A quadratic Abramovich-Bertram formula

Erwan Brugallé; Kirsten Wickelgren

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arXiv · 2506.17854

A quadratic Abramovich-Bertram formula

Erwan Brugallé·Kirsten Wickelgren

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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.

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Erwan Brugallé, Kirsten Wickelgren. 2025-06-21. A quadratic Abramovich-Bertram formula. https://arxiv.org/abs/2506.17854

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