Igusa quartic and Steiner surfaces (Q2895550)

The Igusa quartic is a quartic hypersurface in \(\mathbb{P}^3\) isomorphic to the Satake compactification of the Siegel modular threefold of (full) level~\(2\), i.e.\ the moduli space of principally polarised abelian surfaces with a level-\(2\) structure.NEWLINENEWLINELet \((X,\sigma)\) be a pair consisting of a hypersurface in \(\mathbb{P}^4\) and an involution \(\sigma\in{\text{PGL}}(4,\mathbb{C})\) such that \(\sigma\) lifts to a reflection in \({\text{GL}}(5,\mathbb{C})\). Then the fixed locus of \(\sigma\) on \(\mathbb{P}^4\) consists of a hyperplane and a point. The pair is said to have the Steiner property if the intersection of the fixed hyperplane with \(X\) is a copy \(R\) of Steiner's Roman surface and projection from the fixed point realises \(X/\sigma\) as a double cover of \(\mathbb{P}^3\) branched along the four planes that meet \(R\) in double conics (classically called tropes).NEWLINENEWLINEOf course most quartics do not have the Steiner property, whatever choice of \(\sigma\) we make. The first observation here is that the Igusa quartic does have the Steiner property, for a suitable choice of \(\sigma\). The author gives three different proofs of this, all very short, using respectively Igusa's equation, the alternative equations given by van der Geer, and the classical theory of Kummer surfaces.NEWLINENEWLINEThe main theorem says that the Satake compactification of the Siegel modular variety associated with the subgroup \(\Gamma_1(2)\) is also isomorphic to the Igusa quartic. This is the moduli of abelian surfaces equipped with a Göpel triple, i.e.\ a maximal totally isotropic subspace of the \(2\)-torsion. The essential point is that this variety is a quartic hypersurface in \(\mathbb{P}^4\) that has the Steiner property with \(\sigma\) given by the Fricke involution.NEWLINENEWLINEFor the entire collection see [Zbl 1236.14001].