Quartic del Pezzo surfaces over function fields of curves

Brendan Hassett; Yuri Tschinkel

doi:10.2478/s11533-013-0354-1

Quartic del Pezzo surfaces over function fields of curves

Brendan Hassett, Yuri Tschinkel

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We classify quartic del Pezzo surface fibrations over the projective line via numerical invariants, giving explicit examples for small values of the invariants. For generic such fibrations, we describe explicitly the geometry of spaces of sections to the fibration, and mappings to the intermediate Jacobian of the total space. We exhibit examples where these are birational, which has applications to arithmetic questions, especially over finite fields.

Original language	English (US)
Pages (from-to)	395-420
Number of pages	26
Journal	Central European Journal of Mathematics
Volume	12
Issue number	3
DOIs	https://doi.org/10.2478/s11533-013-0354-1
State	Published - 2014

Keywords

Del Pezzo surfaces
Fibrations
Function fields of curves
Intermediate Jacobians
Rational points

ASJC Scopus subject areas

Mathematics(all)

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10.2478/s11533-013-0354-1

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title = "Quartic del Pezzo surfaces over function fields of curves",

abstract = "We classify quartic del Pezzo surface fibrations over the projective line via numerical invariants, giving explicit examples for small values of the invariants. For generic such fibrations, we describe explicitly the geometry of spaces of sections to the fibration, and mappings to the intermediate Jacobian of the total space. We exhibit examples where these are birational, which has applications to arithmetic questions, especially over finite fields.",

keywords = "Del Pezzo surfaces, Fibrations, Function fields of curves, Intermediate Jacobians, Rational points",

author = "Brendan Hassett and Yuri Tschinkel",

note = "Funding Information: The first author was supported by National Science Foundation Grants 0901645, 0968349, and 1148609; the second author was supported by National Science Foundation Grants 0739380, 0968318, and 1160859. We are grateful to Asher Auel, Jean-Louis Colliot-Th{\'e}l{\`e}ne, and Raman Parimala for useful conversations as well as to Andrew Kresch and Nikita Kozin for comments on the manuscript; we appreciate Jason Starr and Yi Zhu explaining the results of [35] on Abel–Jacobi morphisms for fibrations of rationally connected varieties over curves.",

year = "2014",

doi = "10.2478/s11533-013-0354-1",

language = "English (US)",

volume = "12",

pages = "395--420",

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T1 - Quartic del Pezzo surfaces over function fields of curves

AU - Hassett, Brendan

AU - Tschinkel, Yuri

N1 - Funding Information: The first author was supported by National Science Foundation Grants 0901645, 0968349, and 1148609; the second author was supported by National Science Foundation Grants 0739380, 0968318, and 1160859. We are grateful to Asher Auel, Jean-Louis Colliot-Thélène, and Raman Parimala for useful conversations as well as to Andrew Kresch and Nikita Kozin for comments on the manuscript; we appreciate Jason Starr and Yi Zhu explaining the results of [35] on Abel–Jacobi morphisms for fibrations of rationally connected varieties over curves.

PY - 2014

Y1 - 2014

N2 - We classify quartic del Pezzo surface fibrations over the projective line via numerical invariants, giving explicit examples for small values of the invariants. For generic such fibrations, we describe explicitly the geometry of spaces of sections to the fibration, and mappings to the intermediate Jacobian of the total space. We exhibit examples where these are birational, which has applications to arithmetic questions, especially over finite fields.

AB - We classify quartic del Pezzo surface fibrations over the projective line via numerical invariants, giving explicit examples for small values of the invariants. For generic such fibrations, we describe explicitly the geometry of spaces of sections to the fibration, and mappings to the intermediate Jacobian of the total space. We exhibit examples where these are birational, which has applications to arithmetic questions, especially over finite fields.

KW - Del Pezzo surfaces

KW - Fibrations

KW - Function fields of curves

KW - Intermediate Jacobians

KW - Rational points

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