## Torification and factorization of birational maps.(English)Zbl 1032.14003

The authors prove the “weak factorization theorem” for complete, smooth, algebraic varieties over an algebraically closed field $$\mathbb K$$:
First, it states that every birational equivalence between two varieties can be realized by birational maps originating from a third one. Second, every birational map can be split into a sequence consisting of blowing up in smooth centers – or the inverse of those blowing ups. (Without referring to inverse blowing ups, we would obtain the yet unproven strong factorization theorem.)
The proof uses the notion of a birational cobordism $$B$$ between birational equivalent varieties, developed by the fourth author [J. Włodarczyk, J. Algebr. Geom. 9, No. 3, 425-449 (2000; Zbl 1010.14002)]. Being a variety of one dimension higher and with a $$\mathbb K^\ast$$-action, this notion is the algebro-geometrical analog of the combinatorial cobordism between fans introduced by Morelli to prove the weak factorization theorem in the toric situation.
Proceeding similar to Morse theory (use the fixed points of the $$\mathbb K$$-action instead of the critical points of a Morse function), one obtains, locally in $$B$$, a toric description of the birational transformation going on. However, the embedded torus may vary from point to point. Blowing up so-called torific ideals in $$B$$ and using canonical resolution of singularities, the authors eventually reach a more global situation; the birational map becomes toroidal, and they can apply the weak factorization theorem for the toric case.

### MSC:

 14E05 Rational and birational maps 14M25 Toric varieties, Newton polyhedra, Okounkov bodies 14E15 Global theory and resolution of singularities (algebro-geometric aspects)

Zbl 1010.14002
Full Text:

### References:

 [1] S. Abhyankar, On the valuations centered in a local domain, Amer. J. Math. 78 (1956), 321-348. · Zbl 0074.26301 [2] D. Abramovich and A. J. de Jong, Smoothness, semistability, and toroidal geometry, J. Algebraic Geom. 6 (1997), no. 4, 789 – 801. · Zbl 0906.14006 [3] D. Abramovich and K. Karu, Weak semistable reduction in characteristic 0, Invent. Math. 139 (2000), no. 2, 241 – 273. · Zbl 0958.14006 [4] Dan Abramovich, Kenji Matsuki, and Suliman Rashid, A note on the factorization theorem of toric birational maps after Morelli and its toroidal extension, Tohoku Math. J. (2) 51 (1999), no. 4, 489 – 537. , https://doi.org/10.2748/tmj/1178224717 Kenji Matsuki, Correction: ”A note on the factorization theorem of toric birational maps after Morelli and its toroidal extension” [Tohoku Math. J. (2) 51 (1999), no. 4, 489 – 537; MR1725624 (2000i:14073)] by D. Abramovich, Matsuki and S. Rashid, Tohoku Math. J. (2) 52 (2000), no. 4, 629 – 631. · Zbl 0991.14020 [5] Dan Abramovich and Jianhua Wang, Equivariant resolution of singularities in characteristic 0, Math. Res. Lett. 4 (1997), no. 2-3, 427 – 433. · Zbl 0906.14005 [6] Selman Akbulut and Henry King, Topology of real algebraic sets, Mathematical Sciences Research Institute Publications, vol. 25, Springer-Verlag, New York, 1992. · Zbl 0808.14045 [7] Victor V. Batyrev, Stringy Hodge numbers of varieties with Gorenstein canonical singularities, Integrable systems and algebraic geometry (Kobe/Kyoto, 1997) World Sci. Publ., River Edge, NJ, 1998, pp. 1 – 32. · Zbl 0963.14015 [8] Victor V. Batyrev, Non-Archimedean integrals and stringy Euler numbers of log-terminal pairs, J. Eur. Math. Soc. (JEMS) 1 (1999), no. 1, 5 – 33. · Zbl 0943.14004 [9] Edward Bierstone and Pierre D. Milman, Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant, Invent. Math. 128 (1997), no. 2, 207 – 302. · Zbl 0896.14006 [10] F. Bittner, The universal Euler characteristic for varieties of characteristic zero, preprint math.AG/0111062. [11] L. A. Borisov and A. Libgober, Elliptic Genera of singular varieties, preprint math.AG/0007108. · Zbl 1053.14050 [12] Michel Brion and Claudio Procesi, Action d’un tore dans une variété projective, Operator algebras, unitary representations, enveloping algebras, and invariant theory (Paris, 1989) Progr. Math., vol. 92, Birkhäuser Boston, Boston, MA, 1990, pp. 509 – 539 (French). · Zbl 1008.90042 [13] Chris Christensen, Strong domination/weak factorization of three-dimensional regular local rings, J. Indian Math. Soc. (N.S.) 45 (1981), no. 1-4, 21 – 47 (1984). Chris Christensen, Strong domination/weak factorization of three-dimensional regular local rings. II, J. Indian Math. Soc. (N.S.) 47 (1983), no. 1-4, 241 – 250 (1986). · Zbl 0643.13007 [14] Alessio Corti, Factoring birational maps of threefolds after Sarkisov, J. Algebraic Geom. 4 (1995), no. 2, 223 – 254. · Zbl 0866.14007 [15] Bruce Crauder, Birational morphisms of smooth threefolds collapsing three surfaces to a point, Duke Math. J. 48 (1981), no. 3, 589 – 632. · Zbl 0474.14005 [16] Steven Dale Cutkosky, Local factorization of birational maps, Adv. Math. 132 (1997), no. 2, 167 – 315. · Zbl 0934.14006 [17] Steven Dale Cutkosky, Local monomialization and factorization of morphisms, Astérisque 260 (1999), vi+143 (English, with English and French summaries). · Zbl 0941.14001 [18] S. D. Cutkosky, Monomialization of morphisms from 3-folds to surfaces, preprint math.AG/0010002 · Zbl 1057.14009 [19] Dale Cutkosky and Olivier Piltant, Monomial resolutions of morphisms of algebraic surfaces, Comm. Algebra 28 (2000), no. 12, 5935 – 5959. Special issue in honor of Robin Hartshorne. · Zbl 1003.14004 [20] V. I. Danilov, The geometry of toric varieties, Uspekhi Mat. Nauk 33 (1978), no. 2(200), 85 – 134, 247 (Russian). [21] V. I. Danilov, Birational geometry of three-dimensional toric varieties, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 5, 971 – 982, 1135 (Russian). [22] Jan Denef and François Loeser, Germs of arcs on singular algebraic varieties and motivic integration, Invent. Math. 135 (1999), no. 1, 201 – 232. · Zbl 0928.14004 [23] Igor V. Dolgachev and Yi Hu, Variation of geometric invariant theory quotients, Inst. Hautes Études Sci. Publ. Math. 87 (1998), 5 – 56. With an appendix by Nicolas Ressayre. · Zbl 1001.14018 [24] G. Ewald, Blow-ups of smooth toric 3-varieties, Abh. Math. Sem. Univ. Hamburg 57 (1987), 193 – 201. · Zbl 0648.14016 [25] J. Franke, Riemann-Roch in functorial form, preprint 1992, 78 pp. [26] William Fulton, Introduction to toric varieties, Annals of Mathematics Studies, vol. 131, Princeton University Press, Princeton, NJ, 1993. The William H. Roever Lectures in Geometry. · Zbl 0813.14039 [27] Henri Gillet and Christophe Soulé, Direct images in non-Archimedean Arakelov theory, Ann. Inst. Fourier (Grenoble) 50 (2000), no. 2, 363 – 399 (English, with English and French summaries). · Zbl 0969.14015 [28] H. Hironaka, On the theory of birational blowing-up, Harvard University Ph.D. Thesis 1960. [29] Heisuke Hironaka, An example of a non-Kählerian complex-analytic deformation of Kählerian complex structures, Ann. of Math. (2) 75 (1962), 190 – 208. · Zbl 0107.16001 [30] Heisuke Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II, Ann. of Math. (2) 79 (1964), 109 – 203; ibid. (2) 79 (1964), 205 – 326. · Zbl 0122.38603 [31] Heisuke Hironaka, Flattening theorem in complex-analytic geometry, Amer. J. Math. 97 (1975), 503 – 547. · Zbl 0307.32011 [32] Yi Hu, The geometry and topology of quotient varieties of torus actions, Duke Math. J. 68 (1992), no. 1, 151 – 184. , https://doi.org/10.1215/S0012-7094-92-06806-2 Yi Hu, Erratum to: ”The geometry and topology of quotient varieties of torus actions”, Duke Math. J. 68 (1992), no. 3, 609. · Zbl 0812.14032 [33] Yi Hu, Relative geometric invariant theory and universal moduli spaces, Internat. J. Math. 7 (1996), no. 2, 151 – 181. · Zbl 0889.14005 [34] Y. Hu and S. Keel, A GIT proof of W\lodarczyk’s weighted factorization theorem, preprint math.AG/9904146. [35] Shigeru Iitaka, Algebraic geometry, Graduate Texts in Mathematics, vol. 76, Springer-Verlag, New York-Berlin, 1982. An introduction to birational geometry of algebraic varieties; North-Holland Mathematical Library, 24. · Zbl 0491.14006 [36] A. J. de Jong, Smoothness, semi-stability and alterations, Inst. Hautes Études Sci. Publ. Math. 83 (1996), 51 – 93. · Zbl 0916.14005 [37] K. Karu, Boston University dissertation, 1999. http://math.bu.edu/people/kllkr/th.ps [38] Kazuya Kato, Toric singularities, Amer. J. Math. 116 (1994), no. 5, 1073 – 1099. · Zbl 0832.14002 [39] Yujiro Kawamata, On the finiteness of generators of a pluricanonical ring for a 3-fold of general type, Amer. J. Math. 106 (1984), no. 6, 1503 – 1512. · Zbl 0587.14027 [40] Yujiro Kawamata, Elementary contractions of algebraic 3-folds, Ann. of Math. (2) 119 (1984), no. 1, 95 – 110. , https://doi.org/10.2307/2006964 Yujiro Kawamata, The cone of curves of algebraic varieties, Ann. of Math. (2) 119 (1984), no. 3, 603 – 633. , https://doi.org/10.2307/2007087 János Kollár, The cone theorem. Note to a paper: ”The cone of curves of algebraic varieties” [Ann. of Math. (2) 119 (1984), no. 3, 603 – 633; MR0744865 (86c:14013b)] by Y. Kawamata, Ann. of Math. (2) 120 (1984), no. 1, 1 – 5. · Zbl 0544.14010 [41] Yujiro Kawamata, Crepant blowing-up of 3-dimensional canonical singularities and its application to degenerations of surfaces, Ann. of Math. (2) 127 (1988), no. 1, 93 – 163. · Zbl 0651.14005 [42] G. Kempf, Finn Faye Knudsen, D. Mumford, and B. Saint-Donat, Toroidal embeddings. I, Lecture Notes in Mathematics, Vol. 339, Springer-Verlag, Berlin-New York, 1973. · Zbl 0271.14017 [43] Henry C. King, Resolving singularities of maps, Real algebraic geometry and topology (East Lansing, MI, 1993) Contemp. Math., vol. 182, Amer. Math. Soc., Providence, RI, 1995, pp. 135 – 154. · Zbl 0871.32025 [44] Yujiro Kawamata, Elementary contractions of algebraic 3-folds, Ann. of Math. (2) 119 (1984), no. 1, 95 – 110. , https://doi.org/10.2307/2006964 Yujiro Kawamata, The cone of curves of algebraic varieties, Ann. of Math. (2) 119 (1984), no. 3, 603 – 633. , https://doi.org/10.2307/2007087 János Kollár, The cone theorem. Note to a paper: ”The cone of curves of algebraic varieties” [Ann. of Math. (2) 119 (1984), no. 3, 603 – 633; MR0744865 (86c:14013b)] by Y. Kawamata, Ann. of Math. (2) 120 (1984), no. 1, 1 – 5. · Zbl 0544.14010 [45] M. Kontsevich, Lecture at Orsay (December 7, 1995). [46] Vik. S. Kulikov, Decomposition of birational mappings of three-dimensional varieties outside of codimension 2, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 4, 881 – 895 (Russian). [47] Gilles Lachaud and Marc Perret, Un invariant birationnel des variétés de dimension 3 sur un corps fini, J. Algebraic Geom. 9 (2000), no. 3, 451 – 458 (French, with English and French summaries). · Zbl 0970.14014 [48] M. N. Levine and F. Morel, Cobordisme Algébrique II, C. R. Acad. Sci. Paris 332 (2001), no. 9, 815-820. · Zbl 1009.19002 [49] M. N. Levine and F. Morel, Algebraic cobordism, preprint. · Zbl 1188.14015 [50] E. Looijenga, Motivic measures, preprint math.AG/0006220 · Zbl 0996.14011 [51] Domingo Luna, Slices étales, Sur les groupes algébriques, Soc. Math. France, Paris, 1973, pp. 81 – 105. Bull. Soc. Math. France, Paris, Mémoire 33 (French). · Zbl 0286.14014 [52] K. Matsuki, Introduction to the Mori program, Universitext, Springer Verlag, Berlin, 2001. · Zbl 0988.14007 [53] K. Matsuki, Lectures on factorization of birational maps, RIMS preprint, 1999. [54] J. Milnor, Morse theory, Based on lecture notes by M. Spivak and R. Wells. Annals of Mathematics Studies, No. 51, Princeton University Press, Princeton, N.J., 1963. · Zbl 0108.10401 [55] B. Moishezon, On $$n$$-dimensional compact varieties with $$n$$ algebraically independent meromorphic functions, Amer. Math. Soc. Transl. 63 (1967), 51-177. · Zbl 0186.26204 [56] Robert Morelli, The birational geometry of toric varieties, J. Algebraic Geom. 5 (1996), no. 4, 751 – 782. · Zbl 0871.14041 [57] R. Morelli, Correction to “The birational geometry of toric varieties”, 1997 http://www.math.utah.edu/ morelli/Math/math.html [58] Shigefumi Mori, Threefolds whose canonical bundles are not numerically effective, Ann. of Math. (2) 116 (1982), no. 1, 133 – 176. · Zbl 0557.14021 [59] Shigefumi Mori, Flip theorem and the existence of minimal models for 3-folds, J. Amer. Math. Soc. 1 (1988), no. 1, 117 – 253. · Zbl 0649.14023 [60] D. Mumford, J. Fogarty, and F. Kirwan, Geometric invariant theory, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], vol. 34, Springer-Verlag, Berlin, 1994. · Zbl 0797.14004 [61] Tadao Oda, Torus embeddings and applications, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, vol. 57, Tata Institute of Fundamental Research, Bombay; by Springer-Verlag, Berlin-New York, 1978. Based on joint work with Katsuya Miyake. · Zbl 0417.14043 [62] Tadao Oda, Convex bodies and algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 15, Springer-Verlag, Berlin, 1988. An introduction to the theory of toric varieties; Translated from the Japanese. · Zbl 0628.52002 [63] Rahul Pandharipande, A compactification over \?_{\?} of the universal moduli space of slope-semistable vector bundles, J. Amer. Math. Soc. 9 (1996), no. 2, 425 – 471. · Zbl 0886.14002 [64] Henry C. Pinkham, Factorization of birational maps in dimension 3, Singularities, Part 2 (Arcata, Calif., 1981) Proc. Sympos. Pure Math., vol. 40, Amer. Math. Soc., Providence, RI, 1983, pp. 343 – 371. [65] Michel Raynaud and Laurent Gruson, Critères de platitude et de projectivité. Techniques de ”platification” d’un module, Invent. Math. 13 (1971), 1 – 89 (French). · Zbl 0227.14010 [66] Miles Reid, Canonical 3-folds, Journées de Géometrie Algébrique d’Angers, Juillet 1979/Algebraic Geometry, Angers, 1979, Sijthoff & Noordhoff, Alphen aan den Rijn — Germantown, Md., 1980, pp. 273 – 310. [67] Miles Reid, Minimal models of canonical 3-folds, Algebraic varieties and analytic varieties (Tokyo, 1981) Adv. Stud. Pure Math., vol. 1, North-Holland, Amsterdam, 1983, pp. 131 – 180. · Zbl 0558.14028 [68] Miles Reid, Decomposition of toric morphisms, Arithmetic and geometry, Vol. II, Progr. Math., vol. 36, Birkhäuser Boston, Boston, MA, 1983, pp. 395 – 418. · Zbl 0571.14020 [69] M. Reid, Birational geometry of 3-folds according to Sarkisov, preprint 1991. [70] Judith Sally, Regular overrings of regular local rings, Trans. Amer. Math. Soc. 171 (1972), 291 – 300. , https://doi.org/10.1090/S0002-9947-1972-0309929-3 Judith Sally, Erratum to: ”Regular overrings of regular local rings” (Trans. Amer. Math. Soc. 171 (1972), 291 – 300), Trans. Amer. Math. Soc. 213 (1975), 429. · Zbl 0256.13015 [71] V. G. Sarkisov, Birational maps of standard $${\mathbb{Q} }$$-Fano fiberings, I. V. Kurchatov Institute Atomic Energy preprint, 1989. [72] Mary Schaps, Birational morphisms of smooth threefolds collapsing three surfaces to a curve, Duke Math. J. 48 (1981), no. 2, 401 – 420. · Zbl 0475.14008 [73] David L. Shannon, Monoidal transforms of regular local rings, Amer. J. Math. 95 (1973), 294 – 320. · Zbl 0271.14003 [74] V. V. Shokurov, A nonvanishing theorem, Izv. Akad. Nauk SSSR Ser. Mat. 49 (1985), no. 3, 635 – 651 (Russian). [75] Hideyasu Sumihiro, Equivariant completion, J. Math. Kyoto Univ. 14 (1974), 1 – 28. Hideyasu Sumihiro, Equivariant completion. II, J. Math. Kyoto Univ. 15 (1975), no. 3, 573 – 605. · Zbl 0277.14008 [76] Mina Teicher, Factorization of a birational morphism between 4-folds, Math. Ann. 256 (1981), no. 3, 391 – 399. · Zbl 0445.14005 [77] Michael Thaddeus, Stable pairs, linear systems and the Verlinde formula, Invent. Math. 117 (1994), no. 2, 317 – 353. · Zbl 0882.14003 [78] Michael Thaddeus, Geometric invariant theory and flips, J. Amer. Math. Soc. 9 (1996), no. 3, 691 – 723. · Zbl 0874.14042 [79] Orlando Villamayor, Constructiveness of Hironaka’s resolution, Ann. Sci. École Norm. Sup. (4) 22 (1989), no. 1, 1 – 32. · Zbl 0675.14003 [80] Jarosław Włodarczyk, Decomposition of birational toric maps in blow-ups & blow-downs, Trans. Amer. Math. Soc. 349 (1997), no. 1, 373 – 411. · Zbl 0867.14005 [81] Jarosław Włodarczyk, Birational cobordisms and factorization of birational maps, J. Algebraic Geom. 9 (2000), no. 3, 425 – 449. · Zbl 1010.14002 [82] J. W\lodarczyk, Toroidal Varieties and the Weak Factorization Theorem, preprint math.AG/9904076. [83] Oscar Zariski, Algebraic surfaces, Second supplemented edition, Springer-Verlag, New York-Heidelberg, 1971. With appendices by S. S. Abhyankar, J. Lipman, and D. Mumford; Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 61. · Zbl 0085.36202
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.