On generators of the module of logarithmic 1-forms with poles along an arrangement
type: Item
instance of: scholarly article
title: On generators of the module of logarithmic 1-forms with poles along an arrangement
zbMATH Open document ID: 0826.52017
DOI: 10.1023/A:1022480128534
author: Sergey Yuzvinsky
published in: Journal of Algebraic Combinatorics
publication date: 1995-07-13T00:00:00Z
review text: The author considers elements \(X\) of codimension 2 of the intersection lattice of an arrangement \(A\) of \(n\) hyperplanes in a \(d\)-dimensional linear space over an arbitrary field. Defining differential logarithmic 1-forms \(\omega_X\) for these elements, with poles along \(A\), he describes the class of arrangements for which forms \(\omega_X\) generate the whole module of the logarithmic 1-forms with poles along \(A\). This is done in terms of linear relations among the functionals defining the hyperplanes. Further on, a minimal free resolution of the module generated by \(\omega_X\) is constructed, that in particular defines the projective dimension of this module. For studying relations among \(\omega_X\), free resolutions of certain ideals of a polynomial ring, generated by products of linear forms, are constructed. The author presents certain 3- and 4-arrangements as examples, and he finally discusses two possible directions of generalization.
reviewed by: Horst Martini
Mathematics Subject Classification ID: 52C35
zbMATH DE Number: 773837
zbMATH Keywords: hyperplane arrangements
zbMATH Keywords: logarithmic form
zbMATH Keywords: module
zbMATH Keywords: free resolution
zbMATH Keywords: ideal
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 773837