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

${\displaystyle \sum x_{i}=0}$

${\displaystyle {\big (}\sum x_{i}^{2}{\big )}^{2}=4\sum x_{i}^{4}}$

References

• Dolgachev, Igor V. (2012), Classical Algebraic Geometry: a modern view (PDF), Cambridge University Press, ISBN 978-1-107-01765-8, archived from the original (PDF) on 2014-05-31, retrieved 2016-08-17
• Hunt, Bruce (1996), The geometry of some special arithmetic quotients, Lecture Notes in Mathematics, vol. 1637, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0094399, ISBN 978-3-540-61795-2, MR 1438547
• 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, JSTOR 2372812