Core versus graded core, and global sections of line bundles

We find formulas for the graded core of certain $\mathfrak{m}$-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 a 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.