  In this paper we derive geometric consequences from the presence of a long
strand of linear syzygies in the minimal free resolution of a closed scheme in
projective space whose homogeneous ideal is generated by quadrics. These
consequences are given in terms of intersections with arbitrary linear
subspaces. We use our results to bound homological invariants of some
well-known projective varieties, to give a combinatorial characterization of
quadratic monomial ideals with a long strand of linear syzygies, etc
