Seminario Nacional de Geometría Algebraica en línea

Resumen: 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. They were developed in joint work with Kass, Levine, and Solomon in genus 0 for del Pezzo surfaces. In joint work with Erwan Brugallé and Johannes Rau, we give a complete calculation of these invariants for k-rational del Pezzo surfaces of degree greater than 5.

Moreover, we give these invariants the structure of an unramified Witt invariant for any fixed surface and degree. We then construct a multivariable unramified Witt invariant which conjecturally contains all of these invariants for k-rational surfaces. To prove the conjecture for del Pezzo surfaces of degree greater than 5 and obtain the calculation, we study the behavior of these Gromov-Witten invariants during an algebraic analogue of surgery on del Pezzo surfaces. We obtain a surprisingly simple formula when uncomputable terms cancel out with an identity in (twisted) binomial coefficients in the Grothendieck-Witt group. This is joint work with Brugallé and Rau.