Integral Models of Extremal Rational Elliptic Surfaces

TL;DR

This paper demonstrates that all extremal rational elliptic surfaces classified in characteristic zero have integral models with good reduction everywhere, and that these models can be reduced mod p to obtain all such surfaces over fields of positive characteristic.

Contribution

It establishes the existence of integral models with good reduction for all classified extremal rational elliptic surfaces and their reduction to positive characteristic fields.

Findings

All classified extremal rational elliptic surfaces have integral models with good reduction.

Every extremal rational elliptic surface over an algebraically closed field of characteristic p > 0 can be obtained by reduction.

The exception of type X_{11}(j) surfaces is noted as an exceptional case.

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.