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 \( {\displaystyle {\bar {k}}} \) 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 \( {\displaystyle {\bar {k}}}} \) -algebras:
\( {\displaystyle {\widehat {\mathcal {O}}}_{X,x}\simeq {\widehat {\mathcal {O}}}_{X_{\sigma },t}}} \)
for some affine toric variety \( X_{{\sigma }} with a torus T and a point t such that the above isomorphism takes the ideal of \( {\displaystyle X-U}} \) to that of X\( {\displaystyle X_{\sigma }-T}.} \)
Let X be a normal variety over a field k. An open embedding \( {\displaystyle U\hookrightarrow X}} \) is said to a toroidal embedding if \( {\displaystyle U_{\bar {k}}\hookrightarrow X_{\bar {k}}} } \) is a toroidal embedding.
Examples
Tits' buildings
Main article: Tits' buildings
See also
tropical compactification
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, 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
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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