  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.
