Abstract
Let \(k\) be a field of characteristic \(p\ge 0\) with \(p\ne 2,3\). We prove that there are no geometrically smooth quartic surfaces \(S \subset \mathbb {P}^3_k\) with more than 64 lines. As a key step, we derive the sharp bound that any line meets at most 20 other lines on \(S\).
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Acknowledgments
We are indebted to Wolf Barth for sharing his insights on the subject starting more than 10 years ago. Thanks to Achill Schürmann for helpful discussions on quadratic forms. We are grateful to Igor Dolgachev, Duco van Straten and the anonymous referee for their valuable comments. This project was started in March 2011 when Schütt enjoyed the hospitality of the Jagiellonian University in Krakow. Special thanks to Sławomir Cynk. Funding by ERC StG 279723 (SURFARI) and NCN Grant N N201 608040 (S. Rams) is gratefully acknowledged.
