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On the arithmetic of weighted complete intersections of low degree

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Abstract

A variety is rationally connected if two general points can be joined by a rational curve. A higher version of this notion is rational simple connectedness, which requires suitable spaces of rational curves through two points to be rationally connected themselves. We prove that smooth, complex, weighted complete intersections of low enough degree are rationally simply connected. This result has strong arithmetic implications for weighted complete intersections defined over the function field of a smooth, complex curve. Namely, it implies that these varieties satisfy weak approximation at all places, that R-equivalence of rational points is trivial, and that the Chow group of zero cycles of degree zero is zero.

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Acknowledgements

I would like to thank my advisor, Jason Starr, for proposing the problem, for many invaluable suggestions, and for his constant support and encouragement. I would also like to thank Luigi Lombardi and David Stapleton for many useful conversations.

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  1. Department of Mathematics, University of Michigan, Ann Arbor, MI, 48109, USA

    Cristian Minoccheri

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  1. Cristian Minoccheri
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Correspondence to Cristian Minoccheri.

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Communicated by Wei Zhang.

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Minoccheri, C. On the arithmetic of weighted complete intersections of low degree. Math. Ann. 377, 483–509 (2020). https://doi.org/10.1007/s00208-020-01981-y

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  • DOI: https://doi.org/10.1007/s00208-020-01981-y

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