Related papers: The multiplicity of pairs of modules and hypersurf…

Given a family of pairs of modules parametrised by a smooth space Y, the Multiplicity-Polar Theorem relates the multiplicity of the pair of modules at a special point of the parameter to the multiplicity of the pair at a generic point. This…

Using invariants from commutative algebra to count geometric objects is a basic idea in singularities. For example, the multiplicity of an ideal is used to count points of intersection of two analytic sets at points of non-transverse…

We study the behavior of multidegrees in families and the existence of numerical criteria to detect integral dependence. We show that mixed multiplicities of modules are upper semicontinuous functions when taking fibers and that projective…

Over a regular local ring of dimension two with maximal ideal m, we study the Buchsbaum-Rim multiplicity of a finitely generated module M of finite colength in a free module F. First, we investigate the colength of an m-primary ideal and…

In this work, we consider a pair $(\textbf{X},0)$ and $(\textbf{Y},0)$ of hypersurfaces in $(\mathbb{C}^{n+1},0)$ parametrized by finitely determined, quasihomogeneous map germs $f$ and $g,$ respectively. Zariski asked whether the…

We establish the principle of specialization of integral dependence for submodules of finite colength of free modules, as part of the general algebraic-geometric theory of the Buchsbaum--Rim multiplicity. Then we apply the principle to the…

The notion of multiplicity of a module first arose as consequence of Hilbert's work on commutative algebra, relating the dimension of rings with the degree of certain polynomials. For noncommutative rings, the notion of multiplicity first…

Let $(R,\mathfrak{m})$ be a Noetherian local ring of dimension $d>0$ with infinite residue field. Let $M$ be a finitely generated proper $R$-submodule of a free $R$-module $F$ with $\ell (F/M) < \infty$ and having rank $r$. In this article,…

We show that the relation between multiplier ideals and $V$-filtration on the structure sheaf due to Budur-Musta\c{t}\u{a}-Saito generalizes to singular irreducible varieties, by replacing multiplier ideals with multiplier modules and the…

We give a complete proof of a propagation theorem of multiplicity-free property from fibers to spaces of global sections for holomorphic vector bundles. The propagation theorem is formalised in three ways, aiming for producing various…

Let $A$ be an algebra over any field. We do not assume that $A$ has an identity. The \emph{multiplier algebra} $M(A)$ is a unital algebra associated to $A$. If we require the product in $A$ to be non-degenerate (as a bilinear form), the…

In this paper, we prove the converse of Rees' mixed multiplicity theorem for modules, which extends the converse of the classical Rees' mixed multiplicity theorem for ideals given by Swanson - Theorem \ref{SwansonTheorem}. Specifically, we…

We study the maximal multiplicity locus of a variety $X$ over a field of characteristic $p>0$ that is provided with a finite surjective radical morphism $\delta:X\rightarrow V$, where $V$ is regular, for example, when…

As an attempt to bridge between numerical analysis and algebraic geometry, this paper formulates the multiplicity for the general nonlinear system at an isolated zero, presents an algorithm for computing the multiplicity structure, proposes…

We define a concept which we call multiplicity. First, multiplicity of a morphism is defined. Then the multiplicity of an object over another object is defined to be the minimum of the multiplicities of all morphisms from one to another.…

We consider the notion of mixed multiplicities for multigraded modules by using Hilbert series, and this is later applied to study the projective degrees of rational maps. We use a general framework to determine the projective degrees of a…

We have studied both clusters and bulk systems while investigating amorphous states. We have varied the nature of interaction amongst the particles of the system under consideration in order to reveal the possible presence of universality…

We provide new criteria for the integrality and birationality of an extension of graded algebras in terms of the general notion of polar multiplicities of Kleiman and Thorup. As an application, we obtain a new criterion for when a module is…

For any persistence module $M$ over a finite poset $\mathbf{P}$, and any interval $I$ of $\mathbf{P}$, we give a formula for the multiplicity $d_M(V_I)$ of the interval module $V_I$ in the indecomposable decomposition of $M$ in terms of the…

The problem we are considering came up in connection with the classification of singularities in positive characteristic. Then it is important that certain invariants like the determinacy can be bounded simultaneously in families of formal…