Some examples of Lüroth quartics

In mathematics, a Lüroth quartic is a nonsingular quartic plane curve containing the 10 vertices of a complete pentalateral. They were introduced by Jacob Lüroth (1869). Morley (1919) showed that the Lüroth quartics form an open subset of a degree 54 hypersurface, called the Lüroth hypersurface, in the space P¹⁴ of all quartics. Böhning & von Bothmer (2011) proved that the moduli space of Lüroth quartics is rational.

References

• Böhning, Christian; von Bothmer, Hans-Christian (2011), "On the rationality of the moduli space of Lüroth quartics", Mathematische Annalen, 353 (4): 1–9, arXiv:1003.4635, doi:10.1007/s00208-011-0715-7, ISSN 0025-5831
• Lüroth, J. (1869), "Einige Eigenschaften einer gewissen Gattung von Curven vierter Ordnung" (PDF), Mathematische Annalen, 1: 37–53, doi:10.1007/BF01447385, ISSN 0025-5831, JFM 02.0511.02
• Morley, Frank (1919), "On the Lüroth Quartic Curve", American Journal of Mathematics, 41 (4): 279–282, doi:10.2307/2370287, ISSN 0002-9327, JSTOR 2370287