Elliptic surfaces. (English) Zbl 1216.14036

Schütt, Matthias; Shioda, Tetsuji

Keum, JongHae (ed.) et al., Algebraic geometry in East Asia – Seoul 2008. Proceedings of the 3rd international conference “Algebraic geometry in East Asia, III”, Seoul, Korea, November 11–15, 2008. Tokyo: Mathematical Society of Japan (ISBN 978-4-931469-63-1/hbk). Advanced Studies in Pure Mathematics 60, 51-160 (2010).

A survey is given on the theory of elliptic surfaces with a section.
The first part discusses the general theory of elliptic surfaces, considering the following topics: Kodaira’s classification of singular fibers on an elliptic surface and Tate’s algorithm to determinate the type of the fiber from the Weierstrass equation; the behavior of singular fibers under base change and quadratic twisting; the Mordell-Weil group and the Néron-Severi group of an elliptic surface; torsion in the Mordell-Weil group.
The second part focuses on special results for rational elliptic surfaces and elliptic \(K3\)-surfaces. The following results are discussed in this part of the paper: classification of extremal rational elliptic surfaces; classification of semi-stable rational elliptic surfaces; the Mordell-Weil lattice of a rational elliptic surface; some generalities about elliptic \(K3\)-surfaces (period map, global Torelli, moduli, consequences from lattice theory); singular \(K3\) surfaces (i.e., \(K3\) surfaces with Picard number 20); Shioda-Inose structures; construction of elliptic \(K3\) surfaces with given Mordell-Weil rank; modularity of singular \(K3\) surfaces.
In the final section, the authors discuss the state of the art of the following problem: For which positive integer \(r\) do we know that there exist an elliptic surface with base curve \(\mathbb P^1\) and Mordell-Weil rank \(r\)? In this section, also the Noether-Lefschetz theory for elliptic surfaces is discussed.
For the entire collection see [Zbl 1202.14002].

Reviewer: Remke Kloosterman (Berlin)

MSC:

14J27	Elliptic surfaces, elliptic or Calabi-Yau fibrations
11G05	Elliptic curves over global fields
11G50	Heights
14J20	Arithmetic ground fields for surfaces or higher-dimensional varieties
14J28	\(K3\) surfaces and Enriques surfaces

Keywords:

elliptic surfaces; survey article