Computing the Singularities of Rational Surfaces

TL;DR

The paper introduces an algorithm to decompose the parameter plane of a rational surface's parametrization, identifying points of different multiplicities on the surface except for finitely many base points.

Contribution

It provides a method to compute the singularities of rational surfaces by decomposing the parameter space based on multiplicity, with explicit handling of base points.

Findings

Successfully decomposes parameter plane into regions of constant multiplicity

Identifies all singular points of the surface except finitely many base points

Applicable to a broad class of rational projective surfaces

Abstract

Given a rational projective parametrization $\cP(\ttt,\sss,\vvv)$ of a rational projective surface $\cS$ we present an algorithm such that, with the exception of a finite set (maybe empty) $\cB$ of projective base points of $\cP$, decomposes the projective parameter plane as $\projdos\setminus \cB=\cup_{k=1}^{\ell} \cSm_k$ such that if $(\ttt_0:\sss_0:\vvv_0)\in \cSm_k$ then $\cP(\ttt_0,\sss_0,\vvv_0)$ is a point of $\cS$ of multiplicity $k$.