A Borel open cover of the Hilbert scheme

Abstract

Let $p(t)$ be an admissible Hilbert polynomial in $\PP^n$ of degree $d$. The Hilbert scheme $\hilb^n_p(t)$ can be realized as a closed subscheme of a suitable Grassmannian $\mathbb{G}$, hence it could be globally defined by homogeneous equations in the Plucker coordinates of $\mathbb{G}$ and covered by open subsets given by the non-vanishing of a Plucker coordinate, each embedded as a closed subscheme of the affine space $A^D$, $D=\dim (\mathbb{G})$. 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 $\hilb^n_p(t)$, 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 $\le d+2$ in their natural embedding in $\Af^D$. 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 $\le d+2$. 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.

17 pages. This version contains and extends the first part of version 2 (arXiv:0909.2184v2[math.AG]). A new extended version of the second part, with some new results, is posed at arxiv:1110.0698v3[math.AC]. The title is slightly changed. Final version accepted for publication

References in corpus (4)

• Ideals with an assigned initial ideal
• Homogeneous varieties for Hilbert schemes
• The Hilbert scheme of points and its link with border basis
• A local version of Gotzmann's Persistence