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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

14J25 Special surfaces
14C20 Divisors, linear systems, invertible sheaves
11F27 Theta series; Weil representation; theta correspondences
Full Text: DOI EuDML
[1] Barth,W., Hulek,K., Moore,R.: Shioda’s modular surface S(5) and the Horrocks-Mumford bundle. To appear in: Proceedings of the Tata conference on algebraic vector bundles over algebraic varieties. Bombay 1984
[2] Barth,W., Peters,C., Van de Ven,A.: Compact Complex Surfaces. Springer Verlag, Berlin, Heidelberg, New York 1984 · Zbl 0718.14023
[3] Burns,D.: On the geometry of elliptic modular surfaces and representations of finite groups. In: Springer Lecture Notes in Math., Vol. 1008, 1-29 (1983) · Zbl 0543.14022 · doi:10.1007/BFb0065696
[4] Grauert,H., Remmert,R.: Analytische Stellenalgebren. Grundlehren Math. Wiss. 176, Springer Verlag, Heidelberg 1971 · Zbl 0231.32001
[5] Hirzebruch,F.: Arrangements of lines and algebraic surfaces. In: Progr. in Math. Vol. 36, p.113-140 (1983) · Zbl 0527.14033
[6] Hirzebruch,F.: Chern numbers of algebraic surfaces. An example. Math. Ann. 266, 351-356 (1984) · Zbl 0504.14030
[7] Horrocks,G., Mumford,D.: A rank 2 vector bundle on ?4 with 15.000 symmetries. Topology 12, 63-81 (1973) · Zbl 0255.14017 · doi:10.1016/0040-9383(73)90022-0
[8] Hulek,K.: Projective geometry of elliptic curves. Preprint, Providence 1982 · Zbl 0602.14024
[9] Inose,H.: On certain Kummer surfaces which can be realized as nonsingular quartics in 1c3. Journal Fac. Sc. Tokyo 23, 545-560 (1976) · Zbl 0344.14009
[10] Ishida, M.-N.: Hirzebruch’s examples of surfaces of general type with Hirzebruch’s examples of surfaces of general type with c 1 2 =3c2. In: Springer Lecture Notes in Math., Vol. 1016, 412-431 (1983) · doi:10.1007/BFb0099973
[11] Mumford,D.: Tata lectures on theta I. Progr.in Math. Vol. 28, Birkhäuser Boston 1983 · Zbl 0509.14049
[12] Naruki,I.: Über die Kleinsche Ikosaeder-Kurve sechsten Grades. Math. Ann. 231, 205-216 (1978) · Zbl 0358.14016 · doi:10.1007/BF01420241
[13] Shioda,T.: On elliptic modular surfaces. J. Math. Soc. Japan 24, 20-59 (1972) · Zbl 0226.14013 · doi:10.2969/jmsj/02410020
[14] Shioda,T.: Algebraic cycles on certain K3 surfaces in characteristic p. In: Proc. Int. Congr. on Manifolds, 357-364, Univ. Tokyo Press 1975 · Zbl 0311.14007
[15] Shioda,T., Inose,M.: On singular K3 surfaces. In: Complex Analysis & Algebraic Geometry, 119-136, Cambridge Univ. Press 1977 · Zbl 0374.14006
[16] Piatetcky-Shapiro,I., Shafarevich,I.: A Torelli theorem for algebraic surfaces of type K3. English translation: Math. USSR Izv. 5, 547-588 (1971) · Zbl 0253.14006 · doi:10.1070/IM1971v005n03ABEH001075
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