We consider the question of when a Jacobian of a curve of genus $2g$ admits a $(2,2)$-isogeny to a product of two polarized dimension g abelian varieties. We find that one of them must be a Jacobian itself and, if the associated curve is hyperelliptic, so is the other.

For $g=2,$ this allows us to describe $(2,2)$-decomposable genus $4$ Jacobians in terms of Prym varieties. We describe the locus of such genus $4$ curves in terms of the geometry of the Igusa quartic threefold. We also explain how our characterization relates to Prym varieties of unramified double covers of plane quartic curves, and we describe this Prym map in terms of $6$ and $7$ points in $\mathbb {P}^3$.

We also investigate which genus $4$ Jacobians admit a $2$-isogeny to the square of a genus $2$ Jacobian and give a full description of the hyperelliptic ones. While most of the families we find are of the expected dimension $1$, we also find a family of unexpectedly high dimension $2$.

The aim of this paper is to prove that a $\text{K3}$ surface is the minimal model of the quotient of an Abelian surface by a group $G$ (respectively of a $\text{K3}$ surface by an Abelian group $G$ ) if and only if a certain lattice is primitively embedded in its Néron-Severi group. This allows one to describe the coarse moduli space of the $\text{K3}$ surfaces that are (rationally) $G$ -covered by Abelian or $\text{K3}$ surfaces (in the latter case $G$ is an Abelian group). When $G$ has order 2 or $G$ is cyclic and acts on an Abelian surface, this result is already known; we extend it to the other cases.

Moreover, we prove that a $\text{K3}$ surface ${{X}_{G}}$ is the minimal model of the quotient of an Abelian surface by a group $G$ if and only if a certain configuration of rational curves is present on ${{X}_{G}}$ . Again, this result was known only in some special cases, in particular, if $G$ has order 2 or 3.