Toric varieties are among the simplest objects in algebraic geometry, and they are often a test case for general conjectures. One of the great things about them is that you can completely describe a toric variety by combinatorial data. In this post, I want to explain how this works.
The Projective Plane
To begin with, let’s talk about . I’ll drop the from now on, as we’ll always be thinking over the complex numbers. I’ll write homogenous coordinates on as .
To describe in local coordinates, we need three charts: , and ; we’ll call these , and . On the chart , local coordinates are given by . Similarly, on the chart , the local coordinates are .
Now, of course, there are many ways to choose a system of coordinates on a given chart. Sure, is one coordinate system on , but is another. In algebraic geometry, we remove dependence on this sort of choice by not specifying a system of coordinates, but specifying the ring they generate instead. In the example of , this is the polynomial ring ; this ring depends only on the open set and is denoted .
In the particular case of , all of these rings have bases of monomials. I’ll draw a picture to indicate which monomials are in , and . That dot which lies in all three cones (inside the dark triangle) is the origin. The dot directly to the above of the origin is , and the dot directly to the right of the origin is .
Let’s look at how and are glued together. Write for the intersection . The functions on form the ring . We have a diagram
and is the pushout of this diagram.
Whenever we have a map of spaces , we have a corresponding map of rings from the functions on to the functions on ; this map is simply pullback. So, we have maps of rings
Again, we can draw a picture:
I encourage you all to draw the pictures for all three charts, their pairwise overlaps and the overlap of all three. I won’t draw that picture, because it is a bit cluttered. We can make it more legible by using three tricks: (1) Slide these cones apart onto different regions of the plane, agreeing tacitly that only the shape of the cone, and not its absolute position, matters. (2) Cut off each cone with a circular arc once it gets too far away from the translated origin. (3) Stop drawing the underlying lattice. So here is how I would draw .
Note that the big blob in the center corresponds to the triple overlap; the corresponding cone is the whole of . Right now, the position of the cones in is meaningless, but later we will see why I drew them to line up like this.
In general, a toric variety is an algebraic variety that can be described by drawing a bunch of cones like this, where the cones have to obey certain compatibility conditions.
is blown up at .
Constructing a toric variety from a polytope
Stating the compatibility conditions on the cones is a bit technical, so I’ll skip it for now. Instead, I’ll describe a way to get cones that always will obey the required conditions, although not all toric varieties occur in this way.
As you can see, looks like a triangle to me. In general, if we have any polytope , we can get a collection of cones by the following recipe: There will be one cone for each face of . This cone is the set of all vectors of the form , where , and is a positive scalar. Recall that we allow ourselves to translate cones for ease of drawing; it is usually convenient to draw this cone with the origin in the interior of . For example, if we start with a rectangle, we get , above. Again, this doesn’t describe every toric variety (see above) but it describes a lot of them.
If you are having fun, you might try drawing the following toric varieties. If you’re not having quite that much fine, hopefully this list will convince you that toric varieties provide a fairly versatile class of examples.
(a) , and blown up at .
(b) The total space of the following line bundles on : , and .
(c) blown up along ; blown up first along and then along the proper transform of .
(d) (A real challenge!) Write coordinates on as . Blow up this threefold in the folllowing way: In a neighborhood of , blow up first, and second. In a neighborhood of , blow up first, and second. In a neighborhood of , blow up first, and second. The resulting smooth threefold can not be embedded into projective space (of any dimension) and shows some of the limits of our ability to draw things nicely.