### Toric Varieties

Part of the idea is this: consider a convex polytope in R^{n} whose vertices all have integer coordinates. A simple example is the polytope in R^{2} 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:

x^{0}y^{0}=1,x^{1}y^{0}=x,x^{0}y^{1}=y

Use these to define a mapping from C

^{n}to C

^{m}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 C

^{n}to CP

^{m-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 CP

^{2}. 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

1,x,y,xy

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 CP

^{3}. 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.)

Labels: mathematics

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