Core versus graded core, and global sections of line bundles

Abstract

We find formulas for the graded core of certainm\mathfrak {m}-primary ideals in a graded ring. In particular, ifSSis the section ring of an ample line bundle on a Cohen-Macaulay complex projective variety, we show that under a suitable hypothesis, the core and graded core of the ideal ofSSgenerated by all elements of degrees at leastNN(for some, equivalently every, largeNN) are equal if and only if the line bundle admits a non-zero global section. We also prove a formula for the graded core of the powers of the unique homogeneous maximal ideal in a standard graded Cohen-Macaulay ring of arbitrary characteristic. Several open problems are posed whose solutions would lead to progress on a non-vanishing conjecture of Kawamata.