Rational cuspidal curves with four cusps on Hirzebruch surfaces

Abstract

The purpose of this article is to shed light on the question of how many and what kind of cusps a rational cuspidal curve on a Hirzebruch surface can have. Our main result is a list of rational cuspidal curves with four cusps, their type, cuspidal congurations and the surfaces they lie on. We use birational transformations to construct these curves. Moreover, we find a general expression for and compute the Euler characteristic of the logarithmic tangent sheaf in these cases. Additionally, we show that there exists a real rational cuspidal curve with four real cusps. Last, we show that for rational cuspidal curves with two or more cusps on a Hirzebruch surface, there is a lower bound on one of the multiplicities.