# Toroidal embedding

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{{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](/source/toric_variety). The notion was introduced by Mumford to prove the existence of [semistable reductions](/source/semistable_reduction) of algebraic varieties over one-dimensional bases.

== Definition ==
Let ''X'' be a [normal variety](/source/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](/source/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](/source/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](/source/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](/source/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.](/source/Dan_Abramovich), 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

{{algebraic-geometry-stub}}

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Adapted from the Wikipedia article [Toroidal embedding](https://en.wikipedia.org/wiki/Toroidal_embedding) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Toroidal_embedding?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.