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 [math]\displaystyle{ \bar{k} }[/math] and [math]\displaystyle{ U \subset X }[/math] a smooth open subset. Then [math]\displaystyle{ U \hookrightarrow X }[/math] is called a toroidal embedding if for every closed point x of X, there is an isomorphism of local [math]\displaystyle{ \bar{k} }[/math]-algebras:
- [math]\displaystyle{ \widehat{\mathcal{O}}_{X, x} \simeq \widehat{\mathcal{O}}_{X_{\sigma}, t} }[/math]
for some affine toric variety [math]\displaystyle{ X_{\sigma} }[/math] with a torus T and a point t such that the above isomorphism takes the ideal of [math]\displaystyle{ X - U }[/math] to that of [math]\displaystyle{ X_{\sigma} - T }[/math].
Let X be a normal variety over a field k. An open embedding [math]\displaystyle{ U\hookrightarrow X }[/math] is said to a toroidal embedding if [math]\displaystyle{ U_{\bar{k}}\hookrightarrow X_{\bar{k}} }[/math] is a toroidal embedding.
Examples
Tits' buildings
See also
References
- Kempf, G.; Knudsen, Finn Faye; Mumford, David; Saint-Donat, B. (1973), Toroidal embeddings. I, Lecture Notes in Mathematics, 339, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0070318
- Abramovich, D., Denef, J. & Karu, K.: Weak toroidalization over non-closed fields. manuscripta math. (2013) 142: 257. doi:10.1007/s00229-013-0610-5
External links
 |  |
