# Moscow Mathematical Journal

Volume 22, Issue 4, October–December 2022 pp. 657–703.

On the Cone of Effective Surfaces on $\overline{\mathcal{A}}_3$

**Authors**:
Samuel Grushevsky (1) and Klaus Hulek (2)

**Author institution:**(1) Mathematics Department, Stony Brook University, Stony Brook, NY 11794-3651, USA

(2) Institut für Algebraische Geometrie, Leibniz Universität Hannover, Welfengarten 1, 30060 Hannover, Germany

**Summary: **

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 $g\ge 3$.

2020 Math. Subj. Class. Primary: 14K10; Secondary: 14E30, 14C25.

**Keywords:**Moduli spaces, abelian varieties, effective cycles, extremal rays.

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