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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Concurrent lines on del Pezzo surfaces of degree one
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by Ronald van Luijk and Rosa Winter PDF
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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