  Following the approach in the book "Commutative Algebra", by D. Eisenbud,
where the author describes the generic initial ideal by means of a suitable
total order on the terms of an exterior power, we introduce first the generic
initial extensor of a subset of a Grassmannian and then the double-generic
initial ideal of a so-called GL-stable subset of a Hilbert scheme. We discuss
the features of these new notions and introduce also a partial order which
gives another useful description of them. The double-generic initial ideals
turn out to be the appropriate points to understand some geometric properties
of a Hilbert scheme: they provide a necessary condition for a Borel ideal to
correspond to a point of a given irreducible component, lower bounds for the
number of irreducible components in a Hilbert scheme and the maximal Hilbert
function in every irreducible component. Moreover, we prove that every isolated
component having a smooth double-generic initial ideal is rational. As a
byproduct, we prove that the Cohen-Macaulay locus of the Hilbert scheme
parameterizing subschemes of codimension 2 is the union of open subsets
isomorphic to affine spaces. This improves results by J. Fogarty (1968) and R.
Treger (1989).
