representations and define some Fitting ideals attached to them in Section \[sec:def\]. The main result of this paper is then proved in Section \[sec:main\]. Section \[sec:comp\] is devoted to the computational aspects of our results. In Section \[sec:link\] we will discuss on the link with multiple-points formulas developed in the field of intersection theory. Finally, we will treat the case of parameterizations $\phi$ whose source is ${\mathbb P}^1\times {\mathbb P}^1$ instead of ${\mathbb P}^2$. This type of parameterizations being widely used in geometric modeling (see for instance [@SSV12] for a detailed study of a special case).

Fitting ideals associated to surface parameterizations {#sec:def}
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Let $k$ be a field and $\phi: {\mathbb P}^2_k {\dashrightarrow}{\mathbb P}^3_k$ be a rational map given by four homogeneous polynomials $f_0,f_1,f_2,f_3$ of degree $d$. We will denote by ${\mathscr S}$ the closure of its image. Set $s:=(s_0,s_1,s_2)$, $x:=(x_0,x_1,x_2,x_3)$, and let $S:=k[s]$ and $R:=k[x]$ be the polynomial rings defining ${\mathbb P}^2_k$ and ${\mathbb P}^3_k$ respectively. Finally, define the ideal $I=(f_0,f_1,f_2,f_3)\subset S_d$. In the sequel, we will assume that: $$\label{H}\tag{${\mathcal H}$}
\left.
\begin{array}{l}
- \  {\mathscr S}\textrm{ is a surface in } {\mathbb P}^3_k, \\
- \   \phi \textrm{ is a birational map onto } {\mathcal S}, \\
