4

Let $k$ be a field (if you want, $k=\mathbb C$). The Grothendieck group of varieties is the Abelian group generated by isomorphism classes of $k$-varieties, subject to the relation $[Y]=[X]+[Y\setminus X]$ whenever $X$ is a closed subvariety of $Y$. This group, denoted $K_0(\textrm{Var}_k)$, can be made into a commutative ring by letting $$[X]\cdot [Y]=[X\times_kY].$$ There is a neutral element for the addition, $0=[\emptyset]$, and a neutral element $1=[\textrm{Spec }k]$ for multiplication. It seems like we have all we need to construct the spectrum of this ring.

Question: What is known about the geometry of $\textrm{Spec }K_0(\textrm{Var}_k)$?

The $\mathbb Z$-valued points of $\textrm{Spec }K_0(\textrm{Var}_{\mathbb C})$ are called "generalized Euler characteristics". What about other points?

The question is very broad, but let me show you my ignorance better: what is the dimension, what are its $k$-points, is it singular, how many components does it have, when is it reduced, $\dots$?

In short: is it an interesting ring to study, and why? (Feel free to restrict to $k=\mathbb C$ if you wish to.)

Thanks!

Brenin
  • 14,072
  • As the link in Sergio's answer shows, the ring is not a domain, so its spectrum is not integral; Borisov very recently showed that even $\mathbf L$ is a zero-divisor. Passing to pure speculation based on very little thought, I would be very surprised if (a) it is smooth; (b) it is finite-dimensional; (c) there are more than a handful of nontrivial results about its geometry. –  Jan 26 '15 at 10:18
  • @AsalBeagDubh: I see. Probably a lack of interest is due to the infinite dimensional issue. But still I find it annoying not being able to see what a prime ideal is in this ring... – Brenin Jan 26 '15 at 17:39

1 Answers1

1

I was about to comment that this question would probably fair a lot better on Math Overflow. A quick search on the site leads to:

https://mathoverflow.net/questions/319/spectrum-of-the-grothendieck-ring-of-varieties

The question is not easy to answer as is best described by the answers in the link.

Community
  • 1
Sergio Da Silva
  • 1,970
  • Thank you! That question seems to be about $(K_0(\textrm{Var}_k))[\mathbb L^{-1}]$, but probably not so much changes. Do you know a reference for some of the properties I mentioned? or even those I haven't mentioned? – Brenin Jan 26 '15 at 08:40