Igusa quartic

In algebraic geometry, the Igusa quartic (also called the Castelnuovo–Richmond quartic CR₄ or the Castelnuovo–Richmond–Igusa quartic) is a quartic hypersurface in 4-dimensional projective space, studied by Igusa (1962). It is closely related to the moduli space of genus 2 curves with level 2 structure. It is the dual of the Segre cubic.

It can be given as a codimension 2 variety in P⁵ by the equations

[math]\displaystyle{ \sum x_i=0 }[/math]

[math]\displaystyle{ \big(\sum x_i^2\big)^2 = 4 \sum x_i^4 }[/math]

References

• Dolgachev, Igor V. (2012), Classical Algebraic Geometry: a modern view, Cambridge University Press, ISBN 978-1-107-01765-8, http://www.math.lsa.umich.edu/~idolga/CAG.pdf, retrieved 2016-08-17 
• Hunt, Bruce (1996), The geometry of some special arithmetic quotients, Lecture Notes in Mathematics, 1637, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0094399, ISBN 978-3-540-61795-2 
• Igusa, Jun-ichi (1962), "On Siegel Modular Forms of Genus Two", American Journal of Mathematics (The Johns Hopkins University Press) 84 (1): 175–200, doi:10.2307/2372812, ISSN 0002-9327