In my previous post, I told you how to create a toric variety from a collection of cones. I also told you that the cones had to satisfy certain, unspecified compatibility conditions. (And that, starting from any lattice polytope, you could build a legitimate collection of cones.)
In this post, I want to tell you what these conditions are. I also want to introduce the way that most experts think about these matters, in terms of fans. For most people, fans are less intuitive than the collections of cones I used in the previous post. But, if you are going to want to think about toric varieties, you should eventually learn how to use the fan language.
So, let’s suppose that we have a finite collection of cones in . (To be careful, you want to assume that each of the cones is rational, a hypothesis which excludes cones like .) If we had any collection of cones at all, each cone would give us a variety. Explicitly, for each cone , we take the semigroup in . This corresponds to a subring of , which we’ll call . Then we get a variety, , as .
Gluing the together would give us something. However, there are two problems: First, the space we get will not be Hausdorff (for those who prefer the language of schemes, will not be separated.) Second, the collection will not be recoverable from the scheme we get.
We solve both of these problem by imposing two conditions on the collection .
(a) If is in , and is any vector in , then is also in .
(b) If and are in then there are elements and such that .
Here means and means .
Let’s think about where these conditions come from. In condition (a), suppose that . So is a monomial in . Then is the localization . The translation between rings and varieties tells us that is then an open subvariety of . When we glue into our toric variety, we will also glue in its open subset . Axiom (a) tells us to include this open subset in our data.
Axiom (b) tells us that we have good patches to glue along. Specifically, it says that the overlap between and is an open subset of each, of the kind discussed in the previous paragraph.
Now, these axioms aren’t that bad. (You might enjoy checking that all the collections of cones in the previous post satisfy them.) But someone (Fulton?) had the brilliant realization that they would look much nicer if we dualized.
Let be a cone in . We define the dual cone, to be . Here is the standard inner product on . Properly speaking, and live in dual vector spaces, but I’m going to ignore that. I should also note that you’ll find different conventions on which sign to use in this definition so, if you need that level of precision, check which convention your source is using.
In our previous post, when we considered , there were seven cones: , , , , , and . The corresponding dual cones are , , , , , and .
Notice that we can draw all of the dual cones together without overlaps. In fact, I’ve had to slide them apart a little just so you’ll see that there are 7 cones here. I urge you to compare this figure to the third figure in the previous post.
More precisely, the dual cones are a fan. By definition, a fan is a finite collection of cones in obeying two axioms:
(a’) If is a cone of , and is a face of , then is in .
(b’) If and are cones in , then is a face of both and .
These conditions simply arise by translating axioms (a) and (b) into the dual language.
And, finally, I can tell you the definition of a toric variety. A toric variety is a variety built from a fan by (i) dualizing the cones of that fan and (ii) applying the recipe of the previous post.
Let’s have one more example. In the previous post, I drew blown up at . Let’s see how that looks in the fan picture. This time, I haven’t slid the cones apart, but please remember: there are 6 cones in this picture.
How can we directly see the geometry of the fan from the toric variety? Every toric variety comes equipped with an action of the torus . Intuitively, an element of takes to . To make that precise, one should describe an action of on the ring ; I leave that an an exercise for the reader.
Now, each point in gives a map , taking to . Take a generic point in the toric variety and consider . This limit exists if and only if . Call this limit ; then if and only if and lie in the interior of the same cone of .
So, the direct meaning of the fan is to describe the limiting behavior under the action of the one-parameter subgroups of .
Update: The second figure was accidently rotated; it is correct now.