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

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Let X be a normal variety over an algebraically closed field   and   a smooth open subset. Then   is called a toroidal embedding if for every closed point x of X, there is an isomorphism of local  -algebras:

 

for some affine toric variety   with a torus T and a point t such that the above isomorphism takes the ideal of   to that of  .

Let X be a normal variety over a field k. An open embedding   is said to a toroidal embedding if   is a toroidal embedding.

Examples

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Tits' buildings

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See also

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References

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  • 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:10.1007/BFb0070318, 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
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