We study the subsets V k ( A ) of a complex abelian variety A consisting in the collection of points x ∈ A such that the zero-cycle { x } - { 0 A } is k -nilpotent with respect to the Pontryagin product in the Chow group. These sets were introduced recently by Voisin and she showed that dim V k ( A ) ≤ k - 1 and dim V k ( A ) is countable for a very general abelian variety of dimension at least 2 k - 1 . We study in particular the locus V g , 2 in the moduli space of abelian varieties of dimension g with a fixed polarization, where V 2 ( A ) is positive dimensional. We prove that an irreducible subvariety Y ⊂ V g , 2 , g ≥ 3 , such that for a very general y ∈ Y there is a curve in V 2 ( A y ) generating A satisfies dim Y ≤ 2 g - 1 . The hyperelliptic locus shows that this bound is sharp.