we deduce that $HP_\nu^\pp < \binom{\nu+2}{2}$ for any prime in $\operatorname{Proj}(R)$ if $\nu \geq \nu_0$.

It is worth mentioning two facts that are direct consequences of the above results. First, the embedded components of ${\mathfrak F}^0_\nu(\phi)$ are exactly supported on $V({\mathfrak F}^1_\nu(\phi))\subset V({\mathfrak F}^0_\nu(\phi))={\mathscr S}$. Second, the set-theoretical support of each ${\mathfrak F}^i_\nu(\phi)$, $i$ fixed, stabilizes for $\nu \gg 0$; its set-theoretical support then correspond to those points on ${\mathcal S}$ whose fiber is either a curve and or a finite scheme of degree greater or equal to $i+1$.

Computational aspects {#sec:comp}
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In this section, we detail the consequences of the previous results for giving a computational description of the singular locus of the parameterization $\phi$ of the surface ${\mathcal S}$.

Description of the fiber of a given point on ${\mathscr S}$ {#sec:algo}
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Given a point $\pp$ on ${\mathscr S}$, we summarize the results we obtained with Figure \[figure\]. The blue curve corresponds to a finite fiber, i.e. a point $\pp$ such that $\dim \pi_2^{-1}(\pp)=0$, the red curve corresponds a one dimensional fiber, i.e.  a point $\pp$ such that $\dim \pi_2^{-1}(\pp)=1$, and the black curve represents the Hilbert polynomial of $S$, the coordinate ring of ${\mathbb P}^{2}_{k}$.

![Graph 