대한수학회지 (Journal of the Korean Mathematical Society)

  • 제53권2호
  • /
  • Pages.461-473
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  • 2016
  • /
  • 0304-9914(pISSN)
  • /
  • 2234-3008(eISSN)
대한수학회 (Korean Mathematical Society)
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A BOUND FOR THE MILNOR SUM OF PROJECTIVE PLANE CURVES IN TERMS OF GIT

  • 투고 : 2015.03.10
  • 발행 : 2016.03.01
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초록

Let C be a projective plane curve of degree d whose singularities are all isolated. Suppose C is not concurrent lines. P loski proved that the Milnor number of an isolated singlar point of C is less than or equal to $(d-1)^2-{\lfloor}\frac{d}{2}{\rfloor}$. In this paper, we prove that the Milnor sum of C is also less than or equal to $(d-1)^2-{\lfloor}\frac{d}{2}{\rfloor}$ and the equality holds if and only if C is a P loski curve. Furthermore, we find a bound for the Milnor sum of projective plane curves in terms of GIT.

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참고문헌

  1. I. Cheltsov, Worst singularities of plane curves of given degree, preprint available at http://arxiv.org/abs/1409.6186.
  2. I. Dolgachev, Lectures on Invariant Theory, LMS Lecture Note Series 296, Cambridge University Press, Cambridge, 2003.
  3. T. Fassarella and N. Medeiros, On the polar degree of projective hypersurfaces, J. Lond. Math. Soc. (2) 86 (2012), no. 1, 259-271. https://doi.org/10.1112/jlms/jds005
  4. G.-M. Greuel, C. Lossen, and E. Shustin, Introduction to singularities and deformations, Springer Monographs in Mathematics, Springer, Berlin, 2007.
  5. J. Huh, Milnor numbers of projective hypersurfaces with isolated singularities, Duke Math. J. 163 (2014), no. 8, 1525-1548. https://doi.org/10.1215/00127094-2713700
  6. A. P loski, A bound for the Milnor number of plane curve singularities, Cent. Eur. J. Math. 12 (2014), no. 5, 688-693.