A Borel open cover of the Hilbert scheme

Let p(t) be an admissible Hilbert polynomial in P-n of degree d. The Hilbert scheme Hilb(p(t))(n) can be realized as a closed subscheme of a suitable Grassmannian G, hence it could be globally defined by homogeneous equations in the Plucker coordinates of G and covered by open subsets given by the non-vanishing of a Plucker coordinate, each embedded as a closed subscheme of the affine space A(D), D = dim(G). However, the number E of Plucker coordinates is so large that effective computations in this setting are practically impossible. In this paper, taking advantage of the symmetries of Hillb(p(t))(n), 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(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 <= d + 2. The proofs are constructive and use a polynomial reduction process, similar to the one for Grobner bases, but are term order free. In this new setting, we can achieve explicit computations in many non-trivial cases. (C) 2013 Elsevier B.V. All rights reserved.