Deformation of singular curves on surfaces

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 $C\subset S$ be a complete integral curve on a smooth surface. Let $\tilde C$ be a partial normalization of $C$, and $\varphi\colon \tilde C\to S$ be the induced map. In this paper, we consider deformations of $\varphi$. 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 $\varphi$ has virtually optimal deformation property.