point $P\in{\mathbb P}^3$ is in relation with the fiber of the graph of $\phi$ over $P$. Thus, the Fitting ideals attached to $M(\phi)$ provide a filtration of the surface which is in correspondence with the degree and the dimension of the fibers of the graph of the parameterization $\phi$.

It turns out that this kind of results have already been investigated for different purposes in the field of intersection theory under the name of multiple-point formulas. Given a finite map of schemes $\varphi:X\rightarrow Y$ of codimension one, its source and target double-point cycles have been extensively studied (see e.g. [@KLU96; @KLU92; @Pie78; @Tei77]). Moreover, in [@MP89] Fitting ideals are used to give a scheme structure to the multiple-point loci. Therefore, in the particular case where $\phi:{\mathbb P}^2\rightarrow {\mathbb P}^3$ has no base point (hence is a finite map), our results are partially contained in the above literature. Thus, the main contribution of this paper is to extend these previous works to the case where $\phi$ is not necessarily a finite map under ubiquitous conditions. The theory in this article is thus applicable to most of the parameterizations that appear in applications.

In what follows, we will first briefly overview matrix 