Loïs Faisant

The Grothendieck ring of varieties is a ring defined by generators and relations. Generators are isomorphism classes of varieties over a fixed field. Relations are the well-known scissors relations : if $U$ is an open subset of a variety $X$, then the class of $X$ is the sum of the class of $U$ and the class of its complement in $X$.

This ring provides a relevant framework to study the asymptotic behaviour of moduli spaces of rational curves on a fixed variety, when the degree of such curves becomes infinitely large. This question is the geometric analogue of the asymptotic study of rational points of bounded height, when the bound tends to infinity, known as the Manin conjecture (1989) refined by Peyre (1995).

In this talk, we will start presenting this ring of varieties and providing a bunch of simple and easy examples.
Then we will (try to) state the motivic Manin-Peyre conjecture (early 2000s) in a comprehensible way.
In a second part, we will check that this conjecture is true for the projective space, and state similar results obtained for compactifications of vector spaces.