evaluation of the basis of $S_\nu$ chosen to build $M_{2d-2}(\phi)$ evaluated at the point $(s_0:s_1:s_2)$ is also a nonzero vector in the kernel of the transpose of $M_{2d-2}(\phi)(\pp)$. Hence, from this property it is straightforward to compute the pre-image of $\pp$ if it is unique.

**Algorithm** : Determination of the characters of a fiber $\pi_2^{-1}(\pp)$

**Input:** a parameterization $\phi$ of a surface ${\mathscr S}$ satisfying and a point $\pp \in {\mathbb P}^3$.

**Output:** the hilbert polynomial of the fiber $\pi_2^{-1}(\pp)$.

1.  Pick an integer $\nu\geq \nu_0$ (e.g. $\nu=2d-2$ which is always valid).

2.  Compute a matrix representation $M_{\nu}(\phi)$.

3.  Compute $r_{\nu}:=\corank(M_{\nu}(\phi))(\pp)$.

4.  If $r_{\nu}=0$ then *$\pp$ does not belong to ${\mathscr S}$* (and stop).

5.  If $0<r_{\nu}\leq \nu$ then $\pi_2^{-1}(\pp)$ is a finite set of $r_{\nu}$ points, counted with multiplicity.

6.  If $r_{\nu}>\nu$ then compute $M_{\nu+1}(\phi)$.

7.  Compute $r_{\nu+1}:=\corank(M_{\nu+1}(\phi))(\pp)$.

8.  If $r_{\nu}=r_{\nu+1}$ then $\pi_2^{-1}(\pp)$ is a finite set of $r_{\nu}$ points, counted with multiplicity.

9.  If $r_{\nu}<r_{\nu+1}$ then $\dim(\pi_2^{-1}(\pp))=1$ and $$HP_{\pi_2^{-1}(\pp)}(\nu)=(r_{\nu+1}-r_{\nu})\nu+r_{\nu}-(r_{\nu+1}-r_{\nu})\nu.$$ In other words, $\pi_2^{-1}(\pp)$ is made of a curve of degree $r_{\nu+1}-r_{\nu}$ and of a finite set of $$\binom{r_{\nu+1}-r_{\nu}-1}{2}+r_{\nu}-(r_{\nu+1}-r_{\nu})\nu-1.$$

Pull-back to the parameter space
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So far, we have seen that the fibers of $\pi_2$ can be described from 