The aim of this paper is to present some of the old conjectures and problems related to the classification of irreducible projective plane curves \(C\) in \(\mathbb P^2\) up to the action of the automorphism group \(\text{PGL}(3,\mathbb C)\), together with some results and new conjectures from recent works of the authors, focusing on the case of rational curves. For instance, the classification problem is related to the Nagata-Coolidge problem (whether every rational cuspidal curve can be transformed into a line by a Cremona transformation), and to the determination of the maximal number of cusps for a rational cuspidal plane curve; on this point, the maximal number of cusps known by the authors is 4 and recently K. Tono [Math. Nachr. 278, No. 1–2, 216–221 (2005; Zbl 1069.14029)] proved that it is less than 9. The classification problem is also related to the theory of open surfaces; for example, the open surface \({\mathbb P^2 } \backslash C\) is \(\mathbb Q\)-acyclic if and only if \(C\) is a rational cuspidal curve, but according to the H. Flenner-M. Zaidenberg rigidity conjecture [see Contemp. Math. 162, 143–208 (1994; Zbl 0838.14027)], if such a surface has logarithmic Kodaira dimension 2, then it should be rigid; this would imply that every equisingular deformation of \(C\) in \(\mathbb P^2\) would be projectively equivalent to \(C\).
Among the classification results, the authors show in particular the classification of rational unicuspidal curves, whose cusp has one Puiseux pair, see authors’ paper [in: Real and complex singularities, São Carlos workshop 2004. Papers of the 8th workshop, Marseille, France, July 19–23, 2004. Trends in Mathematics, 31–45 (2007; Zbl 1120.14019)]. From the characterization problem, of the realization of prescribed topological types of singularities, many compatibility properties arise, connecting local invariants of the singular germs \((C,p)\) with some global invariant of the curve \(C\). The authors present their compatibility condition for the characterization problem, [see Proc. Lond. Math. Soc. (3) 92, No. 1, 99–138 (2006; Zbl 1115.14021)], together with some equivalent reformulation; in particular they deal with the semigroup distribution property (connecting the semigroup of the local singularity and the degree of the curve) and they explain its relations (for the unicuspidal case) with the Seiberg-Witten invariant conjecture, formulated by the forth author and L. I. Nicolaescu [Sel. Math., New Ser. 11, No. 3–4, 399–451 (2005; Zbl 1110.14006)], also in connection with the Heegaard-Floer homology, introduced by P. Ozsváth and Z. Szabó [in: Floer homology, Gauge theory, and low-dimensional topology. Proc. Clay Math. Inst. 2004 summer school, Budapest, Hungary, June 5–26, 2004. Clay Math. Proc. 5, 3–27 (2006; Zbl 1107.57022)].
For the entire collection see [Zbl 1111.14001].