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 $$\bar{k}$$ 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 $$\bar{k}$$-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 $$X_{\sigma} - T$$.

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