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 k ¯ {\displaystyle {\bar {k}}} and U ⊂ X {\displaystyle U\subset X} a smooth open subset. Then U ↪ X {\displaystyle U\hookrightarrow X} is called a toroidal embedding if for every closed point x of X, there is an isomorphism of local k ¯ {\displaystyle {\bar {k}}}-algebras:

O ^ X , x ≃ O ^ X σ , t {\displaystyle {\widehat {\mathcal {O}}}_{X,x}\simeq {\widehat {\mathcal {O}}}_{X_{\sigma },t}}

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

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

Examples

Tits' buildings

See also

  • Kempf, G.; Knudsen, Finn Faye; Mumford, David; Saint-Donat, B. (1973), Toroidal Embeddings I, Lecture Notes in Mathematics, vol. 339, Berlin, New York: Springer-Verlag, doi:, ISBN 978-3-540-06432-9, MR
  • Abramovich, D., Denef, J. & Karu, K.: Weak toroidalization over non-closed fields. manuscripta math. (2013) 142: 257. doi:

External links