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Decomposition of toric morphisms. (English) Zbl 0571.14020
Arithmetic and geometry, Pap. dedic. I. R. Shafarevich, Vol. II: Geometry, Prog. Math. 36, 395-418 (1983).
[For the entire collection see Zbl 0518.00005.]
This paper contains an interesting application of Mori’s theory of extremal rays to toric varieties [S. Mori, Ann. of Math., II. Ser. 116, 133-176 (1982; Zbl 0557.14021)]. For a complete survey on toric varieties see V. Danilov [The geometry of toric varieties, Russ. Math. Surv. 33, No. 2, 97-154 (1978); translation from Usp. Mat. Nauk SSSR 33, No. 2(200), 85-134 (1978; Zbl 0425.14013)]. We recall only that a toric variety X is defined by means of a ”fan of cones” F, and a toric morpism \(f: X\to Y\) is related to a ”subdivision” of F. In this note the singularities of X and the extremal rays are interpreted in terms of F. In particular, it is shown that every extremal ray R gives rise to a toric morpism \(\phi_ R: X\to Y\) which is an elementary contraction in the sense of Mori’s theory; moreover, if \(\phi_ R\) is an isomorphism in codimension 1, then \(\phi_ R\) can be decomposed into a product \(\phi_ R=\phi_{-R}\psi_ R\), where \(\psi_ R\) is an elementary transformation. Using these facts the author obtains the following result: Let A be a projective toric variety, and let \(f: V\to A\) be a projective birational toric morphism, where V has Q-factorial terminal singularities. Then f is a composite of toric maps: \offlineqn{027 } where \(V_ i,X\) are suitable varieties, h is a projective birational morphism and, for \(i=1,...,k-1\), the rational map \(V_ i\to V_{i+1}\) is either an elementary contraction \(\phi_{R_ i}\) or an elementary transformation \(\psi_{R_ i}\) associated to an extremal ray \(R_ i\in NE(V_ i/A).\)
This result generalizes a well known situation: in the case of surfaces, f is a resolution of a normal surface, S is the minimal resolution, X is is the relative canonical model and \(V_ 0\to V_ 1\to...\to V_ k\) is the sequence of contractions of exceptional curves which gives the minimal model S in terms of any non-singular model.
Reviewer: L.Picco Botta

MSC:
14J30 \(3\)-folds
14C20 Divisors, linear systems, invertible sheaves
14E15 Global theory and resolution of singularities (algebro-geometric aspects)
14E30 Minimal model program (Mori theory, extremal rays)
14E05 Rational and birational maps