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.
Examples
Tits' buildings
See also
References
- Kempf, G.; Knudsen, Finn Faye; Mumford, David; Saint-Donat, B. (1973), Toroidal Embeddings I, Vol. 339, Lecture Notes in Mathematics, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0070318. ISBN 978-3-540-06432-9. MR 0335518
- Abramovich, D., Denef, J. & Karu, K.: Weak toroidalization over non-closed fields. manuscripta math. (2013) 142: 257. doi:10.1007/s00229-013-0610-5