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

TL;DR

This paper identifies five key extremal rays of the effective surface cone on the compactified moduli space of abelian threefolds and conjectures their generative role in higher genera.

Contribution

It determines specific extremal rays of the effective surface cone on  and proposes a conjecture for their universal applicability across all genera .

Findings

Identified five extremal effective rays on 

Conjectured these rays generate the entire cone of effective surfaces

Proposed the conjecture extends to all  for higher genus

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 toroidal compactification of ${\mathcal A}_g$ for any $g\ge 3$.