A relative orientation for the moduli space of stable maps to a del Pezzo surface

Abstract

We prove orientation results for evaluation maps of moduli spaces of rational stable maps to del Pezzo surfaces over a field, both in characteristic $0$ and in positive characteristic. These results and the theory of degree developed in a sequel produce quadratically enriched counts of rational curves over non-algebraically closed fields of characteristic not $2$ or $3$. Orientations are constructed in two steps. First, the ramification locus of the evaluation map is shown to be the divisor in the moduli space of stable maps where image curves have a cusp. Second, this divisor is related to the discriminant of a branched cover of the moduli space given generically by pairs of points on the universal curve with the same image.