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Core versus graded core, and global sections of line bundles. (English) Zbl 1089.13004

Let \(R\) be a Noetherian \(\mathbb N\)-graded ring, \(I\) a homogeneous ideal of \(R\), and \(M\) a finitely generated graded \(R\)-module. The core of \(IM\), \(\text{core}(IM)\), is the intersection of all submodules \(JM\) of \(M\) where \(J\) runs through all reductions of \(I\), and the graded core of \(IM\), \(\text{gradedcore}(IM)\), is the intersection of all submodules \(JM\) of \(M\) where \(J\) runs through all homogeneous reductions of \(I\). Let \(X\) be a projective scheme over a field and let \(\mathcal L\) be an ample invertible sheaf on \(X\). The section ring of the pair \((X,\mathcal L)\) is the \(\mathbb N\)-graded ring \(S = \bigoplus_{n\geq0}\, H^0(X,\mathcal L^n)\). Then the main result is as follows:
Theorem 3.1. Let \(S\) be an equidimensional section ring of dimension \(d\geq2\) and characteristic zero. Assume that \(\text{Proj}\,S\) is Cohen-Macaulay. Fix \(N \gg 0\), and let \(I = S_{\geq N}\) be the ideal generated by all elements of degree at least \(N\). Then \(\text{gradedcore}(I\omega_S) = \bigoplus_{i \in \mathbb{Z},S_i=0}\, [\omega_S]_{Nd-i}\) as a graded submodule of the graded canonical module \(\omega_S\) of \(S\). Combining this with the formula for the core from [E. Hyry and K. E. Smith, Am. J. Math. 125, 1349–1410 (2003; see the preceding review Zbl 1089.13003)], one has the following theorem:
Let \(X\) be irreducible and characteristic zero. Assume that \(S\) is Gorenstein. Fix \(N \gg 0\) and let \(I = S_{\geq N}\). Then \(\text{gradedcore}(I) = \text{core}(I)\) if and only if \(\mathcal L\) has a non-trivial global section.
In section 4, the following theorem is given:
Let \((R,\mathfrak m)\) be a standard graded reduced Cohen-Macaulay ring of dimension \(d\) over an infinite field of arbitrary characteristic, and let \(a\) denote its \(a\)-invariant. Then \(\text{core}(\mathfrak m^n) = \text{gradedcore}(\mathfrak m^n) = \mathfrak m^{nd+a+1}\) for all \(n\geq1\).
In the final section 5, the authors gather together several commutative algebra questions whose positive solutions would solve the non-vanishing conjecture of Y. Kawamata [Asian J. Math. 4, 173–181(2000; Zbl 1060.14505); see also F. Ambro, J. Math. Sci. 94, 1126–1135 (1999; Zbl 0948.14033)].

MSC:

13A15 Ideals and multiplicative ideal theory in commutative rings
13B22 Integral closure of commutative rings and ideals
13A30 Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics
14B15 Local cohomology and algebraic geometry
14F17 Vanishing theorems in algebraic geometry
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References:

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