Toroidal embedding

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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[edit]

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[edit]

Tits' buildings[edit]

See also[edit]

References[edit]

  • 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

External links[edit]