Abstract
It is well known that a smooth quartic curve has twenty-eight bitangent lines. For a reduced, possibly singular quartic curve, we introduce the notion of weak-bitangent line. This can be considered as a generalization of bitangent lines. In this article, we consider weak-bitangent lines for certain reduced quartic curves from the viewpoint of rational elliptic surfaces. We utilize Mumford representations of semi-reduced divisors in order to deal with equations of weak-bitangent lines for certain reduced quartic curves. As a result, we can give new proofs for some classical results on singular quartic curves and their bitangent lines.
Acknowledgement
The author would like to thank the referee for many helpful comments on the previous version of this article, especially the idea of the proof in Remark 6.3. The author would also like to thank Professor S. Bannai for his valuable advice.
Citation
Ryosuke Masuya. "Geometry of weak-bitangent lines to quartic curves and sections on certain rational elliptic surfaces." Hiroshima Math. J. 54 (1) 1 - 35, March 2024. https://doi.org/10.32917/h2021060
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