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Deformations of Singularities

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Introduction to Singularities
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Abstract

A singularity of dimension higher than 2 is called a higher-dimensional singularity. In this section we mostly discuss higher-dimensional singularities. Unless otherwise stated, singularities are always of dimension nā€‰ā‰„ā€‰2. Varieties are all integral algebraic varieties over \(\mathbb{C}\) and the singularities considered are on such varieties.

Inasmuch as the mathematical theorems are related to reality,

they are not sure;

inasmuch as they are sure, they are not related to reality

(Einstein [Mur, p. 120])

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References

  1. Artin, M.: Lecture on deformations of singularites. Tata Institute of Fund. Research. Springer, Bombay (1976)

    Google ScholarĀ 

  2. Behnke, K., Kahn, C., Riemenschneider, O.: Infinitesimal deformations of quotient surface singularities, in ā€œSingularitiesā€, vol. 20, pp. 31ā€“66. Banach Center Publication, Warsaw (1988)

    Google ScholarĀ 

  3. Behnke, K., Knƶrrer, H.: On infinitesimal deformations of rational surface singularities. Compos. Math. 61, 103ā€“127 (1987)

    MATHĀ  Google ScholarĀ 

  4. Birkar, C., Cascini, P., Hacon, C., McKernan, J.: Existence of minimal models for varieties of log general type. J. Am. Math. Soc. 23, 405ā€“468 (2010)

    ArticleĀ  MATHĀ  MathSciNetĀ  Google ScholarĀ 

  5. Bloch, A.: The Complete Murphyā€™s Law (revised version). Price Stern Sloan, Inc., New York (1991)

    Google ScholarĀ 

  6. Donin, I.F.: Complete families of deformations of germs of complex spaces. Math. USSR Sbornik 18, 397ā€“406 (1972)

    ArticleĀ  Google ScholarĀ 

  7. Elkik, R.: SingularitĆ©s rationelles et dĆ©formations. Invent. Math. 47, 139ā€“147 (1978)

    ArticleĀ  MATHĀ  MathSciNetĀ  Google ScholarĀ 

  8. Esnault, H., Viehweg, E.: Two dimensional quotient singularities deform to quotient singularities. Math. Ann. 271, 439ā€“449 (1985)

    ArticleĀ  MATHĀ  MathSciNetĀ  Google ScholarĀ 

  9. Grauert, H.: Ɯber die Deformation isolierter SingularitƤten analytischen Mengen. Invent. Math. 15, 171ā€“198 (1972)

    ArticleĀ  MATHĀ  MathSciNetĀ  Google ScholarĀ 

  10. Ishii, S.: Small deformations of normal singularities. Math. Ann. 275, 139ā€“148 (1986)

    ArticleĀ  MATHĀ  MathSciNetĀ  Google ScholarĀ 

  11. Ishii, S.: Simultaneous canonical modifications of deformations of isolated singularities. In: Algebraic Geometry and Analytic Geometry, Proceedings of the Satellite Conference of ICM 90, pp. 81ā€“100. Springer, New York (1991)

    Google ScholarĀ 

  12. Kas, A., Schlessinger, M.: On the versal deformation of a complex space with an isolated singularity. Math. Ann. 196, 23ā€“29 (1972)

    ArticleĀ  MATHĀ  MathSciNetĀ  Google ScholarĀ 

  13. Kawakita, M.: Inversion of adjunction on log canonicity. Invent. Math. 167, 129ā€“133 (2007)

    ArticleĀ  MATHĀ  MathSciNetĀ  Google ScholarĀ 

  14. Kawamata, Y.: Deformations of canonical singularities. J. Am. Math. Soc. 12(1), 85ā€“92 (1999)

    ArticleĀ  MATHĀ  MathSciNetĀ  Google ScholarĀ 

  15. Kawamata, Y., Matsuda, K., Matsuki, K.: Introduction to the minimal model problem. In: Algebraic Geometry in Sendai 1985, edited by Oda, Advanced Studies in Pure Mathematics, vol. 10, pp. 283ā€“360. Kinokuniya, Tokyo/North-Holland/Amsterdam/New York/Oxford (1987)

    Google ScholarĀ 

  16. KollĆ”r, J., Shepherd-Barron, N.: Threefolds and deformations of surface singularities. Invent. Math. 91, 299ā€“338 (1988)

    ArticleĀ  MATHĀ  MathSciNetĀ  Google ScholarĀ 

  17. KovƔcs, S., Schwede, K.: Du Bois singularities deform. Adv. Stud. Pure Math. (to appear)

    Google ScholarĀ 

  18. Lipman, J.: Rings with discrete divisor class group: theorem of Danilov-Samuel. Am. J. Math. 101, 203ā€“211 (1979)

    ArticleĀ  MATHĀ  MathSciNetĀ  Google ScholarĀ 

  19. Morrow, J., Kodaira, K.: Complex Manifolds. Holt, Rinehart and Winston, Inc., New York (1971)

    MATHĀ  Google ScholarĀ 

  20. Mumford, D.: Red Book of Varieties and Schemes. Lecture Note in Mathematics, vol. 1358. Springer, New York (1988)

    Google ScholarĀ 

  21. Nakayama, N.: Invariance of the plurigenera of algebraic varieties under minimal model conjecture. Topology 25(2), 237ā€“251 (1986)

    ArticleĀ  MATHĀ  MathSciNetĀ  Google ScholarĀ 

  22. Pinkham, H.C.: Deformation of algebraic varieties with G m action. AstƩrisque 20, 271-301 (1974)

    MathSciNetĀ  Google ScholarĀ 

  23. Riemenschneider, O.: Dihedral singularities: invariants, equations and infinitesimal deformations. Bull. Am. Math. Soc. 82, 725ā€“747 (1976)

    ArticleĀ  Google ScholarĀ 

  24. Schlessinger, M.: Rigidity of quotient singularities. Invent. Math. 14, 17ā€“26 (1971)

    ArticleĀ  MATHĀ  MathSciNetĀ  Google ScholarĀ 

  25. Tjurina, G.N.: Locally semi-universal flat deformations of isolated singularities of complex spaces. Math. USSR. Izv. 3, 976ā€“1000 (1969)

    ArticleĀ  Google ScholarĀ 

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Ishii, S. (2014). Deformations of Singularities. In: Introduction to Singularities. Springer, Tokyo. https://doi.org/10.1007/978-4-431-55081-5_9

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