×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.