Intersection of plane curves and curve constructions with ordinary singularities

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 m(1),...,m(r). 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. (C) 2001 Elsevier Science B.V. All rights reserved.