Projective models for Hilbert squares of $K3$ surfaces

For a very general polarized $K3$ surface S⊂ Pᵍ of genus g≥ 5, we study the linear system on the Hilbert square S[2] parametrizing quadrics in Pᵍ that contain S. We prove its very ampleness for g≥ 7. In the cases of genus 7 or 8, we describe in detail the projective geometry of the corresponding embedding by making use of the Mukai model for S. In both cases, it can be realized as a degeneracy locus on an ambient homogeneous space, in a strikingly similar fashion. In consequence, we give explicit descriptions of its ideal and syzygies. Furthermore, we extract new information on the locally complete families, in a first step towards the understanding of their projective geometry.