Exceptional curves on smooth rational surfaces with $-K$ not nef and of self-intersection zero

Abstract

A (-n)-curve is a smooth rational curve of self-intersection -n, where n is a positive integer. In 1998 Hirschowitz asked whether a smooth rational surface X defined over the field of complex numbers, having an anti-canonical divisor not nef and of self-intersection zero, has (-2)-curves. In this paper we prove that for such a surface X, the set of (-1)-curves on X is finite but non-empty, and that X may have no (-2)-curves. Related facts are also considered.