Application of classical invariant theory to biholomorphic classification of plane curve singularities, and associated binary forms
type: Item
instance of: scholarly article
title: Application of classical invariant theory to biholomorphic classification of plane curve singularities, and associated binary forms
zbMATH Open document ID: 1305.14015
DOI: 10.1134/S0081543812080172
published in: Proceedings of the Steklov Institute of Mathematics
publication date: 2014-10-09T00:00:00Z
full work available at URL: https://arxiv.org/abs/1106.2602
review text: Let \(V_{s,t}\) be the germ at the origin of the curve defined by \[ f_{s,t}:=x^5+sx^4y+tx^3y^2+y^5=0\;,\;s,t\in\mathbb{C}\;. \] The biholomorphically equivalence problem is discussed for the family \(V_{s,t}\). It is proved that \(V_{s_1, t_1}\) and \(V_{s_2, t_2}\) are biholomorphically equivalent if and only if \(j(s_1, t_1)=j(s_2, t_2)\), \(k(s_1, t_1)=k(s_2, t_2)\) and \(l(s_1, t_1)=l(s_2, t_2)\). Here \(j, k, l\) are relatively complicated invariants from classical invariant theory, for example \[ j(s,t)=\frac{5(1440000\cdot 10!)^2(125-3st^2)^2}{256s^5-1600s^3t-27s^2t^4+2250st^2+108t^5+3125}. \]
reviewed by: Gerhard Pfister
Mathematics Subject Classification ID: 14H20
Mathematics Subject Classification ID: 14B05
Mathematics Subject Classification ID: 32S05
zbMATH DE Number: 6353228
zbMATH Keywords: plane curve singularities
zbMATH Keywords: binary forms
zbMATH Keywords: invariant theory
author: A. V. Isaev
MaRDI profile type: Publication
arXiv ID: 1106.2602
cites work: Local normal forms of functions
cites work: On the ubiquity of Gorenstein rings
cites work: Invariant theory, old and new
cites work: Moduli of isolated hypersurface singularities
cites work: Isolated hypersurface singularities and special polynomial realizations of affine quadrics
cites work: Nilpotent algebras and affinely homogeneous surfaces
cites work: Introduction to Singularities and Deformations
cites work: Q4939299
cites work: On the affine homogeneity of algebraic hypersurfaces arising from Gorenstein algebras
cites work: Q4280628
cites work: Q3478699
cites work: Q3686804
cites work: Classification of isolated hypersurface singularities by their moduli algebras
cites work: Q4509277
cites work: Singularities. II: Automorphisms of forms
cites work: Einfach-elliptische Singularitäten
cites work: Analytic equivalence of plane curve singularities yn +xαy +xβa(x) = 0
Recommended article: Associated forms in classical invariant theory and their applications to hypersurface singularities
Recommended article: The Analytic Classification of Irreducible Plane Curve Singularities
Recommended article: On the analytic classification of irreducible plane curve singularities
Recommended article: Homological invariants for classification of kinematic singularities
Recommended article: Q3712431
Recommended article: Q4379990
Recommended article: Q3418280
Recommended article: Associated forms and hypersurface singularities: the binary case
Recommended article: Invariants of plane curve singularities and Plücker formulas in positive characteristic
Recommended article: Q5752536
description: scientific article; zbMATH DE number 6353228