Lurot's quartic (fourth-degree curve of Lurot) is a non-singular flat fourth-degree curve containing 10 vertices of a pentagonal star ( pentagram ), not necessarily correct. The apartments of Lurot were introduced by Jacob Lurot [1] .
Properties
Lurot was the first to notice in 1868 that if a quartic describes a pentagonal star, it describes infinitely many other pentagonal stars [2] .
Morley [3] showed that the Lurot quartics form an open subset of a hypersurface of degree 54, called the Lurot hypersurface , in the space P 14 of all quartics. The equation of this hypersurface is called the Lurot invariant , but it remains unknown [2] . The Lurot hypersurface consists entirely of quartics, so the limits (when the pentagon degenerates) are also now called the Lurot quartics [2] .
Böning and von Botner [4] proved that the moduli space of Lurot's quarts is rational.
The Lurot quartic is closely related to the Clebsch quartic [5] - it is the projective covariant of this curve [6] .
Notes
- ↑ Lüroth, 1869 .
- ↑ 1 2 3 Ottaviani, 2012 , p. 2.
- ↑ Morley, 1919 .
- ↑ Böhning, von Bothmer, 2011 .
- ↑ Ottaviani, 2012 , p. 7.
- ↑ Morley, 1919 , p. 348.
Literature
- Christian Böhning, Hans-Christian von Bothmer. On the rationality of the moduli space of Lüroth quartics // Mathematische Annalen . - Springer Berlin / Heidelberg, 2011. - S. 1–9 . - ISSN 0025-5831 . - DOI : 10.1007 / s00208-011-0715-7 . - arXiv : 1003.4635 .
- Lüroth J. Einige Eigenschaften einer gewissen Gattung von Curven vierter Ordnung // Mathematische Annalen . - Springer Berlin / Heidelberg, 1869 .-- T. 1 . - S. 37–53 . - ISSN 0025-5831 . - DOI : 10.1007 / BF01447385 .
- Frank Morley On the Lüroth Quartic Curve // American Journal of Mathematics . - The Johns Hopkins University Press, 1919 .-- T. 41 , no. 4 . - S. 279–282 . - ISSN 0002-9327 . - DOI : 10.2307 / 2370287 .
- James Joseph Sylvester. The Collected Mathematical Papers of James Joseph Sylvester: Volume IV (1882 - 1897). - Cambridge University Press, 1912.
- Giorgio Ottaviani. A computational approach to Lüroth quartics . - 2012. - arXiv : 1208.1372v1 .