Core versus Graded Core, and Global Sections of Line Bundles

When does an ample line bundle on a smooth projective variety have a nonzero global section? In this paper, we show that this question is equivalent to a fundamental problem in commutative algebra regarding the equality of core and graded core for a certain associated homogeneous ideal. This builds on the project begun in [16], where it was shown that a sufficiently good algebraic understanding of the graded core can be used to show the existence of global sections of line bundles. This paper essentially treats the converse: we show the existence of nonzero sections gives rise to nice formulas for cores. The core of an ideal in a Noetherian commutative ring is the intersection of all its reductions—that is, the intersection of all subideals having the same integral closure. The core first arose in the work of Rees and Sally [25] because of its connection with Briançon-Skoda theorems, and has recently been the subject of active investigation in commutative algebra; see [11, 2, 3, 16, 24, 13]. For a homogeneous ideal in a graded ring, it is also natural to consider its graded core, namely, the intersection of all its homogeneous reductions. The core and graded core of a homogeneous ideal are both homogeneous ideals and there is an obvious inclusion of the core in the graded core. A natural question is: when does equality hold? This question arose in the work [16] in finding sections of line bundles, and has also been considered by Huneke and Trung in [13] for purely algebraic reasons. Quite generally, the core and graded core are equal for homogeneous ideals generated by elements of the same degree (see Lemma 3.3 for a precise statement). On the other hand, there are examples of ideals having so few homogeneous reductions that it is easy to see the core is strictly smaller. But what can be said for ideals having many graded reductions? In this paper we answer this question for a specific type of ideal in a section ring, while providing one answer to the opening question about sections of line bundles. Specifically, we prove a formula for the graded core (see Theorem 3.1 and Corollary 3.10) which, when combined with the formula for core from [16], yields the following result.