Vol. 11, No. 7, 2017

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Rational curves on smooth hypersurfaces of low degree

Tim Browning and Pankaj Vishe

Vol. 11 (2017), No. 7, 1657–1675

We establish the dimension and irreducibility of the moduli space of rational curves (of fixed degree) on arbitrary smooth hypersurfaces of sufficiently low degree. A spreading out argument reduces the problem to hypersurfaces defined over finite fields of large cardinality, which can then be tackled using a function field version of the Hardy-Littlewood circle method, in which particular care is taken to ensure uniformity in the size of the underlying finite field.

rational curves, circle method, function fields, hypersurfaces
Mathematical Subject Classification 2010
Primary: 14H10
Secondary: 11P55, 14G05
Received: 2 November 2016
Revised: 27 March 2017
Accepted: 23 May 2017
Published: 7 September 2017
Tim Browning
School of Mathematics
University of Bristol
United Kingdom
Pankaj Vishe
Department of Mathematical Sciences
Durham University
United Kingdom