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Core versus graded core, and global sections of line bundles

Authors: Eero Hyry and Karen E. Smith
Journal: Trans. Amer. Math. Soc. 356 (2004), 3143-3166
MSC (2000): Primary 13A30; Secondary 13A15, 14B15
Published electronically: November 4, 2003
MathSciNet review: 2052944
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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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Additional Information

Eero Hyry
Affiliation: Department of Mathematics, University of Helsinki, Helsinki, Finland

Karen E. Smith
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1109

Received by editor(s): January 30, 2003
Published electronically: November 4, 2003
Additional Notes: The first author’s research was supported by the National Academy of Finland, project number 48556
The second author’s research was partially supported by the Clay Foundation and by the US National Science Foundation Grant DMS 00-70722.
Article copyright: © Copyright 2003 American Mathematical Society

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