Borel Open Covering of Hilbert Schemes

Let p(t) be an admissible Hilbert polynomial in P of degree d and Gotzmann number r. It is well known that Hilb p(t) can be seen as a closed subscheme of the Grasmannian G(N, s), where N = ` n+r n ́ and s = N − p(r), hence, by Plücker embedding, it becomes a closed subset of a suitable projective space. Let us denote by B the finite set of the Borel fixed ideals in k[X0, . . . ,Xn] generated by s monomials of degree r. We associate to every monomial ideal J ∈ B, a Plücker coordinate pJ . If UJ is the open subset of G(N, s) given by pJ 6= 0, which is isomorphic to the affine space As(N−s), then HJ = Hilb n p(t) ∩ UJ is an open subset of Hilb n p(t) and then can be seen as an affine subvariety of As(N−s). The main results obtained in this paper are the following: i) HJ 6= ∅ ⇔ Proj (k[X0, . . . ,Xn]/J) ∈ Hilb n p(t); ii) if non-empty, HJ parametrizes all the homogeneous ideals I such that the set NJ of the monomials not belonging to J is a basis of k[X0, . . . ,Xn]/I as a k-vector space; iii) the ideal defining HJ as a subvariety of A s(N−s) (i.e. in “local” Plücker coordinates) is generated in degree ≤ d+ 2; iv) HJ can be isomorphically projected into a linear subspace of A s(N−s) of dimension ≤ σ(N−s), where σ is the number of minimal generators of the saturation J of J ; v) up to changes of coordinates in P, the open sets HJ cover Hilb n p(t), namely: Hilb p(t) = G g∈GL(n+1) J∈B g(HJ ).