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.

Produced in cooperation with Akademika publishing. The thesis is produced by Akademika publishing merely in connection with the thesis defence. Kindly direct all inquiries regarding the thesis to the copyright holder or the unit which grants the doctorate. Writing this thesis has been a long lasting and challenging task. At the time of its completion it is time to thank everyone who has contributed. First, I would like to thank Professor Ragni Piene for giving me a solid training in mathematics, for introducing me to the puzzling problem of cuspidal curves, and for providing the ideas, suggestions and advice necessary to complete this thesis. I am very grateful for everything that you have helped me achieve the past ten years and for the support and care you have provided along the way. This thesis could not have been written without the encouragement I have received from my colleagues in the algebra group. In particular, I have to thank Nikolay Qviller for inspirational, instructive and productive cooperation. I am very much indebted to Georg Muntingh for interesting discussions and invaluable contributions, for helping me with images, and for proofreading. Moreover, I would like to thank Heidi Camilla Mork and Professor Kristian Ranestad for motivation and support. I am also very grateful to Inger Christin Borge and Arne Bernhard Sletsjøe for always believing in me and for including me in amazing teaching projects. Additionally, I want to thank Jørgen Vold Rennemo and John Christian Ottem for sharing enthusiasm for mathematics and for impressing me on an everyday basis. I also want to thank Professor Keita Tono for helpful explanations, and Torsten Fenske for sending me a copy of his PhD-thesis. My PhD-position has been funded by the Institute of Mathematics, University of Oslo, and I have been associated to the Centre of Mathematics for Applications (CMA). I thank you for giving me the opportunity to write this thesis. Additionally, I want to thank everyone in the administration and IT-drift. Through the years I have met a vast number of fantastic people in the corridors of the Niels Henrik Abel building. In particular, I would like to thank Agnieszka,

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.

In an article from 1865, Arthur Cayley claims that given a plane algebraic curve there exists an associated 2-Hessian curve that intersects it in its sextactic points. In this paper we fix an error in Cayley's calculations and provide the correct defining polynomial for the 2-Hessian. In addition, we present a formula for the number of sextactic points on cuspidal curves and tie this formula to the 2-Hessian. Lastly, we consider the special case of rational curves, where the sextactic points appear as zeros of the Wronski determinant of the 2nd Veronese embedding of the curve.

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