Factoring birational maps of threefolds after Sarkisov. Appendix: Surfaces over nonclosed fields.

*(English)*Zbl 0866.14007The aim of the birational classification of algebraic varieties is to isolate some distinguished representatives, or standard models, in each birational equivalence class, to be considered simple in some suitable sense, and study the structure of birational transformations between them. This means defining and charting some “elementary” birational maps, and showing that every birational map between standard models can be factored as a chain of elementary ones. Satisfactory answers to these problems have been known in dimension two for over a century. A program to find minimal models in all dimensions, also known as Mori program, was introduced about ten years ago, and completed in dimension three by S. Mori [J. Am. Math. Soc. 1, No. 1, 117-253 (1988; Zbl 0649.14023)]. According to this theory, there are two birationally disjoint classes of “simple” objects: minimal models, that is varieties \(X\) whose canonical classes \(K_X\) are nef, and Mori fiber spaces, or varieties whose anticanonical classes \(-K_X\) are relatively ample for a suitable morphism.

The structure of birational maps between minimal models is, at least in principle, well understood. A birational transformation between minimal models is an isomorphism in codimension one, and is a composition of flops. Flops should be considered elementary birational maps, and in dimension three explicit equations are known for all possible flops [cf. J. Differ. Geom., Suppl. 1, 113-199 (1991) = Proc. Conf., Cambridge 1990, Surv. Differ. Geom., Suppl. J. Differ. Geom. 1, 113-199 (1991; Zbl 0755.14003)]. Also, the set of minimal models in the birational equivalence class of a 3-fold of general type is finite [Y. Kawamata and K. Matsuki, Math. Ann. 276, 595-598 (1987; Zbl 0606.14034)], and its elements correspond to the cells of maximal dimension in a polyhedral decomposition of a rational polyhedral cone defined in term of the canonical model [Y. Kawamata, Ann. Math., II. Ser. 127, No. 1, 93-163 (1988; Zbl 0651.14005)].

V. G. Sarkisov [“Birational maps of standard \(\mathbb{Q}\)-Fano fiberings”, I. V. Kurchatov, Inst. Atom. Energy (preprint 1989)] introduced a notion of elementary map (or elementary link) between Mori fiber spaces, and announced a proof that every birational transformation between 3-fold Mori fiber spaces is a composition of elementary links. Unfortunately, no details have appeared. The Sarkisov program was reviewed by M. Reid [“Birational geometry of 3-folds according to Sarkisov” (preprint 1991)], who also outlined some key ideas involved in the proof. The goal of this paper is to give a complete proof of Sarkisov’s theorem, modelled on the preprint by M. Reid.

This paper is organized as follows: §2 sets up some terminology and gives the construction of extremal blow ups. In §3 the general structure of the Sarkisov program is recalled. §4 deals with Noether-Fano inequalities. This material is very classical, and is treated here in the language of minimal model theory. §§5-6 constitute the heart of this work. In §5, it is shown how to build a chain of elementary maps to factorize a given birational map. In §6, it is shown that this chain eventually terminates in a factorization. – In the appendix, the formalism is applied (mostly without proof) to the class of rational surfaces over arbitrary fields. It is possible in this context to give a full classification of elementary links.

The structure of birational maps between minimal models is, at least in principle, well understood. A birational transformation between minimal models is an isomorphism in codimension one, and is a composition of flops. Flops should be considered elementary birational maps, and in dimension three explicit equations are known for all possible flops [cf. J. Differ. Geom., Suppl. 1, 113-199 (1991) = Proc. Conf., Cambridge 1990, Surv. Differ. Geom., Suppl. J. Differ. Geom. 1, 113-199 (1991; Zbl 0755.14003)]. Also, the set of minimal models in the birational equivalence class of a 3-fold of general type is finite [Y. Kawamata and K. Matsuki, Math. Ann. 276, 595-598 (1987; Zbl 0606.14034)], and its elements correspond to the cells of maximal dimension in a polyhedral decomposition of a rational polyhedral cone defined in term of the canonical model [Y. Kawamata, Ann. Math., II. Ser. 127, No. 1, 93-163 (1988; Zbl 0651.14005)].

V. G. Sarkisov [“Birational maps of standard \(\mathbb{Q}\)-Fano fiberings”, I. V. Kurchatov, Inst. Atom. Energy (preprint 1989)] introduced a notion of elementary map (or elementary link) between Mori fiber spaces, and announced a proof that every birational transformation between 3-fold Mori fiber spaces is a composition of elementary links. Unfortunately, no details have appeared. The Sarkisov program was reviewed by M. Reid [“Birational geometry of 3-folds according to Sarkisov” (preprint 1991)], who also outlined some key ideas involved in the proof. The goal of this paper is to give a complete proof of Sarkisov’s theorem, modelled on the preprint by M. Reid.

This paper is organized as follows: §2 sets up some terminology and gives the construction of extremal blow ups. In §3 the general structure of the Sarkisov program is recalled. §4 deals with Noether-Fano inequalities. This material is very classical, and is treated here in the language of minimal model theory. §§5-6 constitute the heart of this work. In §5, it is shown how to build a chain of elementary maps to factorize a given birational map. In §6, it is shown that this chain eventually terminates in a factorization. – In the appendix, the formalism is applied (mostly without proof) to the class of rational surfaces over arbitrary fields. It is possible in this context to give a full classification of elementary links.