Claudia Polini

Links of prime ideals

We exhibit the elementary but somewhat surprising property that most direct links of prime ideals in Gorenstein rings are equimultiple ideals. It leads to the construction of a bountiful set of Cohen–Macaulay Rees algebras.

The structure of the core of ideals

Abstract. 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. Lipmans 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.

Core and residual intersections of ideals

D. Rees and J. Sally defined the core of an R-ideal I as the in- tersection of all (minimal) reductions of I. However, it is not easy to give an explicit characterization of it in terms of data attached to the ideal. Until recently, the only case in which a closed formula was known is the one of inte- grally closed ideals in a two-dimensional regular local ring, due to C. Huneke and I. Swanson. The main result of this paper explicitly describes the core of a broad class of ideals with good residual properties in an arbitrary local Cohen-Macaulay ring. We also find sharp bounds on the number of minimal reductions that one needs to intersect to get the core.

A formula for the core of an ideal

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.

Sally modules and associated graded rings

We study the depth properties of the associated graded ring of an m-primary ideal I in terms of numerical data attached to the ideal I. We also find bounds on the Hilbert coefficients of I by means of the Sally module S_J(I) of I with respect to a minimal reduction J of I.

Rational normal scrolls and the defining equations of Rees algebras

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 which defines the Rees algebra ℛ = R[It]; so for the polynomial ring S = R[T 1, . . . , Tm ]. We resolve ℛ as an S-module and Is as an R-module, for all powers s. The proof uses the homogeneous coordinate ring, A = S/H, of a rational normal scroll, with . The ideal is 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 because the generators of K (n) are much less complicated then the generators of . We obtain a filtration of K (n) in which the factors are polynomial rings, hypersurface rings, or modules resolved by generalized Eagon–Northcott complexes. The generators of I parameterize an algebraic curve in projective m – 1 space. The defining equations of the special fiber ring ℛ/(x, y)ℛ yield a solution of the implicitization problem for .

Multiplicity of the special fiber of blowups

Let (R, m) be a Noetherian local ring and let I be an m-primary ideal. In this paper we give sharp bounds on the multiplicity of the special fiber ring F of I in terms of other well-known invariants of I. A special attention is then paid in studying when equality holds in these bounds, with a particular interest in the unmixedness or, better, the Cohen-Macaulayness of F.

Cohen-Macaulayness of Special Fiber Rings

Abstract Let (R, 𝔪) be a Noetherian local ring and let Ibe an R-ideal. Inspired by the work of Hübl and Huneke, we look for conditions that guarantee the Cohen-Macaulayness of the special fiber ring ℱ = ℛ/𝔪ℛ of I, where ℛ denotes the Rees algebra of I. Our key idea is to require ‘good’ intersection properties as well as ‘few’ homogeneous generating relations in low degrees. In particular, if Iis a strongly Cohen-Macaulay R-ideal with G ℓand the expected reduction number, we conclude that ℱ is always Cohen-Macaulay. We also obtain a characterization of the Cohen-Macaulayness of ℛ/Kℛ for any 𝔪-primary ideal K. This result recovers a well-known criterion of Valabrega and Valla whenever K = I. Furthermore, we study the relationship between the Cohen-Macaulay property of the special fiber ring ℱ and the Cohen-Macaulay property of the Rees algebra ℛ and the associated graded ring 𝒢 of I. Finally, we focus on the integral closedness of 𝔪I. The latter question is motivated by the theory of evolutions. Dedicated to Steven L. Kleiman on the occasion of his 60th birthday.

Reduction number of links of irreducible varieties

Abstract The reductions of an ideal I give a natural pathway to the properties of I, with the advantage of having fewer generators. In this paper we primarily focus on a conjecture about the reduction exponent of links of a broad class of primary ideals. The existence of an algebra structure on the Koszul and Eagon-Northcott resolutions is the main tool for detailing the known cases of the conjecture. In the last section we relate the conjecture to a formula involving the length of the first Koszul homology modules of these ideals.

The bi-graded structure of symmetric algebras with applications to Rees rings ☆

Abstract Consider a rational projective plane curve C parameterized by three homogeneous forms of the same degree in the polynomial ring R = k [ x , y ] over a field k. The ideal I generated by these forms is presented by a homogeneous 3 × 2 matrix φ with column degrees d 1 ≤ d 2 . The Rees algebra R = R [ I t ] of I is the bi-homogeneous coordinate ring of the graph of the parameterization of C ; and accordingly, there is a dictionary that translates between the singularities of C and algebraic properties of the ring R and its defining ideal. Finding the defining equations of Rees rings is a classical problem in elimination theory that amounts to determining the kernel A of the natural map from the symmetric algebra Sym ( I ) onto R . The ideal A ≥ d 2 − 1 , which is an approximation of A , can be obtained using linkage. We exploit the bi-graded structure of Sym ( I ) in order to describe the structure of an improved approximation A ≥ d 1 − 1 when d 1 d 2 and φ has a generalized zero in its first column. (The latter condition is equivalent to assuming that C has a singularity of multiplicity d 2 .) In particular, we give the bi-degrees of a minimal bi-homogeneous generating set for this ideal. When 2 = d 1 d 2 and φ has a generalized zero in its first column, then we record explicit generators for A . When d 1 = d 2 , we provide a translation between the bi-degrees of a bi-homogeneous minimal generating set for A d 1 − 2 and the number of singularities of multiplicity d 1 that are on or infinitely near C . We conclude with a table that translates between the bi-degrees of a bi-homogeneous minimal generating set for A and the configuration of singularities of C when the curve C has degree six.