A quadratically enriched count of rational curves

TL;DR

This paper introduces a quadratic enrichment of rational curve counts on del Pezzo surfaces, unifying Gromov-Witten and Welschinger invariants through algebraic and motivic methods over various fields.

Contribution

It defines a new quadratic count of rational curves on del Pezzo surfaces that generalizes classical invariants and is compatible with $ ext{GW}(k)$-valued degrees in algebraic topology.

Findings

Recovers Gromov-Witten invariants over $ ext{C}$

Recovers Welschinger invariants over $ ext{R}$

Provides a quadratic count applicable over various fields

Abstract

We define a quadratically enriched count of rational curves in a given divisor class passing through a collection of points on a del Pezzo surface $S$ of degree $\geq 3$ over a perfect field $k$ of characteristic $\neq 2,3.$ When $S$ is $\mathbb{A}^1$-connected, the count takes values in the Grothendieck-Witt group GW(k) of quadratic forms over $k$ and depends only on the divisor class and the fields of definition of the points. More generally, the count is a section of the Grothendieck-Witt sheaf evaluated on $\pi_0^{\mathbb{A}^1}$ of the restriction of scalars of $S$ corresponding to the fields of definition of the points. We also treat del Pezzo surfaces of degree $2$ under certain conditions. The curve count defined in the present work recovers Gromov-Witten invariants when $k = \mathbb{C}$ and Welschinger invariants when $k = \mathbb{R}.$ To obtain an invariant curve count, we define a quadratically enriched degree for an algebraic map $f$ of $n$-dimensional smooth schemes over a field $k$ under appropriate hypotheses. For example, $f$ can be proper, generically finite and oriented over the complement of a subscheme of codimension $2.$ This degree is compatible with F. Morel's GW(k)-valued degree of an $\mathbb{A}^1$-homotopy class of maps between spheres. For $k \subseteq \mathbb{C}$, this produces an enrichment of the topological degree of a map between manifolds of the same dimension.