The maximum number of lines lying on a K3 quartic surface
- 139 Downloads
We show that there cannot be more than 64 lines on a quartic surface with isolated rational double points over an algebraically closed field of characteristic \(p \ne 2,\,3\), thus extending Segre–Rams–Schütt theorem. Our proof offers a deeper insight into the triangle-free case and takes advantage of a special configuration of lines, thereby avoiding the technique of the flecnodal divisor. We provide several examples of non-smooth K3 quartic surfaces with many lines.
Mathematics Subject Classification14J28 14J70 14N10 14N25
First of all, I wish to warmly thank my supervisor Matthias Schütt and Sławek Rams for suggesting the problem and paving the way to solve it. Many new ideas are due to the fruitful discussions with Alex Degtyarev, who made my stay in Ankara extremely pleasant. Thanks to Miguel Ángel Marco Buzunáriz for introducing me to SageMath, to Roberto Laface for carefully proofreading the draft, and to Víctor González Alonso and Simon Brandhorst for their invaluable mathematical help.
- 3.Degtyarev, A., Itenberg, I., Sertöz, A.S.: Lines on Quartic Surfaces. arXiv:1601.04238
- 5.González Alonso, V., Rams, S.: Counting Lines on Quartic Surfaces. arXiv:1505.02018
- 7.Heijne, B.: Picard Numbers of Complex Delsarte Surfaces with Only Isolated ADE-Singularities. arXiv:1212.5006v4
- 8.Liedtke, C.: Algebraic Surfaces in Positive Characteristic. arXiv:0912.4291v4