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Arithmetic of Del Pezzo surfaces

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Birational Geometry, Rational Curves, and Arithmetic

Abstract

This survey is an introduction to the birational, qualitative arithmetic of del Pezzo surfaces over global fields: existence of rational points, weak approximation, Zariski density of points. We begin by reviewing the geometry of these surfaces over separably closed fields. We then show that del Pezzo surfaces of degree at least 5 that have a rational point are rational over their ground field, thereby proving they satisfy weak approximation and have a dense set of rational points. Finally, we discuss Brauer-Manin obstructions and give an example of one on a del Pezzo surface of degree 1.

Mathematics Subject Classification codes (2010): 14 G05, 14 J26, 11 G35.

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Notes

  1. 1.

    Many authors refer only to the Hasse principle in the context of a class \(\mathcal{S}\) of varieties and say that \(\mathcal{S}\) satisfies the Hasse principle if for every \(X \in \mathcal{S}\), the implication X(k v )≠∅ for all \(v\in \Omega_k \implies X(k) \neq \emptyset\) holds.

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Acknowledgments

I thank the conference organizers (Hendrik Lenstra, Cecilia Salgado, Lenny Taelman and Ronald van Luijk) for inviting me to give this minicourse and for their hospitality in Leiden. I also thank the staff at the Lorentz Center for all their help and professionalism. Finally, I thank Jean-Louis Colliot-Thélène for the comments on these notes.

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Authors and Affiliations

  1. Department of Mathematics MS-136, Rice University, Houston, TX, 77005, USA

    Anthony Várilly-Alvarado

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  1. Anthony Várilly-Alvarado
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Correspondence to Anthony Várilly-Alvarado .

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Editors and Affiliations

  1. Department of Mathematics, Courant Institute of Math. Sciences, New York University, Mercer St. 251, New York, 10012, New York, USA

    Fedor Bogomolov

  2. , Department of Mathematics, Rice University, S. Main St. 6100, Houston, 77251-1892, Texas, USA

    Brendan Hassett

  3. Department of Mathematics, Courant Institute of Math. Sciences, New York University, Mercer St. 251, New York, 10012, New York, USA

    Yuri Tschinkel

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Várilly-Alvarado, A. (2013). Arithmetic of Del Pezzo surfaces. In: Bogomolov, F., Hassett, B., Tschinkel, Y. (eds) Birational Geometry, Rational Curves, and Arithmetic. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6482-2_12

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