Toroidal embedding explained

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\subsetX

a smooth open subset. Then

U\hookrightarrowX

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

\bar{k}

-algebras:

\widehat{l{O}}_X,\simeq

\widehat{l{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\hookrightarrowX

is said to a toroidal embedding if

U_\bar{k

}\hookrightarrow X_ is a toroidal embedding.

Examples

Tits' buildings

See main article: Tits' buildings.

References

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

External links

Toroidal embedding