2023/10/21 by Takeo Nishinou, Nishinou, Takeo
Mathematics · Physics and Astronomy · #14J29 #32G10 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.2310.14039
openalex publication_date 2023/10/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we consider deformations of singular complex curves on complex surfaces. Despite the fundamental nature of the problem, little seems to be known for curves on general surfaces. Let C⊂ S be a complete integral curve on a smooth surface. Let C be a partial normalization of C, and φ\colon C→ S be the induced map. In this paper, we consider deformations of φ. The problem of the existence of deformations will be reduced to solving a certain explicit system of polynomial equations. This system is universal in the sense that it is determined solely by simple local data of the singularity of C, and does not depend on the global geometry of C or S. Under a relatively mild assumption on the properties of these equations, we will show that the map φ has virtually optimal deformation property.