# Grothendieck-Pl\"ucker images of Hilbert schemes are degenerate

## TL;DR

This paper investigates the structure of Hilbert schemes through stratifications related to Grassmannian decompositions, proving their degeneracy under certain embeddings and providing geometric proofs for properties of generic initial ideals.

## Contribution

It introduces a stratification of Hilbert schemes based on Schubert decompositions and proves their degeneracy in high-degree Grothendieck-Plücker embeddings, with geometric proofs for generic initial ideals.

## Key findings

- Hilbert schemes admit stratifications with constant initial ideals.
- High-degree Grothendieck-Plücker embeddings cause Hilbert schemes to be degenerate.
- Geometric proofs of generic initial ideals' existence and properties.

## Abstract

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 of nonconstant Hilbert polynomial is embedded by the Grothendieck-Pl\"ucker embedding of a high enough degree, it must be degenerate.