Under the assumptions , for all integer $\nu\geq \nu_0$ the matrix $M(\phi)_\nu$ of the graded map of free $R$-modules $({\mathcal Z}_1)_\nu\rightarrow ({\mathcal Z}_0)_\nu$ is called a *matrix representation* of $\phi$.

For the moment, the integer $\nu_0$ can be taken equal to $2d-2$, but we will come back to its definition in the next section. Here are the main features of the collection of matrix representations $M(\phi)_\nu$ of $\phi$, for $\nu\geq \nu_0$:

-   Its entries are linear forms in $R=k[x_0,\ldots,x_3]$,

-   it has $\binom{\nu+2}{2}$ rows and at least as many columns as rows,

-   it has maximal rank $\binom{\nu+2}{2}$ at the generic point of ${\mathbb P}^3_k$, that is to say $$\mathrm{rank} \, M(\phi)_\nu \otimes_{R}\mathrm{Frac}(R) = \binom{\nu+2}{2},$$

-   when specializing $M(\phi)_\nu$ at a point $P\in {\mathbb P}^3$, its rank drops if and only if $P\in {\mathcal S}$.

Fitting ideals associated to $\phi$
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From the properties of matrix representations of $\phi$, for all $\nu\geq \nu_0$ the support of the Fitting ideal ${\mathfrak F}^0(({\mathcal S}_I)_\nu)$ is ${\mathscr S}$ and hence provides a scheme structure for the closure of the image of $\phi$. Following [@Tei77] (and [@EH00 V.1.3]), we call it the *Fitting image* of $\phi$.

Observe that by definition, the ideals ${\mathfrak F}^0(({\mathcal 