References
S. S. Abhyankar, On the valuations centered in a local domain,Amer. J. Math.,78 (1956), 321–348.
——, Resolution of singularities of arithmetical surfaces, pp. 111–152, inArithmetical Algebraic Geometry, New York, Harper and Row, 1965 (edited byO. F. G. Schlling).
M. Artin, Some numerical criteria for contractability of curves on algebraic surfaces,Amer. J. Math.,84 (1962), 485–496.
——, On isolated rational singularities of surfaces,Amer. J. Math.,88 (1966), 129–136.
N. Bourbaki, Algèbre commutative, chap. 5–6,Act. Sci. et Ind., no 1308, Paris, Humann, 1964.
E. Brieskorn, Über die Auflösung gewisser Singularitäten von holomorphen Abbildungen,Math. Annalen,166 (1966), 76–102.
——, Rationale Singularitäten komplexer Flächen,Inventiones Math.,4 (1968), 336–358.
P. du Val, On isolated singularities of surfaces which do not affect the conditions of adjunction (Part I),Proc. Cambridge Phil. Soc.,30 (1934), 453–459.
(cited [EGA…]),A. Grothendieck andJ. Dieudonné, Éléments de Géométrie algébrique,Publ. Math. Inst. Hautes Études Sci., no8 4, 8, …, 32 (1960, …, 1967).
H. Hironaka, Desingularization of excellent surfaces,Advanced Science Seminar in Algebraic Geometry, Bowdoin College, Brunswick, Maine, 1967.
--, Forthcoming paper on desingularization of excellent surfaces,J. Math. Kyoto Univ.
S. Kleiman, Toward a numerical theory of ampleness,Ann. of Math.,84 (1966), 293–344.
W. Krull, Beiträge zur Arithmetik kommutativer Integritätsbereiche,Math. Z.,41 (1936), 545–577.
S. Lichtenbaum, Curves over discrete valuation rings,Amer. J. Math.,90 (1968), 380–405.
H. T. Muhly andM. Sakuma, Some multiplicative properties of complete ideals,Trans. Amer. Math. Soc.,106 (1963), 210–221.
——, Asymptotic Factorization of Ideals,J. London Math. Soc.,38 (1963), 341–350.
D. Mumford, The topology of normal singularities of an algebraic surface,Publ. Math. Inst. Hautes Études Sci., no 9, 1961.
--, Lectures on Curves on an Algebraic Surface,Ann. of Math. Studies, no 59, Princeton, 1966.
M. Nagata,Local Rings, Interscience, New York, 1962.
F. Oort,Reducible and Multiple Algebraic Curves, Assen, Van Gorcum, 1961.
G. Scheja, Einige Beispiele faktorieller lokaler Ringe,Math. Annalen,172 (1967), 124–134.
I. R. Shafarevich, Lectures on Minimal Models and Birational Transformations of Two-dimensional Schemes, Tata Institute of Fundamental Research,Lectures on Mathematics, no 37, Bombay, 1966.
O. Zariski, Polynomial ideals defined by infinitely near base points,Ann. J. Math.,60 (1937), 151–204.
——, The reduction of the singularities of an algebraic surface,Ann. of Math.,40 (1939), 639–689.
——, The reduction of the singularities of an algebraic variety,Trans. Amer. Math. Soc.,62 (1947), 1–52.
--, Introduction to the Problem of Minimal Models in the Theory of Algebraic Surfaces,Publ. Math. Societ. of Japan, no 4, Tokyo, 1958.
——, andP. Samuel,Commutative Algebra, vol. 2, Princeton, Van Nostrand, 1960.
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Lipman, J. Rational singularities, with applications to algebraic surfaces and unique factorization. Publications Mathématiques de l’Institut des Hautes Scientifiques 36, 195–279 (1969). https://doi.org/10.1007/BF02684604
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DOI: https://doi.org/10.1007/BF02684604