Abstract
In this chapter we consider normal singularities of two-dimensional varieties over \(\mathbb{C}\). A two-dimensional integral algebraic variety is called a surface. A normal singularity on a surface is an isolated singularity and by Corollary 3.5.17 it is a Cohen–Macauley singularity.
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References
Artin, M.: Some numerical criteria for contractability of curves on algebraic surfaces. Am. J. Math. 84, 485–496 (1962)
Artin, M.: On isolated rational singularities of surfaces. Am. J. Math. 88, 129–136 (1966)
Bloch, A.: The Complete Murphy’s Law (revised version). Price Stern Sloan, Inc., New York (1991)
Brieskorn, E.: Rationale Singularitäten komplexer Flächen. Invent. Math. 4, 336–358 (1968)
Chevalley, C.: Invariants of finite groups generated by reflections. Am. J. Math. 77, 778–782 (1955)
Durfee, A.: Fifteen characterizations of rational double points and simple critical points. L’Enseignement XXV 1–2, 131–163 (1979)
Fossum, R.: The Divisor Class Groups of Krull Domain, Ergebnisse der Math, u. Ihrer Grenz., vol. 74. Springer (1973)
Grauert, H.: Über Modifikationen und exzeptionelle analytischen Mengen. Math. Ann. 146, 331–368 (1962)
Hartshorne, R.: Algebraic Geometry. Graduate Texts in Mathematics, vol. 52. Springer, Berlin (1977)
Holmann, H.: Quotienten komplexer Räume. Math. Ann. 142, 407–440 (1961)
Ishii, S.: Du Bois Singularities on a Normal Surface. Advanced Study in Pure Mathematics, Complex Analytic Singularities, vol. 8, pp. 153–163 (1986)
Ishii, S.: Two dimensional singularities with bounded pluri-genera δ m are Q-Gorenstein singularities. In: Proceedings of Symposium of Singularities, Iowa 1986, Contemporary Mathematics, vol. 90, pp. 135–145 (1989)
Klein, F.: Vorlesungen über das Ikosaeder und die Auflösung der Gleihungen vom fünften Grade. Teubner, Leibzig (1884)
Laufer, H.B.: Taut two dimensional singularities. Math. Ann. 205, 131–164 (1973)
Laufer, H.B.: On minimally elliptic singularities. Am. J. Math. 99, 1257–1295 (1977)
Lichtenbaum, S.: Curves over discrete valuation rings. Doctoral dissertation, Harvard (1964)
Matsumura, H.: Commutative Ring Theory. Cambridge Studies in Advanced Mathematics, vol. 8. Cambridge University Press, Cambridge (1986)
Mumford, D., Forgaty, J., Kirwan, F.: Geometric Invariant Theory. Ergebnisse der Math. u. ihrer Grenz. vol. 34. Springer, New York (1994)
Riemenschneider, O.: Deformationen von Quotientensingularitäten (nach zyklischen Gruppen). Math. Ann. 209, 211–248 (1974)
Saito, K.: Einfach elliptische Singularitäten. Univ. Math. 23, 289–325 (1974)
Serre, J-P.: Sur la cohomologie des variétés algebriques. J. de Math. Pures et Appl. 36, 1–16 (1957)
Springer, T.A.: Invariant Theory. Lecture Notes in Mathematics, vol. 585. Springer, New York (1977)
Watanabe, K-i.: Certain invariant subrings are Gorenstein, I, II. Osaka J. Math. 11(1–8), 379–388 (1974)
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Ishii, S. (2014). Normal Two-dimensional Singularities. In: Introduction to Singularities. Springer, Tokyo. https://doi.org/10.1007/978-4-431-55081-5_7
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DOI: https://doi.org/10.1007/978-4-431-55081-5_7
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