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.

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

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

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

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