Counting Plane Rational Curves: Old and New Approaches

These notes are intended as an easy-to-read supplement to part of the background material presented in my talks on enumerative geometry. In particular, the numbers $n_3$ and $n_4$ of plane rational cubics through eight points and of plane rational quartics through eleven points are determined via the classical approach of counting curves. The computation of the latter number also illustrates my topological approach to counting the zeros of a fixed vector bundle section that lie in the main stratum of a compact space. The arguments used in the computation of the number $n_4$ extend easily to counting plane curves with two or three nodes, for example. Finally, an inductive formula for the number $n_d$ of plane degree-$d$ rational curves passing through $3d-1$ points is derived via the modern approach of counting stable maps.