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We study smooth quadric surfaces in the Pfaffian hypersurface inparameterizing 6 × 6 skew-symmetric matrices of rank at most 4, not intersecting its singular locus. Such surfaces correspond to quadratic systems of skew-symmetric matrices of size 6 and constant rank 4, and give rise to a globally generated vector bundle  E  on the quadric. We analyse these bundles and their geometry, relating them to linear congruences in.

Quadric surfaces in the Pfaffian hypersurface in ℙ 14