  In this paper, we study principally polarized abelian varieties $X$ of
dimension $g$ that contain a curve $\nu:C\to X$ such that the class of $C$ is
$m$ times the minimal class. Welters introduced the formalism of stable pairs
to handle this problem in the case $m=2$. We generalize the results of Welters
and construct families of principally polarized abelian varieties for any $m$
and compute the dimension of the locus of these abelian varieties.
