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Free rational curves on low degree hypersurfaces and the circle method

Tim Browning and Will Sawin

Vol. 17 (2023), No. 3, 719–748

We use a function field version of the Hardy–Littlewood circle method to study the locus of free rational curves on an arbitrary smooth projective hypersurface of sufficiently low degree. On the one hand this allows us to bound the dimension of the singular locus of the moduli space of rational curves on such hypersurfaces and, on the other hand, it sheds light on Peyre’s reformulation of the Batyrev–Manin conjecture in terms of slopes with respect to the tangent bundle.

circle method, free rational curve, function field, hypersurface
Mathematical Subject Classification
Primary: 14H10
Secondary: 11D45, 11P55, 14G05, 14J20
Received: 28 June 2021
Revised: 24 February 2022
Accepted: 22 April 2022
Published: 12 April 2023
Tim Browning
Institute of Science and Technology Austria
Am Campus 1
Will Sawin
Department of Mathematics
Columbia University
New York, NY
United States

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