Division flow

I will discuss an algorithm to decide the finite generation of a (multi) graded ring. For $N$-gradings, it is the analog of Erathostenes' sieve method. For $N^2$-gradings, it is the algebraic analog of the Kahler-Ricci flow. The division flow breaks up a finitely generated graded ring into its simplest blocks, a so called linearity fan. I will discuss the properties of the linearity fan. One application is to characterize the chambers and walls in the Log Geography introduced by Shokurov.