Rational cuspidal curves in projective surfaces. Topological versus algebraic obstructions

TL;DR

This paper investigates rational cuspidal curves in projective surfaces, establishing two criteria based on topological and algebraic methods that restrict possible singular point configurations.

Contribution

It introduces two new obstruction criteria for singular configurations on rational cuspidal curves, generalizing previous results and connecting topological and algebraic approaches.

Findings

The two criteria produce similar obstructions upon explicit calculation.

The criteria extend previous results by Fernandez de Bobadilla et al. and Livingston.

The methods combine Bezout theorem and Heegaard Floer homology inequalities.

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.