Gröbner deformations
type: Item
instance of: scholarly article
title: Gröbner deformations
zbMATH Open document ID: 1466.13012
author: Matteo Varbaro
published in: Bollettino dell'Unione Matematica Italiana
publication date: 2021-05-03T00:00:00Z
review text: In this survey the author collects some recent results about connectedness and cohomological properties of the zero loci \(\mathcal Z(I)\subseteq\mathbb C\mathrm P^n\) of homogeneous ideals \(I\) in \(\mathbb C[x_0,x_1,\ldots,x_n]\). These results are mostly concerned with comparing properties and invariants associated to a homogeneous ideal \(I\) and to its initial ideal \(\mathrm{in}(I)\) (the ideal in \(\mathbb C[x_0,x_1,\ldots,x_n]\) generated by the leading monomials of the polynomials in \(I\) w.r.t.\ a fixed monomial ordering). For example, a theorem (which is proved in [\textit{M. Varbaro}, J. Algebra 322, 2492--2507 (2009; Zbl 1200.13029)]) states that if \(\mathcal Z(I)\) is connected, then \(\mathcal Z(\mathrm{in}(I))\) is also connected, and the converse is true if \(\mathrm{in}(I)\) is a radical ideal. In the last part of the survey, the author consideres projective schemes in \(\mathbb C\mathrm \mathbb P^n\) and discusses some related results and conjectures in terms of their cohomological invariants.
reviewed by: Branislav Prvulović
Mathematics Subject Classification ID: 13D99
Mathematics Subject Classification ID: 13F99
zbMATH DE Number: 7342987
zbMATH Keywords: Gröbner basis
zbMATH Keywords: zero locus
zbMATH Keywords: projective scheme
MaRDI profile type: Publication
full work available at URL: s40574-020-00241-w
OpenAlex ID: W4242645006
cites work: Extremal Betti numbers and applications to monomial ideals
cites work: A criterion for detecting m-regularity
cites work: Square-free Gröbner degenerations
cites work: Singularities and radical initial ideals
cites work: Q4748248
cites work: Every algebraic set in n-space is the intersection of n hypersurfaces
cites work: Gröbner deformations, connectedness and cohomological dimension
DOI: 10.1007/S40574-020-00241-W
description: scientific article; zbMATH DE number 7342987