Tuesday, August 16, 2005

Toric Varieties

I keep meaning to read up on this subject but my algebraic geometry ability isn't quite there yet. One of the reasons I'd like to understand toric varieties is that they give you a concrete handle on certain types of algebraic curves and thereby give solid interpretations of results like the Riemann-Roch theorem in terms of counting features of simple geometrical objects.

Part of the idea is this: consider a convex polytope in Rn whose vertices all have integer coordinates. A simple example is the polytope in R2 with vertices (0,0), (1,0) and (0,1). This is also the list of integer points in this polytope. Now write out the monomials in x and y whose exponents are given by the coordinates of the integer points. In this case we get:


Use these to define a mapping from Cn to Cm where m is the number of integer points in the polytope. In this case we get

(x,y) -> (1,x,y)

This defines a map from Cn to CPm-1 by considering (1,x,y) to be homogeneous coordinates. The closure of the image of this map is an example of a toric variety. In this case it's just all of CP2. This looks like a highly contrived definition. But what's neat is that many natural properties of the toric variety are closely related to natural properties of the convex polytope. In particular, a result like Pick's Theorem and its higher dimensional analogues are consequences of the Riemann-Roch theorem applied to the toric variety. We get a dictionary converting statements of algebraic geometry, some of them quite tricky (for me) to grasp, to elementary statements about convex polytopes.

Here's another quick example of a toric variety to illustrate the above process. Consider the convex polytope with corners (0,0), (1,0), (0,1), (1,1). This gives monomials


so we have the mapping

(x,y) -> (p,q,r,s) = (1,x,y,xy)

It's not hard to see that the closure of the image of this map is the surface ps=qr in CP3. So this time we have a non-trivial variety.

But I don't yet understand how Pick's theorem comes from the Riemann-Roch theorem.

I'm also interested in the connection to the ubiquitous number 12. (That is the same 12 that appears in 1+2+3+...=-1/12 in an earlier posting of mine. Again, a result of the Riemann-Roch theorem.)

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