Computing the singularities of rational surfaces

Given a rational projective parametrization $\mathcal {P}(\mathfrak{s},\mathfrak{t},\mathfrak{v})$ of a rational projective surface $\mathcal {S}$ we present an algorithm such that, with the exception of a finite set (maybe empty) $\mathfrak{B}$ of projective base points of $\mathcal {P}$, decomposes the projective parameter plane as ${\mathbb{P}}^2(\mathbb{K})\setminus \mathfrak{B}=\bigcup _{k=1}^{\ell } \mathfrak{S}_k$ such that, if $(\mathfrak{s}_0:\mathfrak{t}_0:\mathfrak{v}_0)\in \mathfrak{S}_k$, then $\mathcal {P}(\mathfrak{s}_0,\mathfrak{t}_0,\mathfrak{v}_0)$ is a point of $\mathcal {S}$ of multiplicity $k$.