Toroidal embedding

In algebraic geometry, a toroidal embedding is an open embedding of algebraic varieties that locally looks like the embedding of the open torus into a toric variety. 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 and $U\subset X$ 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-algebras:

${\widehat {\mathcal {O}}}_{X,x}\simeq {\widehat {\mathcal {O}}}_{X_{\sigma },t}$

for some affine toric variety $X_{\sigma }$ with a torus T and a point t such that the above isomorphism takes the ideal of $X-U$ to that of $X_{\sigma }-T$.

Examples

Tits' buildings

Main article: Tits' buildings

See also

• tropical compactification

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