An Introduction to A1-enumerative geometry

Let $k$ be an arbitrary field and $GW(k)$ its Grothendieck-Witt ring, that is, the group completion of the semi-ring of non-degenerate symmetric bilinear forms over $k$.

New techniques from A1-homotopy theory allow us to count algebro-geometric objects as elements in $GW(k)$. One of the first examples of such enriched counts was the count of lines on a smooth cubic surface by Kass-Wickelgren. The resulting symmetric bilinear form contains information about the count over the chosen base field $k$ and recovers the known classical counts over the complex and real numbers.

In my talk I will explain how to count in $GW(k)$ with the help of some illustrating examples.