Remarks on double points of plane curves (Q2093016)

The authors study double points of plane curves either using their implicit equation or their parametrization. Recall that a singularity oftype \(A_s\) for a plane curve is a double point that can be resolved via \(r\) blown-ups if \(s=2r-\varepsilon\), \(\varepsilon =0,1\) and the desingularization yields two points if \(\varepsilon=1\) and only one point if \(\varepsilon=0\). The authors generalize result from their previous paper [\textit{A. Bernardi} et al., J. Symb. Comput. 86, 189--214 (2018; Zbl 1390.14183)] about type of singularities of points to points on any plane curve. Then there is presented an algorithm which classifies double points of any plane curve. This algorithm is based on studying the osculating curves to a curve at double point. The paper also shows an example which illustrates how to build a plane rational curve with double points of chosen type using projection techniques. This example gives also a counterexample to Lemma 4.2 in [loc. cit.] and this way the authors show that there was a mistake in that paper.