Alterations can remove singularities

Oort, Frans

doi:10.1090/S0273-0979-98-00764-2

Alterations can remove singularities
HTML articles powered by AMS MathViewer

by Frans Oort PDF

Bull. Amer. Math. Soc. 35 (1998), 319-331 Request permission

Abstract:

This note is a review of: A. J. de Jong - Smoothness, semi-stability and alterations. Manuscript 1995, 39 pp.: Publications Mathématiques I.H.E.S. 83, 1996, pp. 51-93.

References

Shreeram S. Abhyankar, Resolution of singularities of arithmetical surfaces, Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ., 1963) Harper & Row, New York, 1965, pp. 111–152. MR 0200272
Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
D. Abramovich & A. J. de Jong - Smoothness, semistability, and toroidal geometry, J. Algebraic Geom. 6 (1997), 789-801.
D. Abramovich & F. Oort - Alterations and resolution of singularities. Notes: Working week on resolution of singularities. September 1997, Obergurgl, Austria. To be published in the proceedings of that conference.
P. Berthelot - Altérations de variétés algébriques [d’après A. J. de Jong], Sém. Bourbaki 1995-96, Exp. 815.
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
Fedor A. Bogomolov and Tony G. Pantev, Weak Hironaka theorem, Math. Res. Lett. 3 (1996), no. 3, 299–307. MR 1397679, DOI 10.4310/MRL.1996.v3.n3.a1
Jean-Luc Brylinski, Propriétés de ramification à l’infini du groupe modulaire de Teichmüller, Ann. Sci. École Norm. Sup. (4) 12 (1979), no. 3, 295–333 (French). With an appendix in English by Ken Baclawski. MR 559345, DOI 10.24033/asens.1368
P. Deligne, Le lemme de Gabber, In: Sém. sur les pinceaux arithmétiques: la conjecture de Mordell (Ed. L.. Szpiro). Astérisque 127 (1985), 131-150.
P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 75–109. MR 262240, DOI 10.1007/BF02684599
B. van Geemen and F. Oort, A compactification of a fine moduli space of curves, University of Utrecht, Preprint 301, 1983. To be published in the Proceedings of Resolutions of Singularities, September 1997, Obergurgl, Austria.
Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157, DOI 10.1007/978-1-4757-3849-0
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 10.2307/1970547
A. J. de Jong - Smoothness, semi-stability and alterations, Preprint 916, Utrecht, June 1995, 32 pp.
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
A. Johan de Jong, Families of curves and alterations, Ann. Inst. Fourier (Grenoble) 47 (1997), no. 2, 599–621. MR 1450427, DOI 10.5802/aif.1575
A. J. de Jong & F. Oort - On extending families of curves, Journ. Algebr. Geom. 6 (1997), 545 - 562.
Finn F. Knudsen, The projectivity of the moduli space of stable curves. II. The stacks $M_{g,n}$, Math. Scand. 52 (1983), no. 2, 161–199. MR 702953, DOI 10.7146/math.scand.a-12001
Serge Lang, Introduction to algebraic geometry, Interscience Publishers, Inc., New York-London, 1958. MR 0100591
Joseph Lipman, Introduction to resolution of singularities, Algebraic geometry (Proc. Sympos. Pure Math., Vol. 29, Humboldt State Univ., Arcata, Calif., 1974) Amer. Math. Soc., Providence, R.I., 1975, pp. 187–230. MR 0389901
M. Mostafa, Die Singularitäten de Modulmannigfaltigkeiten $\overline {M}_g(n)$ der stabilen Kurven vom Geslecht $g \geq 2$ mit $n$-Teilungspunkstruktur, Journ. reine angew. Math. (Crelle), 343 (1983), 81–98.
David Mumford, The red book of varieties and schemes, Lecture Notes in Mathematics, vol. 1358, Springer-Verlag, Berlin, 1988. MR 971985, DOI 10.1007/978-3-662-21581-4
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
Igor R. Shafarevich, Basic algebraic geometry. 1, 2nd ed., Springer-Verlag, Berlin, 1994. Varieties in projective space; Translated from the 1988 Russian edition and with notes by Miles Reid. MR 1328833
M. Spivakovsky - Resolution of singularities I: Local Uniformization, Manuscript, September 1996, 96 pp.
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
Garrett Birkhoff and Morgan Ward, A characterization of Boolean algebras, Ann. of Math. (2) 40 (1939), 609–610. MR 9, DOI 10.2307/1968945
T. Venkatarayudu, The $7$-$15$ problem, Proc. Indian Acad. Sci., Sect. A. 9 (1939), 531. MR 0000001, DOI 10.1090/gsm/058
Leonard Eugene Dickson, New First Course in the Theory of Equations, John Wiley & Sons, Inc., New York, 1939. MR 0000002
Albert Eagle, Series for all the roots of the equation $(z-a)^m=k(z-b)^n$, Amer. Math. Monthly 46 (1939), 425–428. MR 6, DOI 10.2307/2303037