Rational cuspidal curves in projective surfaces. Topological versus algebraic obstructions

Abstract

We study rational cuspidal curves in projective surfaces. We specify two criteria obstructing possible configurations of singular points that may occur on such curves. One criterion generalizes the result of Fernandez de Bobadilla, Luengo, Melle--Hernandez and Nemethi and is based on the Bezout theorem. The other one is a generalization of the result obtained by Livingston and the author and relies on Ozsvath--Szabo inequalities for $d$-invariants in Heegaard Floer homology. We show by means of explicit calculations that the two approaches give very similar obstructions.