3
$\begingroup$

It is well known that a (smooth complete) fan $\Delta$ corresponds to a (smooth proper) toric variety $X= X_\Delta$.

My question is whether there is a relationship between the number of maximal cones in $\Delta$ and a geometric invariant of $X$. (as the number of 1-dimensional cones is the number of torus invariant prime divisor)

Here, we call a cone in $\Delta$ maximal if it is not contained in any other cone in $\Delta$.

Improve this question
$\endgroup$
1
  • $\begingroup$ The reason why I asked this question is that $\# \Delta_{\textrm{max}}$ appeared in the Tamagawa number (in the sense of Peyre) of a universal torsor over the toric variety. $\endgroup$
    – Kazuki Sato
    Commented Mar 23, 2017 at 0:50

1 Answer 1

Reset to default
4
$\begingroup$

Yes, the number of maximal cones is the same as the topological Euler characteristic.

This is the result of the natural generalization of the thing you already pointed out: the number of codimension $r$ irreducible $T$-invariant subvarieties is the number of $r$-dimensional cones in $\Delta$.

So more precisely, maximal cones give the $T$-fixed points. The other $T$-orbits all have topological Euler characteristic zero, and so they contribute zero to the overall Euler characteristic (since for $Y\subset X$ an open subspace of a space $X$, $\chi(X)=\chi(U)+\chi(X\setminus U)$).

Fulton's book explains this all very well.

Improve this answer
$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .