Gr\"obner scheme in the Hilbert scheme and complete intersection monomial ideals

Let $k$ be a commutative ring and $S=k[x_0, \ldots, x_n]$ be a polynomial ring over $k$ with a monomial order. For any monomial ideal $J$, there exists an affine $k$-scheme of finite type, called Gr\"obner scheme, which parameterizes all homogeneous reduced Gr\"obner bases in $S$ whose initial ideal is $J$. Here we functorially show that the Gr\"obner scheme is a locally closed subscheme of the Hilbert scheme if $J$ is a saturated ideal. In the process, we also show that the Gr\"obner scheme consists of complete intersections if $J$ defines a complete intersection.