# nef Cone of a Toric Variety

Is there a systematic/standard way of extracting the nef cone of a toric variety $X$ from its fan (or polytope)? Can I tell from the basis of $A^1(X)$ induced by the one-skeleton, based on coefficients, when a given divisor class is nef?

In particular I am working working with blowups of $\mathbb{P}^n$. I am uncertain if that extra piece of information helps.

I imagine that the answer to my question is yes, but I havent yet found a place where this is cleanly articulated.

Thanks in advance.

• Since you are interested in blow-ups of $\mathbb P^n$, the corresponding fans will have convex support and finite dimension, so their Cartier divisors are nef iff they are basepoint free, in particular this is equivalent to having a convex support function. This is explained in sections 6.1-6.3 of Cox, Little and Schenk. Am I missing something? – Gjergji Zaimi Jan 31 '12 at 10:08
• I meant to say "full dimension" instead of "finite dimension". – Gjergji Zaimi Jan 31 '12 at 10:24
• What kind of blow-ups are you interested in? Along torus invariant subschemes I presume? – J.C. Ottem Jan 31 '12 at 15:53
• Yes the blowups I am interested in are all torus invariant. Gjergji, I will take a closer look at the sections you mentioned. – Dhruv Feb 1 '12 at 17:46
• At least when you are blowing up points, I don't think you need toric geometry to find the nef cone. Just look at strict transforms of hypersurfaces going through the points. As Gjergji points out nef divisors here are base point free, so what you want to look at is linear systems without fixed components that move in large linear systems. e.g., when $n=2$ and you blow-up the 3 torus invariant points, choose $L,E_1,E_2,E_3$ as a basis for $Pic(X)$. – J.C. Ottem Feb 2 '12 at 15:01

## 2 Answers

You can use the fact that a divisor class $D$ on a toric variety is nef if and only if it has non-negative intersection with the finitely many classes of torus invariant curves. If you are using the torus invariant divisors as the basis for $A^1(X)$, then these numbers are easy to compute combinatorially and this will give you a finte set of linear inequalities for the nef cone. See the the book by Cox-Little-Schenck chapter 6 for more details.

As far as I know, there is no explicit description of the rays of the nef cone in terms of the combinatorics of the fan, so this approach with linear inequalities is the best you can do.

What you are interested in is toric Mori theory. This was first written down by Miles Reid back in the 80s (Decomposition of toric morphisms, incidentally to the best of my knowledge the first paper which wrote out the main steps of Mori's programme.) If you google "toric Mori theory", there are plenty of other hits; I checked Wisniewski's nice Toric Mori theory and Fano manifolds which may well do for you.