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?


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).

  • $\begingroup$ In fact, can a toric surface have more than 6 exceptional curves? $\endgroup$
    – M P
    Aug 15 '11 at 10:53
  • $\begingroup$ Yes. What is special about 6? $\endgroup$
    – naf
    Aug 15 '11 at 11:21
  • $\begingroup$ I was thinking of the blow up of the plane at 3 points: I thought that you could only blow up torus invariant points that this introduced new exceptional curves at the expense of changing the previous exceptional to $(-2)$-curves or worse... I guess I was wrong! $\endgroup$
    – M P
    Aug 15 '11 at 11:43
  • 3
    $\begingroup$ Blowing up a torus invariant point creates two new torus invariant points on the exceptional curve! $\endgroup$
    – naf
    Aug 15 '11 at 11:46
  • 2
    $\begingroup$ MP, you're correct that if you blow-up a point on a (-1)-curve, the strict transform of the curve has self-intersection -2. Nonetheless, you can have arbitrarily many (-1)-curves on a toric variety. First, by blowing up repeatedly, you can get arbitrarily many torus fixed points. Second, blow up these fixed points, each of which will yield a (-1)-curve. $\endgroup$ Aug 15 '11 at 21:45

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!


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.


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