Consider the moduli space $\mathcal{A}_{g,n}$ of abelian varieties with some level $n \geq 3$ structure. For simplicity, we just denote it by $\mathcal{A}_{g}$ and drop $n$. Denote the locus of hyperelliptic Jacobians by $H_{g}$ (i.e. the image of hyperelliptic curves under the Torelli map). Now, having a curve $C$ in $\mathcal{A}_{g}$, are there necessary and/or sufficient results to determine whether $C$ intersects $H_{g}$ ?(I am particularly interested in the case where $g=4$)
