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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