Projective structures and neighborhoods of rational curves

We survey on a construction going back to Cartan, and leading, in a paper of Hitchin, to a non linear generalization of the duality between lines and points in the projective plane. Projective structures are defined by second order differential equations, like Painleve equations, while neighborhoods of rational curves have been classified by Grauert and Mishustin. We will explain this duality in details and discuss the existence of meromorphic functions on the neighborhoods. We end by some results recently obtained in collaboration with Maycol Falla Luza.