Hilbert squares of degeneracy loci

Let S be the first degeneracy locus of a morphism vector bundles corresponding to general matrix linear forms in $${\mathbb {P}}^s$$ . We prove that, under certain positivity conditions, its Hilbert square $${{\mathrm{Hilb}}}^2(S)$$ is isomorphic zero global section an irreducible homogeneous bundle on product Grassmannians. Our construction involves naturally associated Fano variety, and explicit description isomorphism.