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On algebraic varieties with finite polyhedral Mori cone. (English) Zbl 1069.14018
Collino, Alberto (ed.) et al., The Fano conference. Papers of the conference organized to commemorate the 50th anniversary of the death of Gino Fano (1871–1952), Torino, Italy, September 29–October 5, 2002. Torino: Università di Torino, Dipartimento di Matematica. 573-589 (2004).
The author reports on known results, obtained mainly by himself and published within the last fifteen years, about projective surfaces and threefolds $$X$$ (with log-terminal or $$\mathbb Q$$-factorial singularities) with finite polyhedral Mori cone NE(X). He explains two fundamental methods of his theory which have their origin in the theory of discrete groups in hyperbolic spaces generated by reflections and with a fundamental domain of finite volume. The method of narrow parts of polyhedra is related to a metric property of a finite closed convex polyhedron in a hyperbolic space [see V. V. Nikulin, Math. USSR, Izv. 16, 573–601(1981; Zbl 0465.22007)]. This method has been applied to algebraic surfaces in his article [Nagoya Math. J. 157, 73–92 (2000; Zbl 0958.14026)] to prove the existence of a very ample divisor of bounded degree. The diagram method was developed for arbitrary reflection groups with fundamental domain of finite volume by È. B. Vinberg [Trans. Mosc. Math. Soc. 1985, 75–112 (1985; Zbl 0593.22007)]. It uses the combinatorial property that the number of faces of highest dimension of a bounded fundamental domain equals the dimension. The diagram method has been applied by the author to del Pezzo surfaces with log-terminal singularities [Math. USSR, Izv. 35, No.3, 657–675 (1990; Zbl 0711.14018)], to Fano threefolds with $$\mathbb Q$$-factorial singularities [J. Math. Kyoto Univ. 34, No.3, 495–529 (1994; Zbl 0839.14030)] and to three-dimensional Calabi-Yau manifolds [in: Higher dimensional complex varieties, Proc. int. conf. Trento 1994, 261–328; appendix 321–328 (1996; Zbl 0957.14030)].
For the entire collection see [Zbl 1051.00013].

##### MSC:
 14E30 Minimal model program (Mori theory, extremal rays) 14J26 Rational and ruled surfaces 14J45 Fano varieties 14J17 Singularities of surfaces or higher-dimensional varieties 14C20 Divisors, linear systems, invertible sheaves
##### Keywords:
narrow parts of polyhedra; diagram method
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