1995/07/07 by Mikhail Zaidenberg, Orevkov, S., Zaidenberg, M.
Mathematics · #14H20 (Primary) 14H10 #14N05 (Secondary) #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.alg-geom/9507005
openalex publication_date 1995/07/07 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
This is an extended, renovated and updated report on a joint work which the second named author presented at the Conference on Algebraic Geometry held at Saitama University, 15-17 of March, 1995. The main result is an inequality for the numerical type of singularities of a plane curve, which involves the degree of the curve, the multiplicities and the Milnor numbers of its singular points. It is a corollary of the logarithmic Bogomolov-Miyaoka-Yau's type inequality due to Miyaoka. It was first proven by F. Sakai at 1990 and rediscovered by the authors independently in the particular case of an irreducible cuspidal curve at 1992. Our proof is based on the localization, the local Zariski--Fujita decomposition and uses a graph discriminant calculus. The key point is a local analog of the BMY-inequality for a plane curve germ. As a corollary, a boundedness criterium for a family of plane curves has been obtained. Another application of our methods is the following fact: a rigid rational cuspidal plane curve cannot have more than 9 cusps.