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**Factorization of birational maps of rational surfaces from the viewpoint of Mori theory.**
*(English.
Russian original)*
Zbl 0914.14005

Russ. Math. Surv. 51, No. 4, 585-652 (1996); translation from Usp. Mat. Nauk 51, No. 4, 3-72 (1996).

This paper gives new proofs for known results on factorization of birational maps between minimal rational surfaces over perfect fields using Mori theory. Starting with the Cremona group of the projective plane, this is subject of classical algebraic geometry, and more recent developments include contributions of the author and many others [cf. the book by Yu. I. Manin, “Cubic forms. Algebra, geometry, arithmetic” (1986; Zbl 0582.14010); translation from the Russian (1972; Zbl 0255.14002) and the following articles: V. A. Iskovskikh and S. L. Tregub, Math. USSR, Izv. 38, No. 2, 251-275 (1992); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 55, No. 2, 254-281 (1991; Zbl 0752.14009), V. A. Iskovskikh, F. K. Kabdykairov and S. L. Tregub, Russ. Acad. Sci., Izv. Math. 42, No. 3, 427-478 (1994); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 57, No. 3, 3-69 (1993; Zbl 0824.14010), Yu. I. Manin and M. A. Tsfasman, Russ. Math. Surv. 41, No. 2, 51-116 (1986); translation from Usp. Mat. Nauk 41, No. 2(248), 43-94 (1986; Zbl 0621.14029) and V. A. Iskovskikh, J. Sov. Mat. 13, 745-814 (1980); translation from Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat. 12, 159-236 (1979; Zbl 0415.14025)]. – The paper under review applies Mori’s method of extremal contractions to obtain a decomposition of birational maps. Such idea to do this for higher dimension goes back to V. G. Sarkisov and was further developed by M. Reid and A. Corti, who gave a proof that for 3-dimensional \(\mathbb{Q}\)-Fano fibre spaces every birational map “decomposes into finitely many elementary links”.

From the author’s summary: In the case of rational surfaces over a perfect field, \(\mathbb{Q}\)-Fano fibre spaces \(\varphi:X\to S\) are precisely the relatively minimal models from the families \(\{\mathbb{D}\}\) and \(\{\mathbb{C}\}\) defined in §1, where \(S=\text{Spec }k\) is a point for surfaces in \(\{\mathbb{D}\}\) and \(S\) is a smooth projective rational curve over \(k\) (that is, either \(\mathbb{P}^1\), or a plane conic \(C\subset \mathbb{P}^2\) having no \(k\)-points) for surfaces in \(\{\mathbb{C}\}\). As we have already said, birational \(k\)-maps between these have been described. Here we prove all these results again from the single new viewpoint of Mori theory. Although the new formalism does not add new results it gives a conceptually transparent understanding of the process of factorization, useful for generalization to higher dimensions. We also make considerable use of the old formalism, particularly for constructing elementary links and describing the relations between them. Comparisons between the two languages are given in the comments.

From the author’s summary: In the case of rational surfaces over a perfect field, \(\mathbb{Q}\)-Fano fibre spaces \(\varphi:X\to S\) are precisely the relatively minimal models from the families \(\{\mathbb{D}\}\) and \(\{\mathbb{C}\}\) defined in §1, where \(S=\text{Spec }k\) is a point for surfaces in \(\{\mathbb{D}\}\) and \(S\) is a smooth projective rational curve over \(k\) (that is, either \(\mathbb{P}^1\), or a plane conic \(C\subset \mathbb{P}^2\) having no \(k\)-points) for surfaces in \(\{\mathbb{C}\}\). As we have already said, birational \(k\)-maps between these have been described. Here we prove all these results again from the single new viewpoint of Mori theory. Although the new formalism does not add new results it gives a conceptually transparent understanding of the process of factorization, useful for generalization to higher dimensions. We also make considerable use of the old formalism, particularly for constructing elementary links and describing the relations between them. Comparisons between the two languages are given in the comments.

Reviewer: M.Roczen (Berlin)

### MSC:

14E07 | Birational automorphisms, Cremona group and generalizations |

14J26 | Rational and ruled surfaces |

14E30 | Minimal model program (Mori theory, extremal rays) |