The Hironaka theorem on resolution of singularities (Or: A proof we always wanted to understand)

Hauser, Herwig

doi:10.1090/S0273-0979-03-00982-0

The Hironaka theorem on resolution of singularities (Or: A proof we always wanted to understand)
HTML articles powered by AMS MathViewer

by Herwig Hauser PDF

Bull. Amer. Math. Soc. 40 (2003), 323-403 Request permission

Abstract:

This paper is a handyman’s manual for learning how to resolve the singularities of algebraic varieties defined over a field of characteristic zero by sequences of blowups.

References

Jose M. Aroca, Heisuke Hironaka, and José L. Vicente, The theory of the maximal contact, Memorias de Matemática del Instituto “Jorge Juan”, No. 29. [Mathematical Memoirs of the Jorge Juan Institute, No. 29], Instituto “Jorge Juan” de Matemáticas, Consejo Superior de Investigaciones Científicas, Madrid, 1975. MR 0444999
José M. Aroca, Heisuke Hironaka, and José L. Vicente, Desingularization theorems, Memorias de Matemática del Instituto “Jorge Juan” [Mathematical Memoirs of the Jorge Juan Institute], vol. 30, Consejo Superior de Investigaciones Científicas, Madrid, 1977. MR 0480502
Bravo, A., Encinas, S., Villamayor, O.: A simplified proof of desingularization and applications. Preprint. Download at http://arXiv.org/archive/math, math.AG/0206244.
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
Edward Bierstone and Pierre D. Milman, Uniformization of analytic spaces, J. Amer. Math. Soc. 2 (1989), no. 4, 801–836. MR 1001853, DOI 10.1090/S0894-0347-1989-1001853-2
Edward Bierstone and Pierre D. Milman, A simple constructive proof of canonical resolution of singularities, Effective methods in algebraic geometry (Castiglioncello, 1990) Progr. Math., vol. 94, Birkhäuser Boston, Boston, MA, 1991, pp. 11–30. MR 1106412, DOI 10.1007/978-1-4612-0441-1_{2}
Edward Bierstone and Pierre D. Milman, Semianalytic and subanalytic sets, Inst. Hautes Études Sci. Publ. Math. 67 (1988), 5–42. MR 972342, DOI 10.1007/BF02699126
Edward Bierstone and Pierre D. Milman, Resolution of singularities, Several complex variables (Berkeley, CA, 1995–1996) Math. Sci. Res. Inst. Publ., vol. 37, Cambridge Univ. Press, Cambridge, 1999, pp. 43–78. MR 1748600
Bierstone, E., Milman, P.: Desingularization algorithms I. The role of exceptional divisors. Preprint. Download at http://arXiv.org/archive/math, math.AG/0207098.
Bodnár, G.: Algorithmic resolution of singularities. Thesis, Univ. Linz, 2001.
Bodnár, G.: Computation of blowing up centers. J. Pure Appl. Alg. 179 (2003), 221–233.
Bodnár, G.: Algorithmic tests for the normal crossings property. Preprint, Univ. Linz.
Gábor Bodnár and Josef Schicho, Automated resolution of singularities for hypersurfaces, J. Symbolic Comput. 30 (2000), no. 4, 401–428. MR 1784750, DOI 10.1006/jsco.1999.0414
Gábor Bodnár and Josef Schicho, Two computational techniques for singularity resolution, J. Symbolic Comput. 32 (2001), no. 1-2, 39–54. Computer algebra and mechanized reasoning (St. Andrews, 2000). MR 1840384, DOI 10.1006/jsco.2001.0452
Gábor Bodnár and Josef Schicho, A computer program for the resolution of singularities, Resolution of singularities (Obergurgl, 1997) Progr. Math., vol. 181, Birkhäuser, Basel, 2000, pp. 231–238. MR 1748621, DOI 10.1007/978-3-0348-8399-3_{7}
A. Bravo and O. Villamayor U., Strengthening the theorem of embedded desingularization, Math. Res. Lett. 8 (2001), no. 1-2, 79–89. MR 1825262, DOI 10.4310/MRL.2001.v8.n1.a9
Bravo, A., Villamayor, O.: A strengthening of resolution of singularities in characteristic zero. Proc. London Math. Soc. 86 (2003), 327–357.
Encinas, S., Hauser, H.: Strong resolution of singularities in characteristic zero. Comment. Math. Helv. 77 (2002), 821–845.
S. Encinas and O. Villamayor, Good points and constructive resolution of singularities, Acta Math. 181 (1998), no. 1, 109–158. MR 1654779, DOI 10.1007/BF02392749
Santiago Encinas and Orlando Villamayor, A course on constructive desingularization and equivariance, Resolution of singularities (Obergurgl, 1997) Progr. Math., vol. 181, Birkhäuser, Basel, 2000, pp. 147–227. MR 1748620
Encinas, S., Villamayor, O.: A proof of desingularization over fields of characteristic zero. Preprint. Download at http://arXiv.org/archive/math, math.AG/0101208.
A. Grothendieck, Travaux de Heisouké Hironaka sur la résolution des singularités, Actes du Congrès International des Mathématiciens (Nice, 1970) Gauthier-Villars, Paris, 1971, pp. 7–9 (French). MR 0414283
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
Heisuke Hironaka, Idealistic exponents of singularity, Algebraic geometry (J. J. Sylvester Sympos., Johns Hopkins Univ., Baltimore, Md., 1976) Johns Hopkins Univ. Press, Baltimore, Md., 1977, pp. 52–125. MR 0498562
Heisuke Hironaka, Desingularization of complex-analytic varieties, Actes du Congrès International des Mathématiciens (Nice, 1970) Gauthier-Villars, Paris, 1971, pp. 627–631 (French). MR 0425170
Kunz, E.: Algebraische Geometrie IV. Vorlesung Univ. Regensburg.
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
Matsuki, K.: Notes on the inductive algorithm of resolution of singularities by S. Encinas and O. Villamayor. Preprint. Download at http://arXiv.org/archive/math, math.AG/0103120.
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
Villamayor, O.: Patching local uniformizations. Ann. Scient. Ec. Norm. Sup. Paris 25 (1992), 629-677.
Villamayor, O.: An introduction to the algorithm of resolution. In: Algebraic Geometry and Singularities (eds. A. Campillo, L. Narváez). Proc. Conf. on Singularities La Rábida. Birkhäuser, 1996.
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
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
S. S. Abhyankar, Resolution of singularities of embedded algebraic surfaces, 2nd ed., Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. MR 1617523, DOI 10.1007/978-3-662-03580-1
Shreeram Shankar Abhyankar, An algorithm on polynomials in one indeterminate with coefficients in a two dimensional regular local domain, Ann. Mat. Pura Appl. (4) 71 (1966), 25–59. MR 207699, DOI 10.1007/BF02413732
Abhyankar, S.: Three dimensional embedded uniformization in characteristic $p$. Lectures at Purdue Univeristy, Notes by M.F. Huang, 1968.
Shreeram S. Abhyankar, Good points of a hypersurface, Adv. in Math. 68 (1988), no. 2, 87–256. MR 934366, DOI 10.1016/0001-8708(88)90015-1
Abhyankar, S.: Current status of the resolution problem. Summer Institute on Algebraic Geometry 1964. Proc. Amer. Math. Soc.
Shreeram S. Abhyankar, Desingularization of plane curves, Singularities, Part 1 (Arcata, Calif., 1981) Proc. Sympos. Pure Math., vol. 40, Amer. Math. Soc., Providence, RI, 1983, pp. 1–45. MR 713043
Shreeram S. Abhyankar, Algebraic geometry for scientists and engineers, Mathematical Surveys and Monographs, vol. 35, American Mathematical Society, Providence, RI, 1990. MR 1075991, DOI 10.1090/surv/035
Shreeram Abhyankar, Ramification theoretic methods in algebraic geometry, Annals of Mathematics Studies, No. 43, Princeton University Press, Princeton, N.J., 1959. MR 0105416, DOI 10.1515/9781400881390
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
Shreeram Shankar Abhyankar, Resolution of singularities of algebraic surfaces, Algebraic Geometry (Internat. Colloq., Tata Inst. Fund. Res., Bombay, 1968) Oxford Univ. Press, London, 1969, pp. 1–11. MR 0257080
Shreeram S. Abhyankar, Resolution of singularities and modular Galois theory, Bull. Amer. Math. Soc. (N.S.) 38 (2001), no. 2, 131–169. MR 1816069, DOI 10.1090/S0273-0979-00-00892-2
D. Abramovich and A. J. de Jong, Smoothness, semistability, and toroidal geometry, J. Algebraic Geom. 6 (1997), no. 4, 789–801. MR 1487237
Shreeram S. Abhyankar and Tzuong Tsieng Moh, Newton-Puiseux expansion and generalized Tschirnhausen transformation. I, II, J. Reine Angew. Math. 260 (1973), 47–83; ibid. 261 (1973), 29–54. MR 337955, DOI 10.1515/crll.1973.260.47
Carlos Biggeri, On abscissas of convergence of the integrals of Laplace and of the Dirichlet series, An. Soc. Ci. Argentina 128 (1939), 65–70 (Spanish). MR 861
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
Brandenberg, M.: Aufblasungen affiner Varietäten. Thesis, Univ. Zürich, 1992.
Geir Ellingsrud and Manfred Lehn, Irreducibility of the punctual quotient scheme of a surface, Ark. Mat. 37 (1999), no. 2, 245–254. MR 1714770, DOI 10.1007/BF02412213
Egbert Brieskorn and Horst Knörrer, Ebene algebraische Kurven, Birkhäuser Verlag, Basel-Boston, Mass., 1981 (German). MR 646612
Bruce Michael Bennett, On the characteristic functions of a local ring, Ann. of Math. (2) 91 (1970), 25–87. MR 252388, DOI 10.2307/1970601
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
Egbert Brieskorn, Über die Auflösung gewisser Singularitäten von holomorphen Abbildungen, Math. Ann. 166 (1966), 76–102 (German). MR 206973, DOI 10.1007/BF01361440
Egbert Brieskorn, Die Auflösung der rationalen Singularitäten holomorpher Abbildungen, Math. Ann. 178 (1968), 255–270 (German). MR 233819, DOI 10.1007/BF01352140
A. Bravo and O. Villamayor U., Smoothness and tangent bundles of arithmetical schemes, Math. Z. 239 (2002), no. 1, 159–182. MR 1879334, DOI 10.1007/s002090100290
Antonio Campillo, Algebroid curves in positive characteristic, Lecture Notes in Mathematics, vol. 813, Springer, Berlin, 1980. MR 584440
Vincent Cossart, Jean Giraud, and Ulrich Orbanz, Resolution of surface singularities, Lecture Notes in Mathematics, vol. 1101, Springer-Verlag, Berlin, 1984. With an appendix by H. Hironaka. MR 775681, DOI 10.1007/BFb0072258
Vincent Cossart, Desingularization of embedded excellent surfaces, Tohoku Math. J. (2) 33 (1981), no. 1, 25–33. MR 613106, DOI 10.2748/tmj/1178229492
Cossart, V.: Polyèdre caractéristique d’une singularité. Thèse d’Etat, Orsay, 1987.
Vincent Cossart, Uniformisation et désingularisation des surfaces d’après Zariski, Resolution of singularities (Obergurgl, 1997) Progr. Math., vol. 181, Birkhäuser, Basel, 2000, pp. 239–258 (French). MR 1748622
V. Cossart, Contact maximal en caractéristique positive et petite multiplicité, Duke Math. J. 63 (1991), no. 1, 57–64 (French). MR 1106937, DOI 10.1215/S0012-7094-91-06303-9
Cossart, V.: Desingularization in dimension two. In: Resolution of surface singularities. Lecture Notes in Math. vol. 1101, Springer, 1984.
F. Cano and R. Piedra, Characteristic polygon of surface singularities, Géométrie algébrique et applications, II (La Rábida, 1984) Travaux en Cours, vol. 23, Hermann, Paris, 1987, pp. 27–42. MR 907920
Eduardo Casas-Alvero, Singularities of plane curves, London Mathematical Society Lecture Note Series, vol. 276, Cambridge University Press, Cambridge, 2000. MR 1782072, DOI 10.1017/CBO9780511569326
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, Geometric and combinatorial aspects of commutative algebra (Messina, 1999) Lecture Notes in Pure and Appl. Math., vol. 217, Dekker, New York, 2001, pp. 165–173. MR 1824226
Steven Dale Cutkosky, Local monomialization and factorization of morphisms, Astérisque 260 (1999), vi+143 (English, with English and French summaries). MR 1734239
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
Steven Dale Cutkosky, Simultaneous resolution of singularities, Proc. Amer. Math. Soc. 128 (2000), no. 7, 1905–1910. MR 1646312, DOI 10.1090/S0002-9939-99-05191-6
Steven Dale Cutkosky, When are the blow ups of two ideals the same scheme?, Comm. Algebra 19 (1991), no. 4, 1119–1124. MR 1102330, DOI 10.1080/00927879108824193
Steven Dale Cutkosky, Symbolic algebras of monomial primes, J. Reine Angew. Math. 416 (1991), 71–89. MR 1099946, DOI 10.1515/crll.1991.416.71
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
Encinas, S.: Constructive resolution of singularities of families of schemes. Thesis, Valladolid, 1996.
David Eisenbud and Joe Harris, The geometry of schemes, Graduate Texts in Mathematics, vol. 197, Springer-Verlag, New York, 2000. MR 1730819
Encinas, S., Nobile, A., Villamayor, O.: On algorithmic equiresolution and stratification of Hilbert schemes. Proc. London Math. Soc. 86 (2003), 607–648.
Jean Giraud, Étude locale des singularités, Publications Mathématiques d’Orsay, No. 26, Université Paris XI, U.E.R. Mathématique, Orsay, 1972. Cours de 3ème cycle, 1971–1972. MR 0409458
Jean Giraud, Sur la théorie du contact maximal, Math. Z. 137 (1974), 285–310 (French). MR 460712, DOI 10.1007/BF01214371
Jean Giraud, Contact maximal en caractéristique positive, Ann. Sci. École Norm. Sup. (4) 8 (1975), no. 2, 201–234 (French). MR 384799, DOI 10.24033/asens.1286
Giraud, J.: Desingularization in low dimension. In: Resolution of surface singularities. Lecture Notes in Math. vol. 1101, Springer, 1984, 51-78.
A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II, Inst. Hautes Études Sci. Publ. Math. 24 (1965), 231 (French). MR 199181
Rebecca Goldin and Bernard Teissier, Resolving singularities of plane analytic branches with one toric morphism, Resolution of singularities (Obergurgl, 1997) Progr. Math., vol. 181, Birkhäuser, Basel, 2000, pp. 315–340. MR 1748626
Hauser, H.: Three power series techniques. Proc. London Math. Soc., to appear.
Herwig Hauser, Seventeen obstacles for resolution of singularities, Singularities (Oberwolfach, 1996) Progr. Math., vol. 162, Birkhäuser, Basel, 1998, pp. 289–313. MR 1652479
Herwig Hauser, Excellent surfaces and their taut resolution, Resolution of singularities (Obergurgl, 1997) Progr. Math., vol. 181, Birkhäuser, Basel, 2000, pp. 341–373. MR 1748627
Hauser, H.: Mysterious resolution in positive characteristic. Preprint, Univ. Innsbruck.
Heisuke Hironaka, Characteristic polyhedra of singularities, J. Math. Kyoto Univ. 7 (1967), 251–293. MR 225779, DOI 10.1215/kjm/1250524227
Heisuke Hironaka, Certain numerical characters of singularities, J. Math. Kyoto Univ. 10 (1970), 151–187. MR 292831, DOI 10.1215/kjm/1250523817
Hironaka, H.: Desingularization of excellent surfaces. Notes by B. Bennett at the Conference on Algebraic Geometry, Bowdoin 1967. Reprinted in: Cossart, V., Giraud, J., Orbanz, U.: Resolution of surface singularities. Lecture Notes in Math. 1101, Springer, 1984.
Heisuke Hironaka, Additive groups associated with points of a projective space, Ann. of Math. (2) 92 (1970), 327–334. MR 269658, DOI 10.2307/1970839
Heisuke Hironaka, Schemes, etc, Algebraic geometry, Oslo 1970 (Proc. Fifth Nordic Summer School in Math.), Wolters-Noordhoff, Groningen, 1972, pp. 291–313. MR 0393028
H. Hironaka, On the presentations of resolution data, Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988) Johns Hopkins Univ. Press, Baltimore, MD, 1989, pp. 135–151. Noted and with an appendix by T. T. Moh. MR 1463700
Hironaka, H.: Bimeromorphic smoothing of a complex analytic space. Mimeographed notes. University of Warwick, 1971.
Heisuke Hironaka, Stratification and flatness, Real and complex singularities (Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Oslo, 1976) Sijthoff and Noordhoff, Alphen aan den Rijn, 1977, pp. 199–265. MR 0499286
Hironaka, H.: Preface to Zariski’s collected papers I, 307-312. MIT Press, 1979.
Heisuke Hironaka, Gardening of infinitely near singularities, Algebraic geometry, Oslo 1970 (Proc. Fifth Nordic Summer School in Math.), Wolters-Noordhoff, Groningen, 1972, pp. 315–332. MR 0393555
Heisuke Hironaka, On resolution of singularities (characteristic zero), Proc. Internat. Congr. Mathematicians (Stockholm, 1962) Inst. Mittag-Leffler, Djursholm, 1963, pp. 507–521. MR 0175898
M. Herrmann, S. Ikeda, and U. Orbanz, Equimultiplicity and blowing up, Springer-Verlag, Berlin, 1988. An algebraic study; With an appendix by B. Moonen. MR 954831, DOI 10.1007/978-3-642-61349-4
Jung, H.: Darstellung der Funktionen eines algebraischen Körpers zweier unabhängiger Veränderlicher $x$, $y$ in der Umgebung einer Stelle $x=a$, $y=b$. J. Reine Angew. Math. 133 (1908), 289-314.
Lê Dũng Tráng and Mutsuo Oka, On resolution complexity of plane curves, Kodai Math. J. 18 (1995), no. 1, 1–36. MR 1317003, DOI 10.2996/kmj/1138043350
Joseph Lipman, Desingularization of two-dimensional schemes, Ann. of Math. (2) 107 (1978), no. 1, 151–207. MR 491722, DOI 10.2307/1971141
Joseph Lipman, Rational singularities, with applications to algebraic surfaces and unique factorization, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 195–279. MR 276239, DOI 10.1007/BF02684604
Joseph Lipman, Topological invariants of quasi-ordinary singularities, Mem. Amer. Math. Soc. 74 (1988), no. 388, 1–107. MR 954947, DOI 10.1090/memo/0388
Joseph Lipman, On complete ideals in regular local rings, Algebraic geometry and commutative algebra, Vol. I, Kinokuniya, Tokyo, 1988, pp. 203–231. MR 977761
Lejeune, M., Teissier, B.: Contribution à l’étude des singularités du point de vue du polygone de Newton. Thèse d’Etat, Paris, 1973.
Moody, J.: The (still open) question about iterating Nash blowups. Preprint, Univ. Warwick.
John Atwell Moody, Divisibility of ideals and blowing up, Illinois J. Math. 45 (2001), no. 1, 163–165. MR 1849991
John Atwell Moody, On resolving singularities, J. London Math. Soc. (2) 64 (2001), no. 3, 548–564. MR 1865549, DOI 10.1112/S0024610701002745
John Atwell Moody, On resolving vector fields, J. Algebra 189 (1997), no. 1, 90–100. MR 1432366, DOI 10.1006/jabr.1996.6853
T. T. Moh, On a stability theorem for local uniformization in characteristic $p$, Publ. Res. Inst. Math. Sci. 23 (1987), no. 6, 965–973. MR 935710, DOI 10.2977/prims/1195175867
T. T. Moh, Quasi-canonical uniformization of hypersurface singularities of characteristic zero, Comm. Algebra 20 (1992), no. 11, 3207–3249. MR 1186705, DOI 10.1080/00927879208824512
T. T. Moh, On a Newton polygon approach to the uniformization of singularities of characteristic $p$, Algebraic geometry and singularities (La Rábida, 1991) Progr. Math., vol. 134, Birkhäuser, Basel, 1996, pp. 49–93. MR 1395176
S. B. Mulay, Equimultiplicity and hyperplanarity, Proc. Amer. Math. Soc. 89 (1983), no. 3, 407–413. MR 715854, DOI 10.1090/S0002-9939-1983-0715854-9
Muhly, H.T., Zariski, O.: The resolution of singularities of an algebraic curve. Amer. J. Math. 61 (1939), 107-114.
R. Narasimhan, Monomial equimultiple curves in positive characteristic, Proc. Amer. Math. Soc. 89 (1983), no. 3, 402–406. MR 715853, DOI 10.1090/S0002-9939-1983-0715853-7
R. Narasimhan, Hyperplanarity of the equimultiple locus, Proc. Amer. Math. Soc. 87 (1983), no. 3, 403–408. MR 684627, DOI 10.1090/S0002-9939-1983-0684627-8
Noether, M. Über die algebraischen Functionen einer und zweier Variabeln. Gött. Nachr. (1871), 267-278.
Tadao Oda, Infinitely very near singular points, Complex analytic singularities, Adv. Stud. Pure Math., vol. 8, North-Holland, Amsterdam, 1987, pp. 363–404. MR 894302, DOI 10.2969/aspm/00810363
Tadao Oda, Hironaka’s additive group scheme, Number theory, algebraic geometry and commutative algebra, in honor of Yasuo Akizuki, Kinokuniya, Tokyo, 1973, pp. 181–219. MR 0472824
Tadao Oda, Hironaka group schemes and resolution of singularities, Algebraic geometry (Tokyo/Kyoto, 1982) Lecture Notes in Math., vol. 1016, Springer, Berlin, 1983, pp. 295–312. MR 726431, DOI 10.1007/BFb0099968
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
Mutsuo Oka, Geometry of plane curves via toroidal resolution, Algebraic geometry and singularities (La Rábida, 1991) Progr. Math., vol. 134, Birkhäuser, Basel, 1996, pp. 95–121. MR 1395177
Vincent Cossart, Jean Giraud, and Ulrich Orbanz, Resolution of surface singularities, Lecture Notes in Mathematics, vol. 1101, Springer-Verlag, Berlin, 1984. With an appendix by H. Hironaka. MR 775681, DOI 10.1007/BFb0072258
Pfeifle, J.: Aufblasung monomialer Ideale. Diplomarbeit, Innsbruck, 1996.
Rosenberg, J.: Blowing up nonreduced toric subschemes of $\mathrm {A}^{n}$. Preprint, 1998.
Rodríguez Sánchez, C.: Good points and local resolution of threefold singularities. Thesis, Univ. Léon, 1998.
Jean-Pierre Serre, Algèbre locale. Multiplicités, Lecture Notes in Mathematics, vol. 11, Springer-Verlag, Berlin-New York, 1965 (French). Cours au Collège de France, 1957–1958, rédigé par Pierre Gabriel; Seconde édition, 1965. MR 0201468, DOI 10.1007/978-3-662-21576-0
Balwant Singh, Effect of a permissible blowing-up on the local Hilbert functions, Invent. Math. 26 (1974), 201–212. MR 352097, DOI 10.1007/BF01418949
Balwant Singh, Formal invariance of local characteristic functions, Seminar D. Eisenbud/B. Singh/W. Vogel, Vol. 1, Teubner-Texte zur Mathematik, vol. 29, Teubner, Leipzig, 1980, pp. 44–59. MR 610062
Mark Spivakovsky, A solution to Hironaka’s polyhedra game, Arithmetic and geometry, Vol. II, Progr. Math., vol. 36, Birkhäuser Boston, Boston, MA, 1983, pp. 419–432. MR 717618
Mark Spivakovsky, A counterexample to Hironaka’s “hard” polyhedra game, Publ. Res. Inst. Math. Sci. 18 (1982), no. 3, 1009–1012. MR 688941, DOI 10.2977/prims/1195183292
Mark Spivakovsky, A counterexample to the theorem of Beppo Levi in three dimensions, Invent. Math. 96 (1989), no. 1, 181–183. MR 981741, DOI 10.1007/BF01393974
Urabe, T.: Resolution of singularities of germs in characteristic positive associated with valuation rings of iterated divisor type. Preprint, 1999.
Villamayor, O.: On equiresolution and a question of Zariski. Acta Math. 185 (2000), no. 1, 123-159.
Villamayor, O.: Tschirnhausen transformations revisited and the multiplicity of the embedded hypersurface. Colloquium on Homology and Representation Theory (Vaquerías, 1998). Bol. Acad. Nac. Cienc. (Córdoba) 65 (2000), 233-243.
Walker, R.J.: Reduction of singularities of an algebraic surface. Ann. Math. 36 (1935), 336-365.
Morgan Ward and R. P. Dilworth, The lattice theory of ova, Ann. of Math. (2) 40 (1939), 600–608. MR 11, DOI 10.2307/1968944
J. Horn, Über eine hypergeometrische Funktion zweier Veränderlichen, Monatsh. Math. Phys. 47 (1939), 359–379 (German). MR 91, DOI 10.1007/BF01695508
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
J. J. Corliss, Upper limits to the real roots of a real algebraic equation, Amer. Math. Monthly 46 (1939), 334–338. MR 4, DOI 10.1080/00029890.1939.11998880
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
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
Zariski, O.: Exceptional singularities of an algebraic surface and their reduction. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., serie VIII, 43 (1967), 135-196.
Oscar Zariski, A new proof of the total embedded resolution theorem for algebraic surfaces (based on the theory of quasi-ordinary singularities), Amer. J. Math. 100 (1978), no. 2, 411–442. MR 489965, DOI 10.2307/2373856
Zariski, O.: Polynomial ideals defined by infinitely near base points. Amer. J. Math. 60 (1938), 151-204.
Oscar Zariski, Algebraic surfaces, Second supplemented edition, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 61, Springer-Verlag, New York-Heidelberg, 1971. With appendices by S. S. Abhyankar, J. Lipman, and D. Mumford. MR 0469915
Garrett Birkhoff and Morgan Ward, A characterization of Boolean algebras, Ann. of Math. (2) 40 (1939), 609–610. MR 9, DOI 10.2307/1968945
Sergio Sispanov, Generalización del teorema de Laguerre, Bol. Mat. 12 (1939), 113–117 (Spanish). MR 3
Oscar Zariski, Collected papers. Vol. I: Foundations of algebraic geometry and resolution of singularities, Mathematicians of Our Time, Vol. 2, MIT Press, Cambridge, Mass.-London, 1972. Edited by H. Hironaka and D. Mumford. MR 0505100
Oscar Zariski, Collected papers. Vol. IV, Mathematicians of Our Time, vol. 16, MIT Press, Cambridge, Mass.-London, 1979. Equisingularity on algebraic varieties; Edited and with an introduction by J. Lipman and B. Teissier. MR 545653
Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335