Moduli of \((1,7)\)-polarized abelian surfaces via syzygies (Q2729623)

In this paper, the authors study the moduli space of polarized abelian varieties, and they construct it as Heisenberg invariant part of a Hilbert scheme. In the particular case of \((1,7)\)-polarized abelian surfaces, they obtain only a birational model of the moduli space. But they obtain a birational parametrization of the moduli space defined over \(\mathbb{Q}\). They prove that the moduli space \(X(1,7)\) of \((1,7)\)-polarized abelian surfaces with canonical level structure is birational to the Fano 3-fold \(V_{22}\) of polar hexagons of the Klein quartic \(\overline{X}(7)\). In particular, they obtain that \(X(1,7)\) is rational with birational map to \(\mathbb{P}^{3}\) defined over \(\mathbb{Q}\). Furthermore they give the equations of the \((1,7)\)-very ample polarized abelian surfaces embedded in \(\mathbb{P}^{6}\).