Related papers: Rees algebras and resolution of singularities
A constructive procedure is given to determine all ideals of a solvable Lie algebra. This is used in determining algorithmically all conjugacy classes of subalgebras of a given solvable Lie algebra.
A new approach is established to computing the image of a rational map, whereby the use of approximation complexes is complemented with a detailed analysis of the torsion of the symmetric algebra in certain degrees. In the case the map is…
The Koszul homology of modules of the polynomial ring $R$ is a central object in commutative algebra.It is strongly related with the minimal free resolution of these modules, and thus with regularity, Hilbert functions, etc. Here we…
We introduce a class of singular log schemes in three dimensions and conjecture that log schemes in this class admit log crepant log resolutions. We provide examples as evidence and relate this conjecture to the conjecture made in [4] and…
Let $G$ be a bipartite graph and $I=I(G)$ be its edge ideal. The aim of this note is to investigate different aspects of the Rees algebra $\mathcal{R}(I)$ of $I$. We compute its regularity and the universal Gr\"obner basis of its defining…
In this paper, we investigate some topics around the closed image $S$ of a rational map $\lambda$ given by some homogeneous elements $f_1,...,f_n$ of the same degree in a graded algebra $A$. We first compute the degree of this closed image…
These lecture notes provide a unified overview of most known canonical desingularization methods in characteristic zero. It starts with discussing the classical method, and then proceeds with the recently discovered ones: logarithmic…
Consider a projective variety $X \subset \mathbb{P}^n$ (over an algebraically closed field of characteristic zero), together with a (reduced) simple normal crossings divisor $E \subset \mathbb{P}^n$, where the degrees of both $X$ and $E$…
Using linear algebra methods we study certain algebraic properties of monomial rings and matroids. Let I be a monomial ideal in a polynomial ring over an arbitrary field. If the Rees cone of I is quasi-ideal, we express the normalization of…
An ideal of a local polynomial ring can be described by calculating a standard basis with respect to a local monomial ordering. However standard basis algorithms are not numerically stable. Instead we can describe the ideal numerically by…
Let C be a clutter and let A be its incidence matrix. If the linear system x>=0;xA<=1 has the integer rounding property, we give a description of the canonical module and the a-invariant of certain normal subrings associated to C. If the…
We give infinite triangularization and strict triangularization results for algebras of operators on infinite dimensional vector spaces. We introduce a class of algebras we call Ore-solvable algebras: these are similar to iterated Ore…
After briefly recalling some computational aspects of blowing up and of representation of resolution data common to a wide range of desingularization algorithms (in the general case as well as in special cases like surfaces or binomial…
Let $K$ be a field and $R=\oplus_{p\in\mathbb{N}}R_p$ an $\mathbb{N}$-graded $K$-algebra, which has an SM $K$-basis (i.e. a skew multiplicative $K$-basis) such that $R$ holds a Gr\"obner basis theory. It is proved that there is a one-to-one…
This paper studies the concept of algorithmic equiresolution of a family of embedded varieties or ideals, which means a simultaneous resolution of such a family compatible with a given (suitable) algorithm of resolution in characteristic…
This paper concerns a generalization of the Rees algebra of ideals due to Eisenbud, Huneke and Ulrich that works for any finitely generated module over a noetherian ring. Their definition is in terms of maps to free modules. We give an…
In this paper we propose the concept of formal desingularizations as a substitute for the resolution of algebraic varieties. Though a usual resolution of algebraic varieties provides more information on the structure of singularities there…
We classify all graphs for which the Rees algebras of their edge ideals are normal and have regularity equal to their matching numbers.
We study the algebraic and geometric properties of the integral closure of different rings of functions on a real algebraic variety : the regular functions and the continuous rational functions.
There exists a well established differential topological theory of singularities of ordinary differential equations. It has mainly studied scalar equations of low order. We propose an extension of the key concepts to arbitrary systems of…