Abstract. We show, under suitable hypothesis which are sharp in a certain sense, that the core of an m-primary ideal in a regular local ring of dimension d is equal to the adjoint (or multiplier) ideal of its d-th power. This generalizes the fundamental formula for the core of an integrally closed ideal in a two dimensional regular local ring due to Huneke and Swanson. We also find a generalization of this result to singular (non-regular) settings, which we show to be intimately related to the problem of finding non-zero sections of ample line bundles on projective varieties. In particular, we show that a graded analog of our formula for core would imply a remarkable conjecture of Kawamata predicting that every adjoint ample line bundle on a smooth variety admits a non-zero section.

No abstract

Abstract. Let T be a multigraded ring defined over a local ring (A, m). This paper deals with the question how the Cohen-Macaulay property of T is related to that of its diagonal subring T ∆ . In the bigraded case we are able to give necessary and sufficient conditions for the Cohen-Macaulayness of T . If I 1 , . . . , Ir ⊂ A are ideals of positive height, we can then compare the CohenMacaulay property of the multi-Rees algebra R A (I 1 , . . . , Ir) with the CohenMacaulay property of the usual Rees algebra R A (I 1 · · · Ir). We also obtain a bound for the joint reduction numbers of two m-primary ideals in the case the corresponding multi-Rees algebra is Cohen-Macaulay. IntroductionLet (A, m) be a local ring. Let T = n∈N r T n be a multigraded ring finitely generated over A by elements in degrees (1, 0, . . . , 0), . . . , (0, . . . , 0, 1). In this paper we are interested in studying the relationship between the Cohen-Macaulay property of T and that of its diagonal subring T ∆ which is the graded ring T ∆ = n∈N T n,...,n . The geometric object associated to a multigraded ring T is the corresponding multiprojective scheme Proj T constructed by means of multihomogeneous localizations. It is easy to see that Proj T is isomorphic to the usual projective scheme Proj T ∆ . From this point of view it is natural to expect that the homological properties of T and T ∆ are closely related. Classically T is the multihomogeneous coordinate ring of a multiprojective variety V defined over a field k and contained in some multiprojective space P (I 1 , . . . , I j−1 ) along the closed subscheme determined by the sheaf of ideals I j O Zj−1 . The diagonal subring is now the usual Rees algebra R A (I 1 · · · I r ) of the product I 1 · · · I r . We shall show in Corollary 2.10 of this paper that if R A (I 1 , . . . , I r ) is Cohen-Macaulay, then so is also R A (I 1 · · · I r ). This is a consequence of our main Theorem 2.5, where we give nec-

Abstract. We find formulas for the graded core of certain 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.

We give two kinds of bounds for the Castelnuovo-Mumford regularity of the canonical module and the deficiency modules of a ring, respectively, in terms of the homological degree and the CastelnuovoMumford regularity of the original ring.