{{Short description|Concept in algebraic geometry}} 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>\bar{k}</math> and <math>U \subset X</math> a smooth open subset. Then <math>U \hookrightarrow X</math> is called a '''toroidal embedding''' if for every closed point ''x'' of ''X'', there is an isomorphism of local <math>\bar{k}</math>-algebras: :<math>\widehat{\mathcal{O}}_{X, x} \simeq \widehat{\mathcal{O}}_{X_{\sigma}, t}</math> for some affine toric variety <math>X_{\sigma}</math> with a torus ''T'' and a point ''t'' such that the above isomorphism takes the ideal of <math>X - U</math> to that of <math>X_{\sigma} - T</math>.
Let ''X'' be a normal variety over a field ''k''. An open embedding <math>U\hookrightarrow X</math> is said to a toroidal embedding if <math>U_{\bar{k}}\hookrightarrow X_{\bar{k}}</math> is a toroidal embedding.
== Examples == === Tits' buildings === {{main|Tits' buildings}}
== See also == *tropical compactification
== References == *{{Citation | last1=Kempf | first1=G. | last2=Knudsen | first2=Finn Faye | last3=Mumford | first3=David | author3-link=David Mumford | last4=Saint-Donat | first4=B. | title=Toroidal Embeddings I | publisher=Springer-Verlag | location=Berlin, New York | series=Lecture Notes in Mathematics | doi= 10.1007/BFb0070318 | mr=0335518 | year=1973 | volume=339| doi-access= | isbn=978-3-540-06432-9 }} *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 == *[https://mathoverflow.net/q/124367 Toroidal embedding]
Category:Algebraic geometry
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