Core versus graded core, and global sections of line bundles

Abstract

We find formulas for the graded core of certain m \mathfrak {m} -primary ideals in a graded ring. In particular, if S S is 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 of S S generated by all elements of degrees at least N N (for some, equivalently every, large N N ) 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.