### October 2008

So, today I discovered something rather nice, that I think could easily get more time in books and the like, but doesn’t. We’ll first have need of a theorem that I don’t want to prove:

Theorem: Let $X$ be a projective variety over $\mathbb{C}$ and let $\mathscr{L}$ be an invertible sheaf on $X$. Then $\mathscr{L}$ is ample if and only if for all $\mathscr{F}$ coherent sheaves, there is a number $n>>0$, such that we have $H^i(X,\mathscr{F}\otimes\mathscr{L}^{\otimes n})=0$ for all $i>0$.

Last time we did a bunch of stuff with fans and polytopes and made a lot of definitions. This time, we get to use that stuff to do some algebraic geometry. First up, we’ll need to define some lattice in some vector space, otherwise nothing we did last time is applicable. Now, Matt’s recently started talking about rational varieties, and I’m going to do him one better. I not only just care about the case where our field is $\mathbb{C}$, but I’m only going to care about rational varieties with special nice group actions.