Abstract:
We study rational cuspidal curves in Hirzebruch surfaces. We provide two obstructions for the existence of rational cuspidal curves in Hirzebruch surfaces with prescribed types of singular points. The first result comes from Heegaard--Floer theory and is a generalization of a result by Livingston and the first author. The second criterion is obtained by comparing the spectrum of a suitably defined link at infinity of a curve with spectra of its singular points.

Abstract:
In this article we give an upper bound for the number of cusps on a cuspidal curve on a Hirzebruch surface. We adapt the results that have been found for a similar question asked for cuspidal curves on the projective plane, and restate the results in this new setting.

Abstract:
In this paper, we consider rational cuspidal plane curves having exactly two cusps whose complements have logarithmic Kodaira dimension two. We classify such curves with the property that the strict transforms of them via the minimal embedded resolution of the cusps have maximal self-intersection number.

Abstract:
Submission on request. This Master thesis from 2008 (University of Oslo, Norway) contains no new results, but it provides an overview of plane rational cuspidal curves, in particular curves of low degree. Please note that new results on this topic and the related topic of rational cuspidal curves on Hirzebruch surfaces have been published after this thesis was written.

Abstract:
We obtain new examples and the complete list of the rational cuspidal plane curves $C$ with at least three cusps, one of which has multiplicity ${\rm deg}\,C - 2$. It occurs that these curves are projectively rigid. We also discuss the general problem of projective rigidity of rational cuspidal plane curves.

Abstract:
We show a counterexample to a conjecture of de Bobadilla, Luengo, Melle-Hern\'{a}ndez and N\'{e}methi on rational cuspidal projective plane curves. The counterexample is a tricuspidal curve of degree 8. On the other hand, we show that if the number of cusps is at most 2, then the original conjecture can be deduced from the recent results of Borodzik and Livingston and (lattice cohomology) computations of N\'emethi and Rom\'an. Then we formulate a `simplified' (slightly weaker) version, more in the spirit of the motivation of the original conjecture (comparing index type numerical invariants), and we prove it for all currently known rational cuspidal curves. We make all these identities and inequalities more transparent in the language of lattice cohomology of surgery 3-manifolds $S^3_{-d}(K)$, where $K$ is a connected sum of algebraic knots $\{K_i\}_i$. Finally, we prove that the zeroth lattice cohomology of this surgery manifold depends only on the multiset of multiplicities occurring in the multiplicity sequences describing the algebraic knots $K_i$. This result is closely related to the lattice-cohomological reformulation of the above mentioned theorems and conjectures, and provides new computational and comparison procedures.

Abstract:
In this short note we give an elementary combinatorial argument, showing that the Conjecture of J. Fern\'andez de Bobadilla, I. Luengo, A. Melle-Hern\'andez, A. N\'emethi follows from the results of M. Borodzik and C. Livingston in the case of rational cuspidal curves with two critical points.

Abstract:
In the open problem of classification of rational cuspidal plane curves it is essential to find good necessary conditions on the type of singularities of a curve C in order C to exit. Motivated by the study of the Seiberg-Witten invariant of normal surface singularities whose link is a rational homology sphere we have found good candidates for such necessary conditions. In this paper we study this compatibility conditions and proved them for all curves whose logarithmic Kodaira dimension is strictly less than 2.

Abstract:
In this paper, we consider rational cuspidal plane curves having at least three cusps. We give an upper bound of the self-intersection number of the proper transforms of such curves via the minimal embedded resolution of the cusps. For a curve having exactly three cusps, we show that the self-intersection number is equal to the bound if and only if the curve coincides with the quartic curve having three cusps.

Abstract:
A characterization of freeness for plane curves in terms of the Hilbert function of the associated Milnor algebra is given as well as many new examples of rational cuspidal curves which are free. Some stronger properties are stated as conjectures.