On the invariants of pencils of binary quintics.
type: Item
instance of: scholarly article
title: On the invariants of pencils of binary quintics.
zbMATH Open document ID: 1062.14060
author: Matthias Meulien
publication date: 2004-12-20T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/116107
review text: A pencil of quintic binary forms is a \(2\)-dimensional subspace of \(\mathbf{S} _{5},\) where \(\mathbf{S}_{5}\) is the \({\text{SL}_{2}}\)-module \( H^{0}\left( \mathcal{O}_{\mathbb{P}^1}\left( 5\right) \right) .\) This paper is a study of the invariants of the pencils of binary quintic forms under the natural action of \({\text{SL}_{2}}.\) As the Grassmannian \( \mathbf{G}_{2}\left( \mathbf{S}_{5}\right) \) is birationally equivalent and \( {\text{SL}_{2}}\)-equivariant to \(\mathbb{P}\left( \mathbf{S}_{8}\right) \) we consider the algebra of invariants of an octic form. Furthermore, the invariants of an octic form corresponds to a Gorenstein algebra of codimension 3. The main result is as follows. For \(B\) the \textit{algèbre du cône} of \(\mathbf{G}_{2}\left( \mathbf{S}_{5}\right) \) and \(R=\mathbb{C}\left[ x_{1},x_{2},x_{3},x_{3}^{\prime },x_{4},x_{5},x_{5}^{\prime },x_{6},x_{7} \right] \) (where the degree of each indeterminate is its index), the graded algebra \(B^{\text{SL}_{2}}\) is isomorphic to \(R/I,\) where \(I\) is an ideal of Pfaffians of order \(4\) of a certain alternating \(5\times 5 \) matrix. Furthermore, \[ 0\rightarrow R\left( -30\right) \rightarrow \bigoplus_{d=16}^{20}R\left( -d\right) \rightarrow \bigoplus_{d=10}^{14}R\left( -d\right) \rightarrow R\rightarrow 0 \] is the minimal projective resolution for \(B^{\text{SL}_{2}}.\)
reviewed by: Alan Koch
Mathematics Subject Classification ID: 14L24
Mathematics Subject Classification ID: 14L30
Mathematics Subject Classification ID: 13A50
Mathematics Subject Classification ID: 15A72
zbMATH DE Number: 2123558
zbMATH Keywords: binary quintic forms
zbMATH Keywords: pencils
zbMATH Keywords: invariant theory
MaRDI profile type: Publication
OpenAlex ID: W2078115365
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DOI: 10.5802/AIF.2009
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 2123558