On invariants of plane curve singularities in positive characteristic
Main Article Content
Abstract
In this survey paper we give an overview on some aspects of singularities of algebraic plane curves over an algebraically closed field of arbitrary characteristic. We review, in particular, classical results and recent developments on invariants of plane curve singularities.
Article Details
Keywords
Primary 14B05; Secondary 14J10
References
[1] V. I. Arnol’d, Normal forms for functions near degenerate critical points, the Weyl groups of Ak, Dk, Ek and Lagrangian singularities, Funct. Anal. Appl. 6 (1972), 254–272.
[2] V. I. Arnol’d, Classification of unimodal critical points of functions, Funct. Anal. Appl. 7 (1973), 230–231.
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[18] A. Hefez,J. H. O. Rodrigues, R. Salom˜ao, The Milnor number of plane branches with tame semigroups of values, Bull. Braz. Math. Soc. (N.S.) 49 (2018), no. 4, 789–809.
[19] J. Herzog, E. Kunz, Die Wertehalbgruppe eines lokalen Rings der Dimension 1, Springer (1971).
[20] H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II, Ann. of Math. (2) 79 (1964), 109–203; 79(1964), 205–326.
[21] A. G. Kouchnirenko, Poly`edres de Newton et nombres de Milnor, Invent. Math. 32 (1976), 1-31.
[22] I. Luengo, The µ-constant stratum is not smooth, Invent. Math., 90 (1987), 139–152.
[23] J. Milnor, Singular points of complex hypersurfaces, Princeton Univ. Press (1968).
[24] A. Melle-Hern´andez; C. T. C. Wall, Pencils of curves on smooth surfaces, Proc. Lond. Math. Soc., III. Ser. 83 (2001), no. 2, 257–278.
[25] H. D. Nguyen, Classification of singularities in positive characteristic, Ph.D. thesis, TU Kaiserslautern (2013). http://www.dr.hut-verlag.de/9783843911030.html
[26] H. D. Nguyen, Invariants of plane curve singularities and Pl¨ucker formulas in positive characteristic, Ann. Inst. Fourier (Grenoble) 66(2016), no.5, 2047–2066.
[27] A. Seidenberg, Elements of the Theory of Algebraic Curves. AddisonWesley Publishing Co., Reading, Mass.- London-Don Mills, Ont. (1968)
[28] R.J. Walker, Algebraic Curves, Dover Publ., New York, 1962.
[29] C. T. C. Wall, Newton polytopes and non-degeneracy, J. reine angew. Math. 509 (1999), 1–19.
[30] O. Zariski, Studies in equisingularity . I. Equivalent singularities of plane algebroid curve, Amer. J. Math. 87 (1960), 507–536.
[31] Zariski, O.; Samuel, P. Commutative Algebra, Vol. I, II, Springer (1960).
[2] V. I. Arnol’d, Classification of unimodal critical points of functions, Funct. Anal. Appl. 7 (1973), 230–231.
[3] V. I. Arnol’d, Local normal form of functions, Invent. Math. 35 (1976), 87–109.
[4] M. Artin, On isolated rational singularities of surfaces, Amer. J. Math. 88 (1966), 129–136.
[5] P. Beelen; R. Pellikaan, The Newton polygon of plane curves with many rational points, Designs, Codes and Cryptography 21(2000), 41-67.
[6] C. Bivi`a-Ausina, Local Lojasiewicz exponents, Milnor numbers and mixed multiplicities of ideals, Math. Z. 262 (2009), no. 2, 389–409.
[7] Y. Boubakri, G.-M. Greuel, and T. Markwig, Invariants of hypersurface singularities in positive characteristic, Rev. Mat. Complut.25(2012), no. 1, 61–85.
[8] E. Brieskorn; H. Kn¨orrer, Plane Algebraic Curves, Birkhaeuser (1986), 721 pages.
[9] A. Campillo, Algebroid Curves in Positive Characteristic, SLN 813, Springer-Verlag (1980).
[10] P. Cassou-Nogu`es, A. Ploski, Invariants of plane curve singularities and Newton diagrams, Univ. Iagel. Acta Math. 49 (2011), 9–34.
[11] P. Deligne, La formule de Milnor, S´em. G´eom. Alg´ebrique du Bois-Marie, 1967-1969, SGA 7 II, Lecture Notes in Math. 340, Expose XVI, (1973), 197–211.
[12] F. Delgado, The semigroup of values of a curve singularity with several branches, Manuscr. math. 59 (1987), 347–374.
[13] E. Garc´ıa Barroso and A. Ploski, An approach to plane algebroid branches, Rev. Mat. Complut. 28 (2015), no. 1, 227–252.
[14] E. Garc´ıa Barroso and A. Ploski, The Milnor number of plane irreducible singularities in positive characteristic. Bulletin of the London Mathematical Society (2016) 48 (1): 94-98.
[15] G.-M. Greuel, C. Lossen and E. Shustin, Introduction to Singularities and deformations, Math. Monographs, Springer-Verlag (2006).
[16] G.-M. Greuel, H. D. Nguyen, Some remarks on the planar Kouchnirenko’s theorem, Rev. Mat. Complut. 25 (2012), no. 2, 557–579.
[17] A. Hefez, Irreducible Plane Curve Singularities. Real and Complex Singularities. Lecture Notes in Pure and Appl. Math., vol. 232, pp. 1–120. Dekker, New York (2003).
[18] A. Hefez,J. H. O. Rodrigues, R. Salom˜ao, The Milnor number of plane branches with tame semigroups of values, Bull. Braz. Math. Soc. (N.S.) 49 (2018), no. 4, 789–809.
[19] J. Herzog, E. Kunz, Die Wertehalbgruppe eines lokalen Rings der Dimension 1, Springer (1971).
[20] H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II, Ann. of Math. (2) 79 (1964), 109–203; 79(1964), 205–326.
[21] A. G. Kouchnirenko, Poly`edres de Newton et nombres de Milnor, Invent. Math. 32 (1976), 1-31.
[22] I. Luengo, The µ-constant stratum is not smooth, Invent. Math., 90 (1987), 139–152.
[23] J. Milnor, Singular points of complex hypersurfaces, Princeton Univ. Press (1968).
[24] A. Melle-Hern´andez; C. T. C. Wall, Pencils of curves on smooth surfaces, Proc. Lond. Math. Soc., III. Ser. 83 (2001), no. 2, 257–278.
[25] H. D. Nguyen, Classification of singularities in positive characteristic, Ph.D. thesis, TU Kaiserslautern (2013). http://www.dr.hut-verlag.de/9783843911030.html
[26] H. D. Nguyen, Invariants of plane curve singularities and Pl¨ucker formulas in positive characteristic, Ann. Inst. Fourier (Grenoble) 66(2016), no.5, 2047–2066.
[27] A. Seidenberg, Elements of the Theory of Algebraic Curves. AddisonWesley Publishing Co., Reading, Mass.- London-Don Mills, Ont. (1968)
[28] R.J. Walker, Algebraic Curves, Dover Publ., New York, 1962.
[29] C. T. C. Wall, Newton polytopes and non-degeneracy, J. reine angew. Math. 509 (1999), 1–19.
[30] O. Zariski, Studies in equisingularity . I. Equivalent singularities of plane algebroid curve, Amer. J. Math. 87 (1960), 507–536.
[31] Zariski, O.; Samuel, P. Commutative Algebra, Vol. I, II, Springer (1960).