A BOUND FOR THE MILNOR SUM OF PROJECTIVE PLANE CURVES IN TERMS OF GIT

Abstract

Abstract. Let C be a projective plane curve of degree d whose singu-larities are all isolated. Suppose C is not concurrent lines. P loski provedthat the Milnor number of an isolated singlar point of C is less than orequal to (d−1) 2 −⌊ d2 ⌋. In this paper, we prove that the Milnor sum of Cis also less than or equal to (d− 1) 2 − ⌊ d2 ⌋ and the equality holds if andonly if C is a P loski curve. Furthermore, we find a bound for the Milnorsum of projective plane curves in terms of GIT. 1. IntroductionLet C = V (f) be a projective plane curve of degree d. In this paper, a planecurve C means a projective plane curve that has at most isolated singularities.Moreover, we assume that C is not concurrent lines. We assume that the basefield k is algebraically closed and char(k)=0. Let f = 0 at [0,0,1]. Then, wedefine its Milnor number at 0 (in the sense of affine chart) byµ 0 (f) = dim k (O 0 /J f ),where O 0 is a function germ of f at the origin and J f = (∂f/∂x,∂f/∂y) isthe Jacobian ideal of f. Since µ