Torification and factorization of birational maps

Abramovich, Dan; Karu, Kalle; Matsuki, Kenji; Włodarczyk, Jarosław

doi:10.1090/S0894-0347-02-00396-X

Torification and factorization of birational maps
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by Dan Abramovich, Kalle Karu, Kenji Matsuki and Jarosław Włodarczyk;

J. Amer. Math. Soc. 15 (2002), 531-572

DOI: https://doi.org/10.1090/S0894-0347-02-00396-X

Published electronically: April 5, 2002

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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.

References

Cahit Arf, Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math. 181 (1939), 1–44 (German). MR 18, DOI 10.1515/crll.1940.181.1
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 10.1007/s002229900024
P. Erdös, On the distribution of normal point groups, Proc. Nat. Acad. Sci. U.S.A. 26 (1940), 294–297. MR 2000, DOI 10.1073/pnas.26.4.294
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 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, DOI 10.1007/978-1-4613-9739-7
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. 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 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 10.1007/s002220050141

The universal Euler characteristic for varieties of characteristic zero

math.AG/0111062

Elliptic Genera of singular varieties

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 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 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

Monomialization of morphisms from 3-folds to surfaces

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 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 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, DOI 10.1007/BF02698859
G. Ewald, Blow-ups of smooth toric $3$-varieties, Abh. Math. Sem. Univ. Hamburg 57 (1987), 193–201. MR 927174, DOI 10.1007/BF02941610

Riemann-Roch in functorial form

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, DOI 10.1515/9781400882526
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, DOI 10.5802/aif.1758

On the theory of birational blowing-up

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 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; 79 (1964), 205–326. MR 199184, DOI 10.2307/1970547
Heisuke Hironaka, Flattening theorem in complex-analytic geometry, Amer. J. Math. 97 (1975), 503–547. MR 393556, DOI 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 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 10.1142/S0129167X96000098

A GIT proof of Włodarczyk’s weighted factorization theorem

math.AG/9904146

Shigeru Iitaka, Algebraic geometry, North-Holland Mathematical Library, vol. 24, Springer-Verlag, New York-Berlin, 1982. An introduction to birational geometry of algebraic varieties; Graduate Texts in Mathematics, 76. MR 637060, DOI 10.1007/978-1-4613-8119-8
A. J. de Jong, Smoothness, semi-stability and alterations, Inst. Hautes Études Sci. Publ. Math. 83 (1996), 51–93. MR 1423020, DOI 10.1007/BF02698644

http://math.bu.edu/people/kllkr/th.ps

Kazuya Kato, Toric singularities, Amer. J. Math. 116 (1994), no. 5, 1073–1099. MR 1296725, DOI 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 10.2307/2374403
Yujiro Kawamata, Elementary contractions of algebraic $3$-folds, Ann. of Math. (2) 119 (1984), no. 1, 95–110. MR 736561, DOI 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 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 335518, DOI 10.1007/BFb0070318
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 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 10.2307/2006964
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

Cobordisme Algébrique II

332

Algebraic cobordism

Motivic measures

math.AG/0006220

Domingo Luna, Slices étales, Sur les groupes algébriques, Supplément au Bull. Soc. Math. France, Tome 101, Soc. Math. France, Paris, 1973, pp. 81–105 (French). MR 342523, DOI 10.24033/msmf.110

Introduction to the Mori program

Lectures on factorization of birational maps

J. Milnor, Morse theory, Annals of Mathematics Studies, No. 51, Princeton University Press, Princeton, NJ, 1963. Based on lecture notes by M. Spivak and R. Wells. MR 163331, DOI 10.1515/9781400881802
Reinhold Baer, Nets and groups, Trans. Amer. Math. Soc. 46 (1939), 110–141. MR 35, DOI 10.1090/S0002-9947-1939-0000035-5
Robert Morelli, The birational geometry of toric varieties, J. Algebraic Geom. 5 (1996), no. 4, 751–782. MR 1486987

Correction to “The birational geometry of toric varieties"

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 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 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, DOI 10.1007/978-3-642-57916-5
Tadao Oda, Torus embeddings and applications, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, vol. 57, Tata Institute of Fundamental Research, Bombay; 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 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, DOI 10.1090/pspum/040.2/713260
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 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 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

Birational geometry of 3-folds according to Sarkisov

Stefan Kempisty, Sur les fonctions à variation bornée au sens de Tonelli, Bull. Sém. Math. Univ. Wilno 2 (1939), 13–21 (French). MR 46

Birational maps of standard ${\mathbb {Q}}$-Fano fiberings

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 10.2307/2373787
V. V. Shokurov, A nonvanishing theorem, Izv. Akad. Nauk SSSR Ser. Mat. 49 (1985), no. 3, 635–651 (Russian). MR 794958
Oswald Teichmüller, Erreichbare Randpunkte, Deutsche Math. 4 (1939), 455–461 (German). MR 49
Mina Teicher, Factorization of a birational morphism between $4$-folds, Math. Ann. 256 (1981), no. 3, 391–399. MR 626957, DOI 10.1007/BF01679705
Michael Thaddeus, Stable pairs, linear systems and the Verlinde formula, Invent. Math. 117 (1994), no. 2, 317–353. MR 1273268, DOI 10.1007/BF01232244
Michael Thaddeus, Geometric invariant theory and flips, J. Amer. Math. Soc. 9 (1996), no. 3, 691–723. MR 1333296, DOI 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 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 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

Toroidal Varieties and the Weak Factorization Theorem

math.AG/9904076

Oscar Zariski, Algebraic surfaces, Second supplemented edition, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], Band 61, Springer-Verlag, New York-Heidelberg, 1971. With appendices by S. S. Abhyankar, J. Lipman, and D. Mumford. MR 469915