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Complete intersections of quadrics and complete intersections on Segre varieties with common specializations
We investigate whether surfaces that are complete intersections of quadrics and complete intersection surfaces in the Segre embedded product \mathbf{P}^1\times \mathbf{P}^k\hookrightarrow \mathbf{P}^{2k+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 …