On rational cuspidal curves, open surfaces and local singularities

Abstract

Let $C$ be an irreducible projective plane curve in the complex projective space ${\mathbb{P}}^2$. The classification of such curves, up to the action of the automorphism group $PGL(3,\mathbb{C})$ on ${\mathbb{P}}^2$, is a very difficult open problem with many interesting connections. The main goal is to determine, for a given $d$, whether there exists a projective plane curve of degree $d$ having a fixed number of singularities of given topological type. In this note we are mainly interested in the case when $C$ is a rational curve. The aim of this article is to present some of the old conjectures and related problems, and to complete them with some results and new conjectures from the recent work of the authors.

References in corpus (3)

• On a class of rational cuspidal plane curves
• Open problems on open algebraic varieties
• Equisingular Deformations of Plane Curve and of Sandwiched Singularities

Cited by in corpus (7)

• Equivalent birational embeddings II: divisors
• Recent progress in the geometry of Q-acyclic surfaces
• Birational geometry of rational quartic surfaces
• Equivalent Birational Embeddings III: cones
• Rational cuspidal curves with four cusps on Hirzebruch surfaces
• Classification of rational unicuspidal curves with two Newton pairs
• Linear systems associated to unicuspidal rational plane curves