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Elementary birational maps between Mori toric fiber 3-spaces. (English. Russian original) Zbl 1094.14011

Math. Notes 78, No. 1, 120-127 (2005); translation from Mat. Zametki 78, No. 1, 132-139 (2005).
Let \(X_i\) be threedimensional \(\mathbb Q\)-factorial projective toric varieties with at most terminal singularities and let \(\phi_i:X_i\rightarrow S_i\) be toric Mori fibrations, \(i=1\), \(2\). This means that \(\phi_i\) is a contraction of a \(K_{X_{i}}\)-negative extremal ray, \(S_i\) is a \(\mathbb Q\)-factorial toric variety and \(\dim S_i\leq 2\). The author is interested in birational maps between \(X_1\) and \(X_2\) which are compatible with \(\phi_1\) and \(\phi_2\). He uses results of Sarkisov who developed an algorithm for decomposing such birational maps into more elementary ones, namely extremal divisorial contractions, their inverses, and \(\log\)-flips [cf. K. Matsuki, Introduction to the Mori program, Universitext (2002; Zbl 0988.14007)]. Three different types of decompositions (“links”) are considered, and lists for all possibilities of any of the three types are given, but there are only outlines of proofs without the combinatorial parts.

MSC:

14E30 Minimal model program (Mori theory, extremal rays)
14E05 Rational and birational maps

Citations:

Zbl 0988.14007
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Full Text: DOI

References:

[1] K. Matsuki, Introduction to the Mori Program, Universitext, Springer-Verlag, New York, 2002. · Zbl 0988.14007
[2] V. I. Danilov, ”The geometry of toric varieties,” Uspekhi Mat. Nauk [Russian Math. Surveys], 33 (1978), no. 2, 85–134. · Zbl 0425.14013
[3] Y. Kawamata, ”Divisorial contractions to 3-dimensional terminal quotient singularities,” in: Higher-Dimensional Complex Varieties (Trento, 1994), de Gruyter, Berlin, 1996, pp. 241–246. · Zbl 0894.14019
[4] A. A. Borisov and L. A. Borisov, ”Singular toric Fano three-folds,” Mat. Sb. [Russian Acad. Sci. Sb. Math.], 183 (1992), no. 2, 134–141. · Zbl 0786.14028
[5] H. Clemens, J. Kollar, and S. Mori, ”Higher-dimensional complex geometry,” Asterisque, 166 (1988).
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