Grothendieck–Plücker Images of Hilbert Schemes are Degenerate

We study the decompositions of Hilbert schemes induced by the Schubert cell decomposition of the Grassmannian variety and show that Hilbert schemes admit a stratification into locally closed subschemes along which the generic initial ideals remain the same. We give two applications. First, we give completely geometric proofs of the existence of the generic initial ideals and of their Borel fixed properties. Second, we prove that when a Hilbert scheme of non-constant Hilbert polynomial is embedded by the Grothendieck–Plücker embedding of a high enough degree, it must be degenerate.