Deformation of holomorphic maps onto the blow-up of the projective plane

Let S be the blow-up of the projective plane at d distinct points and psi: S' -> S be any surjective holomorphic map from a compact complex manifold S'. We will show that all deformations of V, come from automorphisms of S if d >= 3. The result is optimal in the sense that it is not true if d <= 2. The strategy of the proof is to use the infinitesimal automorphisms of the web geometry on S arising from the natural foliations of S induced by the pencils of the lines through the blow-up centers. (c) 2007 Elsevier Masson SAS.