Erwan Brugallé: A quadratically enriched Abramovich-Bertram formula

Speaker

Erwan Brugallé, Nantes Université

Abstract

By interpreting 1 as the unique complex quadratic form \(z->z^2\), some classical enumerations (i.e. with values in \(\mathbb N\)) acquire meaning when the field of complex numbers is replaced with an arbitrary field \(k\). The result of the enumeration is then a quadratic form over \(k\) rather than an integer.
This talk will focus on such enumeration for rational curves in del Pezzo surfaces. In particular I will report on a recent joint work with Kirsten Wickelgren where we generalize a formula originally due to Abramovich and Bertram in the complex setting, that I later extended over the real numbers. This quadratically enriched version of the AB-formula relates enumerative invariants for different \(k\)-forms on the same del Pezzo surfaces.