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

is called a toroidal embedding if for every closed point x of X, there is an isomorphism of local

with a torus T and a point t such that the above isomorphism takes the ideal of

Let X be a normal variety over a field k. An open embedding

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