Abstract
We prove that the moduli space \({\mathfrak{M}_L}\) of Lüroth quartics in \({\mathbb{P}^2}\), i.e. the space of quartics which can be circumscribed around a complete pentagon of lines modulo the action of \({\mathrm{PGL}_3 (\mathbb{C})}\) is rational, as is the related moduli space of Bateman seven-tuples of points in \({\mathbb{P}^2}\).
Article PDF
Similar content being viewed by others
References
Bateman H.: The quartic curve and its inscribed configurations. Am. J. Math. 36, 357–386 (1914)
Bogomolov, F.: Rationality of the moduli of hyperelliptic curves of arbitrary genus (Conf. Alg. Geom., Vancouver 1984). CMS Conf. Proceedings, vol. 6, pp. 17–37. Amer. Math. Soc., Providence (1986)
Böhning Chr., Graf v. Bothmer H.-Chr.: A Clebsch–Gordan formula for \({\mathrm{SL}_3 (\mathbb{C})}\) and applications to rationality. Adv. Math. 224, 246–259 (2010)
Böhning, Chr., Graf v. Bothmer, H.-Chr.: The rationality of the moduli spaces of plane curves of sufficiently large degree. Invent. Math. 179(1). doi:10.1007/s00222-009-0214-6 (2010). preprint available at arXiv:0804.1503
Dolgachev I., Kanev V.: Polar covariants of plane cubics and quartics. Adv. Math. 98, 216–301 (1993)
Dolgachev, I.: Rationality of fields of invariants. In: Proceedings of Symposia in Pure Mathematics. vol. 46, pp. 3–16 (1987)
Dolgachev I.: Lectures on Invariant. Theory London Mathematical Society Lecture Note Series, vol. 296. Cambridge University Press, Cambridge (2003)
Grace, J.H., Young, W.H.: The Algebra of Invariants. Cambridge University Press, Cambridge (1903); reprinted by Chelsea Publ. Co., New York (1965)
Katsylo, P.I.: Rationality of orbit spaces of irreducible representations of SL2. Izv. Akad. Nauk SSSR, Ser. Mat. 47(1), 26–36 (1983); English Transl.: Math USSR Izv. 22, 23–32 (1984)
Katsylo P.I.: Rationality of the moduli spaces of hyperelliptic curves. Izv. Akad. Nauk SSSR Ser. Mat. 48, 705–710 (1984)
Katsylo, P.I.: On the birational geometry of \({(\mathbb{P}^n)^{(m)}/\mathrm{GL}_{n+1}}\), Max-Planck Institut Preprint, MPI/94-144 (1994)
Morley F.: On the Lüroth quartic curve. Am. J. Math. 41, 279–282 (1919)
Ottaviani, G., Sernesi, E.: On the hypersurface of Lüroth quartics, to appear in Michigan Math. J. preprint 2009, arXiv:0903.5149
Acknowledgments
Both authors were supported by the German Research Foundation [Deutsche Forschungsgemeinschaft (DFG)] through the Institutional Strategy of the University of Göttingen.
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Böhning, C., von Bothmer, HC.G. On the rationality of the moduli space of Lüroth quartics. Math. Ann. 353, 1273–1281 (2012). https://doi.org/10.1007/s00208-011-0715-7
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00208-011-0715-7