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.
Ortiz et al. (Thu,) studied this question.