Connected Components in the Hilbert Scheme of hypersurfaces in Grassmannians

TL;DR

This paper investigates the structure of Hilbert schemes of hypersurfaces in Grassmannians, revealing multiple connected components even when elements share the same cohomology class, and establishing bounds on the number of components.

Contribution

It demonstrates the existence of two components in certain Hilbert schemes of hypersurfaces in Grassmannians and provides bounds on the number of components for specific Hilbert polynomials.

Findings

Hilbert scheme has 2 components for d ≥ 3

Elements in both components share the same cohomology class

Hilbert scheme has at most 2 connected components for certain polynomials

Abstract

We show that when $d \geq 3$, the Hilbert scheme $Hilb_{dT+1-\binom{d-1}{2}}(G(k,n))$ has 2 components, even though elements in both components have the same cohomology class. Moreover, we show that the Hilbert scheme associated to the Hilbert polynomial $\binom{T+n}{n}-\binom{T+n-d}{n}$ in Grassmannian has at most 2 connected components.