The twisted homogeneous coordinate ring is one of the basic constructions of the noncommutative projective geometry of Artin, Van den Bergh, and others. Chan generalized this construction to the multi-homogeneous case, using a concept of right ampleness for a finite collection of invertible sheaves and automorphisms of a projective scheme. From this he derives that certain multi-homogeneous rin...

The Goto number of a parameter ideal Q in a Noetherian local ring (R, m) is the largest integer q such that Q : m is integral over Q. The Goto numbers of the monomial parameter ideals of R = k[[x1 , x2 , . . . , xν ]] are characterized using the semigroup of R. This helps in computing them for classes of numerical semigroup rings, as well as on a case by case basis. The minimal Goto number of R...