Gromov--Witten invariants and Welschinger invariants count curves over the complex and real numbers. In joint work with J. Kass, M. Levine, and J. Solomon, we gave arithmetically meaningful counts of rational curves on smooth del Pezzo surfaces over general fields. This talk concerns the behavior of these invariants and generalizations using motivic homotopy theory. With E. Brugallé, we compute how these invariants change under an algebraic analogue of surgery along a Lagrangian sphere. We allow certain del Pezzo surfaces to acquire a -2 curve and use binomial coefficients in the Grothendieck Witt group to give an arithmetic enrichment of a formula due to D. Abramovich and A. Bertram over C and due to E. Brugallé and N. Puignau over R.