of $N_\nu^\pp$ as a function of $\nu$ for a given $\pp \in \operatorname{Proj}(R)$.[]{data-label="figure"}](pic1.pdf)

The fact that $\pp$ belongs or not to the support of ${\mathfrak F}_\nu^i(\phi)$ can be checked by the computation of the rank of a matrix representation $M_\nu(\phi)$ evaluated at the point $\pp$. More precisely, we get the following properties:

-   If $\dim \pi_2^{-1}(\pp)\leq 0$ then for all $\nu\geq \nu_0$ $$\corank(M_\nu(\phi)(\pp))=\deg(\pi_2^{-1}(\pp)).$$

-   If $\dim \pi_2^{-1}(\pp) = 1$ then for all $\nu\geq \nu_0-1$ $$\corank(M_\nu(\phi)(\pp))=\deg(\pi_2^{-1}(\pp))\nu+c$$ where $c\in {\mathbb Z}$ is such that $HP_{\pi_2^{-1}(\pp)}(\nu)=\deg(\pi_2^{-1}(\pp))\nu+c$.

From here, we get an algorithm to determine the Hilbert polynomial of a fiber $\pi_2^{-1}(\pp)$ of a given point $\pp$ by comparing the rank variation of the matrix $M_\nu(\phi)(\pp)$ for two consecutive integers $\nu$ and $\nu+1$, assuming $\nu \geq \nu_0$. Before giving a more precise description in the form of an algorithm, we refine the distinction between finite and non-finite fibers.

Let $\pp$ be a point on ${\mathscr S}$ and choose an integer $\nu$ such that $$\nu_0\leq \nu\leq 2d-2.$$ If $\corank(M_{\nu}(\phi)(\pp))\leq \nu$ then $\pi_2^{-1}(\pp)$ is necessarily a finite fiber.

We know that the Hilbert function of the fiber $\pi_2^{-1}(\pp)$ coincides with its Hilbert polynomial for all $\nu\geq \nu_{0}$. Therefore, to prove our claim it is enough to 