Smooth rational curves on rational surfaces

TL;DR

This paper studies the parameter spaces of rational curves on rational surfaces, proving that certain maps are morphisms and establishing irreducibility and dimension properties of these spaces.

Contribution

It demonstrates that the rational map from morphisms of a non-singular curve to the Chow variety is a morphism and shows irreducibility and expected dimension of the scheme of rational curves on rational surfaces.

Findings

The rational map is a morphism for non-singular curves.

The scheme of rational curves is irreducible on rational surfaces.

The scheme has expected dimension when the class has non-negative self-intersection.

Abstract

Consider the scheme parametrizing non-constant morphisms from a fixed projective curve to a projective surface. There is a rational map between this scheme and the Chow variety of $1$-cycles on the surface. We prove that, if the curve is non-singular, then this rational map is a morphism. As a consequence, we obtain that, if the surface is rational and we fix a divisor class containing a non-singular rational curve, then the scheme parametrizing rational curves on this class is irreducible. Further, if the class has non-negative self-intersection, then the scheme of rational curves has expected dimension.