A Borel open cover of the Hilbert scheme

Abstract

Let p(t) be an admissible Hilbert polynomial in P n of degree d. The Hilbert scheme Hilb n p(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 , 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 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 ≤ 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 Gröbner bases, but are term order free. In this new setting, we can achieve explicit computations in many non-trivial cases.

Double-generic initial ideal and Hilbert scheme

“…≺ ≺ of the Borel ideals. By a direct computation involving marked schemes (see [16,7,9]), using for instance either the Singular library [13,17] or the algorithm described in [8], we obtain that b 4 is contained in two irreducible components Y 2 and Y 3 . Therefore, the point corresponding to b 4 is not smooth for the Hilbert scheme.…”

A general framework for Noetherian well ordered polynomial reductions

“…Actually, this assumption hinders the study of Hilbert scheme; it is well-known [8] that deformations of the Groebner basis of an ideal I in the polynomial ring P are a flat family and can thus be applied for studying geometrical deformations of the scheme X defined by I. However such families of deformations in general cover only locally closed subschemes of Hilbert scheme and are not sufficient to study neighbourhoods of deformations of X , id est opens of Hilbert scheme; such opens can be obtained instead by considering [14] those ideals I of P which share with I a fixed monomial basis of the quotient P/I. In order to determine the family of all such ideals I of P, term-ordering free bases of polynomial ideals were introduced, under the label of marked bases in [13,12,28].…”

Term-ordering free involutive bases

“…A computational description of the whole family Mf (J) is obtained in [1,6] for J strongly stable. These families are optimal for many applications, for instance for an effective study of Hilbert schemes (see [2]). However, the strong stability of the monomial ideal J is a rather limiting condition.…”