대한수학회지 (Journal of the Korean Mathematical Society)

  • 제40권5호
  • /
  • Pages.901-920
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  • 2003
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  • 0304-9914(pISSN)
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  • 2234-3008(eISSN)
대한수학회 (Korean Mathematical Society)
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THEORY OF INFINITELY NEAR SINGULAR POINTS

  • 발행 : 2003.09.01
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초록

The notion of infinitely near singular points, classical in the case of plane curves, has been generalized to higher dimensions in my earlier articles ([5], [6], [7]). There, some basic techniques were developed, notably the three technical theorems which were Differentiation Theorem, Numerical Exponent Theorem and Ambient Reduction Theorem [7]. In this paper, using those results, we will prove the Finite Presentation Theorem, which the auther believes is the first of the most important milestones in the general theory of infinitely near singular points. The presentation is in terms of a finitely generated graded algebra which describes the total aggregate of the trees of infinitely near singular points. The totality is a priori very complex and intricate, including all possible successions of permissible blowing-ups toward the reduction of singularities. The theorem will be proven for singular data on an ambient algebraic shceme, regular and of finite type over any perfect field of any characteristics. Very interesting but not yet apparent connections are expected with many such works as ([1], [8]).

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참고문헌

  1. Proc. Symp. Pure Appl. Math. v.40 Desingularization of plane curves S.Abhuankar
  2. Ann. Inst. Fourier v.1 no.50 Un example effectif de gradue non noetherien associe a une valuation divisorielle V.Cossart;C.Galindo;et O.Piltant https://doi.org/10.5802/aif.1748
  3. Math. Z v.137 Sur la theorie du contact maximal J.Giraud
  4. Ann. Sci. Ecole Norm. Sup. v.8 no.4 Contact maximal en caracteristique positive J.Giraud https://doi.org/10.24033/asens.1286
  5. Proc. Nordic Summer School in Math Gardening of infinitely near singularities H.Hironaka
  6. Consejo Superior de Investiga-ciones Cientificas no.28 Introduction to the theory of infinitely near singular points, Memorias deMatematica del Instituto Jorge Juan H.Hironaka
  7. Algebraic Geometry Idealistic exponents of singularity H.Hironaka
  8. Mem. Amer. Math. Soc. v.433 Newton Polyhedra without coordinates B.Youssin

피인용 문헌

  1. Resolution of singularities of threefolds in positive characteristic. I. vol.320, pp.3, 2008, https://doi.org/10.1016/j.jalgebra.2008.03.032
  2. On the use of naturality in algorithmic resolution vol.324, pp.8, 2010, https://doi.org/10.1016/j.jalgebra.2010.06.008
  3. Hypersurface singularities in positive characteristic vol.213, pp.2, 2007, https://doi.org/10.1016/j.aim.2007.01.006
  4. Resolution of singularities of threefolds in positive characteristic II vol.321, pp.7, 2009, https://doi.org/10.1016/j.jalgebra.2008.11.030
  5. Singularities in positive characteristic, stratification and simplification of the singular locus vol.224, pp.4, 2010, https://doi.org/10.1016/j.aim.2010.01.005
  6. Partial Local Resolution by Characteristic Zero Methods vol.73, pp.1, 2018, https://doi.org/10.1007/s00025-018-0808-7