On the irreducible components of the singular locus of $A_{g}$. II

Abstract:

In our earlier paper it was proved that the singular locus of $A_{g}$ (coarse moduli space of principally polarized abelian varieties over $\mathbb {C}$) is expressed as the union of irreducible varieties $A_{g}(p,\alpha )$ representing abelian varieties with an order $p$ automorphism of fixed entire representation. In this paper we prove that $A_{g}(p,\alpha )$ is an irreducible component of $\text {Sing} A_{g}$ if and only if for a general element of this variety its automorphism group modulo $\{\pm 1\}$, $G_{+}$, satisfies the equivalent conditions: $G_{+}=\langle \alpha \rangle$ or $N_{G_{+}}(\langle \alpha \rangle )=\langle \alpha \rangle$. We illustrate how these results can be used by studying the case $g=4$ and $p=5$.