A quadratic Abramovich–Bertram formula

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 analog 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.