Oliver Club

Inna ZakharevichCornell University
Points, Determinants, and the mystery of K_1(Var)

Thursday, September 2, 2021 - 4:00pm

The Grothendieck ring of varieties is sometimes referred to as "baby motives" because it is one of the first attempts to understand what "motives"---i.e. fundamental pieces of varieties---ought to be.) It is defined to be the free abelian group generated by varieties (over some base field k) modulo the relations [X] = [Y] + [X \ Y]; multiplication is defined via the Cartesian product [X][Y] = [X x Y]. This ring is surprisingly complicated; for example, when k=C the affine line is a zero divisor. It is often analyzed using "motivic measures": ring homomorphisms to other, more tractable, rings. If the base field k is finite there is a motivic measure K_0(Var) --> Z taking each variety to its number of k-points. It turns out that the Grothendieck ring of varieties can be defined as the connected components of a space, called K(Var), and many motivic measures lift to maps out of this space. This implies that motivic measures can also be used to study the higher homotopy groups of this space. In this talk I will give a more detailed introduction to the Grothendieck ring of varieties, discuss possible interpretations of K_1(Var), and show how to detect interesting elements in K_1(Var) using point counting.

Zoom room will open at 3:45 PM for chat before the talk.