Systèmes de courbes planes à singularitésimposées: le cas des multiplicitésinférieuresouégales à quatre

Let P1,…,Pr be r general points of the projective plane over an algebraically closed field. Consider the linear system of plane curves of degree d passing through each point Pi with a prescribed multiplicity mi. It is still an open problem to know whether or not has the expected dimension. Precise conditions for to be regular (i.e. have the expected dimension) are given by a conjecture of Harbourne and Hirschowitz. In this article we prove that this conjecture is true, under the assumption that the prescribed multiplicities are at most 4. The proof is based on the Horace differential method of Alexander and Hirschowitz, and on a careful study of the rational curves arising as base components of the system when the points are specialised so as to lie on a line.