The Proj of the Rees algebra of a graded family of ideals

Abstract: In this article we investigate the condition that the Proj of a Rees algebra of a graded family of ideals in a Noetherian local ring $R$ is Noetherian. In many cases, the Proj will be Noetherian even when the Rees algebra is not. For instance, the Proj of the Rees algebra of a graded filtration of ideals will alway be Noetherian if the analytic spread of the filtration is zero. The Proj of a Rees algebra of a divisorial filtration on a two dimensional normal excellent local ring is always Noetherian, as was proven by Russo and later with a different proof by the author. We give examples in this paper of divisorial filtrations on three dimensional normal excellent local rings whose Proj is not Noetherian, showing that this theorem does not extend to higher dimensions. A consequence of the fact that the Proj of a divisorial filtration over a two dimensional excellent normal local ring is always Noetherian is that the preimage of the maximal ideal of $R$ in the Proj has only finitely many irreducible components. As a consequence, the fiber cone of such a filtration has only finitely many minimal primes. We give an example of a graded filtration of ideals in a two dimensional regular local ring such that the preimage of the maximal ideal in the Proj of the Rees algebra of the filtration has infinitely many irreducible components, so that the Proj is not Noetherian, and the fiber cone of the filtration has infinitely many minimal primes.