Computing the singularities of rational surfaces

Abstract:

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 $k$.