Algebra & Number Theory

Functorial factorization of birational maps for qe schemes in characteristic 0

Dan Abramovich and Michael Temkin

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We prove functorial weak factorization of projective birational morphisms of regular quasiexcellent schemes in characteristic 0 broadly based on the existing line of proof for varieties. From this general functorial statement we deduce factorization results for algebraic stacks, formal schemes, complex analytic germs, Berkovich analytic and rigid analytic spaces, answering a present need in nonarchimedean geometry. Techniques developed for this purpose include a method for functorial factorization of toric maps, variation of GIT quotients relative to general noetherian qe schemes, and a GAGA theorem for Stein compacts.

Article information

Algebra Number Theory, Volume 13, Number 2 (2019), 379-424.

Received: 20 December 2017
Revised: 23 November 2018
Accepted: 4 January 2019
First available in Project Euclid: 26 March 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14E05: Rational and birational maps
Secondary: 14A20: Generalizations (algebraic spaces, stacks) 14E15: Global theory and resolution of singularities [See also 14B05, 32S20, 32S45] 14L30: Group actions on varieties or schemes (quotients) [See also 13A50, 14L24, 14M17] 32H04: Meromorphic mappings

birational geometry blowing up bimeromorphic maps


Abramovich, Dan; Temkin, Michael. Functorial factorization of birational maps for qe schemes in characteristic 0. Algebra Number Theory 13 (2019), no. 2, 379--424. doi:10.2140/ant.2019.13.379.

Export citation


  • D. Abramovich and M. Temkin, “Torification of diagonalizable group actions on toroidal schemes”, J. Algebra 472 (2017), 279–338.
  • D. Abramovich and M. Temkin, “Luna's fundamental lemma for diagonalizable groups”, Algebr. Geom. 5:1 (2018), 77–113.
  • 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. $(2)$ 51:4 (1999), 489–537.
  • D. Abramovich, K. Karu, K. Matsuki, and J. Włodarczyk, “Torification and factorization of birational maps”, J. Amer. Math. Soc. 15:3 (2002), 531–572.
  • D. Abramovich, L. Caporaso, and S. Payne, “The tropicalization of the moduli space of curves”, Ann. Sci. Éc. Norm. Supér. $(4)$ 48:4 (2015), 765–809.
  • M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley, Reading, MA, 1969.
  • V. G. Berkovich, “Étale cohomology for non-Archimedean analytic spaces”, Inst. Hautes Études Sci. Publ. Math. 78 (1993), 5–161.
  • E. Bierstone and P. D. Milman, “Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant”, Invent. Math. 128:2 (1997), 207–302.
  • L. Borisov and A. Libgober, “McKay correspondence for elliptic genera”, Ann. of Math. $(2)$ 161:3 (2005), 1521–1569.
  • M. Brion and C. Procesi, “Action d'un tore dans une variété projective”, pp. 509–539 in Operator algebras, unitary representations, enveloping algebras, and invariant theory (Paris, 1989), edited by A. Connes et al., Progr. Math. 92, Birkhäuser, Boston, 1990.
  • F. Buonerba, “Functorial resolution of tame quotient singularities in positive characteristic”, preprint, 2015.
  • B. Conrad, “Relative ampleness in rigid geometry”, Ann. Inst. Fourier $($Grenoble$)$ 56:4 (2006), 1049–1126.
  • V. Cossart and O. Piltant, “Resolution of singularities of arithmetical threefolds, II”, preprint, 2014.
  • V. Cossart, U. Jannsen, and S. Saito, “Canonical embedded and non-embedded resolution of singularities for excellent two-dimensional schemes”, preprint, 2009.
  • I. V. Dolgachev and Y. Hu, “Variation of geometric invariant theory quotients”, Inst. Hautes Études Sci. Publ. Math. 87 (1998), 5–56.
  • A. Ducros, “Les espaces de Berkovich sont excellents”, Ann. Inst. Fourier $($Grenoble$)$ 59:4 (2009), 1443–1552.
  • A. Ducros, Families of Berkovich spaces, Astérisque 400, Soc. Math. France, Paris, 2018.
  • A. Grothendieck, “Eléments de géométrie algébrique, II: Étude globale élémentaire de quelques classes de morphismes”, Inst. Hautes Études Sci. Publ. Math. 8 (1961), 5–222.
  • A. Grothendieck, “Eléments de géométrie algébrique, III: Étude cohomologique des faisceaux cohérents, I”, Inst. Hautes Études Sci. Publ. Math. 11 (1961), 5–167.
  • A. Grothendieck, “Eléments de géométrie algébrique, IV: Étude locale des schémas et des morphismes de schémas, II”, Inst. Hautes Études Sci. Publ. Math. 24 (1965), 5–231.
  • A. Grothendieck, “Eléments de géométrie algébrique, IV: Étude locale des schémas et des morphismes de schémas, III”, Inst. Hautes Études Sci. Publ. Math. 28 (1966), 5–255.
  • J. Frisch, “Points de platitude d'un morphisme d'espaces analytiques complexes”, Invent. Math. 4 (1967), 118–138.
  • H. Gillet and C. Soulé, “Direct images in non-Archimedean Arakelov theory”, Ann. Inst. Fourier $($Grenoble$)$ 50:2 (2000), 363–399.
  • H. Grauert and R. Remmert, Theory of Stein spaces, Grundlehren der Math. Wissenschaften 236, Springer, 1979.
  • R. Hartshorne, Algebraic geometry, Graduate Texts in Math. 52, Springer, 1977.
  • H. Hironaka, “Flattening theorem in complex-analytic geometry”, Amer. J. Math. 97 (1975), 503–547.
  • L. Illusie and M. Temkin, “Gabber's modification theorem (absolute case)”, pp. [exposé] VIII, pp. 103–160 in Travaux de Gabber sur l'uniformisation locale et la cohomologie étale des schémas quasi-excellents (Palaiseau, 2006–2008), Astérisque 363-364, Soc. Math. France, Paris, 2014.
  • B. Iversen, Cohomology of sheaves, Springer, 1986.
  • K. Kato, “Toric singularities”, Amer. J. Math. 116:5 (1994), 1073–1099.
  • K. Kedlaya, “Algebraic geometry: GAGA”, course notes, Massachusetts Inst. Tech., 2009,
  • G. Kempf, F. F. Knudsen, D. Mumford, and B. Saint-Donat, Toroidal embeddings, I, Lecture Notes in Math. 339, Springer, 1973.
  • J. Kollár, Lectures on resolution of singularities, Ann. of Math. Studies 166, Princeton Univ. Press, 2007.
  • M. Kontsevich and Y. Soibelman, “Affine structures and non-Archimedean analytic spaces”, pp. 321–385 in The unity of mathematics (Cambridge, 2003), edited by P. Etingof et al., Progr. Math. 244, Birkhäuser, Boston, 2006.
  • U. Köpf, Über eigentliche Familien algebraischer Varietäten über affinoiden Räumen, Schr. Math. Inst. Univ. Münster (2) 7, 1974.
  • H. Matsumura, Commutative algebra, 2nd ed., Math. Lect. Note Ser. 56, Benjamin/Cummings, Reading, MA, 1980.
  • R. Morelli, “The birational geometry of toric varieties”, J. Algebraic Geom. 5:4 (1996), 751–782.
  • D. Mumford, J. Fogarty, and F. Kirwan, Geometric invariant theory, 3rd ed., Ergebnisse der Mathematik $($2$)$ 34, Springer, 1994.
  • B. Noohi, “Foundations of topological stacks, I”, preprint, 2005.
  • J. Poineau, “Raccord sur les espaces de Berkovich”, Algebra Number Theory 4:3 (2010), 297–334.
  • D. Popescu, “General Néron desingularization and approximation”, Nagoya Math. J. 104 (1986), 85–115.
  • M. Porta and T. Y. Yu, “Higher analytic stacks and GAGA theorems”, Adv. Math. 302 (2016), 351–409.
  • J.-P. Serre, “Géométrie algébrique et géométrie analytique”, Ann. Inst. Fourier $($Grenoble$)$ 6 (1956), 1–42.
  • M. Baker and S. Payne (editors), “Simons Symposium 2015: open problem sessions”, compilation of submitted problems, 2015,
  • C. Simpson, “Algebraic (geometric) $n$-stacks”, preprint, 1996.
  • M. Spivakovsky, “A new proof of D. Popescu's theorem on smoothing of ring homomorphisms”, J. Amer. Math. Soc. 12:2 (1999), 381–444.
  • M. Temkin, “Desingularization of quasi-excellent schemes in characteristic zero”, Adv. Math. 219:2 (2008), 488–522.
  • M. Temkin, “Functorial desingularization of quasi-excellent schemes in characteristic zero: the nonembedded case”, Duke Math. J. 161:11 (2012), 2207–2254.
  • M. Temkin, “Functorial desingularization over $\mathbb{Q}$: boundaries and the embedded case”, Israel J. Math. 224:1 (2018), 455–504.
  • M. Thaddeus, “Geometric invariant theory and flips”, J. Amer. Math. Soc. 9:3 (1996), 691–723.
  • M. Ulirsch, Tropical geometry of logarithmic schemes, Ph.D. thesis, Brown University, 2015,
  • J. Włodarczyk, “Birational cobordisms and factorization of birational maps”, J. Algebraic Geom. 9:3 (2000), 425–449.
  • J. Włodarczyk, “Toroidal varieties and the weak factorization theorem”, Invent. Math. 154:2 (2003), 223–331.
  • J. Włodarczyk, “Algebraic Morse theory and the weak factorization theorem”, pp. 653–682 in International Congress of Mathematicians, II (Madrid, 2006), edited by M. Sanz-Solé et al., Eur. Math. Soc., Zürich, 2006.
  • J. Włodarczyk, “Simple constructive weak factorization”, pp. 957–1004 in Algebraic geometry, II (Seattle, 2005), edited by D. Abramovich et al., Proc. Sympos. Pure Math. 80, Part 2, Amer. Math. Soc., Providence, RI, 2009.
  • T. Y. Yu, “Gromov compactness in non-Archimedean analytic geometry”, J. Reine Angew. Math. 741 (2018), 179–210.