COMPLETE INTERSECTIONS OF QUADRICS AND COMPLETE INTERSECTIONS ON SEGRE VARIETIES WITH COMMON SPECIALIZATIONS

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