Exceptional curves on rational numerically elliptic surfaces (Q922600)

Let X be a numerically elliptic surface, i.e. X is a smooth surface over an algebraically closed field k with a proper k-morphism \(f:X\to C\) to a smooth curve C such that the generic fiber of f is an integral curve of arithmetic genus 1. X is said to be minimal if no fiber of f contains an exceptional curve of the first kind. The aim of the paper is to determine and enumerate, by combinatorial methods, the exceptional curves on a minimal rational numerically elliptic surface X, having no section of f. The authors obtain a formula for the number of such curves for X ``extremal''. The complete result is somewhat lengthy to state.