A reduced and irreducible curve on a a smooth complex surface \(C\) is called cuspidal if all of its singularities are cusps, that is points \(p\) such that the germ \((C,p)\) is irreducible.
In this paper, the author proves the existence of some types of rational cuspidal curves with four cusps on certain Hirzebruch surfaces. The construction of the curves comes from birational transformations and the corresponding configuration is given. A general expression for the Euler characteristic of the logarithmic tangent sheaf is also obtained. Based in a large number of computations, the author proposes the following extension of the Orekov conjecture: A rational cuspidal curve on a Hirzebruch surface has at most four cusps.