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On the cone of effective surfaces on $\overline{\mathcal{A}}_3$

Samuel Grushevsky^a, Klaus Hulek<sup>b

a</sup> Mathematics Department, Stony Brook University, Stony Brook, NY 11794-3651, USA
^b Institut für Algebraische Geometrie, Leibniz Universität Hannover, Welfengarten 1, 30060 Hannover, Germany

Abstract: We determine five extremal effective rays of the four-dimensional cone of effective surfaces on the toroidal compactification $\overline{\mathcal{A}}_3$ of the moduli space ${\mathcal{A}}_3$ of complex principally polarized abelian threefolds, and we conjecture that the cone of effective surfaces is generated by these surfaces. As the surfaces we define can be defined in any genus $g\ge 3$, we further conjecture that they generate the cone of effective surfaces on the perfect cone compactification $\mathcal A_g^{\mathrm{Perf}}$ for any $g\ge 3$.

Key words and phrases: moduli spaces, abelian varieties, effective cycles, extremal rays.

MSC: Primary 14K10; Secondary 14E30, 14C25

Language: English

DOI: 10.17323/1609-4514-2022-22-4-657-703