##
**Factorization of birational maps in dimension 3.**
*(English)*
Zbl 0544.14005

Singularities, Summer Inst., Arcata/Calif. 1981, Proc. Symp. Pure Math. 40, Part 2, 343-371 (1983).

[For the entire collection see Zbl 0509.00008.]

Here is given a survey over some recent work on the problem of fatorization of proper birational maps between smooth threefolds. At an elementary level the following results are explained: Danilov’s theorem on factorization of toric maps between toric threefolds, the Schaps- Crauder theorem on factorization of birational maps with at most three exceptional divisors meeting transversely [M. Schaps, Duke Math. J. 48, 401-420 (1981; Zbl 0475.14008); B. Crauder, ibid. 589-632 (1981; Zbl 0474.14005)], the Mori theorem on factorization of birational morphisms by elementary contractions [S. Mori, Ann. Math., II. Ser. 116, 133-176 (1982)], Kulikov’s theorem on decomposition of birational maps between smooth threefolds into the product of blow ups and blow downs up to isomorphisms in codimension one. [cf. Math. USSR, Izv. 21, 187-200 (1983); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 46, 881- 895 (1982; Zbl 0544.14024)]. The account contains a lot of interesting examples of birational maps. - In the last section the author states some new results on isomorphisms in codimension one. The exceptional locus of such isomorphism (as well as of the inverse one) is a collection of rational curves. The author gives the complete classification of isomorphisms \(f:X\to Y\) in codimension one under the assumption that this collection of curves in X can be collapsed to a Cohen-Macaulay point \(P\in V\). Under this assumption X and Y are simply two distinct small \((= with\) one-dimensional exceptional locus) resolutions of V at P and thus one can obtain the classification of birational maps f:\(X\to Y\) from the description of all small resolutions of V. Considering V as a one- parametric deformation of its general hyperplane section H through P with a rational Du Val singularity at P the author establishes the 1-1 correspondence between all the small resolutions of V and the Weyl chambers in the divisor class group C\(l({\mathfrak H}_ S)\) where S is the base of the versal deformation of the proper transform \(\tilde H\) of H in X, \({\mathfrak H}_ S={\mathfrak H}_ R\times_ RS,\) R being the versal deformation of H and the action of the Weyl group W arising from the isomorphism \(W=Gal(S/R)\). If the exceptional locus of f is \({\mathbb{P}}^ 1\), then \(| W| =2\) and so there is a unique non-trivial map f:\(X\to Y\), and the author shows, how it decomposes into the product of blow ups and blow downs.

Here is given a survey over some recent work on the problem of fatorization of proper birational maps between smooth threefolds. At an elementary level the following results are explained: Danilov’s theorem on factorization of toric maps between toric threefolds, the Schaps- Crauder theorem on factorization of birational maps with at most three exceptional divisors meeting transversely [M. Schaps, Duke Math. J. 48, 401-420 (1981; Zbl 0475.14008); B. Crauder, ibid. 589-632 (1981; Zbl 0474.14005)], the Mori theorem on factorization of birational morphisms by elementary contractions [S. Mori, Ann. Math., II. Ser. 116, 133-176 (1982)], Kulikov’s theorem on decomposition of birational maps between smooth threefolds into the product of blow ups and blow downs up to isomorphisms in codimension one. [cf. Math. USSR, Izv. 21, 187-200 (1983); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 46, 881- 895 (1982; Zbl 0544.14024)]. The account contains a lot of interesting examples of birational maps. - In the last section the author states some new results on isomorphisms in codimension one. The exceptional locus of such isomorphism (as well as of the inverse one) is a collection of rational curves. The author gives the complete classification of isomorphisms \(f:X\to Y\) in codimension one under the assumption that this collection of curves in X can be collapsed to a Cohen-Macaulay point \(P\in V\). Under this assumption X and Y are simply two distinct small \((= with\) one-dimensional exceptional locus) resolutions of V at P and thus one can obtain the classification of birational maps f:\(X\to Y\) from the description of all small resolutions of V. Considering V as a one- parametric deformation of its general hyperplane section H through P with a rational Du Val singularity at P the author establishes the 1-1 correspondence between all the small resolutions of V and the Weyl chambers in the divisor class group C\(l({\mathfrak H}_ S)\) where S is the base of the versal deformation of the proper transform \(\tilde H\) of H in X, \({\mathfrak H}_ S={\mathfrak H}_ R\times_ RS,\) R being the versal deformation of H and the action of the Weyl group W arising from the isomorphism \(W=Gal(S/R)\). If the exceptional locus of f is \({\mathbb{P}}^ 1\), then \(| W| =2\) and so there is a unique non-trivial map f:\(X\to Y\), and the author shows, how it decomposes into the product of blow ups and blow downs.

Reviewer: D.G.Markushevich