Codimension conjecture for the simple part of the Andreotti–Mayer locus $N_2$

Let ${\mathcal A}_g$ be the moduli space of principally polarized abelian varieties of dimension $g$, and let

$$N_2=\{(X,\Theta_X)\mid \dim \operatorname{Sing}(\Theta_X)\geq 2\}.$$

A principally polarized abelian variety is simple if it is not isogenous to a product of positive-dimensional abelian varieties. The $N_2$ codimension conjecture. Every component of $N_2$ consisting of simple abelian varieties has codimension at least $5$ in ${\mathcal A}_g$. The paper presents this as a weaker anticipated bound than the Ciliberto–van der Geer conjectural bound of $6$ for $N_2$, and it was proposed as a statement expected to follow from the methods developed there.