The core of an R-ideal I is the intersection of all reductions of I . This object was introduced by D. Rees and J. Sally and later studied by C. Huneke and I. Swanson, who showed in particular its connection to J. Lipman's notion of adjoint of an ideal. Being an a priori infinite intersection of ideals, the core is difficult to describe explicitly. We prove in a broad setting that: core(I ) is a finite intersection of minimal reductions; core(I ) is a finite intersection of general minimal reductions; core(I ) is the contraction to R of a 'universal' ideal; core(I ) behaves well under flat extensions. The proofs are based on general multiplicity estimates for certain modules. (2000): 13A30, 13B21, 13H15, 13C40, 13H10 Mathematics Subject Classification
Abstract. Consider a rational projective curve C of degree d over an algebraically closed field k k k. There are n homogeneous forms g 1 , . . . , g n of degree d in B = k k k[x, y] which parameterize C in a birational, base point free, manner. We study the singularities of C by studying a Hilbert-Burch matrix ϕ for the row vector [g 1 , . . . , g n ]. In the "General Lemma" we use the generalized row ideals of ϕ to identify the singular points on C, their multiplicities, the number of branches at each singular point, and the multiplicity of each branch.Let p be a singular point on the parameterized planar curve C which corresponds to a generalized zero of ϕ. In the "Triple Lemma" we give a matrix ϕ ′ whose maximal minors parameterize the closure, in P 2 , of the blow-up at p of C in a neighborhood of p. We apply the General Lemma to ϕ ′ in order to learn about the singularities of C in the first neighborhood of p. If C has even degree d = 2c and the multiplicity of C at p is equal to c, then we apply the Triple Lemma again to learn about the singularities of C in the second neighborhood of p.Consider rational plane curves C of even degree d = 2c. We classify curves according to the configuration of multiplicity c singularities on or infinitely near C. There are 7 possible configurations of such singularities. We classify the Hilbert-Burch matrix which corresponds to each configuration. The study of multiplicity c singularities on, or infinitely near, a fixed rational plane curve C of degree 2c is equivalent to the study of the scheme of generalized zeros of the fixed balanced Hilbert-Burch matrix ϕ for a parameterization of C. Let BalH d = ϕ ϕ is a 3 × 2 matrix; each entry in ϕ is a homogeneous form of degree c from B; and ht I 2 (ϕ) = 2 .
Let R be a Noetherian local ring. We define the minimal j-multiplicity and almost minimal j-multiplicity of an arbitrary R-ideal on any finite R-module. For any ideal I with minimal j-multiplicity or almost minimal j-multiplicity on a Cohen-Macaulay module M, we prove that under some residual assumptions, the associated graded module gr I (M) is Cohen-Macaulay or almost Cohen-Macaulay, respectively. Our work generalizes the results for minimal multiplicity and almost minimal multiplicity obtained
Abstract. The core of an ideal is the intersection of all its reductions. For large classes of ideals I we explicitly describe the core as a colon ideal of a power of a single reduction and a power of I. (2000): Primary 13B22; Secondary 13A30, 13B21, 13C40, 13H10. Mathematics Subject Classification
Abstract. Consider a height two ideal, I, which is minimally generated by m homogeneous forms of degree d in the polynomial ring R = k [x, y]. Suppose that one column in the homogeneous presenting matrix ϕ of I has entries of degree n and all of the other entries of ϕ are linear. We identify an explicit generating set for the ideal A which defines the Rees ring R = R[It]; so R = S/A for the polynomial ring S = R[T 1 , . . . , T m ]. We resolve R as an S-module and I s as an R-module, for all powers s. The proof uses a rational normal scroll ring A = S/H with AA isomorphic to the n th symbolic power of a height one prime ideal K of A. The ideal K (n) is generated by monomials. Whenever possible, we study A/K (n) in place of A/AA because the generators of K (n) are much less complicated then the generators of AA. We obtain a filtration of K (n) in which the factors are polynomial rings, hypersurface rings, or modules resolved by Eagon-Northcott complexes. The generators of I give rise to an algebraic curve C in projective m − 1 space. The defining equations of the fiber ring R/(x, y)R yield a solution of the implicitization problem for C.Introduction.
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