Projective models of Shioda modular surfaces (Q1080492)

The Shioda modular surface S(n) [constructed by \textit{M. Shioda} in J. Math. Soc. Japan 24, 20-59 (1972; Zbl 0226.14013)] may be considered as the universal elliptic curve with level n-structure. This paper investigates some geometrically important projective realizations of S(3), S(4), and S(5). To describe the results let \(L_{ij}\) \((i,j=1,...,n)\) denote the sections of S(n) corresponding to the level n- structure and F a fibre of S(n). For \(n=3\) there is a unique divisor class I in S(3) with \(3I\sim \sum L_{ij}+F\) and the linear system \(| I|\) defines a birational map \(S(3)\Rightarrow {\mathbb{P}}_ 2\) which identifies S(3) with the plane \({\mathbb{P}}_ 2\) blown up in the 9 base points of the Hesse pencil. For \(n=4\) there is a divisor class I in S(4), unique up to 2-torsion, such that 2I\(\sim \sum L_{ij}.\) Hence I determines a 2:1 covering \(\tilde S\to S(4)\) branched along the section \(L_{ij}\). Using this it is shown that S(4) is isomorphic to the Kummer surface associated to \(E\times E\) where E is the unique elliptic curve admitting an automorphism of order 4. The relation to the result of \textit{H. Inose} [cf. J. Fac. Sci., Univ. Tokyo, Sect. IA 23, 545-560 (1976; Zbl 0344.14009)] identifying the Fermat quartic \(F_ 4\) in \({\mathbb{P}}_ 4\) as a certain Kummer surface is outlined: There is an isogeny \(F_ 4\to S(4)\) of degree 4. For \(n=5\) there is a divisor class I in S(5), unique up to 5-torsion, such that 5I\(\sim \sum L_{ij}.\) The linear system \(| I+2F|\) defines an immersion of S(5) as a surface of degree 15 with 30 double points in \({\mathbb{P}}_ 4.\) \(| I+3F|\) defines an embedding of S(5) into \({\mathbb{P}}_ 9\). The image is the transversal intersection of a Grassmannian and a Segre variety of degree 25 in \({\mathbb{P}}_ 9.\) \(| 3I+3F|\) defines a birational map onto a surface of degree 45 in \({\mathbb{P}}_ 9\). The map contracts the 25 sections to isolated singularities and is an isomorphism elsewhere. \(| 3I+5F- \sum P_{ik}|\) defines a birational morphism onto a surface of degree 45 in \({\mathbb{P}}_ 4\). Here \(P_{ik}\) denote the 60 vertices of the singular fibres of S(5). The realisations of S(5) arise in connection with the Horrocks-Mumford bundle on \({\mathbb{P}}_ 4\) [cf. \textit{Barth}, \textit{Hulek} and \textit{Moore}, to appear in: Proc. Tata Conf. on alg. vector bundles over alg. varieties (Bombay 1984)].