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The maximum number of lines lying on a K3 quartic surface

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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.

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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.

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Correspondence to Davide Cesare Veniani.

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Veniani, D.C. The maximum number of lines lying on a K3 quartic surface. Math. Z. 285, 1141–1166 (2017).

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