Complement conjecture for the Coble fourfold moduli locus

Let M4(1)\mathcal M_4^{(1)} be the moduli space under consideration, let MCoble4\mathcal M_{\mathrm{Coble4}}^\circ be the open locus obtained from smooth stable Coble type quartics, and let D2\mathcal D_2, D6\mathcal D_6, D8\mathcal D_8, and D12\mathcal D_{12} be the four HLS Heegner divisors. Complement conjecture. Assuming the stability conjecture, the complement M4(1)MCoble4\mathcal M_4^{(1)}\smallsetminus \mathcal M_{\mathrm{Coble4}}^\circ is the union D2D6D8D12\mathcal D_2\cup \mathcal D_6\cup \mathcal D_8\cup \mathcal D_{12}. The source also recalls that the complement of M4(1)\mathcal M_4^{(1)} in P4(1)\mathcal P_4^{(1)} is D4D16D20\mathcal D_4\cup \mathcal D_{16}\cup \mathcal D_{20}. This describes the expected boundary of the Coble-type moduli locus, conditional on stability.

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Primary source

Benedetta Piroddi, Ángel David Ríos Ortiz, Andrés Rojas and Jieao Song, “Coble type hypersurfaces and hyperkähler fourfolds”, arXiv:2606.10884 (2026).

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