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 reductions of algebraic varieties over one-dimensional bases.

Definition

Let ''X'' be a normal variety over an algebraically closed field $\bar$ 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 $\bar$-algebras:
:$\widehat_ \simeq \widehat_$
for some affine toric variety $X_$ with a torus ''T'' and a point ''t'' such that the above isomorphism takes the ideal of $X - U$ to that of $X_ - T$.

Let ''X'' be a normal variety over a field ''k''. An open embedding $U\hookrightarrow X$ is said to a toroidal embedding if $U_\hookrightarrow X_$ is a toroidal embedding.

Examples

Tits' buildings

See also

*tropical compactification

References

*
*Abramovich, D., Denef, J. & Karu, K.: Weak toroidalization over non-closed fields. manuscripta math. (2013) 142: 257.

External links

Toroidal embedding

Algebraic geometry

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