Schemes of rational curves on Del Pezzo surfaces

The main objects of study in this thesis are schemes parametrizing morphisms from the projective line to projective varieties. The first part of this thesis is dedicated to understand properties obtained from the functor of points of this parameter space, such as the preservation of open an closed immersions of schemes. Furthermore, for any projective variety, there is a natural partition of these schemes on closed subschemes in terms of the degrees of the morphisms. In the second part of this thesis, we use the properties studied on the first part to find a natural partition of the scheme of morphisms from the projective line to the blow-up of projective spaces at finitely many points which refine the aforementioned partition. We fully characterize this refinement on the case of Del Pezzo surfaces obtained by blowing up the projective plane at up to eight points in general position. Moreover, we use this to characterize irreducible components parametrizing rational curves which are resolutions of singularities of plane curves via this blow-up and to compute their dimension.