  We find formulas for the graded core of certain m-primary ideals in a graded
ring. In particular, if S is the section ring of an ample line bundle on a
Cohen-Macaulay complex projective variety, we show that under suitable
hypothesis, the core and graded core of the ideal of S generated by all
elements of degrees at least N (for some, equivalently every, large 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.
