A bound for the Milnor sum of projective plane curves in terms of GIT
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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.
Abstract
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.
참고문헌 (6)
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- G.-M. Greuel, "Introduction to singularities and deformations" (2007), Springer Monographs in Mathematics
- J. Huh, "Milnor numbers of projective hypersurfaces with isolated singularities", Duke Math. J, 163(8), pp.1525~1548 (2014)
- A. P loski, "A bound for the Milnor number of plane curve singularities", Cent. Eur. J. Math, 12(5), pp.688~693 (2014)