Abstract
We prove that the Cox ring of a smooth rational surface with big anticanonical class is finitely generated. We classify surfaces of this type that are blow-ups of \({\mathbb{P}^2}\) at distinct points lying on a (possibly reducible) cubic.
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Acknowledgments
We thank Ana-Maria Castravet, Johan de Jong, David Eisenbud, Brian Harbourne, Brendan Hassett, Seán Keel, Bjorn Poonen, Burt Totaro and Chenyang Xu for helpful conversations during the completion of this work. We also thank the anonymous referee for a thorough report and helpful suggestions.
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D. Testa was partially supported by Jacobs University Bremen, DFG grant STO-299/4-1 and EPSRC grant number EP/F060661/1; the third author is partially supported by NSF grant DMS-0802851.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Testa, D., Várilly-Alvarado, A. & Velasco, M. Big rational surfaces. Math. Ann. 351, 95–107 (2011). https://doi.org/10.1007/s00208-010-0590-7
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DOI: https://doi.org/10.1007/s00208-010-0590-7