On a stratification of the Kontsevich space of the Grassmannian G(2,4) and enumerative geometry

We study the geometry of the Kontsevich compactification of stable maps to the Grassmannian of lines in the projective space. We consider a stratification of this space. As an application we compute the degree of the variety parametrizing rational ruled surfaces with a minimal directix of degree d/2-1 by intersecting divisors in the moduli space of stable maps. For example, there are 128054031872040 rational ruled sextics passing through 25 points in $\mathbb{P}^{3}$ with a minimal directrix of degree 2.