Abstract
Let X be a smooth projective surface and let \({\mathcal {C}}\) be an arrangement of curves on X. The Harbourne constant of \({\mathcal {C}}\) was defined as a way to investigate the occurrence of curves of negative self-intersection on blow ups of X. This is related to the bounded negativity conjecture which predicts that the self-intersection number of all reduced curves on a surface is bounded below by a constant. We consider a geometrically ruled surface X over a smooth curve and give lower bounds for the Harbourne constants of transversal arrangements of curves on X. We also define a global Harbourne constant as the infimum of Harbourne constants for arrangements of a specific type and give a lower bound for it.
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Acknowledgements
We thank Piotr Pokora for reading this paper and giving many useful suggestions. We also thank the referees for making several helpful suggestions which substantially improved the paper.
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The first author was partially supported by DST SERB MATRICS Grant MTR/2017/000243. Both authors were partially supported by a Grant from Infosys Foundation.
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Hanumanthu, K., Subramaniam, A. Bounded negativity and Harbourne constants on ruled surfaces. manuscripta math. 164, 431–454 (2021). https://doi.org/10.1007/s00229-020-01179-1
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DOI: https://doi.org/10.1007/s00229-020-01179-1