Terence Gaffney

Generalized Buchsbaum-Rim Multiplicities and a Theorem of Rees

Abstract Given M, a submodule of Nwhich is in turn a submodule of a free R-module, where Ris a Noetherian local ring which is equidimensional and universally catenary, the theorem of Rees asserts that the integral closures of Mand Nagree provided their multiplicities agree. We extend this result to modules of non-finite colength by introducing a chain of submodules H i (M)which give an increasingly good approximation to the integral closure of M, and a sequence of numbers based on the multiplicity of a pair of modules. Some applications to problems in equisingularity are given. Dedicated to Steven L. Kleiman on the occasion of his 60th birthday.

Fibers of Polynomial Mappings at Infinity and a Generalized Malgrange Condition

Let f be a complex polynomial mapping. We relate the behaviour of f ‘at infinity’ to the characteristic cycle associated to the projective closures of fibres of f. We obtain a condition on the characteristic cycle which is equivalent to a condition on the asymptotic behaviour of some of the minors of the Jacobian matrix of f. This condition generalizes the condition in the hypersurface case known as Malgranges condition. The relation between this condition and the behavior of the characteristic cycle is a partial generalization of Parusinskis result in the hypersurface case. We show that the new condition implies the C∞-triviality of f.

The Multiplicity of Pairs of Modules and Hypersurface Singularities

This paper applies the multiplicity polar theorem to the study of hypersurfaces with non-isolated singularities. The multiplicity polar theorem controls the multiplicity of a pair of modules in a family by relating the multiplicity at the special fiber to the multiplicity of the pair at the general fiber. It is as important to the study of multiplicities of modules as the basic theorem in ideal theory which relates the multiplicity of an ideal to the local degree of the map formed from the generators of a minimal reduction. In fact, as a corollary of the theorem, we show here that for M a submodule of finite length of a free module F over the local ring of an equidimensional complex analytic germ, that the number of points at which a generic perturbation of a minimal reduction of M is not equal to F, is the multiplicity of M.

The Lê numbers of the square of a function and their applications

Le numbers were introduced by Massey with the purpose of numerically controlling the topological properties of families of non-isolated hypersurface singularities and describing the topology associated with a function with non-isolated singularities. They are a generalization of the Milnor number for isolated hypersurface singularities. In this note the authors investigate the composite of an arbitrary square-free f and z2. They get a formula for the Le numbers of the composite, and consider two applications of these numbers. The first application is concerned with the extent to which the Le numbers are invariant in a family of functions which satisfy some equisingularity condition, the second is a quick proof of a new formula for the Euler obstruction of a hypersurface singularity. Several examples are computed using this formula including any X defined by a function which only has transverse D(q, p) singularities off the origin.

Equisingularity of sections, (^{}) condition, and the integral closure of modules

The theory of the integral closure of modules provides a powerful tool for studying strat i cation conditions which are connected with limits of linear spaces It gives an expression which is both algebraic and geometric for these conditions This connection allows one to control many geometric phenomena through associated numerical invariants This pa per will illustrate these points by examining the t conditions which were introduced by Thom and the second author We will show how to apply these conditions to the study of certain families of sections of an analytic space The t conditions deal with the C sections of some strati ed set they were introduced initially by Thom in and developed by the second author from on more recently in collaboration with Kuo and the third author Th Tr Tr Tr Tr Ku Tr T W and were applied to prove various equisingularity results For real and complex analytic sets we show that the t conditions have algebraic formulations in terms of integral closure of modules Our formulation gives a new simple proof for analytic sets of the change in the conditions under Grassmann modi cation proved by Kuo and the second author Ku Tr for subanalytic sets this is used in conjunction with the principle of specialization of integral dependence to give numerical criteria for familes of plane sections of complex complete intersections to be Whitney equisingular Some of the results in this paper were announced by the rst author in the Sao Carlos proceedings G In Section we review the notions of integral closure reduction and strict dependence for submodules of O X x where X x is the germ of a complex analytic set We describe the analogues which are needed for the case of real analytic sets We will apply these concepts to submodules of the Jacobian module JM F where X F We review results from G which use these tools to analyse the limits of tangent hyperplanes to X x and to characterize the Whitney a and Verdier w regularity conditions In Section we de ne jets of transversals and the t conditions The main result of this section is Theorem which is the characterization of condition t in terms of integral closure of modules We note that condition t is in fact w The Grassmann modi cation is a generalization of the blow up in which the projective space is replaced by the Grassmann space of planes of the dimension of the transversals we are using The Grassmann modi cation theorem says that the modi cation improves condition t to condition t This is used at the end of Section and several places in Section for

The theory of integral closure of ideals and modules: Applications and new developments

Many equisingularity conditions such as the Whitney conditions, and their relative versions A f and W f , depend on controlling limiting linear structures. The theory of the integral closure of ideals and modules provides a very useful tool for studying these limiting structures. In this paper we illustrate how these tools are used in three case studies, and describe some of the advances in the theory since the survey article of [12], which was written in 1996.

Equisingularity and the Theory of Integral Closure

This is an introduction to the study of the equisingularity of sets using the theory of the integral closure of ideals and modules as the main tool. It introduces the notion of the landscape of a singularity as the right setting for equisingularity problems.