A Borel open cover of the Hilbert scheme

Let $p(t)$ be an admissible Hilbert polynomial in $\mathbb{P}^n$ of degree $d$. The Hilbert scheme $Hilb^n_{p(t)}$ can be realized as a closed subscheme of a suitable Grassmannian $\mathbb{G}$, hence it could be globally defined by homogeneous equations in the Plücker coordinates of $\mathbb{G}$ and covered by open subsets given by the non-vanishing of a Plücker coordinate, each embedded as a closed subscheme of the affine space $\mathbb{A}^D$, $D = \dim(\mathbb{G})$. However, the number $E$ of Plücker coordinates is so large that effective computations in this setting are practically impossible. In this paper, taking advantage of the symmetries of $Hilb^n_{p(t)}$, we exhibit a new open cover, consisting of marked schemes over Borel-fixed ideals, whose number is significantly smaller than $E$. Exploiting the properties of marked schemes, we prove that these open subsets are defined by equations of degree $\leqslant d + 2$ in their natural embedding in $\mathbb{A}^D$. Furthermore we find new embeddings in affine spaces of far lower dimension than $D$, and characterize those that are still defined by equations of degree $\leqslant d + 2$. The proofs are constructive and use a polynomial reduction process, similar to the one for Gröbner bases, but are term order free. In this new setting, we can achieve explicit computations in many non-trivial cases.