Rational curves on smooth hypersurfaces of low degree

Tim D Browning; Pankaj Vishe

doi:10.2140/ant.2017.11.1657

Rational curves on smooth hypersurfaces of low degree

Tim D Browning, Pankaj Vishe

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Abstract

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.

Original language	English
Pages (from-to)	1657–1675
Number of pages	19
Journal	Algebra and Number Theory
Volume	11
Issue number	7
Early online date	7 Sep 2017
DOIs	https://doi.org/10.2140/ant.2017.11.1657
Publication status	Published - Sep 2017

Keywords

math.AG
math.NT
14H10 (11P55, 14G05)

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Cite this

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N2 - 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.

AB - 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.

KW - math.AG

KW - math.NT

KW - 14H10 (11P55, 14G05)

UR - https://arxiv.org/abs/1611.00553

U2 - 10.2140/ant.2017.11.1657

DO - 10.2140/ant.2017.11.1657

M3 - Article (Academic Journal)

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JO - Algebra and Number Theory

JF - Algebra and Number Theory

SN - 1937-0652

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ER -

Rational curves on smooth hypersurfaces of low degree

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