A Borel open cover of the Hilbert scheme

Let p(t) be an admissible Hilbert polynomial in P of degree d. The Hilbert scheme Hilbnp(t) can be realized as a closed subscheme of a suitable Grassmannian G, hence it could be globally defined by homogeneous equations in the Plücker coordinates of 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 A, D = dim(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 Hilbnp(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 ≤ d + 2 in their natural embedding in A. 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 ≤ 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.