An algorithm for computing torsion differential forms associated with an isolated hypersurface singularity
type: Item
instance of: scholarly article
title: An algorithm for computing torsion differential forms associated with an isolated hypersurface singularity
zbMATH Open document ID: 1476.14008
author: Shinichi Tajima
author: Katsusuke Nabeshima
published in: Mathematics in Computer Science
publication date: 2021-06-25T00:00:00Z
review text: At the beginning, the authors briefly discuss basic relationships between torsion elements of the module of regular differential forms of degree \(n\), logarithmic differential forms and vector fields on an isolated hypersurface singularity of positive dimension \(n\). In fact, similar relations exist in a more general context (see [the reviewer, Adv. Sov. Math. 1, 211--246 (1990; Zbl 0731.32005)]). Then they express the algebraic local cohomology of the structure sheaf of such hypersurface with supports in its singular point in terms of the torsion elements and logarithmic vector fields and describe an algorithm for computing the corresponding objects. Two examples of weighted homogeneous singularities of type \(E_{12}\) and \(U_{12}\) and their 1-parameter semi-quasihomogeneous deformations are computed explicitly. Reviewer's remarks: It should be remarked that both examples are Euler-homogeneous singularities (see [the reviewer, Ann. Global Anal. Geom. 4, No.~2, 225--242 (1986; Zbl 0632.32007)]); above all the first one is a free Saito divisor. Therefore, it is not difficult to analyze such examples directly, even without a computer (see, e.g., [the reviewer, Mat. Sb., Nov. Ser. 137(179), No.~4(12), 554--567 (1988; Zbl 0667.32010). Anyway, it remains unclear how their algorithm will work in more general cases. Moreover, there are much stronger and universal methods for computing syzygies (see, e.g., [\textit{G.-M. Greuel} and \textit{G.~Pfister}, A Singular introduction to commutative algebra. 2nd extended ed., Springer (2007; Zbl 1133.13001]), which can easily be adapted to solve all problems studied in the paper. In the authors' paper [SIGMA, Symmetry Integrability Geom. Methods Appl. 17, Paper 019, 21~p. (2021; Zbl 1472.32013)], the algorithm developed here is adapted to compute representatives of regular meromorphic differential forms.
Mathematics Subject Classification ID: 14B05
Mathematics Subject Classification ID: 32S25
Mathematics Subject Classification ID: 32S05
zbMATH DE Number: 7363382
zbMATH Keywords: isolated hypersurface singularities
zbMATH Keywords: Kähler differentials
zbMATH Keywords: torsion modules
zbMATH Keywords: logarithmic differential forms
zbMATH Keywords: local cohomology
zbMATH Keywords: duality
zbMATH Keywords: polar varieties
zbMATH Keywords: semi-quasihomogeneous singularities
reviewed by: Aleksandr G. Aleksandrov
MaRDI profile type: Publication
full work available at URL: s11786-020-00486-w
OpenAlex ID: W3033287432
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DOI: 10.1007/S11786-020-00486-W
description: scientific article; zbMATH DE number 7363382