A stratification of Hilbert schemes via generic initial ideals

Donghoon Hyeon

Published 2017-03-08Version 1

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 a completely geometric proofs of the existence of the generic initial ideals and of their Borel fixed properties. Secondly, we prove that when a Hilbert scheme is embedded by the Grothendieck-Pl\"ucker embedding of a high enough degree, one of its components must be degenerate.