The moduli space of binary quintics
type: Item
instance of: scholarly article
title: The moduli space of binary quintics
zbMATH Open document ID: 1390.14142
author: C. T. C. Wall
author: Andrew A. du Plessis
published in: European Journal of Mathematics
publication date: 2018-04-25T00:00:00Z
review text: The \(\mathrm{SL}_2\)-invariant ring of binary quintics over \(\mathbb{C}\) is generated by four invariants \(I_4\), \(I_8\), \(I_{12}\), and \(I_{18}\). Their moduli space \(\mathcal M\) (with quintics with cubic factors removed) admits coordinates \((I_4, I_8, I_{12})\). It contains the curve \(\Delta\) of quintics with repeated roots and the curve \(\Gamma_5\) of quintics with nontrivial symmetry. The authors describe the topology of real points of \(\mathcal M\) with prescribed numbers of real roots and consider in more detail five exceptional curves in \(\mathcal M\) that arose in [\textit{C. T. C. Wall}, Math. Proc. Camb. Philos. Soc. 119, No. 2, 257--277 (1996; Zbl 0868.14015)]: There are five special quintics \(H_1\), \(H_2\), \dots, \(H_5\) in \(\mathbb{P}^2\) with one-parametric symmetry group. The intersection points with the corresponding orbits of lines give curves \(\Gamma_1\), \(\Gamma_2\), \dots, \(\Gamma_5\) in \(\mathcal M\). Their equations, intersections with \(\Delta\), mutual intersections, and singularities are computed. In the concluding section, the theory is adapted to characteristic two.
reviewed by: Hans-Peter Schröcker
Mathematics Subject Classification ID: 14L24
Mathematics Subject Classification ID: 14Q05
zbMATH DE Number: 6864590
zbMATH Keywords: binary quintics
zbMATH Keywords: invariants
zbMATH Keywords: moduli space
zbMATH Keywords: symmetries
zbMATH Keywords: stratification
MaRDI profile type: Publication
Wikidata QID: Q59612856
full work available at URL: s40879-017-0187-8
OpenAlex ID: W2766087774
cites work: Highly singular quintic curves
DOI: 10.1007/S40879-017-0187-8
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description: scientific article; zbMATH DE number 6864590