# Automorphism groups of rational elliptic surfaces with section and constant J-map

Open Mathematics (2014)

- Volume: 12, Issue: 12, page 1772-1795
- ISSN: 2391-5455

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topTolga Karayayla. "Automorphism groups of rational elliptic surfaces with section and constant J-map." Open Mathematics 12.12 (2014): 1772-1795. <http://eudml.org/doc/269545>.

@article{TolgaKarayayla2014,

abstract = {In this paper, the automorphism groups of relatively minimal rational elliptic surfaces with section which have constant J-maps are classified. The ground field is ℂ. The automorphism group of such a surface β: B → ℙ1, denoted by Au t(B), consists of all biholomorphic maps on the complex manifold B. The group Au t(B) is isomorphic to the semi-direct product MW(B) ⋊ Aut σ (B) of the Mordell-Weil groupMW(B) (the group of sections of B), and the subgroup Aut σ (B) of the automorphisms preserving a fixed section σ of B which is called the zero section on B. The Mordell-Weil group MW(B) is determined by the configuration of singular fibers on the elliptic surface B due to Oguiso and Shioda [9]. In this work, the subgroup Aut σ (B) is determined with respect to the configuration of singular fibers of B. Together with a previous paper [4] where the case with non-constant J-maps was considered, this completes the classification of automorphism groups of relatively minimal rational elliptic surfaces with section.},

author = {Tolga Karayayla},

journal = {Open Mathematics},

keywords = {Elliptic surface; Rational elliptic surface; Automorphism group; Mordell-Weil group; J map; Singular fiber; elliptic surface; rational elliptic surface; automorphism group; map},

language = {eng},

number = {12},

pages = {1772-1795},

title = {Automorphism groups of rational elliptic surfaces with section and constant J-map},

url = {http://eudml.org/doc/269545},

volume = {12},

year = {2014},

}

TY - JOUR

AU - Tolga Karayayla

TI - Automorphism groups of rational elliptic surfaces with section and constant J-map

JO - Open Mathematics

PY - 2014

VL - 12

IS - 12

SP - 1772

EP - 1795

AB - In this paper, the automorphism groups of relatively minimal rational elliptic surfaces with section which have constant J-maps are classified. The ground field is ℂ. The automorphism group of such a surface β: B → ℙ1, denoted by Au t(B), consists of all biholomorphic maps on the complex manifold B. The group Au t(B) is isomorphic to the semi-direct product MW(B) ⋊ Aut σ (B) of the Mordell-Weil groupMW(B) (the group of sections of B), and the subgroup Aut σ (B) of the automorphisms preserving a fixed section σ of B which is called the zero section on B. The Mordell-Weil group MW(B) is determined by the configuration of singular fibers on the elliptic surface B due to Oguiso and Shioda [9]. In this work, the subgroup Aut σ (B) is determined with respect to the configuration of singular fibers of B. Together with a previous paper [4] where the case with non-constant J-maps was considered, this completes the classification of automorphism groups of relatively minimal rational elliptic surfaces with section.

LA - eng

KW - Elliptic surface; Rational elliptic surface; Automorphism group; Mordell-Weil group; J map; Singular fiber; elliptic surface; rational elliptic surface; automorphism group; map

UR - http://eudml.org/doc/269545

ER -

## References

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- [9] K. Oguiso and T. Shioda, The Mordell-Weil lattice of a rational elliptic surface. Commentarii Mathematici Universitatis Sancti Pauli 40 (1991), 83–99. Zbl0757.14011
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