Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter April 16, 2021

Double cover K3 surfaces of Hirzebruch surfaces

  • Taro Hayashi EMAIL logo
From the journal Advances in Geometry

Abstract

General K3 surfaces obtained as double covers of the n-th Hirzebruch surfaces with n = 0, 1, 4 are not double covers of other smooth surfaces. We give a criterion for such a K3 surface to be a double covering of another smooth rational surface based on the branch locus of double covers and fibre spaces of Hirzebruch surfaces.

  1. Communicated by: I. Coskun

References

[1] M. Artebani, J. Hausen, A. Laface, On Cox rings of K3 surfaces. Compos. Math. 146 (2010), 964–998. MR2660680 Zbl 1197.1404010.1112/S0010437X09004576Search in Google Scholar

[2] P. Comparin, A. Garbagnati, Van Geemen-Sarti involutions and elliptic fibrations on K3 surfaces double cover of ℙ2J. Math. Soc. Japan 66 (2014), 479–522. MR3201823 Zbl 1298.1403810.2969/jmsj/06620479Search in Google Scholar

[3] A. Garbagnati, C. Salgado, Linear systems on rational elliptic surfaces and elliptic fibrations on K3 surfaces. J. Pure Appl. Algebra 223 (2019), 277–300. MR3833460 Zbl 0693411710.1016/j.jpaa.2018.03.010Search in Google Scholar

[4] A. Garbagnati, C. Salgado, Elliptic fibrations on K3 surfaces with a non-symplectic involution fixing rational curves and a curve of positive genus. Preprint 2018, arXiv:1806.03097 [math.AG]10.4171/rmi/1163Search in Google Scholar

[5] T. Hayashi, Galois coverings of the product of the projective lines by abelian surfaces. Comm. Algebra 47 (2019), 230–235. MR3924780 Zbl 0705144910.1080/00927872.2018.1472274Search in Google Scholar

[6] K. Hulek, M. Schütt, Enriques surfaces and Jacobian elliptic K3 surfaces. Math. Z. 268 (2011), 1025–1056. MR2818742 Zbl 1226.1405210.1007/s00209-010-0708-3Search in Google Scholar

[7] R. Kloosterman, Classification of all Jacobian elliptic fibrations on certain K3 surfaces. J. Math. Soc. Japan 58 (2006), 665–680. MR2254405 Zbl 1105.1405510.2969/jmsj/1156342032Search in Google Scholar

[8] R. Miranda, The basic theory of elliptic surfaces. ETS Editrice, Pisa 1989. MR1078016 Zbl 0744.14026Search in Google Scholar

[9] V. V. Nikulin, S. Saito, Real K3 surfaces with non-symplectic involution and applications. Proc. London Math. Soc. (3) 90 (2005), 591–654. MR2137825 Zbl 1078.1405310.1112/S0024611505015212Search in Google Scholar

[10] M. Schütt, T. Shioda, Elliptic surfaces. In: Algebraic geometry in East Asia—Seoul 2008, volume 60 of Adv. Stud. Pure Math., 51–160, Math. Soc. Japan, Tokyo 2010. MR2732092 Zbl 1216.14036Search in Google Scholar

[11] H. Yoshihara, Smooth quotients of bi-elliptic surfaces. Beitr. Algebra Geom. 57 (2016), 765–769. MR3550417 Zbl 1375.1414710.1007/s13366-016-0310-xSearch in Google Scholar

Received: 2018-12-03
Revised: 2019-05-01
Revised: 2019-07-10
Published Online: 2021-04-16
Published in Print: 2021-04-27

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 29.3.2024 from https://www.degruyter.com/document/doi/10.1515/advgeom-2020-0034/html
Scroll to top button