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Projective models of Shioda modular surfaces. (English) Zbl 0599.14035
The Shioda modular surface S(n) [constructed by 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 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. Barth, Hulek and Moore, to appear in: Proc. Tata Conf. on alg. vector bundles over alg. varieties (Bombay 1984)].
Reviewer: H.Lange

##### MSC:
 14J25 Special surfaces 14C20 Divisors, linear systems, invertible sheaves 11F27 Theta series; Weil representation; theta correspondences
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##### References:
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