Core versus graded core, and global sections of line bundles

Hyry, Eero; Smith, Karen

doi:10.1090/S0002-9947-03-03337-3

Core versus graded core, and global sections of line bundles
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by Eero Hyry and Karen E. Smith

Trans. Amer. Math. Soc. 356 (2004), 3143-3166

DOI: https://doi.org/10.1090/S0002-9947-03-03337-3

Published electronically: November 4, 2003

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Abstract:

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.

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