nef Cone of a Toric Variety - MathOverflow

nef Cone of a Toric Variety

2012-01-31T04:37:32Z

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.

Answer by J.C. Ottem for nef Cone of a Toric Variety

2012-01-31T10:08:35Z

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.

Answer by Balazs for nef Cone of a Toric Variety

2012-01-31T11:22:34Z

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.