Toroidal embedding

In algebraic geometry, a toroidal embedding is a normal variety that is locally of a toric variety (torus embedding). The notion was introduced by Mumford to prove the existence of semistable reduction.

Definition

Let X be a normal variety over an algebraically closed field k, $U\subset X$ a smooth open subset. Then $U\hookrightarrow X$ is called a toroidal embedding if for every closed point x of X, there is an isomorphism of local k-algebra

Failed to parse (syntax error): {\displaystyle \widehat{\mathcal{O}_{X, x}} \overset{\simeq}\to \widehat{\mathcal{O}_{X_{\sigma}, t}}

for some affine toric variety $X_{\sigma }$.

References

• Kempf, G.; Knudsen, Finn Faye; Mumford, David; Saint-Donat, B. (1973), Toroidal embeddings. I, Lecture Notes in Mathematics, vol. 339, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0070318, MR 0335518

External links

• http://mathoverflow.net/questions/124367/toroidal-embedding