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

TL;DR

This paper establishes orientation results for evaluation maps in moduli spaces of rational stable maps to del Pezzo surfaces, enabling enriched counts of rational curves over various fields, with implications in algebraic geometry.

Contribution

It provides the first orientation results for evaluation maps on these moduli spaces, linking ramification loci to cusps and discriminants, and develops a theory for quadratically enriched curve counts.

Findings

Orientation results valid in characteristic 0 and positive characteristic.

Ramification locus corresponds to cusps on image curves.

Framework for quadratically enriched counts over non-algebraically closed fields.

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.