Polarized K3 surfaces of genus thirteen and curves of genus three

We describe a general (primitively) polarized K3 surface \((S,h)\) with \((h^2)=24\) as a complete intersection variety with respect to vector bundles on the \(6\)-dimensional moduli space \(\mathcal{N}^-\) of the stable vector bundles of rank two with fixed odd determinant on a curve \(C\) of genus \(3\). If the curve \(C\) is hyperelliptic, then \(\mathcal{N}^-\) is a subvariety of the \(12\)-dimensional Grassmann variety \(\operatorname{Gr}(\mathbf{C}^8,2)\) defined by a pencil of quadric forms. In this case, our description implies that a general \((S,h)\) is the intersection of two (7-dimensional) contact homogeneous varieties of \(\operatorname{Spin}(7)\) in the Grassmann variety \(\operatorname{Gr}(\mathbf{C}^8,2)\).