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We study smooth quadric surfaces in the Pfaffian hypersurface in$\mathbb{P}^{14}$ parameterising $6 \times 6$ skew-symmetric matrices of rankat most 4, not intersecting the Grassmannian $\mathbb{G}(1,5)$. Such surfacescorrespond to quadratic systems of skew-symmetric matrices of size 6 andconstant rank 4, and give rise to a globally generated vector bundle $E$ on thequadric. We analyse these bundles and their geometry, relating them to linearcongruences of lines in $\mathbb{P}^5$.

Quadric surfaces in the Pfaffian hypersurface in $\mathbb{P}^{14}$