Mixed multiplicities and the minimal number of generator of modules

Let (R,m) be a d-dimensional Noetherian local ring. In this work we prove that the mixed Buchsbaum-Rim multiplicity for a finite family of R-submodules of R^p of finite colength coincides with the Buchsbaum-Rim multiplicity of the module generated by a suitable superficial sequence, that is, we generalize for modules the well-known Risler-Teissier theorem. As a consequence, we give a new proof of a generalization for modules of the fundamental Rees’ mixed multiplicity theorem, which was first proved by Kirby and Rees in (1994, [8]). We use the above result to give an upper bound for the minimal number of generators of a finite colength R-submodule of R^p in terms of mixed multiplicities for modules, which generalize a similar bound obtained by Cruz and Verma in (2000, [5]) for m-primary ideals.     

Mixed multiplicities for arbitrary ideals and generalized Buchsbaum-Rim multiplicities

We introduce first the notion of mixed multiplicities for arbitraryideals in a local d-dimensional Noetherian ring (A,m) which,in some sense, generalizes the concept of mixed multiplicitiesfor m-primary ideals. We also generalize Teissier's productformula for a set of arbitrary ideals and extend the notionof the Buchsbaum-Rim multiplicity (BR-multiplicity) of a submoduleof a free module to the case where the submodule no longer hasfinite colength. For a submodule M of A^p, we introduce a sequenceof multiplicities e^k_BR (M), k = 0, . . . , d + p – 1 whichin the case of an ideal (p = 1) coincides with the multiplicitysequence c0(I,A), . . . , c_d(I,A) defined for an arbitrary idealI of A by Achilles and Manaresi. In the case where M has finitecolength in Ap and is totally decomposable, we prove that ourBR-multiplicity sequence essentially falls into the standardBR-multiplicity of M.     

Lê cycles and Milnor classes

The purpose of this work is to establish a link between the theory of Chern classes for singular varieties and the geometry of the varieties in question. Namely, we show that if Z is a hypersurface in a compact complex manifold, defined by the complex analytic space of zeroes of a reduced non-zero holomorphic section of a very ample line bundle, then its Milnor classes, regarded as elements in the Chow group of Z, determine the global Lê cycles of Z; and vice versa: The Lê cycles determine the Milnor classes. Morally this implies, among other things, that the Milnor classes determine the topology of the local Milnor fibres at each point of Z, and the geometry of the local Milnor fibres determines the corresponding Milnor classes.