Quadratic enumerative geometry extends classical enumerative geometry. In this enriched setting, the answers to enumerative questions are classes of quadratic forms and live in the Grothendieck-Witt ring GW(k) of quadratic forms. In the talk, we will compute some quadratic enumerative invariants (this can be done, for example, using Marc Levine's localization methods), for example, the quadratic count of lines on a smooth cubic surface.
We will then study the geometric significance of this count: Each line on a smooth cubic surface contributes an element of GW(k) to the total quadratic count. We recall a geometric interpretation of this contribution by Kass-Wickelgren, which is intrinsic to the line and generalizes Segre's classification of real lines on a smooth cubic surface. Finally, we explain how to generalize this to lines of hypersurfaces of degree 2n − 1 in Pn+1. The latter is a joint work with Felipe Espreafico and Stephen McKean.[-]