Abstract

Gromov--Witten invariants and Welschinger invariants count curves over the complex and realnumbers. In joint work with J. Kass, M. Levine, and J. Solomon, we gave arithmetically meaningfulcounts of rational curves on smooth del Pezzo surfaces over general fields. This talk concerns howthese invariants change under an algebraic analogue of surgery along a Lagrangian sphere. Weallow certain del Pezzo surfaces to acguire a -2 curve and study deformations of curves to give anarithmetic enrichment of a formula due to D. Abramovich and A. Bertram over C and due to E.Brugall and N. Puignau over R. This is joint work with Erwan Brugall.