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R. Callejas-Bedregal

Publications and source records attributed to R. Callejas-Bedregal.

7 recordsLinked to original sources
arXiv2013

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 zero-scheme of a nonzero 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 viceversa: 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.

math.CV↗
arXiv2012

On the Milnor classes of local complete intersections

In this work we study algebraic, geometric and topological properties of the Milnor classes of local complete intersections with arbitrary singularities. We describe first the Milnor class of the intersection of a finite number of hypersurfaces, under certain conditions of transversality, in terms of the Milnor classes of the hypersurfaces. Using this description we obtain a Parusiński-Pragacz type formula, an Aluffi type formula and a description of the Milnor class of the local complete intersection in terms of the global Lê cycles of the hypersurfaces that define it. We consider next the general case of a local complete intersection $Z(s)$ defined by a regular section $s$ of a rank $r$ holomorphic bundle $E$ over a compact manifold $M$, $r \geq 2$. We notice that $s$ determines a hypersurface $Z(\tilde s)$ in the total space of the projectivization $\mathbb{P}(E^{\vee})$ of the dual bundle $E^{\vee}$, and we give a formula expressing the total Milnor class of the local complete intersection $Z(s)$ in terms of the Milnor classes of the hypersurface $Z(\tilde s)$.

math.AG↗
arXiv2011

Mixed multiplicities of arbitrary modules

Let $(R, \mathfrak m)$ be a Noetherian local ring. In this work we extend the notion of mixed multiplicities of modules, given in \cite{Kleiman-Thorup2} and \cite{Kirby-Rees1} (see also \cite{Bedregal-Perez}), to an arbitrary family $E,E_1,..., E_q$ of $R$-submodules of $R^p$ with $E$ of finite colength. We prove that these mixed multiplicities coincide with the Buchsbaum-Rim multiplicity of some suitable $R$-module. In particular, we recover the fundamental Rees's mixed multiplicity theorem for modules, which was proved first by Kirby and Rees in \cite{Kirby-Rees1} and recently also proved by the authors in \cite{Bedregal-Perez}. Our work is based on, and extend to this new context, the results on mixed multiplicities of ideals obtained by Viêt in \cite{Viet8} and Manh and Viêt in \cite{Manh-Viet}. We also extend to this new setting some of the main results of Trung in \cite{Trung} and Trung and Verma in \cite{Trung-Verma1}. As in \cite{Kleiman-Thorup2}, \cite{Kirby-Rees1} and \cite{Bedregal-Perez}, we actually work in the more general context of standard graded $R$-algebras.

math.AC↗
arXiv2011

(FC)-Sequences, Mixed Multiplicities and Reductions of Modules

Let $(R, \mathfrak m)$ be a Noetherian local ring. In this work we use the notion of (FC)-sequences, as defined in \cite{perez-bedregal1}, to present some results concerning reductions and the positivity of mixed multiplicities of a finite collection of arbitrary submodules of $R^p.$ We also investigate the length of maximal (FC)-sequences. We actually work in the more general context of standard graded $R$-algebras.

math.AC↗
arXiv2007

A numerical characterization of reduction for arbitrary modules

Let $(R, \mathfrak m)$ be a $d$-dimensional Noetherian local ring and $E$ a finitely generated $R$-submodule of a free module $R^p.$ In this work we introduce a multiplicity sequence $c_k(E), k=0,..., d+p-1$ for $E$ that generalize the Buchsbaum-Rim multiplicity defined when $E$ has finite colength in $R^p$ as well as the Achilles-Manaresi multiplicity sequence that applies when $E\subseteq R$ is an ideal. Our main result is that the new multiplicity sequence can indeed be used to detect integral dependence of modules. Our proof is self-contained and implies known numerical criteria for integral dependence of ideals and modules.

math.AC↗
arXiv2006

Topological Triviality of Families of Singular Surfaces

We study the topological triviality of families of singular surfaces in ${\mathbb C}^3$ parametrized by $\mathcal A$-finitely determined map germs. We prove that the constancy of the Milnor number of the double point locus characterizes the topological triviality of the family.

math.CV↗