  The Debarre-Voisin hyperk\"ahler fourfolds are built from alternating
$3$-forms on a $10$-dimensional complex vector space, which we call trivectors.
They are analogous to the Beauville-Donagi fourfolds associated with cubic
fourfolds. In this article, we study several trivectors whose associated
Debarre-Voisin variety is degenerate, in the sense that it is either reducible
or has excessive dimension. We show that the Debarre-Voisin varieties
specialize, along general $1$-parameter degenerations to these trivectors, to
varieties isomorphic or birationally isomorphic to the Hilbert square of a K3
surface.
