# Finding a cover of the Hilbert functor, that corresponds to a cover of the Grassmannian

Let $$\mathrm{Grass}_{k+1,n+1}:\mathrm{Sch}^{\circ}\rightarrow\mathrm{Set}$$ be the Grassmann functor, which maps a scheme $$S$$ to the set:

$$\left\{\mathscr{U}\subseteq\mathscr{O}_S^{n+1}:\,\,\mathscr{O}_S^{n+1}/\mathscr{U} \text{ is locally free of rank }n-k\right\}$$

We have a cover of $$\mathrm{Grass}_{k+1,n+1}$$ by open subfunctors $$\mathrm{Grass}_{k+1,n+1}^{I}$$, where $$I\subseteq\{1,...,n+1\}$$ ranges over all subsets with $$n-k$$ elements and $$\mathrm{Grass}_{k+1,n+1}^{I}$$ is defined as

$$\mathrm{Grass}_{k+1,n+1}^{I}(S)=\left\{\mathscr{U}\in\mathrm{Grass}_{k+1,n+1}(S):\,\, \mathscr{U}\oplus\mathscr{O}_S^{I}=\mathscr{O}_S^{n+1}\right\}$$

Let $$P=\binom{m+k}{k}\in\mathbb{Q}[m]$$ and $$h_P:\mathrm{Sch}^{\circ}\rightarrow\mathrm{Set}$$ the Hilbert functor, that associates to any scheme $$S$$ the set of closed subschemes $$Z\subseteq S\times \mathbb{P}_{\mathbb{Z}}^n$$, such that $$Z$$ is flat over $$S$$ and all fibers $$Z_s$$ over $$S$$ have Hilbert Polynomial $$P$$.

Is there are cover of $$h_P$$ by open subfunctors $$h_P^{I}$$, such that $$\mathrm{Grass}_{k+1,n+1}^{I}\cong h_P^{I}$$ ?. Since $$\mathrm{Grass}_{k+1,n+1}\cong h_P$$ , there has to be such a cover, but I am not really able to find one.

For an affine scheme $$S=\mathrm{Spec}(R)$$, I found a map $$\mathrm{Grass}_{k+1,n+1}^{I}(S)\rightarrow h_P(S)$$, which is defined in the following way:

$$\mathscr{U}\in\mathrm{Grass}_{k+1,n+1}(S)$$ is quasicoherent and therefore $$\mathscr{U}=\tilde{M}$$ for an $$R$$-module $$M$$. Now $$\mathscr{U}\oplus\mathscr{O}_S^{I}=\mathscr{O}_S^{n+1}$$ implies $$M\oplus R^{I}=R^{n+1}$$, so that the projection onto $$R^{I}$$ gives a linear map:

$$R^{n+1}\rightarrow R^{I}\cong R^{n-k}$$

Now one can try to take the components $$f_1,...,f_{n-k}$$ of this map and define $$Z=\mathrm{Proj}\left(R[x_0,...,x_{n+1]}/(f_1,...,f_{n-k})\right)$$, where the $$f_i$$ are viewed as linear homogeneous polynomials. Of course we could now define $$h_P^{I}$$ as the image of this morphism, but then it is not clear how this functor is defined for an arbitrary scheme.

I am very thankful for any thoughts on this.