A Borel open cover of the Hilbert scheme

Let p(t) be an admissible Hilbert polynomial in P^n of degree d. It is well known that the Hilbert scheme Hilb_p(t)^n can be seen as a closed subscheme of a Grassmannian, hence, by Plucker embedding, it becomes a closed subset of a suitable projective space P^E. Unluckily, the dimension E of this projective space is generally huge, so effective computations are practically impossible. In this paper, we exhibit an open covering of Hilb_p(t)^n, defined from monomial Borel ideals, made up of "few" open subsets. We prove that each open subset of the covering can be embedded in an affine space of dimension far lower than E using equations of degree <=d+2; furthermore, with no more bound on the degree of the equations, we show that each open subset can be embedded into a linear subspace of even lower dimension. The proofs are constructive and use a polynomial reduction process: it is similar to the one for Groebner bases, but term order free. So in this setting, explicit computations are achievable in many non-trivial cases.