Regular foliations on algebraic surfaces

Given an ordinary differential equation $A(x,y)dx + B(x,y)dy = 0$, its solutions $f(x,y)$ define a decomposition of the plane outside the zeros of $A(x,y)$ and $B(x,y)$ into regular curves. This is a prototype of a foliation, the leaves being the solutions of the given differential equation. In general, a foliation will be a generalization of this concept, i.e. instead of taking one equation, we take a system of equations, and to have solutions we demand an integrability condition. In this talk, I will introduce the concept of holomorphic foliation and give a characterization of regular foliations on rational surfaces.