Number of $(-1)$ curves on toric surfaces Hello.
My question is:
Is it possible that a smooth complete toric surface has infinitely many $(-1)$-curves. I know that there is a blow-up of $\mathbb P^2$ in 9 points containing infintely many $(-1)$-curves, but is it possible to construct a toric surface with this property?
 A: I can't post this as a comment.  A toric surface can have any (finite) number of exceptional curves.  In the fan description, blowing up at a point is just replacing the fan with the fan obtained by adding the sum of two adjacent vectors.  Add adjacent vectors as often as you like, you always get a perfectly good fan!
A: This is rather late since the question has long since been settled, but I wanted to make the following comment (which is too long for the comment box): when it comes to questions of this sort, toric varieties never have infinitely many of anything.
Less flippantly, what I mean is the following: for a variety $X$ of any dimension, generalising the question about the number of $(-1)$-curves, one can ask about either 


*

*the number of extremal rays of the
cone $Eff(X)$ of effective, or

*the number of extremal rays of the cone $Nef(X)$ of nef divisors.
(In the surface case, every $(-1)$-curve spans an extremal ray of $Eff(X)$, and corresponds by duality to a codimension-1 face of $Nef(X)$.)
Now my point is just that for $X$ a toric variety of any dimension, both cones $Nef(X)$ and $Eff(X)$ are known to be closed cones spanned by a finite set of vectors. Indeed, there is the following statement, due to Cox, which lncludes those two statements:
Theorem (Cox): The Cox ring of a toric variety is finitely generated.
A: No, toric surfaces can have only finitely many $(-1)$-curves. Since $(-1)$-curves are rigid, i.e., cannot form a non-trivial family, it follows that any $(-1)$-curve must be contained in the complement of the open torus orbit (which is a finite union of curves).
