We investigate whether surfaces that are complete intersections of quadrics and complete intersection surfaces in the Segre embedded product P1×Pk↪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.
- Complete intersections of quadrics
- Hilbert schemes
- Segre varieties
- local moduli