A Note on Rational Cuspidal curves on $\mathbb{Q}$-Homology Projective Planes

TL;DR

This paper extends known results about rational cuspidal curves from the projective plane to $Q$-homology projective planes, showing similar properties under certain conditions and providing examples with singular surfaces.

Contribution

It generalizes the classification of rational cuspidal curves to $Q$-homology projective planes, demonstrating the results are consistent with the classical case and providing sharpness examples.

Findings

Results match the projective plane case under suitable assumptions

Examples with singular ambient surfaces demonstrate sharpness

The generalization broadens understanding of cuspidal curves on singular surfaces

Abstract

We generalize results by Wakabayashi and Orevkov about rational cuspidal curves on the projective plane to that on $\mathbb{Q}$-homology projective planes. It turns out that the result is exactly the same as the projective plane case under suitable assumptions. We also provide examples which demonstrate sharpness of the results. The ambient surface is singular in these examples.