Integral Models of Extremal Rational Elliptic Surfaces

Abstract

Miranda and Persson classified all extremal rational elliptic surfaces in characteristic zero. We show that each surface in Miranda and Persson's classification has an integral model with good reduction everywhere (except for those of type X ₁₁(j), which is an exceptional case), and that every extremal rational elliptic surface over an algebraically closed field of characteristic p > 0 can be obtained by reducing one of these integral models mod p.

Key Words:

• Elliptic surfaces
• Extremal surfaces
• Integral models

2010 Mathematics Subject Classification:

• 14J20

ACKNOWLEDGMENTS

This article began as a research project by Jeremy Ricks, a BYU undergraduate student, under the direction of Tyler Jarvis. Unfortunately, Jeremy died in a tragic accident in September of 2001, before his work on the project was completed. Both surviving authors were greatly impressed by Jeremy's mathematical ability and his dedication to this project. He produced many of the models given here in a remarkably short amount of time. We are extremely grateful to Jeremy's wife, Melinda, for providing us with Jeremy's notes. Without them, this article would not exist.

We also wish to remember Professor Steven Galovich, who passed away in 2006. Steve was the senior author's mentor at Carleton College (long before the concept of undergraduate mentoring became fashionable) and provided him with an outstanding introduction to the world of algebra and number theory.

The senior author would also like to thank C. Schoen, D. Doud, J. Grout, and K. Rubin for useful conversations on elliptic curves and surfaces, and Brigham Young University for computer support.

Finally, we would like to thank M. Schütt and the referee for their comments on a previous version of this article.

Notes

Communicated by L. Ein.

Dedicated to the memory of Jeremy R. Ricks and Steven Galovich.