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Blow-ups of smooth toric 3-varieties. (English) Zbl 0648.14016

Let \(X_{\Sigma}\) be the toric variety associated to a fan \(\Sigma\) of cones in \({\mathbb{R}}^ n.\) It is known that \(X_{\Sigma}\) is complete if and only if \(\Sigma\) is and \(X_{\Sigma}\) is projective if and only if \(\Sigma\) is obtained by projecting the faces of a convex polytope. The main result of this paper is that any complete smooth toric 3-variety \(X_{\Sigma}\) can be turned into a projective complete smooth toric 3- variety \(X_{\Sigma '}\) by a finite sequence of blow-ups along non singular centers.
The methods used for the proof are completely combinatorial.
Reviewer: L.Picco Botta

MSC:

14J30 \(3\)-folds
14E05 Rational and birational maps
05B99 Designs and configurations
14J10 Families, moduli, classification: algebraic theory
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