## Concurrent lines on del Pezzo surfaces of degree one

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## Abstract:

Let $X$ be a del Pezzo surface of degree one over an algebraically closed field, and $K_X$ its canonical divisor. The morphism $\varphi$ induced by $|-2K_X|$ realizes $X$ as a double cover of a cone in $\mathbb {P}^3$, ramified over a smooth sextic curve. The surface $X$ contains 240 exceptional curves. We prove the following statements. For a point $P$ on the ramification curve of $\varphi$, at most sixteen exceptional curves contain $P$ in characteristic $2$, and at most ten in all other characteristics. Moreover, for a point $Q$ outside the ramification curve, at most twelve exceptional curves contain $Q$ in characteristic $3$, and at most ten in all other characteristics. We show that these upper bounds are sharp, except possibly in characteristic 5 outside the ramification curve.## References

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## Additional Information

**Ronald van Luijk**- Affiliation: Mathematisch Instituut, Niels Bohrweg 1, 2333 CA Leiden, Netherlands
- MR Author ID: 698553
- Email: rvl@math.leidenuniv.nl
**Rosa Winter**- Affiliation: King’s College London, Strand, London WC2R 2LS, United Kingdom
- MR Author ID: 1471783
- ORCID: 0000-0002-5657-3073
- Email: rosa.winter@kcl.ac.uk
- Received by editor(s): April 18, 2020
- Received by editor(s) in revised form: May 4, 2022, and July 10, 2022
- Published electronically: September 12, 2022
- © Copyright 2022 American Mathematical Society
- Journal: Math. Comp.
- MSC (2020): Primary 14J26, 14J45, 14N10
- DOI: https://doi.org/10.1090/mcom/3779