A quadratic Abramovich-Bertram formula

TL;DR

This paper develops a formula for quadratic genus 0 Gromov--Witten invariants on del Pezzo surfaces, enabling arithmetic counts of curves over arbitrary fields and exploring their behavior under algebraic surgeries.

Contribution

It introduces a new Abramovich-Bertram type formula for quadratic Gromov--Witten invariants, extending their computation through algebraic surgeries on del Pezzo surfaces.

Findings

Derived a formula relating invariants of smoothed and nodal surfaces.

Applied results to rational del Pezzo surfaces of degree ≥ 7 and cubic surfaces.

Established invariance under Dehn twists.

Abstract

Quadratic Gromov--Witten invariants allow one to obtain an arithmetically meaningful count of curves satisfying constraints over a field $k$ without assuming that $k$ is the field of complex or real numbers. This paper studies the behavior of quadratic genus $0$ Gromov--Witten invariants during an algebraic analogue of surgery on del Pezzo surfaces. For this, we define and study (twisted) binomial coefficients in the Grothendieck--Witt group, building on work of Serre. We obtain a formula expressing the quadratic genus $0$ Gromov--Witten invariants of surfaces obtained as a smoothing of a given nodal surface in terms of those of the one having the largest Picard group. We give applications to quadratic Gromov--Witten invariants of rational del Pezzo surfaces of degree at least 7, some cubic surfaces, for point constraints defined over quadratic extensions of $k$, as well as an invariance result under a Dehn twist.