We describe smooth rational projective algebraic surfaces over an algebraically closed field of characteristic different from 2 which contain $n \geq b_2-2$ disjoint smooth rational curves with self-intersection −2, where $b_2$ is the second Betti number. In the last section this is applied to the study of minimal complex surfaces of general type with $p_g = 0$ and $K^{\,2} = 8, 9$ which admit an automorphism of order 2.

A new method to approach enumerative questions about rational curves on algebraic varieties is described. The idea is to reduce the counting problems to computations on the Néron-Severi group of a ruled surface. Applications include a short proof of Kontsevich‘s formula for plane curves and the solution of the analogous problem for the Hirzebruch surface F$_3$.

Let S be a smooth, minimal rational surface. The geometry of the Severi variety parametrising irreducible, rational curves in a given linear system on S is studied. The results obtained are applied to enumerative geometry, in combination with ideas from Quantum Cohomology. Formulas enumerating rational curves are found, some of which generalised Kontsevich‘s formula for plane curves.