Complete intersections of quadrics and complete intersections on Segre varieties with common specializations

We investigate whether surfaces that are complete intersections of quadrics and intersection in the Segre embedded product P1×Pk↪P2k+1 can belong to same Hilbert scheme. For k=2 there is a classical example; it comes from K3 projective 5-space degenerate into hypersurface on threefold. show for k≥3 only one more example. It turns out its (connected) scheme has at least two irreducible components. corresponding local moduli problem.