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W. V. Vasconcelos

Publications and source records attributed to W. V. Vasconcelos.

8 recordsLinked to original sources
arXiv2020

Canonical Degrees of Cohen-Macaulay Rings and Modules: a Survey

The aim of this survey is to discuss invariants of Cohen-Macaulay local rings that admit a canonical module. Attached to each such ring R with a canonical ideal C, there are integers--the type of R, the reduction number of C--that provide valuable metrics to express the deviation of R from being a Gorenstein ring. We enlarge this list with other integers--the roots of R and several canonical degrees. The latter are multiplicity based functions of the Rees algebra of C. We give a uniform presentation of three degrees arising from common roots. Finally we experiment with ways to extend one of these degrees to rings where C is not necessarily an ideal.

math.AC↗
arXiv2019

The Bi-Canonical Degree of a Cohen-Macaulay Ring

This paper is a sequel to [8] where we introduced an invariant, called canonical degree, of Cohen-Macaulay local rings that admit a canonical ideal. Here to each such ring with a canonical ideal, we attach a different invariant, called bi-canonical degree, which in dimension 1 appears also in [12] as the residue of a ring. The minimal values of these functions characterize specific classes of Cohen-Macaulay rings. We give a uniform presentation of such degrees and discuss some computational opportunities offered by the bi-canonical degree.

math.AC↗
arXiv2012

Variation of Hilbert Coefficients

For a Noetherian local ring $(\RR, \m)$, the first two Hilbert coefficients, $e_0$ and $e_1$, of the $I$-adic filtration of an $\m$-primary ideal $I$ are known to code for properties of $\RR$, of the blowup of $\spec(\RR)$ along $V(I)$, and even of their normalizations. We give estimations for these coefficients when $I$ is enlarged (in the case of $e_1$ in the same integral closure class) for general Noetherian local rings.

math.AC↗
arXiv2007

On the homology of two-dimensional elimination

We study birational maps with empty base locus defined by almost complete intersection ideals. Birationality is shown to be expressed by the equality of two Chern numbers. We provide a relatively effective method of their calculation in terms of certain Hilbert coefficients. In dimension two the structure of the irreducible ideals leads naturally to the calculation of Sylvester determinants via a computer-assisted method. For degree at most 5 we produce the full set of defining equations of the base ideal. The results answer affirmatively some questions raised by D. Cox.

math.AC↗
arXiv2006

Tangent Algebras

\noindent We study the Zariski tangent cone $T_X\stackrelπ{\lar} X$ to an affine variety $X$ and the closure $\bar{T}_X$ of $π^{-1}({\rm Reg}(X))$ in $T_X$. We focus on the comparison between $T_X$ and $\bar{T}_X$, giving sufficient conditions on $X$ in order that $T_X=\bar{T}_X$. One aspect of the results is to understand when this equality takes place in the presence of the reducedness of the Zariski tangent cone. Our other interest is to consider conditions on $X$ in order that $\bar{T}_X$ be normal or/and Cohen--Macaulay, and to prove that they are met by several classes of affine varieties including complete intersection, Cohen--Macaulay codimension two and Gorenstein codimension three singularities. In addition, when $X$ is the affine cone over a smooth arithmetically normal Calabi--Yau projective variety, we establish when $\bar{T}_X$ is also (the affine cone over) an arithmetically normal Calabi--Yau like (projective) variety.

math.AC↗