Hypersurfaces in P^5 containing unexpected subvarieties

Abstract

Smooth cubic 4-folds in P 5 containing a general pair of 2-planes are known to be rational. They form a family of codimension 2 in P 55 . We find a polynomial which encodes, for all d ≥ 3, the degrees of the loci of hypersurfaces in P 5 of degree d containing some plane-pair.Dedicated to Xavier Gómez-MontÁvalos on the occasion of his 60th birthday.1991 Mathematics Subject Classification. 14N05, 14N15 (Primary).

The Hilbert scheme of a pair of linear spaces

“…Example 2. 25 We can now determine when there is a specialization Z Z in H n n−k,n−k . For any subscheme Z ∈ H n n−k,n−k let W Z = { 1 , .…”

“…Remark 2. 26 In [25], Vainsencher uses the map : Bl 2 Bl 1 Gr(2, 5) 2 → H 5 2,2 to compute the degree of a family of rational cubic fourfolds in P 5 . However, he does not prove the smoothness of H 5 2,2 .…”