A note on plane rational curves and the associated Poncelet surfaces (Q2809906)

In this paper the authors consider the parametrization \((f_0,f_1,f_2)\) of a plane rational curve \(C\) over an algebraically closed ground field \(K\). They want to relate the splitting type of \(C\) (i.e. the second Betti numbers of the ideal \((f_0,f_1,f_2)\subset K[\mathbb{P}^1]\)) with the singularities of the associated Poncelet surface in \(\mathbb{P}^3\). In particular they generalize, for plane curves, a result of \textit{G. Ilardi} et al. [Boll. Unione Mat. Ital. (9) 2, No. 3, 579--589 (2009; Zbl 1197.13013)] and they show that if \(C\) is an Ascenzi curve of degree \(d\) with a point of multiplicity \(m\) and splitting type \((m, d- m)\), then the corresponding Poncelet surface has a particular configuration of \(\binom{m}{3}\) singular points. Moreover they prove that if the Poncelet surface in \(\mathbb{P}^3\) is singular, then it is associated with a curve \(C\) which possesses at least a point of multiplicity \( \geq 3\).