Exceptional curves on rational numerically elliptic surfaces

Abstract

A numerically elliptic surface is a complete smooth algebraic surface X over an algebraically closed field k with a proper morphism f: X-* C to a smooth curve C such that the general fiber off is an integral curve of arithmetic genus 1. If the generic fiber is smooth the surface is called elliptic; otherwise it is called quasi-elliptic. The latter only can occur if k has characteristic 2 or 3, in which case the general fiber is a rational curve with an ordinary cusp [BM]. If no fiber off contains an exceptional cwue (i.e., a smooth irreducible curve isomorphic to [Fp’ and having self-intersection l), then X is said to be minimal; all elliptic surfaces will hereafter be assumed to be minimal. One says that an elliptic surface X is Jacobian if the smooth points of the generic fiber X, comprise the Jacobian curve of X,,. It is equivalent for X, to have a rational point, or for f to have a section. For a rational Jacobian numerically elliptic surface, the exceptional curves are precisely the sections of the libration, which provides a tool by which an enumeration of the exceptional curves on a rational Jacobian numerically elliptic surface can be carried out (cf. [MP, HL, MOP]). Whether Jacobian or not, a rational