Hilbert squares of genus 16 K3 surfaces

We consider the geometry of a general polarized K3 surface $(S,h)$ of genus 16 and its Fourier-Mukai partner $(S^{\mathrm{\prime }},h^{\mathrm{\prime }})$. We prove that $S^{[2]}$ is isomorphic to the moduli space $M_{S^{\mathrm{\prime }}}(2,h^{\mathrm{\prime }},7)$ of stable sheaves with Mukai vector $(2,h^{\mathrm{\prime }},7)$ and describe the embeddings of the projectivization of the stable vector bundle of Mukai vector $(2,-h^{\mathrm{\prime }},8)$ over $S^{\mathrm{\prime }}$ into these two isomorphic hyper-Kähler fourfolds. Following the work of FrédÉric Han in arXiv:2501.16013, we explicitly construct an interesting 3-form $t_1\in {\wedge }^3{V}_{10}^{\ast }$ which potentially gives an isomorphism between $S^{[2]}$ and the Debarre-Voisin fourfold in $G(6,V_{10})$ associated to $t_1\in {\wedge }^3{V}_{10}^{\ast }$. This would provide a geometric explanation of the existence of such an isomorphism, which was proved in arXiv:2102.11622 by a completely different argument.