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Let $C$ be a projective plane curve of degree $d$ whose singularities are all isolated. Suppose $C$ is not concurrent lines. P\l oski 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\l oski curve. Furthermore, we find a bound for the Milnor sum of projective plane curves in terms of GIT.
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참고문헌 (6)
참고문헌 신청
I. Cheltsov / Worst singularities of plane curves of given degree
I. Dolgachev / 2003 / Lectures on Invariant Theory / Cambridge University Press
T. Fassarella / 2012 / On the polar degree of projective hypersurfaces / J. Lond. Math. Soc 86(1):259~271
G.-M. Greuel / 2007 / Introduction to singularities and deformations / Springer Monographs in Mathematics
J. Huh / 2014 / Milnor numbers of projective hypersurfaces with isolated singularities / Duke Math. J 163(8):1525~1548
I. Dolgachev / 2003 / Lectures on Invariant Theory / Cambridge University Press
T. Fassarella / 2012 / On the polar degree of projective hypersurfaces / J. Lond. Math. Soc 86(1):259~271
G.-M. Greuel / 2007 / Introduction to singularities and deformations / Springer Monographs in Mathematics
J. Huh / 2014 / Milnor numbers of projective hypersurfaces with isolated singularities / Duke Math. J 163(8):1525~1548
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