Torification and factorization of birational maps
Authors:
Dan Abramovich, Kalle Karu, Kenji Matsuki and Jarosław Włodarczyk
Journal:
J. Amer. Math. Soc. 15 (2002), 531-572
MSC (2000):
Primary 14E05
DOI:
https://doi.org/10.1090/S0894-0347-02-00396-X
Published electronically:
April 5, 2002
MathSciNet review:
1896232
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Abstract | References | Similar Articles | Additional Information
Abstract: Building on work of the fourth author and Morelli’s work, we prove the weak factorization conjecture for birational maps in characteristic zero: a birational map between complete nonsingular varieties over an algebraically closed field $K$ of characteristic zero is a composite of blowings up and blowings down with nonsingular centers.
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Abhyankar S. Abhyankar, On the valuations centered in a local domain, Amer. J. Math. 78 (1956), 321–348.
- D. Abramovich and A. J. de Jong, Smoothness, semistability, and toroidal geometry, J. Algebraic Geom. 6 (1997), no. 4, 789–801. MR 1487237
- D. Abramovich and K. Karu, Weak semistable reduction in characteristic 0, Invent. Math. 139 (2000), no. 2, 241–273. MR 1738451, DOI https://doi.org/10.1007/s002229900024 Abramovich-Matsuki-Rashid D. Abramovich, K. Matsuki and S. 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. ; Correction: Tohoku Math. J. 52 (2000), 629–631.
- Dan Abramovich and Jianhua Wang, Equivariant resolution of singularities in characteristic $0$, Math. Res. Lett. 4 (1997), no. 2-3, 427–433. MR 1453072, DOI https://doi.org/10.4310/MRL.1997.v4.n3.a11
- Selman Akbulut and Henry King, Topology of real algebraic sets, Mathematical Sciences Research Institute Publications, vol. 25, Springer-Verlag, New York, 1992. MR 1225577
- 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. MR 1672108
- 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. MR 1677693, DOI https://doi.org/10.1007/PL00011158
- 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. MR 1440306, DOI https://doi.org/10.1007/s002220050141 Bittner F. Bittner, The universal Euler characteristic for varieties of characteristic zero, preprint arXiv:math.AG/0111062. Borisov-Libgober L. A. Borisov and A. Libgober, Elliptic Genera of singular varieties, preprint arXiv:math.AG/0007108.
- 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). MR 1103602, DOI https://doi.org/10.1007/s101070100288
- 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). MR 828858
- Alessio Corti, Factoring birational maps of threefolds after Sarkisov, J. Algebraic Geom. 4 (1995), no. 2, 223–254. MR 1311348
- Bruce Crauder, Birational morphisms of smooth threefolds collapsing three surfaces to a point, Duke Math. J. 48 (1981), no. 3, 589–632. MR 630587
- Steven Dale Cutkosky, Local factorization of birational maps, Adv. Math. 132 (1997), no. 2, 167–315. MR 1491444, DOI https://doi.org/10.1006/aima.1997.1675
- Steven Dale Cutkosky, Local monomialization and factorization of morphisms, Astérisque 260 (1999), vi+143 (English, with English and French summaries). MR 1734239 Cutkosky-32 S. D. Cutkosky, Monomialization of morphisms from 3-folds to surfaces, preprint arXiv:math.AG/0010002
- 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. MR 1808613, DOI https://doi.org/10.1080/00927870008827198
- V. I. Danilov, The geometry of toric varieties, Uspekhi Mat. Nauk 33 (1978), no. 2(200), 85–134, 247 (Russian). MR 495499
- V. I. Danilov, Birational geometry of three-dimensional toric varieties, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 5, 971–982, 1135 (Russian). MR 675526
- Jan Denef and François Loeser, Germs of arcs on singular algebraic varieties and motivic integration, Invent. Math. 135 (1999), no. 1, 201–232. MR 1664700, DOI https://doi.org/10.1007/s002220050284
- 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. MR 1659282
- G. Ewald, Blow-ups of smooth toric $3$-varieties, Abh. Math. Sem. Univ. Hamburg 57 (1987), 193–201. MR 927174, DOI https://doi.org/10.1007/BF02941610 Franke J. Franke, Riemann-Roch in functorial form, preprint 1992, 78 pp.
- 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. MR 1234037
- 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). MR 1775354 Hironaka1 H. Hironaka, On the theory of birational blowing-up, Harvard University Ph.D. Thesis 1960.
- Heisuke Hironaka, An example of a non-Kählerian complex-analytic deformation of Kählerian complex structures, Ann. of Math. (2) 75 (1962), 190–208. MR 139182, DOI https://doi.org/10.2307/1970426
- 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. MR 0199184, DOI https://doi.org/10.2307/1970547
- Heisuke Hironaka, Flattening theorem in complex-analytic geometry, Amer. J. Math. 97 (1975), 503–547. MR 393556, DOI https://doi.org/10.2307/2373721
- Yi Hu, The geometry and topology of quotient varieties of torus actions, Duke Math. J. 68 (1992), no. 1, 151–184. MR 1185821, DOI https://doi.org/10.1215/S0012-7094-92-06806-2
- Yi Hu, Relative geometric invariant theory and universal moduli spaces, Internat. J. Math. 7 (1996), no. 2, 151–181. MR 1382720, DOI https://doi.org/10.1142/S0129167X96000098 Hu-Keel Y. Hu and Ś. Keel, A GIT proof of Włodarczyk’s weighted factorization theorem, preprint arXiv:math.AG/9904146.
- 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. MR 637060
- A. J. de Jong, Smoothness, semi-stability and alterations, Inst. Hautes Études Sci. Publ. Math. 83 (1996), 51–93. MR 1423020 Karu-thesis K. Karu, Boston University dissertation, 1999. http://math.bu.edu/people/kllkr/th.ps
- Kazuya Kato, Toric singularities, Amer. J. Math. 116 (1994), no. 5, 1073–1099. MR 1296725, DOI https://doi.org/10.2307/2374941
- 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. MR 765589, DOI https://doi.org/10.2307/2374403
- Yujiro Kawamata, Elementary contractions of algebraic $3$-folds, Ann. of Math. (2) 119 (1984), no. 1, 95–110. MR 736561, DOI https://doi.org/10.2307/2006964
- 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. MR 924674, DOI https://doi.org/10.2307/1971417
- 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. MR 0335518
- 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. MR 1318736, DOI https://doi.org/10.1090/conm/182/02092
- Yujiro Kawamata, Elementary contractions of algebraic $3$-folds, Ann. of Math. (2) 119 (1984), no. 1, 95–110. MR 736561, DOI https://doi.org/10.2307/2006964 Kontsevich M. Kontsevich, Lecture at Orsay (December 7, 1995).
- 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). MR 670170
- 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). MR 1752011 Levine-Morel-cr M. N. Levine and F. Morel, Cobordisme Algébrique II, C. R. Acad. Sci. Paris 332 (2001), no. 9, 815–820. Levine-Morel M. N. Levine and F. Morel, Algebraic cobordism, preprint. Looijenga-motivic E. Looijenga, Motivic measures, preprint arXiv:math.AG/0006220
- Domingo Luna, Slices étales, Sur les groupes algébriques, Soc. Math. France, Paris, 1973, pp. 81–105. Bull. Soc. Math. France, Paris, Mémoire 33 (French). MR 0342523, DOI https://doi.org/10.24033/msmf.110 Matsuki K. Matsuki, Introduction to the Mori program, Universitext, Springer Verlag, Berlin, 2001. Matsuki-notes K. Matsuki, Lectures on factorization of birational maps, RIMS preprint, 1999.
- J. Milnor, Morse theory, Annals of Mathematics Studies, No. 51, Princeton University Press, Princeton, N.J., 1963. Based on lecture notes by M. Spivak and R. Wells. MR 0163331 Moishezon B. Moishezon, On $n$-dimensional compact varieties with $n$ algebraically independent meromorphic functions, Amer. Math. Soc. Transl. 63 (1967), 51–177.
- Robert Morelli, The birational geometry of toric varieties, J. Algebraic Geom. 5 (1996), no. 4, 751–782. MR 1486987 Morelli2 R. Morelli, Correction to “The birational geometry of toric varieties", 1997 http://www.math.utah.edu/~morelli/Math/math.html
- Shigefumi Mori, Threefolds whose canonical bundles are not numerically effective, Ann. of Math. (2) 116 (1982), no. 1, 133–176. MR 662120, DOI https://doi.org/10.2307/2007050
- Shigefumi Mori, Flip theorem and the existence of minimal models for $3$-folds, J. Amer. Math. Soc. 1 (1988), no. 1, 117–253. MR 924704, DOI https://doi.org/10.1090/S0894-0347-1988-0924704-X
- 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. MR 1304906
- 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. MR 546291
- 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. MR 922894
- Rahul Pandharipande, A compactification over $\overline {M}_g$ of the universal moduli space of slope-semistable vector bundles, J. Amer. Math. Soc. 9 (1996), no. 2, 425–471. MR 1308406, DOI https://doi.org/10.1090/S0894-0347-96-00173-7
- 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. MR 713260
- Michel Raynaud and Laurent Gruson, Critères de platitude et de projectivité. Techniques de “platification” d’un module, Invent. Math. 13 (1971), 1–89 (French). MR 308104, DOI https://doi.org/10.1007/BF01390094
- 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. MR 605348
- 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. MR 715649, DOI https://doi.org/10.2969/aspm/00110131
- Miles Reid, Decomposition of toric morphisms, Arithmetic and geometry, Vol. II, Progr. Math., vol. 36, Birkhäuser Boston, Boston, MA, 1983, pp. 395–418. MR 717617 Reid4 M. Reid, Birational geometry of 3-folds according to Sarkisov, preprint 1991. Sally J. Sally, Regular overrings of regular local rings, Trans. Amer. Math. Soc. 171 (1972), 291–300. ; Erratum: Trans. Amer. Math. Soc. 213 (1975), 429. Sarkisov V. G. Sarkisov, Birational maps of standard ${\mathbb {Q}}$-Fano fiberings, I. V. Kurchatov Institute Atomic Energy preprint, 1989.
- Mary Schaps, Birational morphisms of smooth threefolds collapsing three surfaces to a curve, Duke Math. J. 48 (1981), no. 2, 401–420. MR 620257
- David L. Shannon, Monoidal transforms of regular local rings, Amer. J. Math. 95 (1973), 294–320. MR 330154, DOI https://doi.org/10.2307/2373787
- V. V. Shokurov, A nonvanishing theorem, Izv. Akad. Nauk SSSR Ser. Mat. 49 (1985), no. 3, 635–651 (Russian). MR 794958 Sumihiro H. Sumihiro, Equivariant Completion I, II, J. Math. Kyoto Univ. 14, 15 (1974), (1975), 1–28, 573–605. ;
- Mina Teicher, Factorization of a birational morphism between $4$-folds, Math. Ann. 256 (1981), no. 3, 391–399. MR 626957, DOI https://doi.org/10.1007/BF01679705
- Michael Thaddeus, Stable pairs, linear systems and the Verlinde formula, Invent. Math. 117 (1994), no. 2, 317–353. MR 1273268, DOI https://doi.org/10.1007/BF01232244
- Michael Thaddeus, Geometric invariant theory and flips, J. Amer. Math. Soc. 9 (1996), no. 3, 691–723. MR 1333296, DOI https://doi.org/10.1090/S0894-0347-96-00204-4
- Orlando Villamayor, Constructiveness of Hironaka’s resolution, Ann. Sci. École Norm. Sup. (4) 22 (1989), no. 1, 1–32. MR 985852, DOI https://doi.org/10.24033/asens.1573
- Jarosław Włodarczyk, Decomposition of birational toric maps in blow-ups & blow-downs, Trans. Amer. Math. Soc. 349 (1997), no. 1, 373–411. MR 1370654, DOI https://doi.org/10.1090/S0002-9947-97-01701-7
- Jarosław Włodarczyk, Birational cobordisms and factorization of birational maps, J. Algebraic Geom. 9 (2000), no. 3, 425–449. MR 1752010 Wlodarczyk3 J. Włodarczyk, Toroidal Varieties and the Weak Factorization Theorem, preprint arXiv:math.AG/9904076.
- 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. MR 0469915
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Additional Information
Dan Abramovich
Affiliation:
Department of Mathematics, Boston University, 111 Cummington Street, Boston, Massachusetts 02215
MR Author ID:
309312
ORCID:
0000-0003-0719-0989
Email:
abrmovic@math.bu.edu
Kalle Karu
Affiliation:
Department of Mathematics, Harvard University, One Oxford Street, Cambridge, Massachusetts 02139
Email:
kkaru@math.harvard.edu
Kenji Matsuki
Affiliation:
Department of Mathematics, Purdue University, 1395 Mathematical Sciences Building, West Lafayette, Indiana 47907-1395
Email:
kmatsuki@math.purdue.edu
Jarosław Włodarczyk
Affiliation:
Instytut Matematyki UW, Banacha 2, 02-097 Warszawa, Poland
Email:
jwlodar@mimuw.edu.pl
Received by editor(s):
March 14, 2000
Received by editor(s) in revised form:
June 1, 2000
Published electronically:
April 5, 2002
Additional Notes:
The first author was partially supported by NSF grant DMS-9700520 and by an Alfred P. Sloan research fellowship. In addition, he would like to thank the Institut des Hautes Études Scientifiques, Centre Emile Borel (UMS 839, CNRS/UPMC), and Max Planck Institut für Mathematik for a fruitful visiting period.
The second author was partially supported by NSF grant DMS-9700520
The third author has received no financial support from NSF or NSA during the course of this work.
The fourth author was supported in part by Polish KBN grant 2 P03 A 005 16 and NSF grant DMS-0100598.
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