A classification of two-dimensional integrable mappings and rational elliptic surfaces
A Halphen pencil is a pencil of degree \(3m\) plane curves having 9 base points, each with multiplicity \(m\). Some of the 9 base points may be chosen infinitely near. Let \(X\) be the surface obtained by blowing up the 9 base points. Then \(X\) is called a generalized Halphen surface of index \(m\).NEWLINENEWLINEIn this paper automorphisms of Halphen pencils are studied in connection with discrete integrability. This is motivated by the results of \textit{H. Sakai} [Commun. Math. Phys. 220, No. 1, 165--229 (2001; Zbl 1010.34083)], who showed that every discrete Painlevé equation can be obtained as a translational component of an affine Weyl group, acting on a family of generalized Halphen surfaces. A special case of automorphisms of generalized Halphen surfaces are QRT-mappings, which are automorphisms of rational elliptic surfaces, i.e., Halphen surfaces of index one, that respect the elliptic fibration.NEWLINENEWLINEIn this paper the period mapping \(\int_{\alpha} \omega\) on generalized Halphen surfaces is studied, where \(\alpha\) is orthogonal to \(\mathrm{div}(\omega)\). Moreover, examples are given for Halpen surfaces of index 2 with automorphisms that do and that do not respect the elliptic fibration.NEWLINENEWLINEReviewer's remark: At a few spots in this paper the terminology in this paper is somewhat unfortunate. The reviewer was not able to find the difference between the notion of generalized Halphen surfaces and the notion of Halphen surface as presented in [\textit{F. R. Cossec} and \textit{I. V. Dolgachev}, Enriques surfaces. I. Boston, MA etc.: Birkhäuser Verlag (1989; Zbl 0665.14017)]. Moreover, the definition of rational elliptic surface is non-standard and does seems not to yield the cited results in section one. This can be solved by requiring that a rational elliptic surface \(X\) is birational to \(\mathbb P^2\) (instead of requiring that \(X\) is not birational to a product \(E\times \mathbb P^1\)).
scientific article; zbMATH DE number 6031389