Limits of translates of plane curves -- on a paper of Aldo Ghizzetti

We study the limits of PGL(3) translates of an arbitrary plane curve, giving a description of all possible limits of a given curve and computing the multiplicities of corresponding components in the normal cone to the base scheme of a related linear system. This information is a key step in the computation of the degree of the closure of the linear orbit of an arbitrary plane curve. Our analysis recovers and extends results obtained by Aldo Ghizzetti in the 1930's.