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The behavior of rational points in families

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The behavior of rational points in families
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23
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CC Attribution - NonCommercial - NoDerivatives 4.0 International:
You are free to use, copy, distribute and transmit the work or content in unchanged form for any legal and non-commercial purpose as long as the work is attributed to the author in the manner specified by the author or licensor.
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A topic of current interest regards 'how often' a variety has a rational point. This topic was initially studied by Serre who gave upper bounds in the case of families of conics parametrised by a projective space. In the last few years this topic has been significantly enriched by Loughran and others. I will begin by giving an overview of the latest developments and finish by discussing joint work with Erik Visse, where asymptotics are given for a family of conics parametrised by arbitrary smooth hypersurfaces of low degree.