On the rationality of the moduli space of Lüroth quartics
type: Item
instance of: scholarly article
title: On the rationality of the moduli space of Lüroth quartics
zbMATH Open document ID: 1251.14009
author: Christian Böhning
author: Hans-Christian Graf von Bothmer
published in: Mathematische Annalen
publication date: 2012-08-13T00:00:00Z
full work available at URL: https://arxiv.org/abs/1003.4635
review text: A \textit{Lüroth quartic} is a quartic plane curve containing a complete pentagon, i.e. a set of \(10\) points which are the pairwise intersections of \(5\) lines. The locus of Lüroth quartics has a closure \(L\), in the projective space \(\mathbb P^{14}\) of all quartics, which is an irreducible hypersurface of degree \(54\). The quotient of \(L\) modulo the action of SL\({}_3\) is the moduli space \(\mathcal L\) of Lüroth quartics. The authors prove that \(\mathcal L\) is rational. Indeed, they prove that \(\mathcal L\) is birational to the quotient mod SL\({}_3\) of the space of \textit{Clebsch quartics}, i.e. quartics that can be written as a sum of \(5\) powers of linear forms. The rationality of the latter space is proven by using methods of representation theory. The authors also prove that a covering space of \(\mathcal L\), which parametrizes Bateman configurations of seven-tuples of points, is rational.
reviewed by: Luca Chiantini
Mathematics Subject Classification ID: 14E08
Mathematics Subject Classification ID: 14H45
zbMATH DE Number: 6065239
zbMATH Keywords: Plane curves
describes a project that uses: ClebschGordan
describes a project that uses: rationality
MaRDI profile type: Publication
OpenAlex ID: W2040547857
arXiv ID: 1003.4635
cites work: A Clebsch-Gordan formula for \(\text{SL}_3(\mathbb C)\) and applications to rationality
cites work: Rationality of the moduli spaces of plane curves of sufficiently large degree
cites work: Polar covariants of plane cubics and quartics
cites work: Q3808217
cites work: Q4780265
cites work: On the hypersurface of Lüroth quartics
DOI: 10.1007/S00208-011-0715-7
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description: scientific article; zbMATH DE number 6065239