Abstract
In this article, we study the generic curve of the linear system E of plane curves of degree d passing through r generic points with at least prescribed multiplicities m1,...,mr. Assuming that the mi are less than or equal to three and that the degree d is greater than 316 we prove that, if E is not empty, then its generic curve is geometrically irreducible, smooth away from the prescribed singularities and has only ordinary singularities.