pdf

Sous-groupes algébriques de rang maximum du groupe de Cremona

First Principles.- Further Properties of Locally Nilpotent Derivations.- Polynomial Rings.- Dimension Two.- Dimension Three.- Linear Actions of Unipotent Groups.- Non-Finitely Generated Kernels.- Algorithms.- The Makar-Limanov and Derksen Invariants.- Slices, Embeddings and Cancellation.- Epilogue.

Algebraic theory of locally nilpotent derivations

/pdf/propri-et-es-topologiques-des-polyn-omes-de-deux-variables-590e9m578x.pdf

Propri\'et\'es topologiques des polyn\^omes de deux variables complexes, et automorphismes alg\'ebriques de l'espace $C^{2}$

In their paper of this title [1], Abhyankar and Moh have shown that (up to algebraic automorphism) the only embedding of the line in the plane, in characteristic 0, is the Standard one. For the complex numbers, this is a strong sort of unknotting theorem. In this paper, using only some reasonably elementary theory of knots, I give a new proof of the theorem of Abhyankar and Moh; this proof is intrinsically topological, and does not appear merely to be a translation of their original proof using \"high-school algebra\".

Lines in affine toric varieties

Citations

Embedding theorem for flexible varieties

References

The transversality of a general translate

Sous-groupes algébriques de rang maximum du groupe de Cremona

Algebraic theory of locally nilpotent derivations

Propri\'et\'es topologiques des polyn\^omes de deux variables complexes, et automorphismes alg\'ebriques de l'espace $C^{2}$

Embeddings of the line in the plane.

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